---
author:
- Alexandre Gabard
title: |
    Ahlfors circle maps and total reality:\
    from Riemann to Rohlin
---

-15pt

[*Key words.*]{} [Eigenvalues of the Laplacian, bordered Riemannian surface, conformal transplantation. ]{}

Introduction {#sec1}
============

[*Preliminary Warning.*]{} \[13.11.12\]—Despite its exorbitant size, the actual mathematical content of the present text is very limited. It focuses primarily on the Ahlfors map. Neither does our work have the pretence of being the logical sum of all knowledge accumulated in the past, nor will it give an accurate picture of real developments taking shape contemporaneously. Our intention was rather more to delineate a reasonably clear-cut perception of the early branches of the theory as to understand objectively the basic truths making Ahlfors theorem possible. Failing systematically, our pretence converted to that of throwing enough obscurantism on the whole theory as to motivate others to shed fresh lights over the edifice. Even the primary contribution to the field (that of Ahlfors 1950 [@Ahlfors_1950]) has not yet been fully assimilated by the writer (compare optionally Section \[Ahlfors-proof:sec\] for our fragmentary comprehension). We strongly encourage mathematicians having a complete mental picture of Ahlfors proof to publish yet another account helping to clarify the original one. We hope during the next months (or years) to be gradually able to improve the overall organization of this text, in case our understanding of classical results sharpens. All of our ramblings starts essentially in the big-bang of Riemann’s Thesis. It looks almost a triviality alike to expect that subsequent developments will involve a deeper interpenetration between the conformal and algebro-geometric viewpoints. One oft encounters in the field problems requiring serious combinatorial skills or geometric intuition. For instance how does the moduli space of bordered surfaces stratifies along gonalities; Sec.\[sec:profile-histogram\] guesses some scenarios via primitive methods. Riemann surfaces or the allied projective realizations offer an ornithological paradise requiring patience and observational skills from the investigator. This is especially stringent when the complexes are traded against the real number field, and inside this universe of $3g-3$ real dimensions one encounters with probability $1/3$ the so-called real orthosymmetric curves of Felix Klein (1876–1882) subsumed to the paradigm of [*total reality*]{}. This little third is actually all what our topic of the Ahlfors map is about. Last but not least, experimental studies point to a large armada of potential counter-examples menacing the improved bound $r+p$ announced in Gabard 2006 [@Gabard_2006]. It seems safe to declare as an open problem to either corroborate this bound (via other more analytic or algebraic treatments) or to reject it.

\[19.03.13\] The main addition to the present edition (v.2) of our text, is an essay to connect Ahlfors’ theory with Hilbert’s 16th problem (at least the part thereof pertaining to the topology of real plane algebraic curves). Again our dancing queen is the paradigm of total reality (Riemann, Schottky, Klein, Bieberbach, Teichmüller, Ahlfors, Alling-Greenleaf, Geyer-Martens, etc.), but now reoriented as a missile against Hilbert’s 16th (quite akin to an asteroid menacing peaceful life on planet Earth). This trend is not new having been much foreseen in Rohlin’s seminal work 1978 [@Rohlin_1978] effecting a Verschmelzung[^1] between the conceptions of Klein and Hilbert (when it comes to real geometry). Rohlin never refers back to Ahlfors’ work (which he probably ignored?), yet the connection is very vivid through Rohlin’s (conjectural) philosophy that [*schemes*]{} of type I (not in Grothendieck’s highbrow sense, but merely Rohlin’s synonym for a [*distribution of ovals*]{} à la Zeuthen-Harnack-Hilbert) are necessarily [*maximal*]{} in the hierarchy of all schemes of some fixed degree. Through the lines of Rohlin’s text transpires the intuition that what detects (pure) orthosymmetry of schemes is a vertiginous phenomenon of [*total reality*]{} positing existence of adjoint pencils cutting only real points on the given curve. The byproduct is that total reality should act prohibitively upon all schemes enlarging those totally flashed by a pencil, which are so-to-speak already Bézout-saturated. Hence total reality should contribute to Hilbert’s problem (isotopic classification of curves), though this method can hardly be said to have been systematically exploited as yet (apart of course in the prophetical allusions in Rohlin 1978).

This grand vision of Rohlin could benefit from the Klein-Ahlfors theory (which in our opinion has been much neglected in the tradition of the German-US-Italian-Russian school of real geometry involving such [*pointures*]{} as Hilbert 1891, Rohn 1888–1913, Ragsdale 1906, Brusotti 1910–50, Petrovskii 1933/38, Gudkov 1954–69, Arnold 1971, Rohlin 1972–78, Kharlamov, Viro, Marin, Fiedler, Shustin, Itenberg, etc.). Alternatively it could be reassessed through purely synthetical procedures that Rohlin himself envisioned (probably as consequences of deep 4D-topology, notably the type-I-forcing Kharlamov-Marin congruence modulo 8). Alas Rohlin’s synthetical proof even on the simplest prototype of sextics has never been published (and seems now to be lost forever), but was recently (partially) resuscitated in a tour de force of Séverine Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]. So the epoch seems ripe to dream about a big Ahlfors-Rohlin Verschmelzung with direct repercussions upon Hilbert’s 16th byway of prohibitions. It is already breathtaking (for a beginner) to contemplate how the Rohlin-Le Touzé total reality phenomenon for sextics explains nearly all prohibitions observed in Gudkov’s census (1969) of sextics curves (cf. Fig.\[Gudkov-TableTop:fig\]), which after all supplies (nothing less than) the [*complete*]{} solution to Hilbert’s problem in its original formulation (degree $m=6$). Those aspects are addressed in Sec.\[Klein-Rohlin-conj:sec\]&ff. For convenience we wrote a general overview in Sec.\[Hilbert’s16th-PartII:sec\] pointing to some open questions which looks semi-urgent to settle in order to build a more solid theory. All this second part, devoted to Hilbert’s 16th, could not have been written without the constant support and information generously shared by the leading experts (Viro, Marin, Kharlamov, Shustin, Orevkov, Le Touzé, Fiedler), whose instructive e-mails are reproduced in Sec.\[e-mail-Viro:sec\]. Needless to say we have not yet assimilated all their wisdoms and advices, but have reproduced faithfully their messages in the hope that other amateurs of the field can also beneficially profit from their invaluable expertise.

\[11.04.13\] It seems (now) a firm conviction that a big piece of Hilbert’s 16th puzzle still remains to be fixed. This should be a fairly simple matter of assembly between the conceptions of Riemann-Schottky-Klein-Bieberbach-Teichmüller-Ahlfors about total reality and the theory of Hilbert-Rohn-Petrovskii-Gudkov-Arnold-Rohlin aiming to predict the distribution of ovals traced by algebraic curves (a God-given video game). Precisely, the isotopic classification of real plane curves should be regulated by a sole paradigm (total reality) itself piloted by the geometry of the canonical series (adjoint curves of order $m-3$) assigned to visit $(M-3)$ basepoints randomly selected among the most profound ovals of $(M-2)$-curves (alias the [*extended Rohlin-Le Touzé phenomenon*]{}, which in degree 4 boils down to the total reality of the Gürtelkurve quartic with 2 nested ovals). This looks special but wait a moment.

It is conjectured (\[primitive-manifestation-of-tot-real:conj\]) that [*any*]{} [*primitive*]{} manifestation of the phenomenon of total reality on a plane curve is of this sort (i.e. a Rohlin-Le Touzé “adjunction” for $(M-2)$-curves subsumed to the eightfold periodicity $\chi\equiv_8 k^2+4$ of Rohlin-Kharlamov-Marin), except when it comes to the trivial case of $M$-curves, where total reality is nearly completely settled by an extension of (another) Le Touzé’s scholium (\[Le-Touzé-extended-in-odd-degree:scholium\]). The maximality of $M$-schemes being so trivial (Harnack-Klein inequality of 1876), this latter case looks a sterile syllogism not worth paying attention at, but this is probably not so via satellites. If [*not primitive*]{}, this is to say that the scheme is just derived as a [*satellite*]{} replicating the curve up to a certain multiplicity within its tube neighborhood. For instance satellites of a single oval of degree 2 reproduce the infinite series of deep-nests total under a pencil of lines \[by the way highly reminiscent to the “rondelles” of a certain artist known as Markus Schneider-Zeitler, Jura Suisse\]. Such deep-nests are Bézout-saturated hence extremal shapes in the Hilbert-Gudkov hierarchy. More generally, the magic formula reads $A+B=R m c^2$, i.e. Ahlfors plus Bézout implies Rohlin’s maximality conjecture (any scheme of [*type I*]{} kills all its enlargements). At this stage the architecture of higher Gudkov’s pyramids ($m\ge 7$) is completely predestined by (the felicity of) pure orthosymmetry à la Felix Klein, and (optionally) Rohlin’s theorem on the signature of spin $4$-manifolds governing the (Gudkov) $8$-fold periodicity via differential-topology. Paraphrasing, Hilbert’s 16th is virtually solved in [*all*]{} degrees, at least in its qualitative shape (prohibitions). It remains then of course to programme a (patchworking) machine doing all the constructions. This should be merely a matter of passive contemplation, requiring immortality and much patience from the investigator. It is evident that all this programme is not a novel idea, but much—not to say completely—anticipated by Rohlin 1978, safe that he does not seem to have been consciously aware of the Riemann-Ahlfors theory (nor perhaps the conjectural stability under satellites), but seemed rather adventurous enough to rediscover it [*ab ovo*]{} through purely synthetical processes, without any intrusion of analysis or transcendental gadgets, like Abelian integrals. Our messy text is just an invitation to inspect more exactly how the whole process will sediment itself within the next decades, probably via massive usage of Brill-Noether (i.e., Riemann for dummies). Then, the night of ignorance [*(les ténèbres de l’ignorance)*]{} allied to Hilbert’s combinatorial mess about the distribution of 16 ovals (recall that $M_7=11+5=16$) should be completely dissipated. Whether conversely, all this (ancient) geometry can acknowledge in feedback some impact upon 4D-differential-topology, e.g. the question of smooth structures on $S^4$ or $\CC P^2$ (so-called [*smooth Poincaré conjecture*]{}) is merely speculation of longstanding (reminding such names as Arnold, Maxwell, Milnor, Kuiper, Massey, Marin, Akbulut, Donaldson, Taubes, Finashin, Wang, Seiberg, Witten, etc.). But this is another story.

\[22.03.13\] Lastly, we adopted (not deliberately but because we were not clever enough to proceed differently) Arnold’s philosophy of the mushroom (compare Arnold 2004 [@Arnold_2004]). That is, not just presenting overwhelming theorems (arid as they are) but the slow organical eclosion of truths through mistakes, conjectures, historical meanders, etc. The drawback is an intolerable inflation in size, also partly caused by the abundance of pictures, which in our opinion form the true core of any mathematical truth[^2]. We expect in the future to reorganize the material à la Bourbaki as to offer a cleaner view of what happens (after distillation, the factual content should be compressible to ca. 20 pages). Especially crucial is a rectification due to Fiedler of an erroneous theorem of mine that would have proved one-half of the (still open) Ragsdale conjecture for $M$-curves via the Thom conjecture (Sec.\[Thom:sec\]).

$\star$$\star$$\star$

Works by Yamada 1978–2001, Gouma 1998 and Coppens 2011 suggest that fewer sheets than required in Ahlfors’ era is expectable for a clever placement of the basepoint(s) required to pose the extremal problem. E.g., is Coppens’ separating gonality (least degree of a circle map) always sustained by an Ahlfors function? How does the degree of the Ahlfors map fluctuates when the basepoint is dragged through the surface? A list of known applications is tabulated in the hope of guessing future ones. Some (e.g. Fraser-Schoen’s to Steklov eigenvalues) do not require the full swing of Ahlfors’ extremals, raising the hope that some improved control $\le r+p$ (Gabard 2004/06) on the degree of circle maps concretizing surfaces of genus $p$ with $r$ contours prompts some upgrades (e.g. in the corona problem with bounds of Hara-Nakai 1985). As to the foundation of the Ahlfors mapping theory itself, it is our partisan belief that much remains to be clarified both historically and logically. Albeit sembling a retrograde attitude, it is probably not since Ahlfors bound $r+2p$ certainly fails sharpness at least for low values of the invariants $(r,p)$. Apart from an abrupt claim by Teichmüller (1941) that everything (safe bounds) is to be found in Klein (what the writer was unable to certify from printed evidence), it is fair to admit that the bulk of the theory crystallized right after World War II. Several workers like Ahlfors 48/50, Matildi 45/48, Andreotti 50, Heins 50 (perhaps even Courant 39/40, not to mention Grunsky, one of the most brilliant protagonist albeit his work looks confined to the genus $0$ case) offered quite overlapping conclusions, but it seems fair to credit Ahlfors for having first expressed the story in the most clear-cut fashion. Quite shamefully, I confess that Ahlfors argument still escapes me slightly. A non-negligible amount of literature is devoted to reproving Ahlfors’ theorem: Heins 1950/75/85, Garabedian 1950, Kuramochi 1952, Read 1958 (student of Ahlfors), Mizumoto 1960 (topological methods), Royden 1962 (Hahn-Banach like Read), Forelli 1979 (extreme points and Poisson integral), Jenkins-Suita 1979 (Pick-Nevanlinna viewpoint), just to name those addressing the positive genus $p>0$ case. The actual tension between topological and analytic methods (made acute since the $r+p$ bound, granting its correctitude!), is possibly a temporary state of affairs destined to disappear after some examination of Ahlfors’ text (sharpening perhaps its ultimate convex geometry consideration). Another promising route is Meis’ work (1960) validating Riemann’s (semi)intuition of the $[\frac{g+3}{2}]$ gonality of closed genus $g$ surfaces via some Teichmüller-theoretic approach. It is likely that Meis’ approach to be transmutability to the bordered setting, reassessing thereby Ahlfors’ result (probably even with the sharp bound $r+p$). To put it briefly, it seems that the Grötzsch-Teichmüller mode of thinking has not yet fully penetrated the paradigm of the Ahlfors circle maps, more generally that of branched coverings. Dually, it also seems desirable to reprove the Riemann-Meis bound via topological methods (e.g. that used in Gabard 2006, which perhaps is nothing else than Riemann’s parallelogram method). Poincaré (1895) invented “homology” (modulo the Riemann-Betti heritage) with precisely function theory (Abelian functions) as one of the key motivation (beside celestial mechanics and the like). ${\it Warning.}$—This draft is a preliminary version, so avoid printing it for environmental reasons. It contains a list of hopefully clear-cut questions intended to encourage investigators. Several synoptic diagrams (authors names, keywords, etc.) should permit a quick optical scan of the whole content. The article contains no original insights, instead a series of failing attempts to contemplate the theory from different angles. A commented bibliography (of ca. 900 entries) tries to brush a photograph of old and contemporary trends including some ramifications (potential theory, minimal surfaces, spectral theory, analytic capacity, operator theory, Gromov’s filling conjecture, etc.). We have not attempted to reach any overwhelming mathematical density, but rather tried to dilute through historico-philosophical anecdotes, as well as spending some time trying to understand the available sources and their affiliation. Finally, we enjoyed speculating about some mechanical interpretation of the Klein-Ahlfors theory of real orthosymmetric curves (and the allied totally real maps) in terms of gravitational systems, positing a wild extension of Kepler’s planetary motions around ellipses.

This is a prejudiced survey on the Ahlfors (extremal) function and (improvising terminology) the weaker [*circle maps*]{}, effecting the conformal representation upon the disc of an arbitrary differential-geometric membrane, alias [*compact bordered Riemann surface*]{}. Our jargon, borrowed from Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950], translates essentially the term [*Kreisabbildung*]{} used e.g., by Koebe 1915 [@Koebe_1915] and Bieberbach 1914 [@Bieberbach_1914].

Exciting works by Yamada 1978–2001 [@Yamada_1978], [@Yamada_2001], Gouma 1998 [@Gouma_1998] and Coppens 2011 [@Coppens_2011] suggest that fewer sheets than required in Ahlfors’ era is expectable, for a clever placement of the basepoint(s) required to pose the extremal problem. E.g., is Coppens’ (absolute) gonality of a membrane always sustained by an Ahlfors function? We also started tabulating a list of known applications in the hope of guessing future ones. Some applications (e.g. Fraser-Schoen’s recent one to Steklov eigenvalues [@Fraser-Schoen_2011]) do not require the full punch of Ahlfors’ extremals, raising the hope that the improved control $r+p$ on the degree of circle maps (predicted in Gabard’s Thesis 2004/06 [@Gabard_2006] for surfaces of genus $p$ with $r$ contours) could imply some ‘automatic’ upgrades (e.g. in the corona problem with bounds, as studied by Hara-Nakai 1985 [@Hara-Nakai_1985]).

As to the foundation of the Ahlfors mapping theory itself, the issue that the naive qualitative approach (used in Gabard 2004/06 [@Gabard_2006]) affords a bound, $r+p$, quantitatively stronger than Ahlfors’ original $r+2p$ is somewhat surprising. It results a certain psychological tension between topological and analytical methods, which hopefully is just a superficial and temporary state of affairs destined to disappear after renewed examination of Ahlfors’ argument. The latter seems indeed to leave some free man[œ]{}uvring room, in its ultimate convex geometry portion (cf. Sec.\[Ahlfors-proof:sec\] for some strategy).

It is our partisan belief that much remains to be clarified both historically and logically in the theory of the Ahlfors map. Albeit sembling a retrograde attitude, it is probably not since Ahlfors bound $r+2p$ certainly fails sharpness, at least for low values of the invariants $(r,p)$. (Consider for instance the topological type of Klein’s Gürtelkurve; i.e. $(r,p)=(2,1)$ where a projective realization (of the Schottky double) as a plane quartic with 2 nested ovals prompts existence of a total map of degree $3$ via projection from the inner oval. This beats by one unit Ahlfors’ bound $r+2p=4$.)

Apart from an abrupt claim by Teichmüller 1941 [@Teichmueller_1941], that everything (safe bounds) is to be found in Klein (what the writer was unable to certify from printed evidence), it is fair to admit that the bulk of the theory crystallized right after World War II. Several workers like Ahlfors 1948/50 [@Ahlfors_1950], Matildi 1945/48 [@Matildi_1945/48], Andreotti 1950 [@Andreotti_1950], Heins 1950 [@Heins_1950] (perhaps even Courant 1939/40 [@Courant_1939], not to mention Grunsky 1937–40–41–42 [@Grunsky_1937], one of the most brilliant protagonist albeit his work looks confined to the genus $0$ case) offered quite overlapping conclusions. It seems fair however to give full credit to Ahlfors for having first expressed the story in the most clear-cut fashion. Quite shamefully, I confess that Ahlfors argument still escapes me slightly. A non-negligible amount of literature is devoted to reproving Ahlfors’ theorem: Heins 1950/75/85 [@Heins_1950] [@Heins_1975] [@Heins_1985-Extreme-normalized-LIKE-AHLF], Garabedian 1950 [@Garabedian_1950], Kuramochi 1952 [@Kuramochi_1952], Read 1958 [@Read_1958_Acta] (student of Ahlfors), Mizumoto 1960 [@Mizumoto_1960] (topological methods), Royden 1962 [@Royden_1962] (Hahn-Banach like Read), Forelli 1979 [@Forelli_1979] (extreme points and Poisson integral), Jenkins-Suita 1979 [@Jenkins-Suita_1979] (Pick-Nevanlinna viewpoint), just to name those authors addressing the positive genus case ($p>0$).

Another promising route is Meis’ work 1960 [@Meis_1960] validating Riemann’s (semi)intuition of the $[\frac{g+3}{2}]$ gonality of closed genus $g$ surfaces via some Teichmüller-theoretic background. It is likely that Meis’ approach is transmutable to the bordered setting, reassessing thereby Ahlfors’ result (probably even with the sharp bound $r+p$ in case the latter is reliable). To put it briefly, it seems that the Grötzsch-Teichmüller mode-of-thinking (of the [*möglichst konform*]{} mapping) has not yet fully penetrated the paradigm of the Ahlfors circle map, more generally that of branched coverings, except of course in Meis’ memoir (alas notoriously difficult to access). Dually, it also seems desirable to reprove the Riemann-Meis bound via topological methods (e.g. that used in Gabard 2006 [@Gabard_2006], which perhaps is nothing else than Riemann’s parallelogram method). Poincaré’s “Analysis Situs” (1895 [@Poincare_1895-Analysis-Situs]) invented “homology” (modulo the Riemann-Betti=Brioschi \[sic!\] heritage) with precisely function theory (Abelian functions) as one of the key motivation (beside celestial mechanics and the like). This, jointly with the subsequent work of Brouwer, gives the basic conceptual framework for implementing such topological methods.

[*User guide.*]{}—This draft is a preliminary version, so avoid printing it for environmental reasons. A list of hopefully clear-cut questions is given in Sec.\[sec:question\]. This is intended to challenge investigators. Several synoptic diagrams scattered as figures through the text should permit a quick optical scan of the whole content. More specifically, those includes:

$\bullet$ an [*exhaustive*]{} list (Fig.\[Map:fig\]) of [*all*]{} articles supplying (or claiming to supply) a proof of Ahlfors theorem (existence of circle maps),

$\bullet$ a list of keywords (Fig.\[Keyword:fig\]) tabulating concepts traditionally related to the Ahlfors map,

$\bullet$ a comprehensive map (Fig.\[Geneal:fig\]) of authors involved in the theory (at least those cited in the bibliography).

This essay, as already said, contains no original insights, instead a series of attempts to contemplate the theory from different angles. A commented bibliography (of ca. 900 entries) tries to brush a panorama of trends related to the Ahlfors map. This includes topics like Riemann surfaces, algebraic curves, conformal mapping, potential theory, Green’s functions, Dirichlet’s principle, Riemann mapping theorem, Kreisnormierung, parallel slit-maps, Bieberbach’s least-area map interpretation of the Riemann map, Bergman and Szegö kernels, minimal surfaces, Plateau’s problem, spectral theory, analytic capacity, removable singularities, corona problem, operator theory, Gromov’s filling conjecture, etc.). We have not attempted to reach any overwhelming mathematical density, but rather tried to dilute through historico-philosophical anecdotes.

There is some interplay between Ahlfors maps and total reality of Klein’s orthosymmetric curves which gives rise to the gallery of pictures mentioned in the abstract. For a tourist view, browse the string of figures starting from Fig.\[Pencil:fig\] up to Fig.\[Fcubic3:fig\]. For “do-it-yourself” purposes, it is probably more valuable to describe the general recipe used to manufacture such pictures. Take any configuration of simple objects like lines and conics, and smooth it in an orientation-preserving sense to get a dividing curve (one is free to keep certain nodes unsmoothed). (Rohlin’s eminent student Thomas Fiedler (1981 [@Fiedler_1981]) ensures for us that the smoothed curve is dividing, alias orthosymmetric in the sense of Klein.) According to Ahlfors theorem there must be a totally real pencil of auxiliary curves cutting only real points on the given curve (plus maybe some imaginary conjugate basepoints). Geometric intuition usually tells us where to locate such a total pencil, roughly by assigning basepoints among the [*deepest*]{} ovals (in the sense of D. Hilbert’s 16th Problem). Albeit this is just a Plato cavern style extrinsic manifestation of Ahlfors theorem, the possibility of finding [*always*]{} such a total pencil reveals strikingly (in our opinion) some of the depth of Ahlfors theorem. (Incidentally it is not to be excluded that a deep understanding of extrinsic algebraic geometry (say à la Brill-Noether) could reprove the full Ahlfors theorem from within the Plato cavern.) In philosophical terms, [*real orthosymmetric curves behave on the reals as if they were complex varieties*]{}: all intersections prompted by Bézout are visible over the reals. This phenomenon is what we (and others, e.g. Geyer-Martens 1977 [@Geyer-Martens_1977]) call the paradigm of [*total reality*]{}. It seems evident that a global study of such pencils bears some close connection with Poincaré index theory, foliations à la Poincaré-Kneser-Ehresmann-Reeb, etc., and that both experimentally and theoretically much remains to be explored along the way. In particular we failed to make such totally real pictures for an $M$-quintic (Sec.\[sec:Total-reality-Harnack-max-case\]). This could be a challenging problem of computer visualization.

As to our speculation about a mechanical interpretation of the Klein-Ahlfors theory of real orthosymmetric curves (and the allied totally real maps) in terms of gravitational systems, see Sec.\[sec:gravitation\]. This posits a broad extension of Kepler’s planetary motions around ellipses, enabling virtually all algebraic curves (and not just conic sections) to arise as the trajectories of a perfectly stable and periodic motion. Of course if such a grandiose connection between Klein-Ahlfors and Kepler-Newton-Coulomb-Poincaré is not verifiable, this may just be interpreted as a metaphoric language describing the dynamics of totally real morphisms prompted by Ahlfors theorem. In fact rather than mere gravitation, it is really a “[*dynamique de l’éléctron*]{}” which seems to be involved; for a toy example on the Gürtelkurve compare Fig.\[FGuert:fig\]. The resulting metaphysics is quite akin to Lord Kelvin’s speculations about the ultimate constitution (and stability) of matter (via the vortex atom geometrized by a knot), except that in our story the dancing queen is rather a (naked) bordered Riemann surface.

Our partisan belief even in its most basic aspects the theory of the Ahlfors function (even when confined to the finite bordered realm) is far from being a closed (and transparent) chapter of geometry despite its sexagenary oldness, and the absence of significative achievements since Ahlfors 1950. Historical continuity invited us to rely massively upon primary sources, and this contributed to the prohibitive size of our output. Looking for logical alternatives, we oft rambled into dubious ‘historical revisionism’, wondering if other sources (maybe Klein—as cryptically suggested by Teichmüller 1941 [@Teichmueller_1941]—or perhaps Courant 1939/40 [@Courant_1939], [@Courant_1940-Acta]) do not anticipated the Ahlfors map (1948/50), at least at the qualitative level of circle maps. The Italian school (Cecioni and his students, including Matildi 1945/48 [@Matildi_1945/48] and the related work Andreotti 1950 [@Andreotti_1950]) should not be neglected (Ahlfors himself meticulously indexed those Italian references in the Ahlfors-Sario book (1960 [@Ahlfors-Sario_1960])). We warn again the present text is far from streamlined, so please avoid printing it for environmental reasons. It is rather intended to be an interactive browser, hopefully more structured than generic electronic databases, albeit the latter were of invaluable assistance to the present compilation. Human feedback is most welcome if some sloppy distortions look worth rectifying or the bibliography lacunary. Alas, we were soon confined to brush the large scale structure of the story without keeping close control over the logical details making the edifice. The field anyway is reputed for its overwhelming organical mode of growth, for Koebe proclaimed (im September 1921, Jena, DMV Jahresversammlung): “Es gibt viele Gebiete in der Mathematik, wo man sich durch Entdecken neuer Ergebnisse verdient machen kann. Es sind meistens lange und steile Gebirgshänge für meckernde Ziegen. Die Funktionentheorie ist aber mit einem saftigen Marschland zu vergleichen, besonders geeignet für dickes Rindvieh!” As usual, the paper is probably best read from the ending references, if one is patient enough. Else several synoptic diagrams (scattered through the text giving authors names and keywords) should permit a 5 minutes glimpse of the whole text. As the writer has zero analytic capacitance (=competence) this text should best be seen as a removable singularity in the realm of the literature devoted to the Ahlfors function, except possibly for our attempt to compile an up-to-date bibliography.

Trying to wet some appetite out of the blue
-------------------------------------------

Our aim is to discuss the following Riemann surface, giving an extremally biased view of some conformal mapping theory centering the discussion around the Ahlfors function.

A long time ago (ICM 1908 Rome), Poincaré argued that in mathematics we need a strong principle of economy of thoughts by conceptualizing such notions as ‘uniform convergence’ as if the sole naming process would spare us repeating long intricate arguments. On the other hand, Felix Klein, asserted boldly “die Franzosen unhistorisch wie Sie sind” (exercise recover the source) and liked the motto “Zurück zur Natur, sie bleibt die grö[ß]{}te Lehrmeisterin”. Beside all those psychological tips of the masters of geometrization, we can safely agree with both of them that science requires—as a matter of conciliating the principle of economy with that of historical continuity (of course not so structurally incompatible as neo-expressionism seems to assess) —a certain amount of respectfulness about wisdoms accumulated during the past. This explains our ca. 900 references (albeit the explosion was mainly caused by my lack of internet connection occasioning a manual references chasing).

In these notes we propose a (poorly guided) tour of some geometric function theory (GFT). The field is an old fashioned one, lying quite dormant with its old mysteries and legends (e.g., Koebe’s Kreisnormierungsprinzip, the exact determination of the Bloch constant in quasi-stagnation since Ahlfors-Grunsky 1937 [@Ahlfors-Grunsky_1937], etc.). Function-theory seems a volcano alike awaiting anxiously the next explosive eruption, whose pyroclastic rejections turned out to act (in the past at least) as a powerful fertilizer over neighbouring areas (like Riemannian and algebraic geometry, spectral theory, etc.). Actually Koebe had a more picturesque description, when proclaiming (im September 1921, Jena, Jahresversammlung der DMV[^3]): “Es gibt viele Gebiete in der Mathematik, wo man sich durch Entdecken neuer Ergebnisse verdient machen kann. Es sind meistens lange und steile Gebirgshänge für meckernde Ziegen. Die Funktionentheorie ist aber mit einem saftigen Marschland zu vergleichen, besonders geeignet für dickes Rindvieh!”

The field itself (GFT) seems to be a strange cocktail of qualitative-flexible versus quantitative tricks, or as Gauss puts it [*geometria situs*]{} versus [*geometria magnitudinis*]{}. If topological methods look a priori quite foreign to the discipline, it was probably Riemann who first revealed:

$\bullet$ the reactivity of the underlying topological substratum (anticipated maybe by Abel 1826 [@Abel_1826], who first introduced the [*genus*]{} (under a different name and the transcendant disguise of differentials of the first kind). \[The word [*Geschlecht*]{} is first coined in Clebsch 1865 [@Clebsch_1865 p.43]; and the allied [*Geschlechtsverkehr*]{}[^4] must have originated about the same period\] $\bullet$ the amazing plasticity (inherited from potential-theoretic considerations) of 2D-conformal mappings, leaving out moduli spaces of finite dimensionality after conformal evaporation of all metrical incarnation of a given surface. \[Gromov wrote in 1999 [@Gromov_1999]: [*Shall we ever reach spaces beyond Riemann’s imagination?*]{}\]

Our text will soon be biased toward a single obsession, the so-called [*Ahlfors function*]{}, which is one (among several other possible) generalisation of the [*Riemann mapping theorem*]{} (RMT) to configurations of higher topological structure than the disc. Such configurations (compact bordered surfaces) are topologically determined by the number $r$ of boundary contours and the genus $p$ (number of handles) (see Fig.1a), as is well-known since the days of Möbius 1860/63 [@Moebius_1863] and Jordan 1866 [@Jordan_1866] (and of course very implicit in all of Riemann’s work).

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The possibility of mapping any bordered surface to the disc conformally was pioneered by such towering figures as:

$\bullet$ Riemann 1857/76 [@Riemann_1857_Nachlass] (manuscript not published during his lifetime), in which circular domains (hence $p=0$) of finite connectivity are mapped upon the disc. This fragment was edited by H. Weber and appeared in print only in 1876 in the first edition of Riemann’s Werke. The date of 1857 follows some oral tradition (Schwarz–Schottky), compare Bieberbach 1925 (Quote \[quote:Bieberbach-1925\] below), but conflicts slightly with Summer 1858 as estimated by Klein (cf. Quote \[Klein-1923:quote:Riemann-1858\]). \[11.08.12\] To pinpoint more about the exact date, should we recall that Riemann himself reports in the introduction of “Theorie der Abel’schen Functionen” 1857 [@Riemann_1857 p.116] his involvement with the topic of conformal mapping of multi-connected “surfaces” (Flächen) right after his Thesis (Fall 1851–Begin 1852), but was then sidetracked to another subject ([*ward aber dann durch einen andern Gegenstand von dieser Untersuchung abgezogen*]{}).

$\bullet$ Schottky 1875–77 [@Schottky_1877] (=Dissertation under Weierstrass, Berlin, 1875), where a similar mapping is obtained for general real analytic contours. At first sight, it is natural to speculate that Schottky knew about Riemann’s Nachlass, but Schottky himself describes his trajectory as independent (cf. Quote \[quote:Schottky-1882\]). Apparently, it was Weierstrass’ special pupil, namely H.A. Schwarz who made Schottky aware of this connection, as reported in Bieberbach 1925 [@Bieberbach_1925], compare Quote \[quote:Bieberbach-1925\]. Albeit independent of Riemann’s, Schottky’s work was likewise physically motivated as emphasized by Klein 1923 [@Klein-Werke-III_1923 p.579]=Quote \[Klein-1923:quote:Riemann-1858\] below, or via Schottky’s own recollections (1882)=Quote \[quote:Schottky-1882\].

$\bullet$ Bieberbach 1925 [@Bieberbach_1925], found some elementary arguments (or just modernization) of the same Riemann–Schottky result, while emphasizing the trivial fact that the degree bound is optimum (apparently Schottky gave no bound),

$\bullet$ Grunsky 1937–41 [@Grunsky_1937; @Grunsky_1941_KA], 1940–42–49 [@Grunsky_1940; @Grunsky_1942; @Grunsky_1950], who in a first series of papers rederived Bieberbach’s result and then switched to an extremal interpretation of the mapping problem. This terrible quantitative/competitive weapon (with historical precedents to be soon discussed) culminated, finally, in:

$\bullet$ Ahlfors 1947 [@Ahlfors_1947], but it remained until Ahlfors 1950 [@Ahlfors_1950], to prove a generalization capable of including positive genera ($p>0$), superseding thereby quite dramatically the planarity (Schlichtartigkeit) where all previous efforts were perpetuated. (We shall attempt to ponder this absolute originality of Ahlfors, by comparing with others writers (e.g., Courant), but only with limited success due to my moderate competence with minimal surfaces and Plateau.)

$\bullet$ Subsequent ramifications in the West (corona, operator theory, etc.), in Russia with Golusin, and Havinson (domains of infinite connectivity), and in Japan.

For an overall picture of the roots plus some ramifications of Ahlfors, the reader may glance at the following map (Fig.\[Map2:fig\]) showing some of the links we are going to explore in this survey. We have opted for a Riemann surface style depiction of this histogram so as to give a quick-view of the varied [*troncs vivaces*]{} (in A. Denjoy’s prose when alluding to history of mathematics). Such trunks or handles are attached whenever some philosophical dependence (citation) is detected. Alas, it resulted a prolix accumulation of links creating a somewhat chaotical picture. For sharper pictures of the “Riemann galaxy”, we recommend Neuenschwander 1981 [@Neuenschwander_1981], Gray 1994 [@Gray_1994] and Remmert 1998 [@Remmert_1998].

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[—The own contribution of the writer (Gabard 2006 [@Gabard_2006]) predicting an improved control $r+p$ upon Ahlfors’ degree $r+2p$ is enormously exaggerated, especially if it turns out to be false. Other distortions only reflects the writer’s poor understanding of this tentacular topic. For a more extensive compilation of authors involved in the theory, cf. Fig.\[Geneal:fig\]. If you are not cited on it, please send me an e-mail. ]{}

As already said, our central hero will be Ahlfors, especially his paper of 1950 [@Ahlfors_1950]. In retrospect, it is not quite impossible that Riemann himself (or disciples like Schwarz, Schottky, Klein, Hurwitz, Koebe, Hilbert, Grötzsch, Teichmüller, etc., or also Bieberbach, Grunsky, Wirtinger, Courant, while not forgetting in Italy, Cecioni, Matildi 1945/48 [@Matildi_1945/48], Andreotti 1950 [@Andreotti_1950]) could have succeeded in proving such a version. Such speculations look not purely science-fictional especially in view of Ahlfors’ elementary argument in [@Ahlfors_1950 pp.124–126], which involves primarily only classical tricks (no deep extremal problem), like annihilating all the periods to ensure single-valuedness of the conjugate potential, and basic potential functions arising from the Green-Gauss-Dirichlet era. All these tricks are standard since Riemann’s days (cf. e.g. Riemann 1857 [@Riemann_1857 p.122], “[*so bestimmen da[ß]{} die Periodicitätsmoduln sämmtlich $0$ werden.*]{}”). Remember also, despite sembling dubious historical revisionism, that Teichmüller 1941 [@Teichmueller_1941](=Quote \[quote:Teichmueller-1941\]) seems to have possessed a clear-cut conception of the result at least without precise bound, while ascribing the assertion even back to Klein.

$\bullet$ Mizumoto’s topological argument in 1960 [@Mizumoto_1960];

$\bullet$ Gabard’s topological argument in 2006 [@Gabard_2006].

However it took ca. 91 years—say from Riemann’s 1857/58 Nachlass up to the 1948 Harvard lecture held on the topic by Ahlfors, cf. Nehari’s Quote \[Nehari-1950:quote\] of 1950—until somebody puts it on the paper and it turned out to be no less an authority than Lars Valerian Ahlfors[^5].

It is true that Ahlfors moved in considerably deeper waters by solving as well a certain [*extremal problem*]{}. This extremal viewpoint is more punchy, yet arguably the corresponding extremals (so-called Ahlfors functions) are only circle maps of a special character. We gain in punch but loose in flexibility. The extremal functions do not substitute to—nor are substituted by—circle maps. Deciding which viewpoint is more useful is another question, probably premature to answer except for guessing a complementary nature depending on the problem at hand. Incidentally in Ahlfors paper (1950 [@Ahlfors_1950]), existence of circle maps is required as a preliminary step toward posing (non-nihilistically) the extremal problem. Ahlfors’ extremal problem stemmed surely not out of the blue, but was patterned along a tradition, whose first steps should probably be located in the following works. (We acknowledge guidance by Remmert’s book 1991 [@Remmert_1991 p.160–2, p.170–2], to which we refer for sharper historical details.)

$\bullet$ Koebe’s elementary proof 1907, 1909, 1912 [@Koebe_1912], 1915 [@Koebe_1915] of the (RMT); ([*Quadratwurzeloperationen*]{}, [*Schmiegungsverfahren*]{}, etc.)

$\bullet$ Carathéodory 1912 [@Caratheodory_1912]: similar iterative methods and convergence of his sequence via Montel’s theorem. This revitalized Koebe’s interest (cf. again Remmert’s description [@Remmert_1991 p.160, p.172]); in Carathéodory 1914 [@Caratheodory_1914] full details of the method were given in the Schwarz-Festschrift;

$\bullet$ Fejér and F. Riesz 1922 obtain the Riemann mapping via an extremal problem for the derivative (published in Radó 1922/23 [@Rado_1922-3]). Montel’s normal families are also used, plus a tedious derivative computation eradicated in: $\bullet$ Carathéodory 1928 [@Caratheodory_1928] and Ostrowski 1929 [@Ostrowski_1929], where (independently) ultimate simplifications are provided.

Carathéodory wrote about these developments:

\[quote:Caratheodory-1928\]

Nachdem die Unzulänglichkeit des ursprünglichen [*Riemann*]{}schen Beweises erkannt worden war, bildeten für viele Jahrzehnte die wunderschönen, aber sehr umständlichen Beweismethoden, die [*H.A. Schwarz*]{} entwickelt hatte, den einzigen Zugang zu diesem Satze. Seit etwa zwanzig Jahren sind dann in schneller Folge eine gro[ß]{}e Reihe von neuen kürzeren und besseren Beweisen \[von ihm selbst und von Koebe (Remmert’s addition); in the original Lindelöf 1916 is also quoted\] vorgeschlagen worden; es war aber den ungarischen Mathematikern [*L. Fejér*]{} und [*F. Riesz*]{} vorbehalten, auf den Grundgedanken von [*Riemann*]{} zurückzukehren und die Lösung des Problems der konformen Abbildung wieder mit der Lösung eines Variationsproblems zu verbinden. Sie wählten aber nicht ein Variationsproblems, das, wie das [*Dirichlet*]{}sche Prinzip, au[ß]{}erordentlich schwer zu behandeln ist, sondern ein solches, von dem die Existenz einer Lösung feststeht. Auf diese Weise entstand ein Beweis, der nur wenige Zeilen lang ist, und der auch sofort in allen neueren Lehrbüchern aufgenommen worden ist. \[Footnote 2: Siehe [*L. Bieberbach*]{}, Lehrbuch der Funktionentheorie, Bd.2 S.5.\] Mein Zweck ist nun zu zeigen, da[ß]{} man durch eine geringe Modifikation in der Wahl des Variationsproblems den [*Fejér-Riesz*]{}chen Beweis noch wesentlich vereinfachen kann.

Let us quote thrice Ahlfors in this connection (the second of which occurred while celebrating the centennial of Riemann’s Thesis, 1851):

\[Ahlfors-1961\]

In complex function theory, as in many other branches of analysis, one of the most powerful classical methods has been to formulate, solve, and analyze extremal problems. This remains the most valuable tool even today, and constitutes a direct link with the classical tradition.

\[Ahlfors-1953\]

Very important progress has also been made in the use of variational methods. I have frequently mentioned extremal problems in conformal mapping, and I believe their importance cannot be overestimated. It is evident that extremal mappings must be the cornerstone in any theory that tries to classify conformal mappings according to invariant properties.

\[Ahlfors-1958\]

Es ist mir zugefallen, eine Übersicht über die Extremalprobleme in der Funktionentheorie zu geben. Seit der Formulierung des Dirichletschen Prinzips ist es klar gewesen, dass die Cauchy-Riemannschen Gleichungen nichts anderes sind als die Eulerschen Gleichungen eines Variationsproblems, und in diesem Sinne ist alle Funktionentheorie mit Extremaleigenschaften verbunden. Aber es ist nicht immer von vornherein klar, wie diese Probleme gestellt werden sollen, damit sie in wesentlicher Weise die tiefen Eigenschaften der analytischen Funktionen abspiegeln. Es gibt natürlich unzählige Maximaleigenschaften, etwa in der konformen Abbildung, die ganz nahe an der Oberfläche liegen. Von da aus soll man zu schwierigeren Problemen aufsteigen. Das geschieht nicht etwa so, dass man ein beliebiges, wenn auch verlockendes, Extremalproblem ins Auge fasst und es zu lösen versucht. Im Gegenteil, die Entwicklung ist so vor sich gegangen, dass man die Aufgaben stellt, die man lösen kann. Dadurch ist ein reiches Erfahrungsmaterial entstanden, und die Aufgabe des heutigen Funktionentheoretikers besteht darin, dieses Material zu klassifizieren und dadurch weiter zu entwickeln.

\[…, and on page 7, of the same philosophical paper\]

Carathéodory sagte einmal, dass er immer wieder zur Funktionentheorie zurückkehrt, weil man gerade dort die verschiedensten und verblüffendsten Methoden verwenden kann. Das ist sicher wahr, und eben deshalb ist die Funktionentheorie kein eng spezialisierter Zweig der Mathematik. Im Gegenteil, die Funktionentheorie scheint fast wie ein Miniaturbild der gesamten Mathematik, denn es gibt kaum eine Methode in der Geometrie, der Algebra und der Topologie, die nicht früher oder später in der Funktionentheorie wichtige Anwendung findet. \[…\]

Such wisdoms cultivating the extremal philosophy—in particular as a growing mode for conformal mappings—presumably capture the deepest telluric part of the mushroom, out of which everything derives effortlessly. Alas, our survey is far from this ideal conception. In fact, we would be quite challenged if we were demanded to list a single application of Ahlfors’ extremal property, except of course in the planar case where one can easily mention all the activities centering around Painlevé’s problem.

Applications
------------

The writer’s interest in the topic was recently revived by the article of Fraser-Schoen 2011 [@Fraser-Schoen_2011], where the Ahlfors function received a clear-cut interaction with spectral theory (Steklov eigenvalue) with a view toward minimal surfaces.

At a more remote period of time, in the early 1950’s, when classification theory of open Riemann surfaces was a hot topic (especially in the Finnish and Japanese schools), Kusunoki 1952 [@Kusunoki_1952] proposed an application to the type problem, in the analytic sense of Nevanlinna’s Nullrand (null boundary). A (somewhat misleading but frequently used) synonym is [*parabolic type*]{} (not to be confused with the geometric sense of uniformization theory). This (analytic) sense of parabolicity is the one related to the transience of the Brownian motion (Kakutani, etc.)

In view of the extremal rôle played by the (round) hemisphere as a vibrating membranes (compare Hersch 1970 [@Hersch_1970], and less relevantly Gabard 2011 [@Gabard_2011]), the author speculatively expected—yet failed dramatically to establish (Summer 2011)—the following:

[(Gabard, April 2011, ca. 300 pages of sterile hand-written notes, unpublished)]{} There is a mysterious connection between the Ahlfors function and the (still open) [*filling area conjecture*]{} (FAC) of [Gromov 1983 [@Gromov_1983]]{}, whose genus zero case follows from the Thesis of [Pu 1952 [@Pu_1952]]{}, under Loewner 1949. More precisely, the filling area conjecture is true for all genus $p\ge 0$, and the proof will employ an Ahlfors map, at least as one of the ingredients \[others being Schwarz’s inequality, and group theoretical tricks à la Hurwitz–Haar–Loewner like in the $p=0$ case\]. The basic link is of course that conformal maps supply isothermic coordinates, yielding a way to compute areas via the infinitesimal calculus (of Newton–Leibniz, etc.).

The best available result on FAC is still the hyperelliptic case handled by Bangert-Croke-Ivanov-Katz 2004 [@Bangert_2004], implying the full conjecture for $p=1$ (as in this case the double is of genus $g=2$, hence automatically hyperelliptic). Remember the formulation of the FAC problem: among all compact bordered (orientable?) Riemannian surfaces bounding the circle without shortening its intrinsic distance, the round hemisphere has the least possible area.

The above “Ahlfors$\Rightarrow$Gromov” conjecture flashed my attention, after completing the note (Gabard 2011 [@Gabard_2011]) in view of the striking analogy between the isoperimetric rôle of the hemisphere both acoustically (spectral theory, like in Hersch 1970 [@Hersch_1970]) and geometrically in the Löwner-Pu-Gromov isosystolic ($\approx$filling) problem. Of course this analogy is already explicit in Gromov 1983 [@Gromov_1983], where Hersch 1970 () is cited. Incidentally, Gromov’s account also let play to Jenkins, Ahlfors’ student and Grötzsch’s admirator, a predominant logical rôle via the notion of “extremal length”. After more immature thinking (August 2012), it seems safer to formulate a relaxed version of the conjecture where the impulse does not necessarily come from the Ahlfors map but from some more ancestral source like the Green’s function (or the allied Gauss-Riemann isothermic coordinates). Also the (Lorenz-)Weyl’s asymptotic law enabling to “hear” the area of a drum from high-vibratory modes could be involved as well in FAC. When Marcel Berger describes Gromov’s systolic exploits (1983 ), he insinuates (surely with right) of them as lying at a much higher level of sophistication than 2D-conformal geometry (à la Gauss-Riemann, etc.). This acts as an optimism killer against anything like the above conjecture. Of course our conjecture or its relaxed variant “Conformal$\approx$Isothermic$\Rightarrow$Gromov” is far from prophetical, but only the expectation that the traditional methods (conformal theory and uniformization) which settled low-genus cases (Loewner 1949, Pu 1952 [@Pu_1952]) will extend soon or later to $p\ge 2$. Yet, who knows?

Remember that even Marcel Berger, once validated (or at least quoted) an erroneous proof (ca. 1998) of the 2D-case of the filling conjecture in question. Compare his brilliant “Panoramic view” (2002 [@Berger_2002-A-Panoramic-view-of-RG]), or rather his likewise excellent survey in JDMV (1998 [@Berger_1998-JDMV p.147]): “[*The simplest filling volume, namely that for the circle $S^1$, was only obtained in (\[N.\] Katz, 1998).*]{}”, where the reference is given as (cf. p.196) “[*Katz, N. (1998). Filling volume of the circle.*]{}” This work has apparently never been published and probably turned out to contain a gap. This reference is still quoted in the “Panoramic view” (2002 [@Berger_2002-A-Panoramic-view-of-RG p.790]) modulo a puzzling shift of authorship from Neil N. Katz to Mikhail G. Katz: “Entry \[794\]=[M.G. Katz]{}, Filling volume of the circle, to appear, 1998.” In the text of “A Panoramic view…” this reference is apparently not cited, and at any rate on p.367 we read “[*Today there is not a single manifold whose filling volume is known, not even the circle (for which Gromov conjectures the value \[is\] $2\pi$).*]{}”

Of course, probably no better guide than Ahlfors himself for listing applications of his method would have been desired. Alas it seems that the latter was suddenly sidetracked in the stratosphere of Teichmüller theory in the early 1950’s, leaving the Ahlfors map topic in some standby “in absentia” status. An exception is the later paper Ahlfors 1958 [@Ahlfors_1958], where Ahlfors discusses again extremal problems, though in a more philosophical way. Also the work of his student Read 1958 [@Read_1958_Acta] is described, which supplies another existence-proof of circle maps via a more abstract viewpoint (Hahn-Banach) inspired by other works like Macintyre-Rogosinski 1950 [@Macintyre-Rogosinski_1950], Rogosinski-Shapiro 1953 [@Rogosinski-Shapiro_1953], Rudin, etc. This Teichmüller shift in Ahlfors activities seems to coincide with the 100 years celebration of Riemann’s Thesis (in 1951), where L. Bers cames up with his list of urgent questions about Riemann surfaces. As a partial consolation, Grunsky worked out a brilliant book (1978 [@Grunsky_1978]) where much of the historical continuity is supplied.

[*Quoting some first-hand sources.*]{}— We shall have to reproduce several quotations from primary sources as an attempt to observe the mutual influences among the variety of viewpoints. It resulted some inflation in size, but hopefully excusable as the information of some relevance to our topic is otherwise dispatched through a vast amount of literature. Those are given in the self-explanatory format [**Quote (Author, year)**]{}.

Beside the historical aspect, the ultimate desideratum would be a logically optimal reconstruction of the theory. In the case at hand, our first task was just to list, locate (and sometimes understand) the several arguments already available.

—We shall essentially touch the following aspects (all in reference to the Ahlfors mapping):

\(1) Origins, background: prehistory of Ahlfors (Sec.\[Sec:Prehistory-Ahlfors\]); potential precursors (Sec.\[Sec:Precusors\]);

\(2) How the writer came across this topic? (via Klein); cf. Sections \[Sec:Klein\] and \[Sec:Biased-recollections-of-Gabard\];

\(3) Potential theory vs. extremal problems (both from the same variational soup);

\(4) Applications (Sec.\[Sec:Applications-of-the-Ahlf-map\]): equilibrium of electricity Riemann 1857, Painlevé’s problem, type problem, Carathéodory metric, corona problem, quadrature domains, spectral theory (Steklov or Dirichlet-Neumann);

\(5) Open problems fictionally related to the Ahlfors function (Sec.\[Sec:Virtual-applications-Ahlf-map\]);

\(6) (Partial) assimilation of Ahlfors or other works (logical reconstruction); via Green in Sec.\[Green:sec\] and via Ahlfors in Sec.\[Ahlfors-proof:sec\];

\(7) Sharpening Ahlfors work (for circle maps not necessarily subjected to the extremal problem).

Roughly speaking our text splits as follows. A first half is devoted to historical aspects, while a second half (initiated by Sec.\[Sec:Starting-from-zero\] titled “Starting from zero knowledge”) is more “logical”, or rather liberal and futurist. This second part tries to explore what sort of mathematics lies beyond Ahlfors theorem. Of course it is hard going beyond Ahlfors without having digested his own work, and consequently much energy is spent to the original account. His result affords considerable information, especially the realizability of all gonalities lying above Ahlfors bound $r+2p$. (The [*gonality*]{} $\gamma$ is the least degree of a circle map tolerated by the given bordered surface.) Classically, some (episodic) penetrations beyond Ahlfors occurred by Garabedian, Heins, Royden, etc., and more recently in the spectacular progresses made by Yamada, Gouma on the extremal function. In the dual direction (of circle maps), Coppens’ work on the gonality is likewise penetrating deep behind the line fixed by Ahlfors, and raises several questions of primary importance. This includes that of describing how the moduli space of bordered surfaces (with fixed topological type $(r,p)$) stratifies along gonalities. Calculating dimensions of the varied strata is a first step toward quantifying by how much and how frequently one can expect to improve Ahlfors bound. We obtain so the [*gonality profile*]{}, that is, the function assigning to each gonality $\gamma$ (in the Coppens range $r\le
\gamma \le r+p$, or outside it in case Gabard is wrong) the dimension of the moduli strata with prescribed gonality $\le
\gamma$ (Section \[sec:profile-histogram\]). Describing this gonality profile appears to me a challenging (but hopefully reasonably accessible) problem. Another “futurist” problem is the one of describing the list of all degrees of circle maps tolerated by a given surface. This we call the [*gonality sequence*]{}. It is full above Ahlfors bound $r+2p$, but what can be said below? These are perhaps two typical kind of problems hinting at what sort of games we may encounter “beyond Ahlfors”.

It seems also evident that a good understanding of (real) algebraic geometry could shed some light upon Ahlfors viewpoint. This interaction is alas quite rare to observe in the twenty century literature which is mostly split into specialized viewpoints.

Bibliographic and keywords chart
--------------------------------

The following chart (Fig.\[Map:fig\]) focuses on the tabulation of several articles where an existence-proof of Ahlfors circle maps is given. Such items are marked by full black circular symbols with eventual decorations. Applications are marked by triangular symbols. All entries of the picture (e.g. “Ahlfors 1950”) can unambiguously be located in the bibliography at the end of the paper. One counts essentially ca. 13 papers addressing the existential question of circle maps. Those includes: Ahlfors 1950 [@Ahlfors_1950], Garabedian 1950 [@Garabedian_1950], Heins 1950 [@Heins_1950], 1975 [@Heins_1975], 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF] and in the same spirit Forelli 1979 [@Forelli_1979]. Another trend is Nehari 1950 [@Nehari_1950] and Tietz 1955 [@Tietz_1955] (alas those works are a bit confusing, Tietz criticizes Nehari and is in turn attacked subsequently in Köditz-Timmann 1975 [@Koeditz-Timmann_1975]). The latter work (KT1975) actually offers an alternative existence-proof without degree control. In Japan we have Kuramochi 1952 [@Kuramochi_1952] and Mizumoto 1960 [@Mizumoto_1960]. (One should probably add several works of Kusunoki from the early 1950’s, but those are often confusing with subsequent errata, etc.) Another mouvance is the usage of Hahn-Banach in the papers Read 1958 [@Read_1958_Acta] and Royden 1962 [@Royden_1962]. Finally there is a work by the writer, Gabard 2006 [@Gabard_2006], which even claim a better control $r+p$ upon the degree of circle maps. Of course this work should still be better understood and its result should be either disproved or consolidated by alternative techniques.

To this obvious list one can add some more telluric flows or possible forerunners:

$\bullet$ Teichmüller’s claim (1941 [@Teichmueller_1941]) that everything is already in Klein.

$\bullet$ Courant’s works starting say with Courant 1939 [@Courant_1939] where a Plateau-style approach à la Douglas is asserted to reproduce the Bieberbach-Grunsky “schlichtartig” case of Ahlfors.

$\bullet$ Italian workers: Matildi 1945/48 [@Matildi_1945/48] and Andreotti 1950 [@Andreotti_1950].

If one has a good memory or a glouton working mode it is rather impressive to see the high level of branching and complexity of several neighboring fields. It will be soon flagrant that the writer’s competence will quickly break down in front of our too ambitious survey project. Again the writer would be enormously acknowledgeable if some specialists wants to tell their own vision of this tentacular circle of ideas.

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—Let us now put Ahlfors 1950 [@Ahlfors_1950] at the center of the universe, while trying to describe the portion of the cosmos visible from this perspective. Picturing in the non-Euclidean crystal, we obtain something like the following chart of keywords (Fig.\[Keyword:fig\]): a nebulosity of sidereal dusts gravitating in the immediate conceptual vicinity of the Ahlfors map.

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Mathematical questions {#sec:question}
----------------------

In this section we collect questions raised by our text. Most of our questions are of the retrograde sort “Can we reprove Ahlfors via …”, yet striving toward a perfect crystallography, where each result of the theory is certified by all methods ever imagined (compare optionally the kaleidoscopic Fig.\[Kaleidoscope:fig\] much below).

$\bullet$ [**Klein $\Rightarrow$ Ahlfors?**]{} \[reported 04.11.12\] Is it possible to reprove existence of Ahlfors circle maps via Klein’s Rückkehrschnitttheorem (RST) (cf. Klein 1882 [@Klein_1882_Ruckkehrschnitt] or Klein 1923 Ges. Math. Abh. III [@Klein-Werke-III_1923 p.622–626])? This paradigm RST may be conceived as a positive genus case of the Kreisnormierung (of Koebe, but implicit in the Latin version of Schottky’s Thesis, cf. Klein’s Quote \[Klein-1923:quote:Riemann-1858\]). Further recall that Riemann (1857 [@Riemann_1857_Nachlass]) was able to produce circle maps for domains bounded by circles, and by analogy it seems plausible that Klein’s RST implies (modulo some work à la Riemann) the Ahlfors circle map. Of course Klein himself may not have been able to prove rigorously his RST, but the result was completed via some Brouwer-Koebe techniques ca. 1911/12 [@Klein-Brouwer-Koebe_1912]. (For a few more details about this strategy, cf. Sec.\[sec:Ruckkehrschnittthm\].) \[18.11.12\] An allied historical question is whether Teichmüller’s accreditation to Klein (1941 [@Teichmueller_1941]) of circle maps is based on the same stratagem (RST) as we are just suggesting.

$\bullet$ [**Witt or Geyer $\Rightarrow$ Ahlfors?**]{} Can we reprove the theorem of Ahlfors via a purely algebraic method (say Abel, or Riemann-Roch) as Witt 1934 [@Witt_1934], Geyer 1964/67 [@Geyer_1964-67] or Martens 1978 [@Martens_1978] succeeded to do for the Witt mapping (of 1934)? For more on this, cf. Sec.\[sec:Witt\].

$\bullet$ [**Plateau $\Rightarrow$ Ahlfors?**]{} Can we reprove the theorem of Ahlfors via the method based on the Plateau problem (as Courant 1939 [@Courant_1939] did for the Riemann-Schottky-Bieberbach-Grunsky theorem, i.e. the schlichtartig case $p=0$ of Ahlfors). (See Sections \[sec:Courant\] and \[sec:Douglas\] for historical precedents (i.e., Douglas 1931 [@Douglas_1931-Solution]), and precise references about contemporary workers attacking related questions (Jost, Hildebrandt, von der Mosel). A closely related historical question is whether the works of Courant do not already contain (more-or-less explicitly) an existence-proof of Ahlfors circle maps.

$\bullet$ [**Bergman $\Rightarrow$ Ahlfors?**]{} Idem via the method of the Bergman kernel function. This seems implicit in the literature (say especially by Bell, e.g. Bell 2002 [@Bell_2002], the great specialist of the technique), but to the writer’s knowledge no pedestrian account is available to the mathematical public (in the positive genus case). Compare Sec.\[sec:Bergman\] for some links to the literature. Of course behind Bergman 1922 [@Bergman_1922] one finds Bieberbach’s characterization (1914 [@Bieberbach_1914]) of the Riemann map via an extremal problem involving least area. This problems should be in some duality with Ahlfors extremal problem, more about this soon.

$\bullet$ [**Behnke-Stein $\Rightarrow$ Ahlfors?**]{} \[reported 05.11.12\] The article (of Köditz-Timmann 1975 [@Koeditz-Timmann_1975 Satz 3, p.159]) seems to contain a qualitative version of Ahlfors’ theorem based upon an “Approximationssatzes von Behnke u. Stein”, yet without any bound on the degree. Can one improve the argument to get a quantitative control? As to Behnke-Stein 1947/49 [@Behnke-Stein_1947/49] (the famous paper going back to 1943), it contains the result that any open Riemann surface (arbitrary connectivity and genus) admits a non-constant analytic function. Is it possible conversely to deduce this theorem from Ahlfors theorem by exhaustion while pasting together various circle maps defined over a system of expanding compact subregions?

$\bullet$ [**Other techniques?**]{} Koebe’s iteration, circle packings (cf. Rodin-Sullivan 1987 [@Rodin-Sullivan_1987]), Ricci flow, etc. Virtually any technique involved in the proof of the RMT (=Riemann mapping theorem) or the allied uniformization is susceptible to reprove the Ahlfors circle map.

$\bullet$ [**Does Ahlfors imply Ahlfors?**]{} \[02.09.12\] This repetition is intentional and intended to emphasize that the writer was not able to digest Ahlfors argument in full details (compare Sections \[Green:sec\] and \[Ahlfors-proof:sec\]). If one remembers the proof of Koebe’s Kreisnormierung (say as implemented in Grunsky 1978 [@Grunsky_1978] or Golusin 1952/57 [@Golusin_1952/57]), then upon making abstraction of Koebe’s proof by iterative methods, it may be noticed that ultimately the proof depends on a topological principle (namely Brouwer’s invariance of domain). In comparison, Ahlfors’ proof of a circle map (1950 [@Ahlfors_1950]) makes no use of any topological principle, reducing rather to considerations of convex geometry (cf. Ahlfors 1950 [@Ahlfors_1950]). Should one deduce that the Ahlfors function lies somewhat less deep than Koebe’s Kreisnormierung? If not then maybe Ahlfors’ argument lacks a global topological character, and perhaps its validity needs to be reevaluated. (Of course this is only a superficial objection arising from my own frustration in not being able to catch the substance of Ahlfors text.)

$\bullet$ [**Does Brill-Noether ($+$ Harnack’s trick) imply Ahlfors?**]{} \[26.10.12\] Upon using projective models of Riemann surfaces, especially birational models in the plane, it is common practice to understand the geometry on a curve via auxiliary pencils living on the ambient plane. Of particular importance are the so-called adjoint series passing through the singularities of the model which have the distinctive feature of cutting economical series of points on the curve. Such pencils are thus involved in the description of low-degree pencils living on the (abstract) smooth curve, hence morphisms to the line. Adapting this methodology to orthosymmetric curves one can evidently hope to reprove Ahlfors theorem, provided one is able to ensure total reality of the corresponding morphism. Details look quite formidable to implement. If such a proof exists it will probably be a happy hour for its discoverer. For more vague ideas about this strategy, see Sec.\[sec:Brill-Noether-approach-to-Ahlfors\].

$\bullet$ [**Does Ahlfors imply Gabard?**]{} \[09.09.12\] Upon using Ahlfors’ original argument in [@Ahlfors_1950] for the existence of a circle map of degree $r+2p$, it seems evident that one could append to Ahlfors argument a sharper geometric lemma which could produce a better control than Ahlfors’. Ideally one would like to recover Gabard’s bound $r+p$. For some evidence of why this should be possible compare Sec.\[Red’s-function:sec\].

$\bullet$ [**Gabard true? If, yes analytifiable?**]{} \[June 2012\] Is the bound $r+p$ predicted by the writer on the degree of a circle map true? And if yes is it accessible to more conventional analytical methods? Remember that the derivation in Gabard use some topological methods combined with the classical Abel theorem.

$\bullet$ [**Gonality profile.**]{} \[June 2012\] Can we compute the dimension of the moduli spaces of membranes having fixed gonality $\gamma\le r+p$. (The [*gonality*]{} is the least degree of a circle map from the given bordered surface.) The similar question in the case of complex curves is well-known and easily predicted by a simple Riemann-Hurwitz count (but established rigorously much later). Slightly more on this in Sec.\[sec:profile-histogram\].

$\bullet$ [**Ahlfors extremals as economic as Gabard?**]{} \[March 2012\] Can the degree of the Ahlfors extremal function be made as economical as $r+p$, the circle map degree predicted by the writer, for a suitable location of the two points required to pose the extremal problem (resp. of a single point when considering the derivative maximizing variant of the problem)?

$\bullet$ [**Ahlfors extremals as super-economic as Coppens?**]{} \[March 2012\] Same question for the sharper [*(separating) gonality*]{} introduced by Coppens 2011 [@Coppens_2011], that is, the minimum sheet-number required to concretize the bordered Riemann surface as a (holomorphic) branched cover of the disc.

$\bullet$ [**Topology$\Rightarrow$Riemann-Meis complex gonality?**]{} \[21.06.12\] Can the topological method (irrigation) used in Gabard 2006 [@Gabard_2006] be adapted to prove that any complex curve of genus $g$ is $\le [\frac{g+3}{2}]$-gonal, meaning that there is always a morphism to ${\Bbb P}^1$ of degree $\le$ than the specified bound. (Perhaps this is already answered in the lectures of Gunning 1972 [@Gunning_1972], who uses Mattuck’s topological description of the symmetric powers of the curve).

Conversely, there is a dual problem:

$\bullet$ [**Grötzsch-Teichmüller-Meis$\Rightarrow$Ahlfors-Gabard separating gonality?**]{} \[16 June 2012\] According to secondary sources (e.g. Kleiman-Laksov 1974 [@Kleiman-Laksov_1974]), Meis’ proof (1960 [@Meis_1960]) of the complex gonality $\le
[\frac{g+3}{2}]$ of genus-$g$ curves, is eminently Teichmüller-theoretic. By analogy, it should therefore be possible to prove the $(r+p)$-gonality of membranes (cf. Gabard 2006 [@Gabard_2006]) by using the same (Teichmüller-style) method as Meis. This would incidentally give an “analytic” proof (or if you prefer, a “geometria magnitudinis” proof of Gabard 2006 [@Gabard_2006]). Notice the fighting interplay between topology and analysis (or geometry) since Teichmüller amounts essentially to the “möglichst konform” map of Grötzsch.

$\bullet$ \[05 June 2012\] Ozawa 1950 [@Ozawa_1950] presents a genuine extension of the Schwarz lemma to multiply-connected domain. Can we do the same job for a membrane of positive genus?

$\bullet$ [**Ahlfors$\Rightarrow$Gromov?**]{} \[Mai 2011\] Does Ahlfors (or perhaps the non-orientable variant of Witt 1934 [@Witt_1934]) implies Gromov’s filling area conjecture? Any solution to this puzzling problem is rewarded by 50 Euros by Mikhail Katz (cf. his home web-page). Perhaps, some other ingredients than Ahlfors are required. We (already) loosely suggested, Weyl’s asymptotic law (acoustic proof) or perhaps a sort of duality between “Ahlfors” extremal problem and that of Bieberbach 1914 [@Bieberbach_1914] (more widely known for its connection to Bergman). Added \[02.09.12\], maybe it is enough to consider the isothermic coordinate generated by a single Green’s function (or a dipole avatar) instead of an Ahlfors function.

$\bullet$ [**Gromov non orientable**]{} (Easier?) \[June 2011\] Is the Gromov filling conjecture also true (and meaningful) for non-orientable membranes? Can it be generalized to several contours (desideratum J. Huisman 2011, oral e-mail communication).

We may also drift to related problems like KNP (Kreisnormierungsprinzip). This asserts that any domain (or planar Riemann surface) is conformally diffeomorphic to a domain bounded by circles (we suppose finite connectivity for simplicity).

$\bullet$ [**Extremal problem$\Rightarrow$KNP?**]{} Inspired by the paper Schiffer-Hawley 1962 [@Schiffer-Hawley_1962], where (Koebe’s) Kreisnormierung (in finite connectivity) is derived from a minimum problem of the Dirichlet type, one may wonder if a suitable variant of Ahlfors extremal function may not be used to reprove the Kreisnormierung. More about this is Sec.\[sec:KNP\] (related to works by Grötzsch, and others.).

$\bullet$ [**Bieberbach’s (least area) minimum problem.**]{} Bieberbach 1914 [@Bieberbach_1914] considers in a simply-connected domain $B$ the problem of minimizing the integral $\int\!\! \int_B \vert f(z)\vert^2 d\omega$ amongst analytic functions $f\colon B \to {\Bbb C}$ normed by $f'(t)=1$ at some fixed point $t\in B$ of the domain. He shows that the minimum gives the Riemann map. (It is well-known that this problem constitutes the origin of the Bergman kernel theory, cf. besides Bergman’s original paper of 1922 [@Bergman_1922], e.g. Behnke’s BAMS review of Bergman’s 1950 book [@Bergman_1950].) The naive question is what sort of maps are obtained when this problem is formulated on a multiply connected domain? Do we obtain a circle map? And if yes, does this $\beta$-function coincides with the Ahlfors map? Can the problem be extended to Riemann surfaces? More on this is discussed in Sec.\[Sec:Bieberbach-Bergman\]. Of course this is closely allied to the Bergman kernel, and was treated by several authors, cf. e.g. Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950]. However as far as the writer browsed the literature, the qualitative feature of this $\beta$-map appear to have not been explicitly described. In fact it seems that ultimately the answer is a bit disappointing in the sense that the least-area map may lack single-valuedness. This is well-explained in papers by Maschler (1956–59, e.g. [@Maschler_1956]), and was probably known earlier by Bergman, Schiffer, etc.

$\bullet$ [**Heins’ proof?**]{} \[28.06.12\] Heins 1950 [@Heins_1950] proposes another existence-proof of circle maps à la Ahlfors, by using some theory of Martin and concepts from convex geometry (minimal harmonic functions and extreme points of convex bodies). Unfortunately, he does not keep a quantitative control upon the degree of the map so obtained. However, on p.571 Heins introduces the number $m$ (of loops generating the fundamental group), which is easily estimated as $2p+(r-1)$ for a surface of genus $p$ with $r$ contours. \[E.g., imagining contours as punctures, the first perforation liberates a free group of rank $2p$ (twice the genus), and each additional perforation creates a new generator.\] Since this must be augmented by one (cf. Heins’ lemma on p.568, i.e. essentially the issue that each point of a convex body in Euclidean $m$-space is expressible as a barycentric sum of $m+1$ extreme points of the body spanning an $m$-simplex) it seems probable that Heins’ proof reproduces the bound $r+2p$ of Ahlfors. More about this in Sec.\[sec:Heins\]. (Actually, Heins’ convex geometry argument looks quite akin to the one used “subconsciously” by Ahlfors 1950 [@Ahlfors_1950].)

$\bullet$ \[22.10.12\] [**The gonality sequence.**]{} An emerging question of some interest is that of calculating for a given bordered surface $F$ (of type say $(r,p)$) the list of all integers arising as degrees of a circle map defined on the given surface. We call this invariant the [*gonality sequence*]{} of $F$. As a noteworthy issue Ahlfors upper bound $r+2p$ is always effectively realized, in sharp contrast to Gabard’s one $r+p$ which can fail to be. For some messy and premature thoughts on this problem cf. Sec.\[sec:gonality-sequence\]. Of course the problem looks a bit insignificant combinatorics, yet studying it properly seems to require both experimental contemplation of concrete Riemann surfaces and sharp theoretical analysis of the existence-proofs available presently. Asking fine quantitative questions should aid clarifying the qualitative existence theorems.

$\bullet$ \[03.11.12\] [**Generalized Keplerian motions via Klein-Ahlfors?**]{} It is well known that the motion of a single planet around a star describes an orbit which is a certain algebraic curve, namely an ellipse (other conics do occur for cold comets escaping at infinity without periodicity). To visualize Ahlfors circle maps on real plane algebraic curves of dividing type (Klein’s orthosymmetry), one can contemplate totally real pencils of curves sweeping out the given curve along totally real collections of points. The prototypical example is the Gürtelkurve (quartic with two nested ovals) swept out by a pencil of lines whose center of perspective is located inside the deepest oval. All such lines cut the quartic in 4 [*real*]{} points (cf. Fig.\[FGuert:fig\]). This paradigm of total reality is the exact algebro-geometric pendant of Ahlfors theorem, and suggests looking at real dividing curves as orbits of planetary systems with dynamics governed by a total pencil. For instance the Gürtelkurve could occur as the orbit of a system of 4 electrons gravitating around a proton with electric repulsive forces explaining the special shape of the Gürtelkurve (cf. again Fig.\[FGuert:fig\] below). In Sec.\[sec:electrodynamics\] we explore the (overambitious?) idea positing that the real locus of [*any*]{} real orthosymmetric curve (in the Euclid plane or space) arises as the orbital structure of an electrodynamical system obeying Newton-Coulomb law’s of attraction/repulsion via a dynamics controlled by an Ahlfors circle map (incarnated by a totally real pencil). This gives quite an exciting interpretation affording plenty of periodic motions to the $n$ body problem. This idea probably requires to be better analyzed. Even if physically irrelevant, one can (by Ahlfors) trace for any orthosymmetric real curve (in the plane) a totally real pencil generating usually quite intriguing figures, especially when members of the pencil are varied through the full color spectrum to create some rainbow effect. Depictions of such totally real rainbows are given in Sec.\[sec:electrodynamics\], but we failed drastically to make serious pictures for Harnack-maximal curves. This represents perhaps a certain challenge for computer graphics?

$\bullet$ \[27.12.12\] [**Green-Riemann imply Schoenflies?**]{} This question is quite outside our main track of 2D-conformal geometry, belonging really to highbrow unsettled differential topology. Remember first that the Riemann mapping theorem (and the closely allied Green’s function measuring the proliferation of a bacteria in a nutritive medium) is essentially the best approach toward the (topological) Schoenflies theorem (ca. 1906) stating that any plane Jordan curve bounds a disc. Compare the contributions of Osgood and Carathéodory 1912 [@Caratheodory_1912], plus the recent discussion in Siebenmann 2005 [@Siebenmann_2005]. When it comes to high-dimensional versions of Schoenflies, we know in the smooth category by combination of the topological version of Mazur-Brown with Smale’s $h$-cobordism theorem giving uniqueness of the smooth structure on high-dimensional balls ($\dim\ge 5$). However remind that presently differential-topologic methods failed to prove the (so-called) smooth Schoenflies conjecture in dimension $4$, that any smoothly embedded $S^3$ in ${\Bbb R}^4$ bounds a $4$-ball with its usual smoothness structure. It is tempting to wonder if the classical tools of potential theory (especially the Green’s function) are able to reprove at least the high-dimensional cases of Smooth-Schoenflies, and if so, if it is able to crack the residual remaining exceptional case resisting all efforts of topologists so far. More details and references in Sec.\[Schoenflies:sec\]. The intuition behind all this is that the bacteria expand from any given interior point as concentric circles (resp. spheres) in the infinitesimally small but soon realize where there is more free vital room for expanding more quickly in those directions (cf. Fig.\[Green:fig\]). In particular all bacteria reach the boundary spheroid simultaneously. Mathematically this is formalized by considering the Green’s function $G(z,t)$, where $z\in {\Bbb R}^n$ and $t$ is an interior point of the bounded component of $\Sigma$ the $S^{n-1}$ embedded in ${\Bbb R}^n$, defined as $\log \vert z -t
\vert - u$ where $u$ is the unique harmonic function with boundary values given by $\log \vert z -t \vert$ on the boundary $\Sigma$. Studying the Green’s lines,that is the trajectory orthogonal to the levels of $G(z,t)$ should (possibly) enable one to establish the required diffeomorphism between the (sealed) interior of $\Sigma$ with the ball $B^{n}$ endowed with its usual smooth structure. (It is unknown if $B^4$ supports an exotic differential structure but that this another question a priori much harder to decide.)

Some vague answers
------------------

This section tried to report question which looks exciting, and to which I tried some premature answer. It requires to be polished drastically and reorganized seriously. Hence it is probably safer to skip, but maybe readers fluent with techniques like Ahlfors extremals, Teichmüller extremal quasi-conformal maps, Plateau’s problem, etc. may find useful to clarify our vague ideas.

$\bullet$ [**Quantum fluctuations of Ahlfors’ degree**]{} \[20.09.12\] The following problem is somewhat ill-posed, yet it is just an attempt to excite the imagination. Suppose given a compact bordered Riemann surface $F$ with $r\ge 1$ contours and of genus $p\ge 0$. For each interior point $a\in \rm{int} (F)$ there is a uniquely defined analytic Ahlfors function $f_a$ solving the extremal problem of making the derivative $f'(a)$ as large as it can be, while keeping this magnitude positive real and the range inside the unit disc. This extremal function is uniquely defined and independent of the local uniformizer used to compute the derivative. It is known by Ahlfors 1950 that each $f_a$ is a circle map of degree somewhere in the range from $r$ to $r+2p$, that is a (surjective) branched cover of the disc. According to Coppens 2011 [@Coppens_2011] the generic bordered surface has gonality $r+p$ so that one can considerably squeeze the Ahlfors range to the interval $r+p$ to $r+2p$. One would like to understand in geometric term (if possible?) what phenomena is responsible of the fluctuation of the Ahlfors degree. Of course, if $p=0$ there is no fluctuation just because of the Ahlfors squeezing: i.e. $\deg f_a$ is constant when the center of expansion $a$ is dragged throughout the surface. However if $p>0$, it is likely that some jump must occur albeit I know no argument. Gabard 2006 only showed that there is a circle map of degree $\le
r+p$, but a priori there is no reason forcing such low degree maps to be realized as Ahlfors maps. Following Coppens we may define the gonality $\gamma$ of $F$ as the least degree of a circle map on $F$. By Gabard (2006 [@Gabard_2006]) $\gamma\le r+p$ (and trivially $r\le \gamma$). Coppens tell us that all intermediate values of $\gamma$ are realized (modulo the trivial exception that when $r=1$ and $p>0$, $\gamma=1$ cannot be realized). This gonality invariant infers a sharpened variability for the Ahlfors degrees, namely $r\le \gamma\le \deg f_a \le r+2p$, where $\gamma
\le r+p$. A priori all intermediate values could be visited (between $\gamma$ and $r+2p$). However this scenario is incompatible with the case of hyperelliptic membranes studied in Yamada and Gouma, where the effective Ahlfors degrees are either maximal $r+2p$ or minimal (i.e. $2$). Those examples still indicate that despite a sparse repartition the degree distribution is in some sense extremal, occupying the maximum space at disposition. Is this a general behavior? This is the maximum oscillation (Schwankung) conjecture (MOC). If true, then Coppens gonality would always be sustained by an Ahlfors map and also Ahlfors upper bound $r+2p$ would be sharp for any surface, whatsoever its differential-geometric granularity. MOC displays the most naive scenario for the fluctuation of Ahlfors degree, and it would be a little miracle if it is correct. If not, then what can be said? A very naive idea idea would be that there is a sort of conservation law like in the Gauss-Bonnet theorem: whatsoever you bend the surface the Curvatura integra keeps constant. (Of course this holds for a closed surface but not for a bordered one, unless the geodesic curvature of the boundaries is controlled, e.g. by making it null.) The vague idea would be that if we think of the Ahlfors degree $\deg f_a$ as a sort of discrete curvature $\delta(a)$ assigned to the point $a$ then maybe $\int_F \delta
(a) d\omega$ keeps a constant value (independent of the conformal structure). If so then at least in the cases where there is a hyperelliptic model (i.e. $r=1$ or $2$) one could conclude that the Ahlfors degree are somehow balanced. Yet recalling Yamada-Gouma’s investigations it seems that the maximum degree $r+2p$ occurs very sporadically for the center $a$ located on the finitely many Weierstrass points of the membrane, hence high values have little weight. So in the hyperelliptic case (with few contours $r=1$ or $2$) the Ahlfors degree are constantly very low $2$ with exceptional jump taking place on a finite set of points. Maybe this suggests a low energy scenario valid in general: given any (finite) bordered surface $F$ the Ahlfors degree is always equal to the gonality safe for some jump occurring on a finite set of points. Of course this must be perhaps refined suitably by saying that there is a stratification (decomposition) in pieces, where the lowest degree (i.e. the gonality) is always nonempty and containing the contours, and then as we penetrate more deeply inside the surface the degree may increase (eventually always reaching the extremum value $r+2p$?).

$\bullet$ [**Quasiconformal doodlings**]{} \[02.10.12\] As is well known, Teichmüller 1939 [@Teichmueller_1939] exploited the flexibility of quasiconformal maps to put Riemann’s intuition of the moduli of conformal classes of differential-geometric surfaces (Riemannian surfaces) on a sound footing. The idea is both soft and flexible, yet with the devil of capitalism (geometria magnitudinis) cached just behind for one counts the distortion effected upon infinitesimal circles into ellipses. Using Grötzsch idea of the möglischt konform map relating two configurations produces an extremal map relating both configurations, and this least distortion gives the Teichmüller metric (a first step to endow the moduli “set” of a genuine space structure). Maybe this methodology is also fruitful in the theory of the (Ahlfors) circle maps. The first desideratum is to show existence of circle maps, and then the game refines in finding best possible bounds (over the degree of such maps).

The framework is as follows (aping again Grötzsch-Teichmüller): given a finite bordered surface (and maybe also a mapping degree $d\ge r$) we look at all quasiconformal map (not necessarily schlicht), i.e. (full) branched cover of the disc (with the same topological feature as circle maps of taking the boundary to the boundary and the interior to the interior). Following Grötzsch’s idea we may look at the “möglischt konform” map, i.e. the most conformal quasiconformal map in the family (hoping eventually to find a beloved conformal one). Measuring distortion (largest eccentricity of the ellipses images of infinitesimal circles) one gets a numerical invariant $\varepsilon(F, d) \ge 0$, namely the infimum of the dilation among the class of all (differentiable) maps from the bordered surface $F$ to the disc. This invariant $\varepsilon(F, d)$ vanishes precisely when $F$ admits a (conformal) circle-map of degree $d$. Hence it vanishes if $d\ge
r+2p$ by Ahlfors 1950 [@Ahlfors_1950 p.124–126], and even as soon as $d\ge r+p$ if one believes in Gabard 2006 [@Gabard_2006], where as usual $p$ is the genus and $r$ the number of contours of $F$. However we are rather interested to use the Grötzsch-Teichmüller theory to rederive an independent existence-proof. Of course in contrast with the classical setting of Teichmüller’s approach to the moduli problem, where one considers exclusively schlicht(=injective) maps, we tolerate now multivalent mappings, but this should not be an insurmountable obstacle.

Our intuition is that it is not just a matter of measuring that is required, but one must somehow explore the pretzel underlying the surface to get an existence proof. Yet the flexible-quantitative viewpoint of measuring eccentricity probably gives an interesting numerical invariant which is now not a metric (Teichmüller metric), but rather a (potential) function on the moduli space. In fact we assign to a given (bordered) surface $F$ a series of number $\varepsilon(F,d)$ for $r\le d\le r+p$ (larger values of $d$ give $0$ by Gabard 2006 [@Gabard_2006]), which is probably decreasing (after eventually modifying the original problem by permitting all maps of degree $\le d$ instead of those having degree exactly $d$). So we get attached to $F$ a series of dilations $\varepsilon(F,r)\ge\varepsilon(F,r+1)\ge \dots \ge
\varepsilon(F,r+p)=0$. Of course the sequence can crash to zero before the $r+p$ bound and indeed do so as soon as Coppens’ gonality $\gamma$ is reached (that is, the least degree of a circle map for the fixed $F$). \[Of course in the exact degree $d$ variant of the problem one can imagine more romantic behaviors with oscillation down to zero and then becoming positive again (touch-and-go phenomenology).\] Those $p$ invariants would refine Coppens gonality in a continuous fashion, yet fails to be “moduli” since there are $3g-3$ of them (Riemann-Klein) where $g$ is the genus of the double (that is $2p+(r-1)$), hence giving a total of $3g-3=3(2p+(r-1))-3=6p+3r-3$ free parameters which exceeds of course our $p$ parameters.

But coming back to the basic existence problem, one can get started by observing that any topological type of membrane admits a circle map. One trick is to use symmetric membranes (cf. Chambéry section \[sec:Chambery\] below). This amounts to imagine a membrane in $3$-space symmetric under rotation by 180 degree so that the quotient as genus zero (cf. Fig.\[Chambery:fig\] below). Once the handles are killed one is reduced to the simple (planar) case of Ahlfors due to Bieberbach-Grunsky (and largely anticipated by Riemann, Schottky (no bound by Schottky?), and Enriques-Chisini (via Riemann-Roch and a continuity argument, cf. e.g. Gabard 2006 [@Gabard_2006 Sec.4]). The degree of the resulting map is easily computed (and of degree essentially equal to $(r/2) \cdot
2=r$ the minimum possible value, for the rotation identifies pairs of contours and gyrate all handles over themselves, cf. again Fig.\[Chambery:fig\], below). Thinking in the moduli space $M$ we have shown that the set $C$ of all circle-mappable surfaces is nonempty, and using the connectedness (of $M$) it would suffice to show that $C$ is [*clopen*]{} (i.e., closed and open). Checking openness, certainly requires enlarging the mapping degree to larger values. Now given an arbitrary bordered surface $F$ we can quasiconformally map it to our symmetric model $S$ and then compose with the circle-map. The dilatation is then controlled in term of the Teichmüller distance from $F$ to $S$, giving an upper bound over the eccentricity invariant $\varepsilon$ (for the appropriate degree). Of course this is still miles away from reproving even Ahlfors but maybe the idea is worth pursuing.

In fact what is truly interesting is that we get for each $d$ a numerical function $\varepsilon_d$ (defined as $\varepsilon_d(F):=\varepsilon(F,d)$) on the moduli space $M_{p,r}$ of membranes of genus $p$ with $r$ contours, that vanishes precisely when $F$ has gonality $\le d$. Of course this sequence of functions is monotone decreasing when the index increases, and $\epsilon_d\equiv 0$ is identically zero (for $d\ge
r+p$). According to Coppens result each of these functions (let us call them the Teichmüller potentials) vanishes somewhere. It is then perhaps interesting to look at the gradient flow $\varphi_d$ (w.r.t. Teichmüller metric) of these functions $\varepsilon_d$ affording a dynamical system (=flow) in which each bordered surface evolves in time to a sort of best possible surfaces for the prescribed gonality. (Morally each surface tries to improve its gonality along the trajectory of steepest descent.) If the global dynamics is simplest (say each trajectory finishes its life on a surface of gonality $d$) it is therefore reasonable to expect that the whole Teichmüller space is retracted by deformation to a sort of spine consisting of surfaces having the prescribed gonality $d$. Maybe one can deduce that the global topology of this spine is that of a cell (like the full Teichmüller space). Further it seems probable that the flows preserve the stratification by the gonality of $M_{p,r}$ since if $F$ has gonality say $d$ then its future $F_t$ has lower gonality. \[The situation looks analog to some works of René Thom (isotopy lemma, vector fields preserving a stratification, and “fonction tapissante” as it arise in the Thom-Mather problem of the stability of polynomial mappings??\]

\[03.10.12\] Of course the above can be adapted to the case of closed (non-bordered) surfaces of genus say $g$, by replacing the target disc by the (Riemann) sphere. Likewise we define Teichmüller potentials $\varepsilon_d$, measuring the dilatation of the “möglichst konform” map of a fixed degree $d$ from the surface $F$ to $S^2$, and ideally one can imagine that the theory is able to reprove the famous (Riemann-Brill-Noether) bound $[\frac{g+3}{2}]$ first proved by Meis 1960 [@Meis_1960]. Hence all what we are trying to do is surely already well-known (alas I was never able to find a copy of Meis’ work, which is Teichmüller-theoretic according to other sources). Hence if Meis theory is just a sort of Teichmüller theory for branched covers of the sphere, with the ultimate miracle that Teichmüller not only affords a solution to Riemann’s moduli problem but also to the gonality question. A priori Meis’ theory should adapt to the bordered setting and arguably lead to another proof of the Ahlfors map, and optimistically with the sharp bound predicted in Gabard 2006 [@Gabard_2006]. Sharpness of the bound is due to Coppens 2011 [@Coppens_2011]. Recall that, Teichmüller himself was close to this (bordered) topic in the article Teichmüller 1941 [@Teichmueller_1941], yet the details (as well as exact bounds) are probably missing.

$\bullet$ [**Ahlfors inflation/injection and generalized Ahlfors maps taking values outside the disc (alias, circle)**]{} \[09.10.12\] The theory of the Ahlfors function is primarily based upon the paradigm of maximizing the derivative (its modulus) within the family of maps with range confined to a (compact) container namely the unit disc. So it is primarily an inflation/injection (or pressurization) procedure (by opposition to the dual deflation/suction approach of Bieberbach-Bergman amounting to minimize the area among maps normed by $f'(z_0)=1$). Ahlfors 1950 [@Ahlfors_1950] showed that if the source object is any compact bordered Riemann surface and the target the unit disc then the Ahlfors (inflating) map turns out to be a circle map, i.e. a full covering of the unit circle taking boundary to boundary. This behavior is not surprising since maximizing the distortion (scaling factor) at a given basepoint forces the whole surface to be maximally stretched over the target, like an elastic skin pushed to its ultimate limit (in the Hollywoodian context of aesthetical surgery). The existence of Ahlfors maps relies on a Montel normal family argument, in substance inherited from the compactness of the disc. This suggests replacing the target disc by any compact bordered Riemann surface. We formulate then the following extremal problem:

Given two finite bordered Riemann surfaces $F$ and $G$ and a given point $a \in F$ and $b\in G$, we look inside the family of all analytic maps $f\colon F \to G$ taking $a$ to $b$ at the map maximizing the modulus of the derivative $f'(a)$ computed w.r.t. local parameters introduced at $a$ and $b$.

By analogy with the Ahlfors et ali theory, we expect that the extremal function exist (compactness of the receptacle $G$), is unique (this is either less evident or false for in the classical case $G=\Delta$ the argument relied heavily on the Schwarz lemma for the disc, so that our only hope in favor of uniqueness is that what actually counts is the universal covering). Arguably, even if lacking uniqueness extremals could still be interesting. Finally it is reasonable to expect that extremals are not oversensitive to the choice of local uniformizers. So we can speak of the map $f_{a,b}$ of extremum dilatation at $a,b$. Finally we are interested about knowing if the extremals are total maps in the sense of taking boundary to the boundary, as do the classical Ahlfors map in the circle/disc-valued case. Before proceeding to examples let us perhaps observe that in the special case where $F$ is given as a subsurface of $G$ and both points $a=b$ coincide, then the (complex) tangent space are readily identified so that $f'(a)$ has an intrinsic meaning as scaling factor of this complex line. Another special case of interest is when $G$ is a plane subregion, in which case the tangent bundle is trivialized so that one can consider a relaxed form of the problem without the constraint $f(a)=b$, in which no point $b$ is given but the sole extremalization of $f'(a)$ will actually dictate where $a$ has to be mapped.

Albeit all we are saying looks a bit messy and unnatural (?), it should be noted that the whole game can be drastically simplified by just looking at avatars of circle maps, that is given two finite Riemann surfaces $F$ and $G$ when does there exist a total map (taking boundary to boundary) from the first to the second. (Of course this question is quite standard yet probably hard to answer precisely, cf. Landau-Osserman 1960 [@Landau-Osserman_1960], and Bedford 1984 [@Bedford_1984].) As we shall soon explain a vague answer is readily supplied by “algebraic geometry”, namely when the target $G$ is not the disc, and if $F$ has general moduli then in general there in not a single total map from $F$ to $G$. The moral is that circle maps enjoy a certain privilege due to their unconditional existence (by Ahlfors 1950 precisely).

A basic obstruction arises from the Riemann-Hurwitz formula. Indeed given $f\colon F \to G$ a total map, it has no ramification along the boundary and is a full covering surface (cf. e.g. Landau-Osserman 1960 [@Landau-Osserman_1960 p.266, Lemma 3.1]). Denoting by $d\ge 1$ the degree of the map, we have $\chi(F)=d \chi (G)-b$, where $b\ge 0$ counts the branch points. When $d=1$, there is no branching and the topological types must agree. Another constraint says roughly that a total map can only simplifies the topology, precisely $\chi(F)=d
\chi (G)-b\le d \chi (G)\le \chi(G)$, when $\chi(G)\le 0$.

If $G$ is not the disc then the existence of a bordered map $f\colon F \to G$ implies that the Euler characteristic satisfies $\chi(F) \le \chi(G)$. (Of course the conclusion persists when $G$ is the disc for it maximizes the Euler characteristic among bordered surfaces.)

Another simple constraint comes from the fact that a total map $f\colon F \to G$ induces a covering of the boundary $\partial
f\colon \partial F \to \partial G$. Hence if $G$ has $r'$ contours then $F$ has at most $d \cdot r'$ contours, i.e. $r\le  d \cdot
r'$ where $r$ is the number of contour of $F$. On the other hand as $\partial f$ is onto, the surjection induced by $\partial f$ on the $\pi_0$ (=the arc-wise connected component functor from TOP to SET) implies that $r\ge r'$.

Then there is a little zoology of cases to study.

(Z1) Let us first suppose that the [*source*]{} is just the disc, then who is the (“Ahlfors”) extremal map? So we assume $F=\Delta$ and $G$ any bordered surface marked at $a=0$, $b\in G$ respectively. By uniformization (Koebe-Poincaré 1907) we know that the universal cover of the interior of any finite bordered surface is the disc. Now the extremal map $f_{a,b}\colon \Delta
\to G$ (maximizing the distortion) may be lifted to the universal cover as say $F \colon \Delta \to \Delta$. Now by the Schwarz-Pick principle of hyperbolic contraction for analytic maps, the latter map contracts the hyperbolic metric implying the universal projection to effect a greater dilatation than the presumed extremal $f_{a,b}$. It follows that $F$ must be the identity (up to rotation) and the extremal function get identified to the universal cover. (Actually, works by Carathéodory and Grunsky actually manage to prove uniformization via the (Ahlfors) extremal problem, whereas we assumed it.)

(Z2) Now consider the situation were both source and target have complicated topology. For instance the source is any bordered surface and the target an annulus. One may expect to get analogues of circle maps, i.e. [*total maps*]{} taking boundary to boundary (sometimes known as proper maps). (Such maps are called [*boundary preserving*]{} in Jenkins-Suita 1988 [@Jenkins-Suita_1988], cf. also Landau-Osserman 1960 [@Landau-Osserman_1960 p.265] who speak of maps “which takes the boundary into the boundary”, while ascribing to Radó 1922 [@Rado_1922-Z-Theorie-mehr] the basic result that such maps are full coverings taking each value of the image surface a constant number of times). Unfortunately, there is severe obstructions to boundary preservation of such (generalized) Ahlfors maps. One way to argue is via algebraic geometry and the Jacobians. It is indeed classic that a generic closed Riemann surface tolerates only nonconstant maps to the sphere (ruling out the trivial identity map or automorphisms available incidentally only for surfaces with specialized moduli). Assuming the Ahlfors map of $F$ to an annulus to be total, its symmetric extension to the Schottky-Klein double is a map from a closed surface to the torus, which for general moduli cannot exist at all! Of course all this requires better proofs, but is fairly well-known and classical (cf. e.g. Griffiths-Harris 1980 [@Griffiths-Harris_1980], who argue as follows (p.236–237): “[*A general curve $C$ of genus $g\ge 2$ cannot be expressed as a multiple cover of any curve $C'$ of genus $g'\ge 1$.*]{} This is readily seen from a count of parameters: the curve $C'$ will depend on $3g'-3$ parameters, and the $m$-sheeted covering $C\to
C'$ depends on $b$ parameters, where \[$\chi (C) = m \chi(C')-b$, that is\] $$b=2g-2-m(2g'-2)$$ is the number of branch points of the cover. Thus if $m\ge 2$, $C$ will depend on $$b+(3g'-3)=b+\frac{3}{2}(2g'-2)
=2g-2-\underbrace{\bigl(m-\frac{3}{2}\bigr)}_{\ge
1/2}\underbrace{(2g'-2)}_{\ge 0}\le 2g-2<3g-3$$ parameters, and so cannot be general.” (Another argument is given in the exercises of Arbarello-Cornalba-Griffiths-Harris 1985 [@Arbarello-Cornalba-Griffiths-Harris_1985-BOOK p.367, Ex.C-6], which of course we were not able to solve!)

(Z3) Finally one can imagine a bordered surface embedded in a slightly larger one (say of the same topological type). Then the inclusion map is permissible in the extremal problem, so the extremal map will have distortion $\ge 1$ at some basepoint, and naively should expand the small surface into the larger one. However by the argument of (Z2) in general it is unlikely that the extremal will be total, and also a priori it not even clear that a true expansion can occur (try to lift the map to the universal cover a get maybe a conflict with the Schwarz-Pick principle of contraction??) But of course this looks dubious for when the subsurface is a disc expansion is possible.

$\bullet$ [**Cyclotomic Riemann surfaces**]{} \[09.10.12\] (but similar examples in Chambéry Talk ca. 20 December 2004) At this stage we can do perhaps the following sort of experiment. As is well-known (Riemann-Prym-Klein 1882 [@Klein_1882]) a Riemann surface structure can also be defined in the most simplest way to visualize, namely as differential-geometric surface in $3$-space with metric (hence conformal structure) inherited by the Pythagorean/Euclidean line element. Consider a hemisphere in Euclidean 3-space surmounted by $m$ handles cyclotomically distributed as on Fig.\[Cyclo:fig\], joining themselves above the north pole.

-5pt0

-5pt0

Ignoring the south hemisphere, we obtain so a bordered surface $F$ with one contour ($r=1$) and of genus $p=m-1$. (Notice here the standard psychological aberration that the genus is one less than the “handles”, for the first handle is not yet coupled to another one to create a real handle!) On rotating by angle $\frac{2\pi}{m}$ the configuration $F$ upon itself we obtain a map from $F$ to the disc (hence a circle map), because the fundamental domain of the rotation is glued over itself to give a disc. The circle map so obtained has degree $m=p+1$. This matches with the general bound $r+p$ predicted in Gabard 2006 [@Gabard_2006]. Let us now assume that a bubbling, i.e. an Euclid-Riemannian deformation of the metric takes place at one of the handle (yet not on the remaining ones) then the rotational symmetry is killed and it becomes much nontrivial that a circle map of same degree is still persistent. This experiment seems to damage the truth of Gabard 2006 [@Gabard_2006] (but hopefully is not?) A naive parade would be to use the (Riemann-Schwarz) uniqueness of the conformal structure on the closed 2-cell to resorb the cancerigenic bubbling. Yet this looks cavalier (for we are not living in the soft smooth $C^{\infty}$ category) and this would not settle the case of less localized cancerigenic degenerations not supported over a disc, but along a subregion having itself moduli. Then one cannot repair easily the deformation by a simple surgical lifting. At this stage we see that the result of Gabard 2006 [@Gabard_2006], if true at all, looks quite formidable for it should resist all those plastic deformations within the flexibility of conformal maps.

Perhaps this might be not so surprising in view of the Riemann-Schwarz super-flexibility theorem telling us in particular that any simply-connected bordered surface (hence topologically the disc) is conformally the same as the disc or the hemisphere. Yet in our context where no simple connectivity is assumed an incarnation of the Riemannian miracle (highly counterintuitive but hopefully true)!

It could be interesting (by adapting Yamada-Gouma) to study the degree of the Ahlfors map of such cyclotomic Riemann surfaces, especially when the basepoint is situated on the 3 fixed points of the rotation.

$\bullet$ [**Special triangulations**]{} \[10.10.12\] Given a circle map of a bordered surface $F$, one can post-compose it with the map taking conformally the disc to an equilateral triangle (in the Euclid plane ${\Bbb C}$). (Recall that this can be done for any three point prescribed along the boundary). Upon subdividing the triangle in a mesh of equilateral triangles, and lifting via the conformal map we generate certain triangulations of $F$ which are almost equilateral. In fact if the mesh size is chosen so that all ramification points lye in the interior of the tiny triangles then the inverse image of such ramified triangles will be small hexagons. Try to study the differential geometry and specialize to Gromov’s Filling conjecture, or try to find a link with Belyi-Grothendieck (a Riemann surface is defined of $\overline{{\Bbb Q}}$ iff it admits an equilateral triangulation).

Another special triangulation of the disc is the hyperbolic tessellation depicted on the front cover of Grunsky’s Collected papers (by equilateral triangles with angles $\pi/6$). \[This tessellation is supposed via the Ahlfors-Grunsky conjecture (1937 [@Ahlfors-Grunsky_1937]) to play an extremal rôle in the Bloch schlicht radius of maps $\Delta \to {\Bbb C}$ for it dominates the densest circle packing of the Euclidean plane.\] Try to understand if it is useful (or aesthetical) to lift this tessellation to the bordered surface via a circle map.

$\bullet$ [**Plateau heuristics $\Rightarrow$ Ahlfors maps?**]{} \[17.10.12\] Soap film experiments of the Belgian physicist have a certain existential convincing power, albeit the rigorous mathematical existence proof (Douglas/Radó ca. 1930/31) required circa 30 years more delay than the allied Dirichlet principle (Hilbert 1900) itself interpretable at the equilibrium temperature distribution in a heat-conducting plate with assigned boundary values. Now Douglas 1931 [@Douglas_1931-Solution] observed that the Riemann mapping theorem (RMT) may be derived by specializing Plateau’s problem to the case where the contour degenerate to the plane, and Courant pushed the remark further so as to include the Riemann-Bieberbach-Grunsky theorem (=planar case of the Ahlfors map). On the other hand Douglas 1936 [@Douglas_1936-Some-new-results] envisaged the so-called Plateau-Douglas problem (PDP, or just PP) for membranes of higher topological structure. It should thus follow (either logically or intuitively) a physico-chemical existence proof of the Ahlfors map.

Let $F$ a finite bordered Riemann surface of genus $p$ with $r\ge
1$ contours, and suppose also given a fixed circle in the plane interpreted as the prescribed wire frame of PP. More generally one can imagine a collection of $r$ contours to be given, and we look at the special case where all these coincide with the unit circle. Now cultivating the right intuition about PP it should be possible to deduce the existence of Ahlfors maps perhaps even with the degree control $r+p$ of Gabard. In fact it should even be possible to study wide extensions where not all frames are coincident with the unit circle. Can one take any frame prescription (e.g. disjoint round circles)? For instance take two unit circles with centers lying distance ten apart ($\vert z \vert=1$ and $\vert
z-10 \vert=1$). Suppose the membrane to have the topological type of an annulus ($r=2$ and $p=0$). Then the minimal surface is something like a flat catenoid, where the inside of each circle is covered once and a certain tube connecting both circles is covered twice by the map. Yet notice that the apparent contour (where the map is folded) of such a film violates the local behavior of holomorphic maps. As we just saw the folding obstruction makes unlikely to span contours consisting of disjoint maximal circles. (Circles being ordered by inclusion of their interior in the plane.) In contrast a nested configuration of circles should cause no trouble to holomorphy. Thus it should be possible to render Ahlfors intuitively obvious via soap film experiments. Of course this was essentially done in Courant’s book (1950 [@Courant_1950]), yet the exact juncture with Ahlfors result probably deserves some extra working. Of course the real challenge would be to investigate if Plateau-style approaches are susceptible to vindicate the degree bound $r+p$ advanced by Gabard 2006 [@Gabard_2006].

Another idea is to imagine a Plateau problem with “wind” blowing through 3-space in some prescribed way (along a given vector field). For instance a soap film spanning a planar disc at rest could deform under a perpendicular wind into say a hemispherical membrane. Try to connect this with Gromov’s filling conjecture, yet unlikely due to the embedded nature of Plateau. Another more plausible connection would be with Gottschalk’s conjecture on flows in 3-space (no vector fields in 3-space having only dense trajectories). This is probably one of the most alienating open problem in the qualitative theory of dynamical systems.

$\bullet$ [**(Ahlfors) circle maps of minimal degree**]{} \[19.10.12\] Given a finite bordered Riemann surface $F$ of genus $p$ with $r$ contours, there is always (by Ahlfors) a circle map. The set of (positive) integers being well-ordered there is a circle map of minimal degree. Call perhaps such maps [*minimal circle maps.*]{} We may ask to which extent such a map is unique and if not can we describe the “moduli space” of such maps. Of course in the most trivial case where $p=0$ and $r=1$ (topologically a disc) the Riemann map is essentially unique ignoring automorphisms of the disc. Likewise uniqueness holds for surfaces with hyperelliptic double provided the latter is not Harnack-maximal. Such hyperelliptic membranes have $r=1$ or $r=2$ and the hyperelliptic involution induces a totally real morphism of degree $2$. Our uniqueness assertion follows of course from the uniqueness for complex curves of the hyperelliptic involution when $g\ge 2$ and thus holds in our context provided $p\ge 1$ (recall that $g=(r-1)+2p$). When $p=0$ and $r=2$ uniqueness fails, for then the double has genus one and may be concretized as a smooth plane cubic with two circuits: one being a genuine “oval” bounding a disc in ${\Bbb P}^2({\Bbb R})$, the other being termed a pseudo-line. Projecting from any point located on the oval gives a totally real morphism of degree $2$, and correspondingly a circle map when restricted to the semi-Riemann surface. Another example is the Gürtelkurve, i.e. any smooth quartic with two nested ovals. Then the minimal degree of a circle map (for the half of the curve) is 3 (argue with the complex gonality of smooth plane curves), and such maps arise by projecting the curve from a real point located on the innermost oval of the nest. Hence there $\infty^1$ circle maps of minimum degree, those being parameterized by a circle $S^1$. Of course the problem does not depend only on the topology: the half of the Gürtelkurve belongs to the topological type $r=2$ and $p=1$, which contains also hyperelliptic representatives, those being circle mappable in a unique fashion via a map of degree 2.

When $F$ is planar ($p=0$) then the double is Harnack-maximal and either the argument of Enriques-Chisini or that of Bieberbach-Grunsky shows that any divisor with one point on each oval moves in a linear system which is totally real (cf. e.g. Gabard 2006 [@Gabard_2006]). So we have now essentially a torus of dimension $r$ ($r$=number of contours) of circle maps of minimum degree. A details description is not so evident for such a divisor $D$ moves in a linear system of dimension $\dim \vert D
\vert \ge \deg D-g$ (Riemann’s inequality, a direct consequence of Abel), where $g=r-1$ is the genus of the double. Thus $\dim \vert
D \vert \ge r-(r-1)=1$ so that $D$ does not necessarily determines unambiguously a totally real pencil. Despite this difficulty it seems reasonable to assert that the set of circle maps for a planar membrane is a torus perhaps of dimension only $r-1$ for one has to unite divisors lying in the same pencil. Extrapolating such examples, we may wonder about structural properties of the set of (minimal) circle maps. Is it always compact? Always a manifold? Perhaps even always a torus. Is it always connected? Of course there are various way to formulate the question and there probably basic experiments giving quick answers to the naive connectedness assumption. Another question is to understand how the global degree $d$ of the circle map splits (partitioned) into the bordered degrees of the restriction to each contours. For instance in the case of the Gürtelkurve, albeit both ovals are perfectly equivalent from the viewpoint of analysis situs, it seems that on the Riemann surface the one corresponding to the inner oval can be mapped with degree 1 whereas the other is less “economic” requiring a wrapping of degree 2. Of course it would be nice to understand this in some intrinsic fashion? But how? (Perhaps via the uniformizing hyperbolic metric and the length of the corresponding ovals???)

Let us try a naive approach to the connectedness problem (by actually trying to corrupt it). Consider in the “abstract quadric surface” ${\Bbb P}^1\times {\Bbb P}^1$ a configuration of bidegree say $(3,3)$. We have chosen both degrees equal so that both projections have the same degree. Imagine 3 lines in each ruling and smooth out the corresponding line arrangement to create a smooth curve $C_{3,3}$ (cf. Fig.\[Cyclo:fig\]b). Actually we have performed sense-preserving smoothings (cf. again the figure) so that the resulting curve is dividing (Fiedler type argument [@Fiedler_1981]). Contemplating the figure we count $r=3$ “ovals”. Both projections on the factors are totally real morphisms of degree 3 (the minimum possible degree in view of the trivial lower bound $\deg f \ge r$). However it seems unlikely that one can continuously deform one map into the other (while keeping its degree minimum). Hence this may give some evidence that the space of minimal circle maps (for the corresponding bordered surface, namely the half of the orthosymmetric Riemann surface underlying our dividing curve $C_{3,3}$) is not connected. However our argument is quite sloppy, having equally well applied to bidegree $(2,2)$ in which case the corresponding curve is Harnack-maximal \[recall that $g=(a-1)(b-1)$ for bidegree $(a,b)$\], hence subsumed to the connectivity principle. Of course it is probable that some basic complex algebraic geometry (gonality of complex curves) suffices to complete the above argument. Is it true that a smooth curve of bidegree $(n,n)$ is $n$-gonal in only two fashions (provided $n\ge 3$) via the natural projections? Of course the assertion is false for $n=2$, for then $g=1$ and a smooth plane cubic model creates $\infty^{1}$ pencils of degree 2. For another plane example seeming to violate the connectivity principle of minimal maps see Fig.\[Coppens:fig\](code 313).

Some historical puzzles
-----------------------

$\bullet$ The most scorching question is whether Klein really anticipate Ahlfors as suggested in Teichmüller 1941 [@Teichmueller_1941]? (Compare Sec.\[sec:Teichmueller\], especially Quote \[quote:Teichmueller-1941\].) Of course the question bears not only historical interest, but has some didactic importance for if a Kleinian argument ever existed (and not just in Teichmüller’s imagination), it is quite likely to be more geometric than Ahlfors’ (decidedly analytic) account. As we already said, it is possible that the Klein-Teichmüller proof rest upon the Rückkehrschnitt intuition of Klein. Even in case Klein himself never anticipated the Ahlfors circle maps, one may wonder from where Teichmüller derived it? In turn one may wonder if Ahlfors took inspiration by Teichmüller 1941 [@Teichmueller_1941]? Of course Ahlfors himself never quoted this Teichmüller work, except in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960], where also all the Italian workers are carefully listed (especially Matildi 1945/48 [@Matildi_1945/48] and Andreotti 1950 [@Andreotti_1950]).

$\bullet$ Does Courant’s paper of 1939 [@Courant_1939] (and the somewhat earlier announcement of 1938 [@Courant_1938], plus the later book treatment of 1950 [@Courant_1950]) presage (modulo a suitable interpretation) any anticipation over the circle map result of Ahlfors 1950 [@Ahlfors_1950]? (For more, compare Sec.\[sec:Courant\].)

The province of Felix Klein {#Sec:Klein}
===========================

Felice Ronga and Felix Klein’s influence
----------------------------------------

In fact the writer himself came across (a weak version of) the Ahlfors function topic from a somewhat different angle, namely via Klein’s theory of [*real algebraic curves*]{} (spanning over the period 1876–92). For Klein this was probably just a baby case of his paradigm of the Galois-Riemann Verschmelzung (Erlanger Program 1873, friendship with Sophus Lie, Ikosaheder and its relation to quintic in one variable, etc.). Yet, real curves surely deserved special (Kleinian) attention as it provided a panoramic view (through the algebro-geometric crystal) of the just emerging topological classification of surfaces (Möbius 1863 [@Moebius_1863], Jordan 1866 [@Jordan_1866], etc.). This offered also a bordered (even possibly non-orientable) avatars of Riemann surfaces, as shown in the somewhat grandiloquent title chosen by Klein “[*Über eine neue Art der Riemannschen Flächen*]{}” (=title of 1874 [@Klein_1874], 1876 [@Klein_1876]) . Those works of Klein were probably not extremely influential (and still today represent only a marginal subbranch of the giant tree planted by Riemann).

Klein himself lamented at several places his work not having found the quick impact he expected from. In 1892 [@Klein-Werke-II_1922 p.171] (ten years after his systematic theory presented in 1882 [@Klein_1882]), he writes: “[*Inzwischen hat noch niemand, so viel ich wei[ß]{}, die hier gegebene Fragestellung seither aufgegriffen, [\[…\]]{}*]{}”. About the same period in his lectures of 1891/92 [@Klein_1892_Vorlesung-Goettingen p.132], he wrote: [*Was ich bislang von diesen Theoremen publicirt habe (so die Einteilung der symmetrischen Flächen in meiner Schrift von 1881), hat nur wenig Anklang gefunden. Ich meine aber, da[ß]{} das nicht am Gegenstande der Untersuchung liegt, der mir viel mehr das grö[ß]{}te Intere[ß]{}e zu verdienen scheint, sondern an der knappen Form, mit der ich meine Resultate darstellte.* ]{}

Of course this impact was first limited to his direct circle of students, where we count Harnack 1876 [@Harnack_1876], Weichold 1883 [@Weichold_1883] and Hurwitz 1883 [@Hurwitz_1883] (also a student of Weierstrass). Klein was also very proud that his results on real moduli supplied a natural answer to questions addressed (but not solved) at the end of Riemann Thesis. Klein insists twice on this issue in 1882 [@Klein_1882]=[@Klein-Werke-III_1923 p.572, §24] and in his subsequent lectures 1891/92 [@Klein_1891--92_Vorlesung-Goettingen p.154], where he writes: “[*Mit dieser Abzählung ist implicite die entsprechende Frage für beantwortet, was darum ein gewi[ß]{}es Intere[ß]{}e hat, weil diese Frage von Riemann in seiner Di[ß]{}ertation aufgeworfen, aber nicht zu Ende discutirt wird. Riemann denkt natürlich nur an berandete Flächen (nicht an Doppelflächen; dem deren Existenz wurde erst zehn Jahre später von Moebius bemerkt und wohl erst in meiner Schrift für funktionentheoretische Zwecke herangezogen).*]{}”

From the very beginning 1876 [@Klein_1876]=[@Klein-Werke-II_1922 §7,p.154], Klein noticed that real curves are subjected to the dichotomy of being dividing or not, where the former case amounts to a separation of the complex locus through its real part (consisting of [*ovals*]{}, a jargon immediately suggesting Hilbert’s 16th problem, yet used much earlier, e.g. by Zeuthen 1874 [@Zeuthen_1874]). Zeuthen’s work seems to have much inspired Klein’s investigation on real curves, starting circa 1876, just two years later (cf., e.g. Klein 1892 [@Klein-Werke-II_1922 p.171]: “[*Ich hatte 1876 den Ausgangspunkt unmittelbar von den Kurven genommen. Das war bei $p=3$ möglich, wo ich zahlreiche geometrische Vorarbeiten, insbesondere diejenigen des Herrn Zeuthen \[…\] (1874), benutzen konnte.*]{}”)

Perhaps, the more tenacious followers of Klein’s viewpoint came somewhat later and the real demographic explosion of the subject took place much later, say perhaps in the 1970’s. Here is a little chronology: $\bullet$ del Pezzo 1892 [@del-Pezzo_1892], where Klein’s trick of assigning the unique real point of an imaginary tangents is taken as the starting point of a study of curves of low genus.

$\bullet$ Berzolari 1906 [@Berzolari_1906], who in an encyclopedia article surveyed in few pages Klein’s achievements and virtually coined the term “Klein surfaces” (Kleinsche Flächen) as a way to designate possibly non-orientable and eventually bordered avatar of Riemann surfaces. To say the least, this terminology was dormant during several decades until Alling-Greenleaf managed in 1969 [@Alling-Greenleaf_1969] a resurrection of Berzolari’s coinage, and since then the nomenclature gained in popularity.

$\bullet$ Koebe 1907 [@Koebe_1907_UrAK] who studied uniformization of real algebraic curves taking advantage of Klein’s distinction orthosymmetric vs. diasymmetric.

$\bullet$ Severi 1921 [@Severi_1921-Vorlesungen-u-alg.-Geom-BUCH p.230–6], who devotes some few pages of his book to Klein’s theory of real curves, \[Note: there Severi writes down the same formula as one used by Courant in his approach to conformal circle maps, ascribing it to Cauchy\].

$\bullet$ Comessatti 1924-25 [@Comessatti_1924/26] in Italy (full of admiration for Klein), who pushed the philosophy up to include a study of real abelian varieties, rational varieties, etc. (For this ramification we refer to the remarkable survey by Ciliberto-Pedrini 1996 [@Ciliberto-Pedrini_1996].)

$\bullet$ several works of Cecioni in the late 1920’s ([@Cecioni_1929], [@Cecioni_1933], [@Cecioni_1935]), and his students (Li Chiavi 1932 [@Stella-li-Chiavi_1932]) makes direct allusion to Klein’s works.

$\bullet$ In France, the work of Klein found a little echo in some passages of the book by Appell-Goursat whose second tome (1930) was apparently mostly written by Fatou. There, Klein’s orthosymmetry occurs at several places [@Appell-Goursat-Fatou_1930 p.326–332 and p.513–521].

$\bullet$ Witt 1934 [@Witt_1934], where a general existence theorem for [*invisible*]{} real algebraic curves (those with empty real locus like, e.g. $x^2+y^2=-1$) was established. This will be discussed later (Sec.\[sec:Witt\]), and is somehow quite akin to the Ahlfors function. Witt’s work makes explicit mention of Klein, and was subsequently elaborated by Geyer 1964/67 [@Geyer_1964-67], who arranged a purely algebraic interpretation of Weichold’s work. His pupil G. Martens, managed (1978 [@Martens_1978]) to determine the lowest possible degree of the Witt mapping;

$\bullet$ (Jesse) Douglas 1936–39 makes a systematic use of Klein’s symmetric surfaces in his study of Plateau’s problem for configuration of higher topological structure. (We shall have to come back to this in Sec.\[sec:Douglas\].)

$\bullet$ A marked influence of Klein upon Teichmüller 1939 [@Teichmueller_1939], 1941 [@Teichmueller_1941]. We shall try to explore this connection in greater detail later (Section \[sec:Teichmueller\]).

Then different events occurred at a rather rapid pace with several schools penetrating into Klein’s reality paradigm through different angles:

$\bullet$ Ahlfors 1950 [@Ahlfors_1950], who never quotes Klein. Probably with Lindelöf and Nevanlinna as teachers one is more inclined toward hard analysis à la Schwarz, than innocent looking geometry à la Klein. Of course Ahlfors quotes instead Schottky, as typified by the terminology Schottky differential, etc. used in Ahlfors 1950 (). It may then appear as a little surprise that Ahlfors’ result affords a purely algebraic (in term of real function fields) characterization of Klein’s orthosymmetric curves. However to my knowledge, this connection—as trivial as it is—was never emphasized in print until much later, namely in Alling-Greenleaf 1969 [@Alling-Greenleaf_1969].

$\bullet$ Schiffer-Spencer’s book 1954 [@Schiffer-Spencer_1954] (outgrowing from Princeton lectures held during the academic year 1949–50) where the book is started by recalling how Klein assimilated the full Riemannian concept after a 1874 discussion with Prym revealing him the ultimate secret of Riemann’s function theory developed over arbitrarily curved surfaces not necessarily spread over the plane. The original source reads as follows:

\[quote:Klein-Prym\] [Ich wei[ß]{} nicht, ob ich je zu einer in sich abgeschlossenen Gesamtaufassung gekommen wäre, hätte mir nich Herr Prym vor längeren Jahren (1874) bei gelegentlicher Unterredung eine Mitteilung gemacht, die immer wesentlicher für mich geworden ist, je länger ich über den Gegenstand nachgedacht habe. Er erzählte mir,]{} [*da[ß]{} die Riemannschen Flächen ursprünglich durchaus nicht notwendig mehrblättrige Flächen über der Ebene sind, da[ß]{} man viel mehr auf beliebig gegebenen krummen Flächen ganz ebenso komplexe Funktionen des Ortes studieren kann, wie auf den Flächen über der Ebene.* ]{}

From circa 1970 upwards, the study of so-called Klein surfaces (jargon of Berzolari [@Berzolari_1906]) [*per se*]{} enjoyed a rather exponential rate of growth as if the simple naming of them was a stimulus for a big expansion of the topic. After two decades an impressive body of knowledge has been accumulated (cf. e.g. the rich bibliography compiled in Natanzon’s survey 1990 [@Natanzon_1990/90]). Those developments can be roughly ranged into 3 main axes:

$\bullet$ [*Foundational aspects.*]{}—Alling-Greenleaf 1971 [@Alling-Greenleaf_1971], and also in Romania with the numerous contribution of Andreian Cazacu (1986–88 [@Andreian-Cazacu_1986], [@Andreian-Cazacu_1988-Interior]) about the structure of morphism between them (interior influence of Stoilow).

$\bullet$ [*Symmetry, automorphisms and NEC(=non-Euclidean crystallography).*]{}—This is especially active in the Spanish school but started somewhat earlier with Singerman 1971–88 (5 items), May 1975–88 (9 items), Bujalance 1981–89 (29 items) Costa, etc.

$\bullet$ [*Moduli spaces of Klein surfaces.*]{} This starts of course in Klein 1882 [@Klein_1882], to reach a certain climax in Teichmüller 1939 [@Teichmueller_1939] and the Ahlfors-Bers school, Earle 1971, Seppäla 1978–89 (6 items on Teichmüller and real moduli), Silhol 1982–89 (Abelian varieties and Comessatti), Costa, Huisman 1998+, etc.

All those works contributed to feel virtually as comfortable with real curves as with their complex grand sisters. We just mention one result of Seppälä 1990 (revisited by Buser-Seppälä-Silhol 1995 [@Buser-Seppala-Silhol_1995] and Costa-Izquierdo 2002 [@Costa-Izquierdo_2002]) to the effect that the moduli space of real curves is connected. (This sounds almost like a provocation to anybody familiar with the bio-diversity of topological types of symmetric surfaces listed by Klein). Of course the trick, here, is that those authors regard this moduli space projected down in that of complex curves (by forgetting the real involution). In other words we may deform the structure until new anti-conformal symmetries appears and switch from one to the other. Hence the subject is sometimes hard to grasp (due to varying jargon) and more seriously is full of real mysteries allied to the real difficulty of the subject.

$\bullet$ [*Geometry of real curves*]{}. Here much of the impulse—very much in Klein’s tradition—came through the paper of Gross-Harris 1981 [@Gross-Harris_1981]. In this or related direction, we may cite authors like Natanzon, Ballico, Coppens, G. Martens, Huisman, Monnier, etc. This area proved very active since the 2000’s up to quite recently and a remarkable variety of difficult question are addressed giving the field arguably some maturity soon comparable to the complex hegemony.

Of course, another line of thought is the interest aroused by Hilbert’s 16th problem (on the mutual disposition of circuits of real algebraic varieties esp. curves) especially among the early German, Italian and then mostly the Russian annexion of the subject. This captured and probably contributed to mask Klein’s more intrinsic viewpoint for a while. This axis includes the following workers (precise references listed in Gudkov 1974 [@Gudkov_1974/74]):

$\bullet$ Hilbert 1891–1900–09, Rohn 1886–1911–11–13; (it is interesting to note that Hilbert’s first 1891 paper on the subject is quite synchronized with Klein’s lectures of 1891/92, which conjecturally may have stimulated Hilbert’s interest, yet not a single allusion to Klein in this paper, and recall also that Hilbert was still in Königsberg at that time).

$\bullet$ Brusotti 1910–13–14–14–15–16–16–16–16–17–21–28–38/39–40–44/45–46–50/51–52–55–55 (characterized by “[*la piccola variazione*]{}”, i.e. the method of small perturbation permitting to construct real algebraic curves with controlled topology. The writer is indebted to Felice Ronga for this method, which of course has some historical antecedents older that Brusotti. In Klein 1873, footnote 2 in [@Klein-Werke-II_1922 p.11] the principle is traced at least back to Plücker 1839 [@Plücker_1839]. However Brusotti 1921 [@Brusotti_1921] may have been the first—modulo its reliance over work of Severi—to notice that the Riemann-Roch theorem admits as extrinsic traduction the possibility of smoothing independently the nodes of a plane curve. The main issue (as transmitted by Felice) is that the nodes a plane curve with nodal singularities impose independent conditions on curves of the same degree. Hence when the curve is being imagined as a point in the (projective) space of all curves, it sits on the discriminant hypersurface (parameterizing all singular curves) and nearby our nodal curve we see several transverse smooth branches crossing transversally. (In French or Italian, there are better synonyms like “falde analytiche” or “nappe”.) The net effect of transversality is that one can leave at will certain strata, while staying on others. This implies the independency of smoothing crossings, and thereby a rigorous foundation to the small perturbation method. (The resulting graphical flexibility of algebraic curves is a pleasant way to create Riemann surfaces, and we shall exploit it later in this text as a way to explore degrees of Ahlfors circle maps.)

$\bullet$ Comessatti (more in the spirit of Klein) 1924–25–27/28–31–32–33, etc.

$\bullet$ Petrovskii 1938–49 ([@Petrowsky_1938]), etc. many joint with Oleinik (real algebraic (hyper)surfaces and Betti numbers).

$\bullet$ Gudkov 1954–54–62–62–62–65–66–69–69–73 (those works include in particular the spectacular discovery of a sextic whose oval configuration was expected to be impossible by Hilbert).

$\bullet$ Arnold 1971–73.

$\bullet$ Rohlin 1972–72–73.

$\bullet$ Finally the long awaited (?) reunification of forces (call it maybe the Klein-Hilbert Verschmelzung) came in the work of Rohlin (himself apparently inspired by Arnold). Surprisingly, Rohlin took notice of Klein’s work quite late, ca. 1978 (compare Rohlin 1978 [@Rohlin_1978]).

$\bullet$ Then real algebraic geometry exploded through the work of Kharlamov, Viro, Fiedler, Nikulin 1979, Orevkov, Finashin, etc. and in the west Risler, Marin, and many others gave a new golden age to a discipline reaching a certain popularity.

Sometimes the real theory seems only to adapt over ${\Bbb R}$ whatever has been achieved over ${\Bbb C}$, yielding usually a kaleidoscopic fragmentation of truths into a real zoology. Thus for instance the Castelnuovo-Enriques classification of (algebraic) surfaces can be pushed through reality: K3 (Nikulin-Kharlamov), Abelian surfaces (Comessatti-Silhol), elliptic surfaces, etc. The topic is then strongly allied to deep methods in differential topology, Galois cohomology, symplectic geometry, Gromov-Witten, enumerative problems, tropical geometry, etc. The present number of active workers is so impressive and the recent connections so amazing (Okounkov, etc.) that we prefer to stop here our impressionist touristic overview of real algebraic geometry.

Digression about Hilbert’s 16th problem (Klein 1922, Rohlin 1974, Kharlamov-Viro ca. 1975, Marin 1979, Gross-Harris 1981) 
--------------------------------------------------------------------------------------------------------------------------

The connection between Klein’s theory (especially the ortho- and diasymmetric dichotomy) with Hilbert’s 16th problem (plane curves in the projective plane ${\Bbb P}^2$) were profoundly investigated by the Russian school in the early 1970’s especially Arnold, Rohlin, Viro, Kharlamov, etc. Klein himself always dreamed of such a relationship , without really being able to formulate its precise shape. Here is a quote which Klein added (ca. 1922) to his Werke (cf. [@Klein-Werke-II_1922 p.155, footnote]):

\[Klein-1922-immer-vorsgeschwebt:quote\]

Es hat mir immer vorgeschwebt, dass man durch Fortsetzung der Betrachtungen des Textes Genaueres über die Gestalten der reellen ebenen Kurven beliebigen Grades erfahren könne, nicht nur, was die Zahl ihrer Züge, sondern auch, was deren gegenseitige Lage angeht. Ich gebe diese Hoffnung auch noch nicht auf, aber ich muss leider sagen, dass die Realitätstheoreme über Kurven beliebigen Geschlechtes (welche ich aus der allgemeinen Theorie der Riemannschen Flächen, speziell der “symmetrischen” Riemannschen Flächen ableite) hierfür nicht ausreichen, sondern nur erst einen Rahmen für die zu untersuchenden Möglichkeiten abgeben. In der Tat sind ja die doppelpunktslosen ebenen Kurven $n$-ten Grades für $n>4$ keineswegs die allgemeinen Repräsentanten ihres Geschlechtes, sondern wie man leicht nachrechnet, durch $(n-2)(n-4)$ Bedingungen partikularisiert. Da man über die Natur dieser Bedingungen zunächst wenig weiss, kann man noch nicht von vornherein sagen, dass alle die Arten reeller Kurven, die man gemäss meinen späteren Untersuchungen für $p={ n-1 \cdot n-2 \over 2}$ findet, bereits im Gebiete besagter ebener Kurven $n$-ter Ordnung vertreten sein mü[ß]{}ten, auch nicht, da[ß]{} ihnen immer nur [*eine*]{} Art ebener Kurven entspräche. K.

It took several decades until the experimentally obvious conjecture (possibly anticipated by Klein, though he left no trace in print) that dividing curves in the plane have at least as many ovals as the half value of its degree found place in a paper of Gross-Harris 1981 [@Gross-Harris_1981 p.177, [*Note*]{}]. In fact, in a paper by Alexis Marin 1979/81 [@Marin_1979] this is stated as a corollary of a Rohlin formula (1978 [@Rohlin_1978]), involving intersection of homology classes deduced from the halves of the dividing curve capped off by the interiors of ovals in ${\Bbb P}^2({\Bbb R})$). In the case of $M$-curves (=the Russian synonym of Harnack-maximal coined by Petrowskii 1938 [@Petrowsky_1938]), this technique occurred earlier in Rohlin 1974/75 [@Rohlin_1974/75]. Moral: the tool missing to Klein was intersection theory of homology classes developed by Poincaré, Lefschetz, etc. In the little note Gabard 2000 [@Gabard_2000] it is verified that this Rohlin-Marin obstruction ($r\ge \frac{m}{2}$) is the only one, settling thereby completely the Klein-Gross-Harris question. This (simple) fact was known to Rohlin’s students Kharlamov and Viro which were familiar with this result as early as the middle 1970’s (as they both kindly informed me by e-mail). Of course the crucial ideas are due to Rohlin.

A long unnoticed tunnel between Klein and Ahlfors (Alling-Greenleaf 1969, Geyer-Martens 1977)
---------------------------------------------------------------------------------------------

More importantly, for our present purpose is to keep the abstract viewpoint of Klein (by opposition to the embedded Hilbert’s 16th problem), and to make the following observation.

[(Klein?, Teichmüller 1941?, Ahlfors 1948/50, Matildi 1945/48?, Andreotti 1950?, who else?)]{} Dividing curves are precisely those admitting a real morphism (i.e., defined over the ground field ${\Bbb R}$) to the projective line ${\Bbb P}^1$ such that all fibers over real points consist entirely of real points.

The non-trivial implication of this fact follows precisely from Ahlfors 1950 [@Ahlfors_1950] (but is made very explicit only in Alling-Greenleaf 1969 [@Alling-Greenleaf_1969], see also Geyer-Martens 1977 [@Geyer-Martens_1977]). To my actual knowledge there is no record in print of this fact prior to Ahlfors’ intervention, modulo the cryptical allusion in Teichmüller 1941 [@Teichmueller_1941] that the result was implicit in Klein’s works. Another related works are those of Matildi 1948 [@Matildi_1945/48] and Andreotti 1950 [@Andreotti_1950]. As we shall recall later Ahlfors’ result was exposed at Harward as early as 1948 (cf. Nehari 1950 [@Nehari_1950] reproduced here as Quote \[Nehari-1950:quote\]).

It is however picturesque to notice that an analog result stating that a real curve without real points maps through a real morphism upon the empty curve $x^2+y^2=-1$ (or projectively $x_0^2+x_1^2+x_2^2=0$) was established as long ago as Witt 1934 [@Witt_1934]. Perhaps both problems are of comparable difficulty, and the method employed by Witt—namely Abelian integrals—turns out to be likewise relevant to the Ahlfors context (i.e. dividing curves). Hence in our opinion, there were no technological obstruction to Ahlfors result being discovered much earlier, say by Witt in the 1930’s, or by Bieberbach in 1925 [@Bieberbach_1925], or by Klein in the 1876–80’s, or even by Riemann in the late 1850’s (especially in view of his [*Parallelogramm methode/Figuren*]{}, cf. e.g., Haupt 1920 [@Haupt_1920]), and ultimately why not by Abel himself? (Of course all these peoples were probably involved with more urgent tasks, like some [*flüchtigen Versuche*]{} about the Riemann hypothesis, or regarding Klein the [*Grenzkreistheorem*]{} (in his health taking contest with Poincaré), which later became known as the uniformization theorem. The list of competent workers coming also very close to the paradigm ultimately discovered by Ahlfors could easily be elongated: especially Schwarz, Hurwitz (esp. in 1891 [@Hurwitz_1891-Uber-Riemannsche-Flachen]), Koebe, Courant (esp. in 1939 [@Courant_1939], 1940 [@Courant_1940-Acta] or 1950 [@Courant_1950]).

As to the interesting result of Witt 1934 (on invisible real curves), we will try to discuss it later in more details (Sec.\[sec:Witt\]).

Albeit, the writer lacks strong (analytical) competence, we shall attempt to delineate several schools (and teams of workers) scattered through the planet, which are still exploring this or related topics.

Motivation (better upper bounds exist)
--------------------------------------

Even though Ahlfors’ result is approaching 65 years (a venerable age for retirement) the basic result looks still grandiose, and mysterious enough if one wonders about the exact distribution of Ahlfors’ degrees (as suggested in Yamada-Gouma’s penetrating study (1978–1998–2001), discussed in Sec.\[Yamada-Gouma:subsec\]).

The writer published a paper in 2006 [@Gabard_2006] where a circle map with fewer sheets (viz. $\le r+p$) than that proposed by Ahlfors (namely $\le r+2p$) is exhibited. This quantitative improvement is the motivation for much of this survey, and will hopefully excuses the bewildering variety of topics addressed. An obvious game is to renegotiate known application of the Ahlfors’ mapping involving a controlled degree in the hope to upgrade the bound. As tactically simple as it may look, we were not very successful in this game as it often already requires analytical skills beyond the competence of the writer. Yet we shall mostly content to list some articles where some upgrade could be expected (e.g., Hara-Nakai’s quantitative version of the corona with bounds [@Hara-Nakai_1985] looks to be a challenging place to test). Of course for this [*bound upgrading procedure*]{} to work it requires that the application in question does not use the full strength of the Ahlfors function, but only its qualitative property of being a circle map. A concrete instance were this was accomplished is Fraser-Schoen’s paper 2011 [@Fraser-Schoen_2011].

Alternatively we can dream of certain high powered applications requiring the full extremal power of the Ahlfors mapping. In this case it is known a priori (Yamada-Gouma) that we cannot lower the degree of the Ahlfors function, except possibly for very particular choices of base-points.

So the main philosophical issue is roughly the following point:

Is the Ahlfors extremal property truly required in applications, or just the arcane residue of those attempt to salvage the Dirichlet principle via extremal methods. Put differently, is the extremal problem just an artefact of the proof or something really worth exploiting in practice?

Full coverings versus Ahlfors’ extremals
----------------------------------------

To avoid any confusion, one must from the scratch relativize strongly the importance of the recent contribution on the $r+p$ bound (Gabard 2006 [@Gabard_2006]) for several reasons.

First the result is quite recent and probably not sufficiently verified as yet. In later sections when looking at explicit curves from the experimental viewpoint it seems that there is a large armada of potential counterexamples flying at high altitudes (flying fortresses).

Next, Ahlfors’ upper bound $r+2p$ is known to be sharp within the realm of the extremal problem it solves. Indeed, Yamada 1978 [@Yamada_1978] has a rather simple argument showing that the Ahlfors function centered at the Weierstrass points of a hyperelliptic membrane has degree precisely $r+2p$ (and not less). Maybe it is an open question whether a similar sharpness holds for all membranes.

Hence, one must keep in mind a subtle distinction between Ahlfors’ deep extremal problem (involving hard analysis via the paradigm of extremality) and the writer’s soft version ([@Gabard_2006]) which leads to a sharper bound but is based only upon (soft) topological methods, i.e. the Brouwerian degree and the allied criterion of surjectivity. To put it briefly, we must distinguish Ahlfors’ extremal function from the mere [*circle map*]{}, defined as follows (nomenclature borrowed from Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950 p.182]):

A circle map is an analytic function from a compact bordered Riemann surface to the disc, expressing the former as a (generally branched) cover of the disc, say $f\colon W \to D=\{\vert z\vert \le 1\}$. Each interior points maps to an interior points of the disc (otherwise there is a problem as infinitesimally the mapping is a power map $z\mapsto z^n$, $n\ge 1$). Thus, the restricted covering $\partial W \to
\partial D=S^1$ is unramified, whereupon it follows that $r\le \deg (f)$ (i.e. the number of contours is a trivial lower bound for the degree of a circle map).

Varied synonyms (or closely allied designations) are used throughout the literature (here is a little sampling with citation of the relevant sources):

$\bullet$ $n$ fach ausgebreitete Fläche, $n$ fach bedeckende Fläche (Riemann 1857–Weber 1876 [@Riemann_1857_Nachlass p.473]);

$\bullet$ Schottky 1877 no clear cut terminology, and re-reading it (25.06.12) in details I realize that the statement about existence of circle maps is in fact not really proved (thus much of the written is somewhat biased), note that Bieberbach somewhat wrongly ascribe the result as well to Schottky, but that remains to be elucidated... In contrast, Grunsky never (?) credits Schottky, but rather Bieberbach 1925 [@Bieberbach_1925];

$\bullet$ mehrfach bedeckte Kreisscheibe, $n$-blättrige Kreisscheibe (Bieberbach 1925 [@Bieberbach_1925 p.6]);

$\bullet$ mehrblättrige Kreise, $n$-blättrige Kreisscheibe (Grunsky 1937 [@Grunsky_1937 p.40]);

$\bullet$ ein endlichvielblättriges Flächenstück über der oberen $z$-Halbebene mit endlich vielen Windungspunkten, das durch Spiegelung an der reellen Achse eine symmetrische geschlossene Riemannsche Fläche ergibt (Teichmüller 1941 [@Teichmueller_1941]);

$\bullet$ cerchio multiplo (Matildi 1945/48 [@Matildi_1945/48 p.82], a student of Cecioni);

$\bullet$ full covering surface of the unit circle (Ahlfors 1950 [@Ahlfors_1950 p.124, p.132]);

$\bullet$ $(2g+m)$-sheeted unbounded covering surface of the unit disc (Encyclopedic Dictionary of Mathematics 1968/87 [@EDM_1968/87 p.1367]);

$\bullet$ unbounded finitely sheeted covering surfaces of the unit disk (Nakai 1983 [@Nakai_1983 p.164]);

$\bullet$ Schottky functions (Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949 p.214], Kühnau 1967 [@Kuehnau_1967 p.96], and earlier (yet without this appellation) in several works of Picard, e.g. Picard 1913 [@Picard_1913] and Cecioni, e.g. Cecioni 1935 [@Cecioni_1935]);

$\bullet$ $p$-times covered unit-circle (Bergman 1950 [@Bergman_1950 p.87, line 5]);

$\bullet$ $n$-times covered circle, multiply-covered circle (Nehari 1950 [@Nehari_1950 p.256, resp. p.267], Stanton 1971 [@Stanton_1971 p.289 and 293] Aharonov-Shapiro 1976 [@Aharonov-Shapiro_1976 p.60]);

$\bullet$ Ahlfors mapping (Nehari 1950 [@Nehari_1950 p.256, p.267], Stanton 1971 [@Stanton_1971 p.289 and 293];

$\bullet$ Ahlfors function (Aharonov-Shapiro 1976 [@Aharonov-Shapiro_1976 p.60]);

$\bullet$ Ahlfors map (Alling 1966 [@Alling_1966 p.345–6], Stout 1967 [@Stout_1967-Interpolation p.274], and then in many papers by Bell);

$\bullet$ Ahlfors type function (Yakubovich 2006 [@Yakubovich_2006 p.31]);

$\bullet$ Einheitsfunktionen (Carathéodory 1950 [@Caratheodory_1950_Buch_Funktionentheorie vol.II, p.12]), translated as:

$\bullet$ [*unitary*]{} function in Heins 1965 [@Heins_1965 p.130], a jargon also adhered to by Fay 1973 [@Fay_1973 p.108, 111, etc.];

$\bullet$ unimodular function (Douglas-Rudin 1969 [@Douglas-Rudin_1969], Fisher 1969 [@Fisher_1969-BAMS-convex-combination-unimodular-fct], Gamelin 1973 [@Gamelin_1973-BAMS], Lund 1974 [@Lund_1974]);

$\bullet$ many-sheeted disc (A. Mori 1951 [@Mori_1951]);

$\bullet$ multi-sheeted circle (Havinson 1953 [@Havinson_1953]);

$\bullet$ finitely sheeted disks (Hara-Nakai 1985 [@Hara-Nakai_1985]);

$\bullet$ Vollkreisabbildung (Meschkowski 1951 [@Meschkowski_1951 p.121]);

$\bullet$ (volle) $n$-blättrige (Einheits)Kreisscheibe (Golusin 1957 [@Golusin_1952/57 p.240, 412], as translated by Grunsky or Pirl);

$\bullet$ interior mappings (Stoïlow, Beurling);

$\bullet$ inner functions (Beurling 1949 [@Beurling_1949], Hoffman 1962 [@Hoffman_1962] (esp. p.74, where Beurling is credited of the coinage), Rudin 1969 [@Rudin_1969], Stout 1972 [@Stout_1972 p.343], Černe-Forstnerič 2002 [@Cerne-Forstneric_2002 p.686]). This concept usually refers to analytic functions with modulus a.e. equal to one along the boundary, but some writers corrupted this sense to mean a circle map, cf. Stout 1966/67 [@Stout_1966/67] which is followed by Fedorov 1990/91 [@Fedorov_1991 p.271].

$\bullet$ boundary preserving maps (Jenkins-Suita 1984 [@Jenkins-Suita_1988]); maps taking the boundary into the boundary (Landau-Osserman 1960 [@Landau-Osserman_1960]).

$\bullet$ complete covering surfaces (cf. Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.41–42, §21A]), i.e. one such that any point in the range has a neighborhood whose inverse image consists only of compact components; complete Klein coverings (Andreian Cazacu 2002 [@Andreian-Cazacu_2002]) (a direct extension of the former concept shown to be equivalent in the case of finite coverings to the next conception of Stoïlow).

$\bullet$ total Riemann coverings (Stoïlow 1938 [@Stoilow_1938-Lecons]), i.e. one such that any sequence tending to the boundary has an image tending to the boundary.

$\bullet$ unlimited covering surfaces (Nakai 1988 [@Nakai_1988], EDM=Japanese encyclopedia 1968/87 [@EDM_1968/87], Minda 1979 [@Minda_1979-hyperbolic-metric-and-coverings])

$\bullet$ proper (holomorphic) maps (onto the unit disc) (e.g., Bedford 1984 [@Bedford_1984 p.159], Bell 1999 [@Bell_1999-Ahlfors-maps p.329], Černe-Flores 2007 [@Cerne-Flores_2007], Fraser-Schoen 2011 [@Fraser-Schoen_2011]).

$\bullet$ distinguished map (Jurchescu 1961 [@Jurchescu_1961-A-maximal])

$\bullet$ Myrberg surface over the unit disc (Stanton 1975 [@Stanton_1975 p.559, §2] uses this terminology for a Riemann surface $W$ admitting an analytic function $z\colon
W \to \Delta$ realizing $W$ as an $n$-sheeted, branched, full covering surface of the unit disc $\Delta$).

As no ramification appears along the boundary, explains the naming:

$\bullet$ Randschlicht mapping (Köditz-Timann 1975 [@Koeditz-Timmann_1975]).

In fact the writer came across this concept through real algebraic geometry where I used (2006 [@Gabard_2006]) the term [*saturated*]{}, whereas Coppens 2011 [@Coppens_2011] proposes the term [*separating*]{} morphism. In the same context, Geyer-Martens 1977 [@Geyer-Martens_1977] coined:

$\bullet$ “total reell Morphismus”=totally real morphism/map, to which we shall adhere as it seems to be the most convenient terminology, especially when abridged just as “total maps”, which is quite in agreement with Stoïlow’s jargon.

We shall attempt to reserve the designation [*Ahlfors maps/functions*]{} for those solving the extremal problem formulated in Ahlfors 1950 [@Ahlfors_1950]. The latter are known (since Ahlfors 1950 [@Ahlfors_1950]) to be circle maps, but the converse is wrong. Indeed, circle maps may have arbitrarily large degrees (post-compose with a power map $z\mapsto z^n$ for some large integer $n$), whereas Ahlfors maps have degrees $\le
r+2p$ (in view of the deep result in Ahlfors 1950 [@Ahlfors_1950]).

Are circle maps of degree compatible with Ahlfors’ bound always realizable via an Ahlfors map? The answer seems to be in the negative, at least if attention is restricted to infinitesimal Ahlfors maps. This follows from Gouma’s restriction (1998 [@Gouma_1998]) in the hyperelliptic case. Indeed consider a 2-gonal membrane, then post-composing with $z\mapsto z^n$ we get circle maps of degrees ranging through all multiples $2n$, whereas only 2 and $r+2p$ are realized as degrees of Ahlfors maps, by a result of Gouma 1998 [@Gouma_1998]. Note that Gouma restricts to ponctual Ahlfors maps and our claim is only firmly established in this context.

A somewhat deeper question is whether any (or at least one) circle map of smallest degree arises via an Ahlfors map. We were not able to settle this question, but in a tour de force Yamada 2001 [@Yamada_2001] proved this in the hyperelliptic case. It amounts to know if the Ahlfors map is flexible enough to capture a circle map of the lowest possible degree (alias the gonality). Let us optimistically pose the conjecture, amounting to say that we can essentially take out the best of the two worlds:

\[gonality:conj\] Any (or at least one) conformal mapping realizing the gonality arises as an Ahlfors extremal function $f_{a,b}$ (perhaps for coalescing two points yielding then the Ahlfors map $f_a$ maximizing the modulus of the derivative at $a$).

Recent work by Marc Coppens 2011 [@Coppens_2011] supplies a sharp understanding of the gonality $\gamma$ as spreading through all permissible values $r\le \gamma \le r+p$ when the membrane is varied through its moduli space.

Paraphrased differently the conjecture wonders if a suitable Ahlfors map always realizes the gonality. As yet we lack evidence, but the vague feeling that Ahlfors’ method is the best possible (being distilled by the paradigm of extremality) inclines one to believe that its economy should be God given. In contradistinction, it may be argued that Ahlfors maps depend on so few parameters (essentially one or two points on the surface), that they are perhaps not flexible enough to explore the full room of all circle maps. Such simple minded question exemplifies that the old subject of the Ahlfors’ map still deserves better understanding. A fine understanding of the Ahlfors map would truly be worth studying if we had some clear-cut applications in mind (taking full advantage of the extremal property of the map). In practice, one is often content with the weaker notion of circle maps, but in the long run it is likely that more demanding applications requires the full punch of the Ahlfors map.

Sorting out applications: finite vs. infinite/compact vs. open
--------------------------------------------------------------

As to applications (of the Ahlfors map), there are several ramifications, which —at the risk of oversimplification—may be ranked in two headings ([*in finito*]{} vs. [*in infinito*]{}). By this we have in mind essentially the sharp opposition between compact and non-compact Riemann surfaces. The later were intensively approached by several schools (mostly Finnish, Japanese and US), but the theory is certainly less complete than for compact surfaces, which from our viewpoint already represent a serious challenge. Furthermore it is evident that there is essentially one and only one road leading from the finite to the infinite namely the exhaustion process affording a cytoplasmic expansion of a compact bordered Riemann surface in some ambient open surface. Now let us enumerate such applications.

\(A) [**Lifting truth from the disc via conformal transplantation.**]{} A reliable philosophy is roughly that a result known to hold good in the disc is lifted via the Ahlfors map to configurations of higher topological type. This is the strategy used by Alling 1964 [@Alling_1964] to transplant the corona of Carleson 1962 [@Carleson_1962] to Riemann surfaces. (The corona theorem amounts to say that the Riemann surface is dense in the maximal ideal space of its algebra of bounded analytic functions.)

In spectral theory this method (systematically utilized by Polyá-Szegö) is known as “[*conformal transplantation*]{}”. Subsequent elaborations arose through the work of Hersch 1970 [@Hersch_1970] and Yang-Yau 1980 [@Yang-Yau_1980] (where branched covering are admitted, thereby diversifying widely the topology).

Recently Fraser-Schoen 2011 [@Fraser-Schoen_2011] applied the Ahlfors mapping to spectral theory (Steklov eigenvalues). (This inspired a note of the writer [@Gabard_2011] extending Hersch 1970’s study of Dirichlet and Neumann eigenvalues on spherical membranes to arbitrary (compact) bordered surfaces.) Another spectacular work is due to Girouard-Polterovich 2012 [@Girouard-Polterovich_2012] where Fraser-Schoen’s work is extended to higher eigenvalues.

\(B) [**Exhaustion and infinite avatars.**]{} Another philosophy (Nevanlinna, Ahlfors, etc.) is to exploit the fact that (infinite, i.e. open Riemann surface) may be exhausted by compact subregions (reminding somehow the finitistic slogan of André Bloch, [*“Nihil est in infinito...”*]{}) offering thereby a wide range of application of compact bordered Riemann surfaces to the more mysterious realm of open Riemann surfaces. This ramifies quickly to the so-called classification theory of Riemann surface (Nevanlinna 1941 [@Nevanlinna_1941], Ahlfors 46, Sario 46–49, Parreau 1951 [@Parreau_1951], Royden 1952 [@Royden_1952], etc.) much completed by the Japanese school (Tôki 1951, A. Mori, Kuramochi, Kuroda, etc.). Several books attempt to give a coherent account of this big classification theory, e.g. Ahlfors-Sario 1960 [@Ahlfors-Sario_1960], Sario-Nakai 1970 [@Sario-Nakai_1970], where the guiding principle (due to Sario 1946) is to classify surfaces according to the force of their ideal boundary.

In another infinite direction, S.Ya. Havinson 1961/64 [@Havinson_1961/64] was the first (with Carleson 1967 [@Carleson_1967-book]) to extend the theory of the Ahlfors function to domains of infinite connectivity , and was followed by S. Fisher 1969 [@Fisher_1969], which propose some simplifications.

The Slovenian school of complex geometry (Černe, Forstnerič, Globevnik, etc.) are also employing the Ahlfors function, often in connection with the open problem (Narasimhan, Bell, Gromov, etc.) of deciding if any open Riemann surface embeds properly in ${\Bbb C}^2$. In Forstnerič-Wold 2009 [@Forstneric-Wold_2009] reduced the full problem to a finitary question as to whether each compact bordered Riemann surface embeds holomorphically in the plane ${\Bbb C}^2$. (Maybe this is achievable by a suitable of Ahlfors functions, or more sophisticated variant thereof like (?) in the broader Pick-Nevanlinna context). As suggested by those authors, it is maybe enough to embed one representant in each topological type (this is possible, compare Černe-Forstnerič 2002 [@Cerne-Forstneric_2002 Theorem 1.1]) and try to use a continuity argument through the Teichmüller (moduli) space.

Biased recollections of the writer {#Sec:Biased-recollections-of-Gabard}
==================================

Klein’s viewpoint: real curves as symmetric Riemann surfaces (as yet another instance of the Galois-Riemann Verschmelzung)
--------------------------------------------------------------------------------------------------------------------------

If the writer is allowed to recollect his own memories about his involvement with this circles of ideas, it started as follows. Maybe a natural point of departure is the (basic) algebraic geometry of curves. While reading Shafarevich’s Basic algebraic geometry (ca. 1998) one encounters some nice drawings of the real locus of a plane cubic into its complex locus materialized by a torus (as we know since time immemorial: Euler?, Abel?, Jacobi, Riemann, etc.). A torus of revolution reflected across a plane cutting the torus along two circles yields a plausible visualization of the embedding of $C({\Bbb R})$ into $C({\Bbb C})$ (even with the symmetry induced by the complex conjugation).

Of course there are also real cubic curves whose real loci possess only one component. How to visualize the corresponding embedding? Lee Rudolph quickly helped us by just realizing that the Galois action (complex conjugation) acts over the torus $S^1\times S^1$ just by exchanging the two factors $(x,y)\mapsto (y,x)$ fixing thereby the diagonal (circle) $\{(x,x)\}\approx S^1$.

More generally how to picture out the topology of a real curve? The first observation is that the complex locus $C({\Bbb C})$ is acted upon by complex conjugation $\sigma$ relative to some ambient projective space ${\Bbb P}^n({\Bbb C})$ (where after all the concrete curve is embedded). Therefore to each real curve $C$ is assigned a [*symmetric surface*]{} $(C({\Bbb C}),
\sigma)=(X,\sigma)$ consisting of a pretzel $X$ together with an orientation reversing involution $\sigma\colon X\to X$. (For aesthetical reasons all of our algebraic curves are projective and non-singular, prompting thereby compactness of the allied Riemann surfaces.) With the invaluable assistance of (overqualified scholars) Claude Weber and Michel Kervaire, I learned how to classify such objects, according to the invariants $(g,r,a)$ where $g$ is the genus of $X$, $r$ the number of “ovals” (fixed under $\sigma$), and $a$ is the invariant counting mod 2 the number of components of $X-{\rm
Fix}(\sigma)$. In other words $a=0$ corresponds to the separating (or dividing) case where ${\rm Fix} (\sigma)$ disconnects $X$, whereas $a=1$ means that the fixed locus does not induce a morcellation of the surface.

I soon realized thanks to the paper Gross-Harris 1981 [@Gross-Harris_1981], that all this material was a well-known game for Felix Klein, who was essentially the first to classify symmetric surfaces taking advantage of the just established classification of compact bordered surfaces (Möbius 1863 [@Moebius_1863], Jordan 1866 [@Jordan_1866], etc.). The key trick is of course the yoga assigning to $(X,\sigma)$ its quotient $X/\sigma=:Y$ by the involution, and moving upward again via the orientation covering supplied by local orientations. If the point lies on the boundary then there is no duplication of the point by local orientations (alias “indicatrix” in older literature).

[(Klein 1876 [@Klein_1876]=[@Klein-Werke-II_1922 p.154], explicit in Klein 1882 [@Klein_1882], Weichold 1883 [@Weichold_1883])]{} There is one-to-one correspondence between symmetric surfaces and compact bordered surfaces. Moreover the correspondence extends to the realm of conformal geometry, i.e. Riemann surfaces or Klein surfaces, if you prefer.

[Modernized treatments of this Klein correspondence—say compatible with Weyl–Radó’s (1913/1925) abstract conception of the Riemann surface—are plenty, compare, e.g. Teichmüller 1939 [@Teichmueller_1939 p.99–101, Die Verdoppelung, §92, 93]=[@Teichmueller_1982], Schiffer-Spencer 1954 [@Schiffer-Spencer_1954 p.29–30, §2.2], Alling-Greenleaf 1971 [@Alling-Greenleaf_1971]. ]{}

Via this dictionary, it is plain that the dividing case corresponds precisely to the orientable case. \[As a matter of terminology, Klein used (since Wintersemester 1881/82) the jargon [*orthosymmetrisch*]{} versus [*diasymmetrisch*]{} corresponding to the dividing respectively nondividing case. For instance Weichold 1883 [@Weichold_1883 p.322] writes:

\[Weichold-1883:quote\]

Was ferner die symmetrischen Riemann’schen Flächen anbelangt, deren Betrachtung die Grundlage der folgenden Untersuchung bildet, so sind auch diese schon mehrfach behandelt worden, wenn auch zum Theil unter ganz anderen Gesichtspunkten. Es hat sich nämlich Herr Professor Klein in den Bänden VII und X der Mathem. Annalen in den Aufsätzen mit dem Titel: “Über eine neue Art von Riemann’schen Flächen” mit diesen Flächen eingehender beschäftigt und daselbst auch schon die Hauptunterscheidung derselben in orthosymmetrische und diasymmetrische Flächen aufgestellt. Diese Bezeichnung findet sich allerdings noch in keiner Publication angewendet; sie wurde zuerst in einem in Wintersemester 1881/82 von Herrn Professor Klein abgehaltenen Seminar eingeführt, in welchem derselbe auch die weiter unten erwähnte weitergehende Classification mittheite und bei welchem auch der Verfasser die unmittelbare Anregung für die vorliegende Arbeit empfing.

Perhaps it is worth tracking down further Klein’s motivation for this “savant” terminology; for this we supply the following extract:

\[Klein-1923:quote\]

Die Benennungen “diasymmetrisch” und “orthosymmetrisch” für die beiden Klassen symmetrischer Flächen wurden später von mir gerade wegen der im Text berührten Verhältnisse eingeführt; siehe Bd. 2 dieser Ausgabe, S.172. Vgl. auch Fu[ß]{}note $^{
58)}$ auf S.565/566 im vorliegenden Bande. K.

So this brings us at other places, the first cross-reference leads us to the following quote (whereas Fu[ß]{}note $^{ 58)}$ is merely a text written by Vermeil, not really worth reproducing here):

\[Klein-1892/22:quote\]

Reelle algebraische Kurven ergeben [*symmetrische*]{} Riemannsche Flächen und können umgekehrt allgemein gültig von letzteren aus defieniert werden, das ist der hier fundamentale Satz, den ich in §21 meiner Schrift entwickelte. Ich bezeichne dabei eine Riemannsche Fläche als symmetrisch, wenn sie durch eine konforme Abbildung zweiter Art von der Periode 2 in sich übergeführt wird (i.e. durch eine konforme Abbildung, welche die Winkel umlegt). Die symmetrischen Riemannschen Flächen eines gegebenen $p$ zerfallen, wie ich ebendort angab und Herr Weichold a.a.O. eingehender ausgeführt hat, nach der Zahl und Art ihrer “Symmetrielinien” in $[\frac{3p+4}{2}]$ Arten. Wir haben erstlich $[\frac{p+2}{2}]$ Arten [*orthosymmetrischer*]{} Flächen bez. mit $p+1, p-1, p-3, \dots$ Symmetrielinien; das sind solche symmetrische Flächen, welche längs ihrer Symmetrielinien zerschnitten, in zwei (zueinander symmetrische) Hälften zerfallen; — das einfachste (zu $p=0$ gehörige) Beispiel ist eine Kugel, welche durch “orthogonale” Projektion auf sich selbst bezogen ist —. Wir haben ferner $(p+1)$ Arten [*diasymmetrischer*]{} Flächen bzw. mit $p, p-1, \dots, 1, 0$ Symmetrielinien; das sind Flächen, die längs ihrer Symmetrielinien zerschnitten gleichwohl noch ein zusammenhängendes Ganzes vorstellen; — man vergleiche bei $p=0$, die durch eine “diametrale” Projektion auf sich selbst bezogene Kugel. —

Hence to summarize this explanation of Klein, the fundamental dichotomy seems to be motivated by the basic case of genus $0$ (the sphere), which may be acted upon in two fashions by a sense-reversing involution (orthogonal vs. diametral). This basic motivation is even more emphasized in Klein’s lectures, worth reproducing (despite its very elementary character):

\[quote:Klein-1891/92-ortho/dia\]

Wir beginnen damit, anzugeben, auf wieviel verschiedene Weisen eine Kugel mit sich selbst symmetrisch sein kann (d.h. durch eine $\Sigma$ von der Periode $2$ in sich selbst übergehen kann). Das ist offenbar auf $2$ wesentlich verschiedene Arten möglich: das eine Mal bezieht man die Kugel auf sich selbst durch eine Centralprojection, deren Centrum au[ß]{}erhalb liegt:

($1,1';2,2'; \dots$ sind entsprechende Puncte), das zweite mal durch eine Centralprojection, deren Centrum sich innererhalb der Kugel befindet.

Im ersten Falle giebt es auf der Kugel eine sogenannte , deren Puncte bei der Umformung sämmtlich festbleiben, das ist der Schnitt der Kugel mit der Polarebene des Projectionscentrums; im $2^{\rm ten}$ Falle giebt es eine solche Symmetrielinie nicht. Wir haben damit dasjenige Unterscheidungsmerkmal, nach welchem wir sogleich die symmetrischen Flächen einteilen: nach Erwähnen wir da gleich die Terminologie, welche ich anlä[ß]{}lich der Figuren 1 und 2 in Vorschlag gebracht habe. Figur 1 kann insbesondere so gezeichnet werden, da[ß]{} das Projectionscentrum unendlich weit liegt. Die Polarebene wird dann eine Diametralebene und die zugehörige Centralprojection eine orthogonale Projection. Ich sage dementschprechend überhaupt von der Figur 1, die Kugel sei bei der selben auf sich selbst bezogen. Die bei Figur 2 vorliegende Beziehung aber nenne ich , insofern bei ihr das Projectionscentrum, insbesondere in den Mittelpunkt der Kugel rücken kann, worauf je zwei diametrale Puncte der Kugel zusammengeordnet erscheinen. Diese Benennungen “orthosymmetrisch” u. “diasymmetrisch” übertrage ich dann demnächst in noch zu erklärender Weise auf die Flächen eines beliebigen $p$.

Ahlfors result precisely affords a deeper function-theoretical propagation of this Kleinian paradigm: orthosymmetric surfaces are precisely those mapping in totally real way to the orthosymmetric sphere!

The Russian school (Gudkov, Rohlin, Kharlamov, Viro, etc.) uses the (less imaginative) nomenclature Type I versus Type II, whose labelling is pure convention vintage; yet still a heritage from Klein’s initial nomenclature of 1876 [@Klein_1876]=[@Klein-Werke-II_1922 p.154] reproduced in the follwing:

\[Klein-1876:quote\]

Andererseits ergibt sich für die Kurven, deren Zügezahl $C>0$, $C<p+1$ eine bemerkenswerte Einteilung in zwei Arten.

[*Die Kurven der ersten Art haben die Eigenschaft, da[ß]{} ihre Riemannsche Fläche, längs der $C$ Züge zerschnitten, zerfällt: bei den Kurven der zweiten Art findet ein solches Zerfallen nicht statt.*]{}

Rohlin 1978 [@Rohlin_1978 p.90] refers explicitly to Klein as follows:

\[Rohlin:quote\]

Following Klein (see \[4\], p.154), we say that $\alpha$ belongs to type I if $A$ splits ${\Bbb C}A$ and to type II if $A$ does not split ${\Bbb C}A$. For example, $M$-curves obviously belong to type I.

It is worth recalling that Rohlin made a surprisingly late discovery of Klein’s work as shown by the following extract:

\[Rohlin2:quote\]

As I learned recently, more than a hundred years ago, the problem of this article occupied Klein, who succeeded in coping with curves of degree $m\le 4$ (see \[4\], p.155). I do not know whether there are publications that extend Klein’s investigations.

It is concomitant to speculate that the infamous [*Klein bottle*]{} (=[*Kleinsche Fläche*]{} which traversed the Atlantic as a “Flasche”) probably originated during Klein’s study of real curves. It just amounts to have a real curve of genus one without real points, whose complex locus will be a torus (of revolution) acted upon by a diametral involution $(x,y,z)\mapsto (-x,-y,-z)$.

Criterion for Klein’s orthosymmetry=Type I, in Russian (Klein 1876–82; Rohlin 1978, Fiedler 1978 vs. Alling-Greenleaf 1969, Geyer-Martens 1977)
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Klein’s dichotomy for symmetric surfaces prompts for criterion detecting the dividing character of a real curve.

The writer knows of essentially two methods: the first being genetic and the other qualifiable of synthetic. Despite their simplicity those criterions where overlooked by Klein, who relied upon more complicated arguments (cf. the following optional remark).

[Besides, there are several other original methods due to Klein. One involves the dual curve, and more specifically a representation assigning to each imaginary point of the curve the real line passing through it and its conjugate. When the points becomes real the limiting position of this secant becomes the tangent. In this way Klein manages to visualize the complex locus of a plane curve living in the 4D-space ${\Bbb P}^2({\Bbb C})$ onto a the 2D real projective plane as a multiple cover, and to guess the type of the curve. Beautiful pictures are to be found in vol. II of his Ges. math. Abhandl. [@Klein-Werke-II_1922]). Another brilliant argument of Klein involves a degeneration to the hyperelliptic case.]{}

[**Genetic method.**]{} This is essentially a [*surgery*]{} (if we may borrow the jargon of Thom, Milnor, etc.), and applies primarily to curves gained by small perturbation of two curves whose type is known. Maybe it is best explained on a specific example. Consider the [*Gürtelkurve*]{} as a small deformation of two conics having two nested ovals. ([*Gürtel*]{} means “belt”, a nomenclature coined by Klein in 1876 [@Klein_1876_Verlauf]=[@Klein-Werke-II_1922 p.111], presumably as a translation of the term “[*quartique annulaire*]{}” used by Zeuthen in 1874 [@Zeuthen_1874 p.417+Tafel I.,Fig.1].) Each conic corresponds to an equatorial sphere, and each smoothing amounts attaching a handle. During the process one can keep track of the two real braids to make a global drawing of the surface (compare right part of Figure \[Guertel-genetic:fig\]). Some contemplation of the drawing shows that when all smoothings are dictated by orientations then the resulting curve is dividing. Thus the Gürtelkurve is dividing. Indeed in this case all handles contains twisted braids and thus when travelling in the imaginary locus, say starting from position $A$ in the north (top) hemisphere of the left sphere and moving to the right sphere via an handle we reach position $B$ in the south hemisphere of the right sphere. Coming back to the left sphere, the twisting forces a return to the north hemisphere. We are thus never able to visit the south hemisphere of the left sphere.

-5pt0

-5pt0

[**Synthetic method.**]{} Another way to see the dividing character of the Gürtelkurve involves looking at the pencil of lines through a point lying deepest inside the two nested ovals (Figure \[Guertel-saturated:fig\]). Since each real line of this pencil cuts the quartic $C_4$ along a totally real collection of points, this induces a map between the imaginary loci $C_4({\Bbb
C})-C_4({\Bbb R})\to {\Bbb P}^1({\Bbb C})-{\Bbb P}^1({\Bbb R})$. It follows that $C_4$ is dividing since ${\Bbb P}^1$ is. (Just use the fact that the continuous image of a connected set is connected.)

-5pt0

-5pt0

More generally, this argument gives the following criterion (which quite curiously seems to have escaped Felix Klein’s attention, cf. e.g. his lectures notes 1891–92 [@Klein_1892_Vorlesung-Goettingen p.168–69], where in our opinion Klein draws the orthosymmetric character of the Gürtelkurve from more complicated arguments than those just given):

\[saturated:lemma\] If a real curve permits a morphism to the line whose fibers over real points are exclusively real, then the curve is dividing.

Conversely, one may wonder if any dividing curve is expressible as such a totally real cover of the line. I clearly remember having asked this question at several experts (ca. 1999), yet without receiving clear-cut answers, and so decided to embark on a self-study of this question. Being a slow and superficial worker, I needed circa 2 years of work until getting an answer, which turned to be positive:

\[saturated:Gabard\] [(Gabard 2001, first published in 2004)]{} Any dividing real curve admits a totally real morphism to the line. Moreover the degree of such a morphism can always be chosen $\le g+1$, where $g$ is the genus of the curve.

Having completed this work, I started some detective work, and via papers of Geyer-Martens 1977 [@Geyer-Martens_1977] and Alling-Greenleaf 1969 [@Alling-Greenleaf_1969] (probably located via the bibliography of a survey by Natanzon 1990 [@Natanzon_1990/90]) realized that L.V. Ahlfors already proved this result in 1950 (and even exposed his results at Harvard in 1948 as reported in Nehari 1950 [@Nehari_1950]). This was a great deception, or rather more my first contact with the (glamorous) L.V. Ahlfors.

[**Very anecdotic details:**]{} However as Ahlfors’ result was not fairly well-known (among the real algebraic geometry community) I received a nice invitation to expose this re-discovery in a RAAG-conference at Rennes in 2001. It was a great pleasure to meet for the first time great specialists like Johannes Huisman, Natanzon, Finashin, Viro, etc. My original proof involved an argument with incompressible fluids and Abel’s theorem to prove (\[saturated:Gabard\]). Some one week after the talk (or maybe even during the week of that conference yet preceding my talk), I confusedly realized that my argument was probably vicious, and reworked it completely to find a topological parade, amounting to the paragraphs 5,6 of Gabard 2006 [@Gabard_2006]. This argument looked more tangible and I was again invited to Rennes in 2001–2002 (by J. Huisman) to present it at a specialized seminar. At this stage I started to believe that one could improve the bound $g+1$ into $\frac{r+g+1}{2}$, which is the mean value of the number of ovals $r$ and the so-called [*Harnack bound*]{} $r\le g+1$. (In the abstract setting is truly a remark of Klein directly reducible to Riemann’s definition of the genus as the maximal number of retro-sections practicable on the pretzel without disconnecting, compare Klein 1876 [@Klein_1876 §7].) I needed some weeks (or months?) to establish this sharper version which gave a relative progress over Ahlfors.

\[saturated\_new:Gabard\] [(Gabard 2002, published 2004, 2006 [[@Gabard_2006]]{})]{} Any dividing real curve admits a totally real morphism to the line ${\Bbb P}^1$ of degree $\le \frac{r+g+1}{2}$, where $g$ is the genus of the curve and $r$ the number of “ovals” (=reellen Züge).

Using the Schottky(-Klein) double of a compact bordered Riemann surface (whose genus is visually seen to be $g=(r-1)+2p$) this can be translated as

\[saturated\_new\_bordered:Gabard\] Any compact bordered Riemann surface with $r$ contours of genus $p$ is conformally representable as full covering of the disc of degree $\le r+p$.

Dirichlet’s principle (Überzeugungskraft vs. mathematical comedy) {#Sec:Dirichlet}
=================================================================

This section (with parenthetical title derived from jokes by Hilbert 1905 [@Hilbert_1905] and Monna 1975 [@Monna_1975] resp.) recalls the early vicissitudes of a principle supported by strong physical evidence (as early as Green 1828 [@Green_1828] in print), which Riemann placed as the grounding for the edification of the theory of conformal mappings (and the allied Abelian integrals). This section can be skipped without any further ado, but it fixes the context out of which emerged (simpler?) variational problems more suited to pure function-theoretical purposes. However, Dirichlet’s principle (after Hilbert’s resurrection) pursued his life (especially in the fingers of Courant) and merged again to our main topic of the Ahlfors mapping (at least in the schlichtartig situation handled by Riemann-Schottky-Bieberbach-Grunsky). Of course, this “Dirichlet” line of thought is very active today, e.g., by Hildebrandt and his collaborators. In short, Dirichlet’s principle flourished above any expectation by Riemann, was “killed” by Weierstrass, but resurrected by Hilbert, yet re-marginalized by extremal methods (Fejér-Riesz, Carathéodory, Ostrowski, Grunsky, up to Ahlfors) and re-flourished by Douglas and Courant as a (reliable) instrument for the existence of conformal mappings.

Chronology (Green 1828, Gauss 1839, Dirichlet ca. 1840, Thomson 1847, Kirchhoff 1850, Riemann 1851–57, Weierstrass 1859/70, etc.)
---------------------------------------------------------------------------------------------------------------------------------

Apart from a early contribution of Gauss 1825 [@Gauss_1825] about local isothermic parameters (conformal mappings in the small), the “global” theory of such mappings emerged from Riemann’s Thesis 1851 [@Riemann_1851] and his subsequent work 1857 [@Riemann_1857] on abelian functions. A landmark is the [*Riemann mapping theorem*]{} (RMT) (cf. Riemann 1851 [@Riemann_1851], and Riemann 1857 [@Riemann_1857]), derived from the so-called [*Dirichlet principle*]{}. This was apparently formulated by Dirichlet as long ago as the early 1840’s (lectures in Berlin, attended by Riemann in 1847/49). (The Göttingen 1856/57 version of those were published by Grube in 1876 as [@Dirichlet_1840-1876].) Independent formulations (or utilizations) of this principle are due to Gauss 1839 [@Gauss_1839], Thomson 1847 [@Thomson_1847] (popularizing the long neglected work of Green 1828 [@Green_1828]) and Kirchhoff 1850 [@Kirchhoff_1850]. It is known that Riemann knew all those works (when exactly in another question) from a manuscript estimated 1855/60 reproduced below (source=Neuenschwander 1981 [@Neuenschwander_1981 p.225]). Riemann does not cite Thomson and Kirchhoff in 1857 [@Riemann_1857].

Mit dem Namen des Dirichlet’schen Princip’s habe ich eine Mehode bezeichnet, um nachzuweisen, da[ß]{} eine Function durch eine partielle Differentialgleichung und geeignete lineare Grenzbedingungen völlig bestimmt ist, d.h.da[ß]{} die Aufgabe, eine Function diesen Bedingungen gemä[ß]{} zu bestimmen, eine Lösung und zwar nur eine einzige Lösung zulä[ß]{}t. Es ist diese Methode von William Thomson in seiner Note Sur une équation aux différences (Liouville.T.12.p.493.) und von Kirchhoff in seiner Abhandlung über die Schwingungen einer elastischen Scheibe angewandt worden, nachdem Gau[ß]{} schon vorher eine Aufgabe, welche als ein specieller Fall dieser Aufgabe betrachtet werden kann, ähnlich behandelt hatte (Allgemeine Lehrsätze.Art.29–34.) Ich habe diese Methode nach Dirichlet benannt, da ich von Hrn Professor Dirichlet erfahren hatte, da[ß]{} er sich dieser Methode schon $\langle$seit dem Anfang der vierziger Jahre (wenn ich nicht irre) \[Bl.66r\]$\rangle$ in seinen Vorlesungen bedient habe.

There is also a letter of Riemann dated 30.Sept.1852 (cf.Neuenschwander 1981 [@Neuenschwander_1981-lettres]), where it is reported that Dirichlet supplied some references to Riemann. Here is the relevant extract, out of which we may speculate that Riemann learned the ref. to Thomson and Kirchhoff at this occasion (through Dirichlet).

[Am Freitag Morgen, um in meinem Berichte fortzufahren, suchte Dirichlet mich in meinem Zimmer auf. Ich hatte ihn bei meiner Arbeit um Rath gefragt und er gab mir nun die dazu nöthigen Notizen so vollständig, da[ß]{} mir dadurch die Sache sehr erleichtert ist. Ich hätte nach manchen Dingen auf der Bibliothek sonst lange suchen können. D.\[irichlet\] war überhaupt äu[ß]{}erst nett theilte mir mit, womit er sich in den letzten Jahren beschäftigt hatte, ging meine Dissertation mit mir durch; und so hoffe ich, da[ß]{} er mich auch später nicht vergessen und mir seine Theilnahme schenken wird. ]{}

As we know the principle was disrupted by the (non-fatal) Weierstrass’ critique 1870 [@Weierstrass_1870], but resuscitated by Hilbert in 1900-1 [@Hilbert_1900] [@Hilbert_1901/04] [@Hilbert_1905], after partial results by Neumann 1870, 1878 [@Neumann_1878], and 1884 [@Neumann_1884] Schwarz 1869/70 [@Schwarz_1869-70_Zur-Theorie-der-Abbildung], 1870 [@Schwarz_1870], 1872 [@Schwarz_1872] ([*alternierendes Verfahren*]{}) and Poincaré for fairly general boundary contours.

Dirichlet’s principle (as Riemann christened it in 1857 [@Riemann_1857-DP]) amounts to solve the first boundary value problem for the Laplacian $\Delta u=0$ by minimizing the Dirichlet integral $$\int\!\!\int \Big\{ \bigl(\frac{\partial u}{\partial x}\bigr)^2+\bigl(\frac{\partial
u}{\partial y}\bigr)^2\Big\} dx dy.$$ As a such the paradigm of [*extremality*]{} entered the arena of geometric function theory since its earliest day, and governed much of the subsequent developments.

Other noteworthy hot spots in this realm are:

$\bullet$ [*The Bieberbach conjecture*]{} (1916 [@Bieberbach_1916-BC]) $\vert a_n \vert \le n$ on the coefficients of schlicht (=univalent=injective) functions from the disc $\Delta=\{ \vert z \vert <1 \}$ to the (finite) plane ${\Bbb C}$ with Koebe’s function $k(z)=\frac{z}{(1-z)^2}=z+2z^2+3z^3+\dots$ as unique extremals among those satisfying the normalization $f(0)=0,f'(0)=1$. Completely solved by de Branges 1984.

$\bullet$ Grötzsch-Teichmüller extremal quasi-conformal mappings (1928–1939 [@Teichmueller_1939]), i.e. the search of the “möglichst konform” mapping relating two configurations. This gave a sound footing to Riemann’s liberal study of the moduli spaces (1857 [@Riemann_1857]), and paved the way to the modern theory of deformation of complex structures (Kodaira-Spencer).

Early suspicions about the Dirichlet principle (Weierstra[ß]{} 1859/70, Schwarz 1869, Prym 1871, Hadamard 1906)
---------------------------------------------------------------------------------------------------------------

Weierstrass seems to have been the first to express doubts about the Dirichlet principle, pivotal to Riemann’s theory. Weierstrass lectured on his critique in 1870, and this appeared in print as late as 1894 in his Werke. However it is known that a meeting between Riemann and Weierstrass took place in Berlin, 1859, where this issue was discussed. Klein reports upon Riemann’s reaction at several places:

Er \[Riemann\] erkannte die Berechtigung und Richtigkeit der Weierstra[ß]{}chen Kritik zwar voll an; sagte aber, wie mir Weierstra[ß]{} bei Gelegenheit erzählte: “er habe das Dirichletsche Prinzip nur als ein bequemes Hilfsmittel herangeholt, das gerade zur Hand war—seine Existenztheoreme seien trotzdem richtig.” Weierstra[ß]{} hat sich dieser Meinung wohl angeschlossen. Er veranla[ß]{}te nämlich seinen Schüler H.A. Schwarz, sich eingehend mit den Riemannschen Existenzsätzen zu befassen und andere Beweise dafür zu suchen, was durchaus gelang.

Ich erinnere mich, da[ß]{} Weierstrass mir bei Gelegenheit erzählte, Riemann habe auf die Gewinnung seiner Existenzsätze durch das “Dirichletsche Prinzip” keinerlei entscheidenden Wert gelegt. Daher habe ihm auch seine (Weierstrass’) Kritik des “Dirichletschen Prinzips” keinen besonderen Eindruck gemacht. Jedenfalls ergab sich die Aufgabe, die Existenzsätze auf andere Art zu beweisen. Diese dürfte dann Weierstrass seinem Spezialschüler Schwarz übertragen haben, bei dem er die erforderliche Verbindung geometrisch-anschaulichen Denkens mit der Fähigkeit, analytische Konvergenzbeweise zu führen, bemerkt hatte.

A more detailed chronology is roughly as follows (cf. Elstrodt-Ullrich 1999 [@Elstrodt-Ullrich_1999 p.285–6]):

$\bullet$ In the late 1850s Weierstrass notices some gap in the Dirichlet principle (DP), and presents his objection to Riemann in 1859, who is not tremendously affected claiming that his existence theorems keep however their truths.

$\bullet$ Thieme 1862, who met Riemann and requested from him some elucidations about his theory of Abelian functions, and the conversation turned to the foundation of the Dirichlet’s principle. This is materialized by a letter of Thieme to Dedekind of 1878 (reproduced in Elstrodt-Ullrich 1999 [@Elstrodt-Ullrich_1999 p.270–1], or as Quote \[quote:Thieme\] below)

$\bullet$ Kronecker 1864, in a discussion with Casorati, also exposes some criticism of the (DP). This is materialized by notes taken by Casorati, and published by Neuenschwander 1978

$\bullet$ Schwarz 1869 [@Schwarz_1869-Ueber-einige-Abbildungsaufgaben p.120] expresses for the first time in print doubts about (DP) (compare Quote \[quote:Schwarz-1869\] below).

$\bullet$ Heine February 1870 [@Heine_1870 p.360] also puts in print the reserves expressed by Weierstrass and Kronecker, specifically their objections to the assumption that a minimum must exist.

$\bullet$ Weierstrass July 1870 [@Weierstrass_1870] presents a variational problem where the minimum is not attained. This note, however, appeared in print only in 1895 in the second volume of Weierstrass’s Werke [@Weierstrass_1870].

$\bullet$ Prym 1871 [@Prym_1871 p.361–4] gives the first (published) counterexample to the (DP) (as formulated, e.g. in Grube’s text 1876 [@Dirichlet_1840-1876] based upon Dirichlet’s lectures). Prym gives a continuous function on the boundary of the unit disc such that the Dirichlet integral for the associated harmonic solution to the Dirichlet problem is infinite. However Prym expressly emphasizes that Riemann never stated such a naive version of DP corrupted by Prym’s example. In fact Prym’s example seems rather to attack a vacillating attempt by Weber 1871 [@Weber_1870] to rescue the Dirichlet principle.

\[quote:Schwarz-1869\]

Dass es stets möglich ist, die einfach zusammenhängende Fläche, welche von einer aus Stücken analytischer Curven bestehenden einfachen Linie begrenzt ist, auf die Fläche eines Kreises zusammenhangend und in den kleinsten Theilen ähnlich abzubilden, hat [*Riemann*]{} mit Zuhülfenahme des sogenannten [*Dirichlet*]{}schen Principes zu beweisen gesucht.

Da gegen die Zulässigkeit dieses Principes bei einem Existenzbeweise hinsichtlich der Strenge gegründete Einwendungen geltend gemacht worden sind, war es wünschenswerth, ein Beweisverfahren zu besitzen, gegen welches die bezüglich des [*Dirichlet*]{}schen Principes geltend gemachten Bedenken nicht erhoben werden konnten.

\[quote:Thieme\]

Vielleicht werden Sie sich meiner noch erinnern, als ich mich im Sommer 1862 in Göttingen aufhielt um bei Riemann Aufklärung über seine Theorie der Abel’schen Funct. zu erbitten. Ich traf Sie damals in der Krone, wo wir beide abgestiegen waren, und das Gespräch kam auf die, meiner damaligen Meinung nach (was seitdem vielseitig anerkannt), nicht ganz stichhaltige Begründung des Dirichlet’schen Princips, welches in der Riemann’sche Theorie fundamental ist.

Philosophical remarks {#Sec:Philosophical}
=====================

Flexibility of 2D-conformal maps
--------------------------------

Maybe one way to enlarge slightly the discussion at the philosophical level is to observe some unifying plasticity in conformal maps. The underlying principle is roughly as follows:

If there is no topological obstruction to a mapping problem, then a conformal mapping exist.

This idea is very close to Koebe’s allgemeines Uniformisierungsprinzip in Koebe 1908 [@Koebe_1908_UbaK3], which is stated as follow [*Jedes Problem der im Sinne der Analysis situs eine Lösung hat kann auch funktiontheoretisch verwirklicht werden.*]{}

Of course this is not quite true in view of say Riemann’s moduli for closed (or non closed) Riemann surfaces. However seminal instances where it works are the (RMT), the uniformization theorem (UNI) \[any simply-connected Riemann surface is biholomorphic to the sphere, the plane or the disc\], and the more general Koebe schlicht theorem to the effect that a schlichtartig Riemann surface is schlicht. Here the topological condition of “Schlichtartigkeit” (i.e. any Jordan curve divides) implies the stronger conformal embeddablility in the Riemann sphere.

[(Koebe 1908 [@Koebe_1908_UbaK3], 1910 [@Koebe_1910_UAK2])]{} Any dichotomic[^6] (=schlichtartig) Riemann surface (i.e. one divided by any Jordan curve) embeds conformally in the Riemann sphere.

Since simply-connected implies dichotomic this implies the (UNI) via (RMT).

An even stronger assertion is Bochner 1928 that any Riemann surface of finite connectivity embeds in a closed Riemann surface.

Ahlfors’ result about circle maps likewise illustrates the above principle (CP), especially if we interpret it in Klein’s theory of symmetric surfaces (compare Lemma \[saturated:lemma\]).

Free-hand pictures of some Riemann-Ahlfors maps
-----------------------------------------------

It would be nice if some general methodology for picturing such mappings could be developed. Let us try a naive look for domains (Riemann surface are harder but not hopeless). Maybe first a comment by Poritsky 1949–52 [@Poritsky_1949-52:Book p.21]:

From the above it is clear that analytical methods, at least as developed thus far, have only limited power in solving the complicated field problems arising in electric machines. Electrical engineers have resorted extensively to the use of “flux plotting” or [*free-hand drawing*]{} of the flux lines and equipotentials. As is well known, these curves, when drawn for constant equal increments $\Delta \varphi=\Delta \psi$, form a curvilinear set of [*small squares*]{}. A certain aptitude, somewhat between mechanical drawing ability and artistic drawing, is required for successful flux plotting, and with practice people possessing such aptitude can learn to draw flux plots for a great variety of cases with relative ease.

The picture below (Fig.\[Riemann:fig\]) is supposed to depict the pullback of the radial-concentric bi-foliation of the disc via a conformal representation of this 4-ply connected circle domain by a Riemann map to the disc. (Usually the term “Riemann map” is reserved for the simply-connected case, but recall that Riemann was the first to prove the existence of such maps, cf. Riemann 1857/76 Nachlass [@Riemann_1857_Nachlass]). Physically one may try to interpret it at the galvanic current generated by 4 batteries (electric charge) situated on a conducting plate. Whenever the potential generated by two charge enter in conflicts some saddle type singularity is generated (those can be counted via an Euler characteristic argument à la Riemann-Hurwitz). In the present case there is 6 saddles. In general $\chi D=d \chi
(\Delta) -\deg (R)$, and $\chi D=2-r$ (holed sphere) and the degree $d$ is $d=r$, hence there is $\deg(R)=2r-2$ ramification points. (This was of course well-known to Riemann, compare his Nachlass, or Quote \[quote:Riemann\]). Dashed lines are equipotentials.

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This sort of picture as mystical as it is (the reader confesses to have had some trouble to generate it without grasping completely the possible physico-chemical interpretation) gives the impression of grasping slightly Riemann’s title to his Nachlass (Gleichgewicht der Electricität), i.e. equilibrium potential of electricity. Our figure is pure free-hand drawing without much scientific understanding. Thus it would be nice if the computer can do better pictures, maybe via the Bergman kernel (an eminently computable object, compare e.g. Bell papers). In particular albeit it looks physically obvious, it is not clear if the charge may be placed arbitrarily. (For instance it is not clear why the corresponding divisor should be linearly equivalent to its conjugate, compare Lemme 5.2 in Gabard 2006 [@Gabard_2006].)

\[Some related references: Henrici Computational conformal map, Gaier, Konstruktive methoden in Konformen Abbildungen, etc... Or maybe Crowdy via the Klein’s prime\]

Extracting some global understanding in the non-schlicht case of such isothermic coordinates may be of some relevance to Gromov’s filling conjecture.

Further for less contours we may do similar pictures, and we then obtain the following figures (Fig.\[Annulus:fig\]). The fact that the boundary contours are circles is not crucial (but convenient for simple depiction). First we draw the electrical forces in the case of an annulus. Then we made two pictures for triply-connected with symmetrically disposed battery (electrical charge). Geometrically those are supposed to be the pull-back of the origin under the Riemann-Ahlfors map. Finally we would like to make a similar picture in the case where the charge distribution is not symmetric. Then the picturing becomes very difficult. Already in the symmetric cases it is hard to be convinced that what we are doing is really serious. There is a sort of subconscious algorithm to make such pictures: (1) first draw the thick black lines where the particles enter in collision, (2) draw at angle $\pi/4$ the dual saddle at those collision point, and then the filling by thin lines is essentially a matter of artistic feeling. Of course it is not always easy to arrange such that all lines meet perpendicularly, but experience gives some sort of algorithm to do this. Of course it is quite convenient to do such pictures on a computer rather than on the paper, as one can adjust trajectories by successive approximations. As we used a software Adobe Illustrator; with Bézier curves, thus the mathematical faithfulness of all this picture is highly questionable, but we hope that the picture are still of some qualitative value to help visualize such mappings, and to feel some sort of physical interpretation. One guess is that it amounts to have some positive electric charge at the marked point plus perhaps a distribution of such charges on the border. Then each positive particle is rejected by the charge and the border. Thus the particle move faster when there is much free-room in the plate. Alternatively one may have a biological interpretation where the source are bacteroides and the black line show the progression of the growing population which is faster in those direction where there is much vital room (imagine herbivores). Then the saddle amounts to junctions between various ethnical population, and at time one the full universe is explored. In this interpretation the proliferation of species can be slowed down either by proximity to the border (limitation of resources), or by vicinity of a competing population.

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Hard problems and the hyperelliptic claustrophobia
--------------------------------------------------

Another unifying theme when it comes to hard problems regarding Riemann surfaces is the following constat:

Several problems are fully settled in the hyperelliptic case, but horribly complicated otherwise.

This is a paradigm well known since time immemorial. Probably one of the first problem were it came acute was Jacobi’s inversion problem occupying Jacobi, then Göpel and Rosenhain (hyperelliptic case) and only Weierstrass and above all Riemann 1857 [@Riemann_1857] could handle the general case. (Weierstrass never managed to put in print his own approach probably due to the extreme difficulty to follow an arithmetized path.) Another place is Klein’s trick of degeneresence to the hyperelliptic configurations (cf. Klein 1892 [@Klein_1892_Realitaet]).

In some more contemporary problems we already addressed briefly this hyperelliptic barrier also delineate the current frontier of knowledge regarding:

\(1) The Gromov filling area conjecture (compare the work by Bangert et al. [@Bangert_2004] where the conjecture is established in the hyperelliptic case, hence in particular for membranes of genus $p=1$).

\(2) The Forstnerič-Wold conjecture [@Forstneric-Wold_2009] that compact bordered Riemann surface embeds in ${\Bbb C}^2$ (this is also known in the hyperelliptic case).

\(3) The exact determination of Ahlfors degrees à la Yamada-Gouma (this is also settled in the hyperelliptic case, but not much seems to be known beyond those configurations).

Topological methods
-------------------

We started our Introduction by claiming that topological methods have some relevance to the field of function theory, Riemann surfaces, and the allied fields.

The experience of the writer in this realm is rather modest and essentially reduces to his lucky stroke in Gabard 2006 [@Gabard_2006], about lowering the degree of a circle map upon the prediction made by Ahlfors for his extremal function.

Such topological methods are quite common in function theory (Riemann, Klein, Poincaré, Brouwer, Koebe, etc.) albeit occupying a marginal place in comparison to potential-theoretic consideration or the allied quantitative extremum problems. Let us list some contributions using qualitative topological methods in the realm of classical function theory:

\(1) The most famous (and probably important) example is the continuity method of Klein-Poincaré related to the uniformization problem. (Prior to this we may detect earlier trace of the continuity method, as one learns by reading Koebe 1912 [@Koebe_1912_BdKm], in the work of Schwarz-Christoffel and Schläfli.)

\(2) The intuitions of Klein-Poincaré were put on a firm footing by Brouwer 1912 [@Brouwer_1912_Modulmannig], [@Brouwer_1912_top-Schwierig], using invariance of domain which he was the first able to prove via combinatorial topology.

\(3) Closer to our main topic, we cite Garabedian 1949 [@Garabedian_1949] who also relies heavily on combinatorial topology to select appropriately certain auxiliary parameters.

\(4) Mizumoto 1960 [@Mizumoto_1960], who reproves the existence of (Ahlfors-type) circle map of degree $r+2p$ (i.e. like Ahlfors bound).

\(5) Gabard 2006 [@Gabard_2006], where the degree is lowered to $r+p$ (also via topological methods).

Roughly, the philosophy is that Riemann surfaces are volatile objects when fluctuating through their moduli spaces, so that practically nothing is observable outside the inherent topological substratum which turns out to behave rather stably say w.r.t. the Abel-Jacobi mapping. At least this is philosophical substance of the proof in Gabard 2006 [@Gabard_2006].

Bordered Riemann surfaces and real algebraic curves
---------------------------------------------------

It may seem at first that bordered surfaces are a bit borderline deserving less respectableness than the temple of closed Riemann surfaces. Likewise real algebraic geometry always appears like a provincial subdiscipline of pure complex algebraic geometry of the best stock.

Perhaps, less is true. The rehabilitation of reality within algebraic geometry is in good portion, especially regarding the connection with bordered (possibly) non-orientable surfaces, the credit of Felix Klein (especially in 1882 [@Klein_1882]). Moreover, independently of the algebro-geometric analytic correspondence (à la Riemann, etc.) there is another simple reason for which bordered (Riemann) surfaces took gradually more-and-more importance during the 20th century, especially under the fingers of the Finnish and Japanese schools. This pivotal rôle of compact bordered objects results indeed merely from their intervention as building elements of general open surfaces. The latter being always exhaustible through such compact elements as follows easily from Radó’s triangulability theorem of 1925 [@Rado_1925]. Once again this illustrates basically the philosophy “Nihil est in infinito…”.

Remind also that the device of exhaustion by finite (=compact) surface is somewhat older than those schools. It may have first occurred in Poincaré 1883 [@Poincare_1883] (where analytic curves or open Riemann surfaces are first taken seriously and subsumed to the uniformization paradigm) and then Koebe 1907 [@Koebe_1907_UbaK1] in same context. To caricature a bit Koebe’s proof, it amounts to use the RMT for compact discs (in a version cooked by Schwarz) and expand in the large. The exhaustion device is again used in Nevanlinna 1941 [@Nevanlinna_1941], where via exhaustions one constructs the corresponding so-called harmonic measure solving the Dirichlet problem for boundary values $0$ and $1$ on the initial resp. expanding contours of the exhaustion $F_n$, yielding the “Nullrand” dichotomy according to whether the $\omega_n$ flatten to $0$ or converge to a positive function. Ahlfors 1950 [@Ahlfors_1950] also uses (or planned to use) a similar technique for other problem. This was enough to launch the big classification programme of open Riemann surfaces.

This philosophy is surely so well-known that it was probably not worth insisting on it so vaguely. Of course, in practice this is not a simple game but one which has experimented certain successes in the past.

Lebesgue versus Riemann
-----------------------

\[11.10.12\] This paragraph is free-style philosophical lucubration coming to me right after reading the fantastic paper Forelli 1978 [@Forelli_1979]. From a narrow minded viewpoint (the writer having zero measure theoretic knowledge) it seems that modernism, especially along the “capitalistic” line of thought involving measure theory, albeit initially quite concomitant with the (older complex) function theory, ultimately may have drifted a vast body of the vital fluid in a somewhat arid valley. (For a somewhat related diagnostic cf. Morse-Heins 1947 [@Morse-Heins_1947].)

Let us be more specific. Circa 1898 the way was paved toward measure theory starting from function theoretic preoccupations (not to mention the earlier “Cantorism” starting from Fourier series). We have of course in mind E. Borel 1898 [@Borel_1898], and then the stream along Lebesgue 1902 [@Lebesgue_1902], Fatou 1906 [@Fatou_1906], the old brother F. Riesz 1907 (Fischer-Riesz effecting an Hilbert-Lebesgue unification, etc.). All those grandiose efforts/achievements may have polluted the pureness of (Riemann’s) geometric conceptions by charging the theory with complicated pathological paradigms not truly inherent to its geometric substance (at least in its finitistic aspects, which are not completely elucidated yet, e.g. Gromov’s filling conjecture). Of course the antagonism we are speaking about goes back to older generations, e.g. already acute in the Hermite vs. Jordan opposition, who were resp. anti- and pro-Lebesgue[^7]). This tension is also felt when it comes to prove existence of circle maps, where say proofs like Ahlfors’ 1950 [@Mizumoto_1960], Mitzumoto’s 1960 [@Mizumoto_1960], and many others (maybe even Gabard’s 2006 [@Gabard_2006]) proceeds along essentially classical lines, often emphasizing the soft topological category (very implicit by Riemann-Klein-Poincaré-Brouwer) instead of measure theory (again Borel-Lebesgue-Fatou-Riesz). Of course initially topology also arose from capitalism over the real line, namely the notion of metric (distance function). Yet ultimately the theory (be it axiomatically Bolzano-Cantor-Hilbert-Fréchet-Riesz-Hausdorff-Weyl or through educated intuition Riemann-Klein-Poincaré-Brouwer-Thurston) reached some higher romantic stratosphere producing some lovely science essentially the most remote from capitalistic preoccupation we were able to produce. Alas or fortunately, Grisha Perelman (and precursors Thurston/Yau-Hamilton) showed us that the likewise pleasant Riemannian geometry (albeit slightly more quantitative) turned to have some important topological impact (typically over Poincaré’s conjecture).

In a survey article by Lebesgue (ca. 1927, easy to locate), a rather primitive mercantile metaphor is appealed upon to argue that his theory of integration supersedes Riemann’s. Lebesgue argues that when a huge amount of money (delivered as a chaotic mixture of pieces and bills) requires enumeration, his theory amounts to count things properly by first enumerating what has highest value and then paying attention to the more negligible money pieces. This procedure is tantamount to subdividing rather the range of the function as do Lebesgue instead of its domain as did Cauchy or Riemann. The bulk of the US production (Rudin, Gamelin, Forelli and many others) in the 1950-1970’s is much influenced by measure flavored analysis, and the art-form continues to prosper with deep paradigms allied to Painlevé’s problem (fully solved in Tolsa 2003 [@Tolsa_2003]).

In contrast, some older workers, e.g. Koebe (cf. Gray’s 1994 paper [@Gray_1994]) as well as Lindelöf (cf. Ahlfors’ 1984 [@Ahlfors_1984-The-Joy]) (and probably more recent ones) were never full partisans of Lebesgue’s integral. Of course the latter theory added a mass of grandiose contributions, yet in some finitary problems like the one at hand (Ahlfors circle maps) its significance can probably be marginalized, or completely eliminated. So measure theory exists, but does it really capture the quintessence of the problematic we are interested in, which is more likely to be first of a [*qualitative*]{} nature (coarse existence theory). Arguably, the next evolution step is the [*quantitative*]{} phase (e.g. Ahlfors extremal problem, which is essentially solved modulo fluctuating incertitudes about degree variations of such maps). Finally any theory should culminate in the [*algorithmic*]{} era, that is claustrophobic (computer ripe) era. Remind that Riemann precisely disliked Jacobi’s approach, finding it too algorithmic and not conceptual enough (according to some forgotten source, try maybe Klein’s history [@Klein_1926-Vorlesungen-über-die-Entwicklung]: “Jacobi war ihm zu algorithmisch.” \[quoted by memory\]). At such a stage it is safer to let computers do the work, but of course it remains to find the algorithms. Will the machine not quickly be more fluent in this game as well? (Compare the little green men survey by David Ruelle in Bull. Amer. Math. Soc. ca . 1986, who tabulated on the imminence of machines cracking theorems with more ease than we are able to do. Hopefully so, since the goal of any science (indeed any living being) is to reach immortality.

So if measure theory and general open (=non-compact) Riemann surfaces inclines much to Lebesgue (and the like), it seems evident that still much work must be clarified at the more basic (combinatorial) geometric level of simpler objects, e.g. super classical algebraic geometry should be cultivated again to penetrate more deeply in a variety of problems still unsolved.

Prehistory of Ahlfors {#Sec:Prehistory-Ahlfors}
=====================

This section attempts a fairly exhaustive tabulation of works antedating Ahlfors 1950 [@Ahlfors_1950], bearing more-or-less direct connection to it. In some critical cases, some of those may also be considered as (vague?) anticipations of the Ahlfors mapping by other “pretenders”. In chronological order, we shall discuss contributions of Riemann 1857–58–76, Schottky 1875–77, Klein ca. 1876–82–92, Koebe ca. 1907, Bieberbach 1925, Grunsky 1937–41–49, Courant 1937–39–50, Teichmüller 1941.

Our history is not intended to be a smoothly readable account inclining to passive somnolence, but rather one inviting to further active searches to clarify several puzzling aspects, where in our opinion historical continuity is violently lacking. Historical turbulences arise mostly from several links hard to track down due to poor cross-referencing (especially in the case of Teichmüller 1941 [@Teichmueller_1941], who seems to credit Klein for a sort of qualitative version of the Ahlfors circle map, yet without bound upon the degree). In contrast, the first steps, i.e. the affiliation Riemann-Schottky-Bieberbach-Grunsky is well documented (but confined to planar surfaces, hence inferior to Ahlfors’ work). Courant’s contribution is more in the trend Dirichlet-Riemann-Plateau-Hilbert, but ultimately a bit sketchy when it comes to compare with Ahlfors.

Regarding Koebe, he was quite influenced by Klein’s orthosymmetry (which bears a direct connection to Ahlfors’ conformal circle map via the algebro-geometric viewpoint), but was more involved with uniformization (in particular of real curves) and the [*Kreisnormierungsprinzip*]{} (rooted back in Schottky, if not Riemann). Koebe’s work concentrates more upon conformal diffeomorphisms than branched covers. Perhaps an exception concerns his later works ca. 1910 influenced by Hilbert, where he comes to investigate more closely non-schlichtartig surfaces. However in the overall we could not find (as yet) in the torrential series of Koebe’s papers a clear-cut anticipation of Ahlfors’ result. (Relevant works of Koebe will in fact rather be surveyed in the next section.)

To summarize we have located essentially 3 potential forerunners of the Ahlfors circle map:

\(1) Klein, through a citation (or rather allusion) of Teichmüller in 1941 (supplied without precise reference!) and to which we were not able to supply sound footing (despite long searches through Klein’s collected papers, plus his harder-to-find Göttingen lectures in 1891–92 [@Klein_1891--92_Vorlesung-Goettingen], [@Klein_1892_Vorlesung-Goettingen]). In case no trace is to be found in Klein’s work, it is conceivable that Teichmüller distorted somehow his memory about Klein, in which case Teichmüller should be regarded as the genuine forerunner. It may be imagined that a micro-tunnel (=logical wormhole) links Klein to Ahlfors, and this may have existed in Teichmüller’s brain (but as far as I know no proofs are to be found in print).

\(2) Courant who makes a vague claim that the result of Riemann-Schottky-Bieberbach-Grunsky extend to configurations of higher genus. If Courant’s claim is correct, it would be of extreme interest to present the details, especially if it is possible to write down the bound arising from Courant’s argument (inspired from Plateau’s problem).

\(3) Matildi 1945/48 [@Matildi_1945/48] and Andreotti 1950 [@Andreotti_1950].

Tracing back the early history (Riemann 1857, Schwarz, Schottky 1875–77, H. Weber 1876, Bieberbach 1925, Grunsky 1937–50, Wirtinger 1942)
-----------------------------------------------------------------------------------------------------------------------------------------

From Grunsky’s papers (1937 [@Grunsky_1937], 1941 [@Grunsky_1941_KA], both cited in Ahlfors 1950’s paper) one can trace down the early history of Ahlfors theorem back to the very origin (i.e. Riemann) as follows. Grunsky was Bieberbach’s student. The latter proved a version of this theorem (yet without the extremal interpretation) for planar (schlichtartig membrane, i.e. $p=0$) in Bieberbach 1925 [@Bieberbach_1925]. In this paper, one detects an early influence of Schottky’s Dissertation (Berlin 1875, under Weierstrass) published 1877 [@Schottky_1877], as well as a Nachlass of Riemann estimated of 1857 (which was published in his Werke ca. 1876). Riemann apparently only handles the case of a [*Kreisbereich*]{} (circular domain), yet it seems that Heinrich Weber—who edited this Riemann’s Nachlass—may have considerably amputated the original manuscript. (Of course it would be a first class Leistung if some specialist of Riemann’s work would undertake the difficult project of producing a more all-inclusive account.) Let us reproduce the introduction of Bieberbach 1925 [@Bieberbach_1925]:

\[quote:Bieberbach-1925\] [Es handelt sich in dieser Arbeit um die Abbildung eines mehrfach zusammenhängenden schlichten Bereiches auf eine mehrfach bedeckte Kreisscheibe. Insbesondere stelle ich mir die Aufgabe, zu beweisen, da[ß]{} ein $n$-fach zusammenhängender Bereich stets auf eine $n$-blättrige Kreisscheibe abgebildet werden kann. Die erste im Druck erschienene Arbeit, die sich mit diesen Fragen beschäftigt, ist die Dissertation von Schottky (Berlin 1875), die im 83. Bande des Crelleschen Journal abgedruckt ist. Die Frage nach der kleinstmöglichen Blätterzahl ist dort nicht behandelt, aber die Analogie und die Beziehung zur Theorie der algebraischen Funktionen und ihrer Integrale liegt den Betrachtungen zugrunde, und auch die Beziehung zur Theorie der linearen Differentialgleichungen 2. Ordnung kommt zum Vorschein. Wie mir Herr Schottky erzählte, machte bald darauf H.A. Schwarz darauf aufmerksam, da[ß]{} sich Riemann im Sommer 1857 bereits mit der eingangs erwähnten Frage beschäftigte. In der von H. Weber bearbeiteten Darstellung dieses Teils des Riemannschen Nachlasses findet sich freilich keine volle Erledigung der Frage. Ich finde, da[ß]{} auch nicht alle Gedanken des Riemannschen Manuskriptes zur Verwendung kamen. (Vrgl. Riemanns Werke 2.Auflage S.440–444) Riemann knüpft bei seinen Überlegungen an die Theorie der linearen Differentialgleichungen an. Die Theorie der algebraischen Funktionen wird nach der Weberschen Darstellung zur Lösung des Abbildungsproblems nicht herangezogen. Dagegen scheinen mir die Riemannschen Notizen zu lehren, da[ß]{} Riemann auch einen über die Theorie der Abelschen Integralen führenden Weg unabhängig von dem bei Weber dargestellten erwogen hat. Welcher von beiden Wegen der frühere ist, vermag ich nicht zu entscheiden. ]{}

Hence the tension between Abelian integrals and potential theory seems to have always been surrounded by a little ring of mysteriousness, even in the passage of Bieberbach 1925’s article just quoted. Furthermore, after Grunsky completed in 1941 his series of papers on the question, it looked desirable to Wirtinger to publish 1942 [@Wirtinger_1942] his own interpretation of Riemann’s Nachlass which he probably knew since ca. 1899 during his duties as publisher of the second edition of Riemann’s Werke.

\[Wirtinger:quote\]

Die Abhandlung des Hrn. Helmut Grunsky, welche in diesen Berichten, Jahrgang 1941, Nr. 11, unter dem Titel “Über die konforme Abbildung mehrfach zusammenhängender Bereiche auf mehrblättrige Kreise II” erschienen ist, bringt mir Überlegungen wieder gegenwärtig, welche unmittelbar an die klassische Dissertation von F. Schottky (Berlin 1875) anschlie[ß]{}en, welche noch vor dem Bekanntwerden des Riemannschen Fragmentes über das Gleichgewicht der Elektrizität auf Zylindern von kreisförmigem Querschnitt (1876) erschienen ist. Zusammen mit dem dort entwickelten Symmetrieprinzip reicht die Theorie der algebraischen Funktionen vollkommen aus, um zu beweisen, da[ß]{} ein von $p+1$ Randkurven, welche völlig getrennt verlaufen und von denen keine sich auf einen Punkt reduziert, begrenzter Bereich sich konform auf die $p+1$fach überdeckte Halbebene der Variabeln $z=x+iy, y\ge 0$ abbilden lä[ß]{}t, wobei noch auf jeder Linie der dem Punkte $z=\infty$ entsprechende beliebig vorgegeben werden kann.

In the above quote, Bieberbach also mentions that H.A. Schwarz was well acquainted with this Riemann’s Nachlass. In this connection, it can be reminded that the whole trend connected to the so-called [*Schwarz lemma*]{} involving Schwarz 1869–70 [@Schwarz_1869-70_Zur-Theorie-der-Abbildung p.109], Carathéodory 1907 [@Caratheodory_1907], 1912 [@Caratheodory_1912] (where the coinage “Schwarz lemma” is first used), Pick 1916, Ahlfors 1938, with some intermediate steps due to E. Schmidt ca. 1906 (as acknowledged in Carathéodory 1907 [@Caratheodory_1907]) is well known to have been another inspiring source for Ahlfors’ extremal problem.

To be even more mystical, Carathéodory mentions—in his 1936 laudation to Ahlfors’ reception of the (chocolate) Fields medal (ICM 1936)—a certain “[*Ölfleckmethode*]{} of Schwarz, which seems to be related to all this. This intriguing terminology, probably refers to the common “Ölfleck” experiment consists of taking any “oil” region in a water recipient while exciting it slightly or even strongly with a thin instrument, yet preferably without causing a rupture of its connectedness. Observationally, one can then contemplate with which determination and structural stability the possibly highly distorted “Ölfleck” restores to the round circle-shape even if there are thin necks in the initial position. This seems to be one of the most beautiful way to visualize the Riemann mapping theorem in nature. Mathematically this Öfleck experiment bears perhaps more analogy to the normal curvature flow (Huisken, etc.), than the levels of the Riemann mapping function. One can wonder if there is an identity between the curvature flow and RMT.

Schottky 1875–77
----------------

All sources indicate that Schottky discovered the circle mapping for multiply-connected domains independently of Riemann’s Nachlass. Compare the next 3 quotes of Schottky (\[quote:Schottky-1882\]) and Klein (\[Klein-1923:quote:Schottky\]), (\[Klein-1923:quote:Riemann-1858\]). In 1882, during the hot Klein-Poincaré “competition” on automorphic functions vs. Fuchsian functions, Schottky’s Thesis came again to the forefront, with Klein asking its writer for some precision about its genesis. Besides, Schottky rectified some (historically) inaccurate statement made by Klein. It resulted a letter published 1882 in Math. Annalen [@Schottky_1882_Brief], which we reproduce in part:

\[quote:Schottky-1882\]

Dass übrigens Riemann bereits die mit dieser Figur in Zusammenhang stehenden Functionen und ihre Differentialgleichungen entdeckt hat, wird durch die Stelle pag. 413–416 seiner gesammelten Werke bewiesen. \[$\bigstar$Footnote: Gleichgewicht der Electricität auf Cylindern mit kreisförmigem Querschnitt und parallelen Axen.—Herr Weber fügt als Herausgeber diesem Aufsatze die Bemerkung zu: “Von dieser und den folgenden Abhandlungen \[des Riemann’schen Nachlasses\] liegen ausgeführte Manuscripte von Riemann nicht vor. Sie sind aus Blättern zusammengestellt, welche ausser wenigen Andeutungen nur Formeln enthalten.”$\bigstar$\] Indess möchte ich betonen, dass meine Dissertation ein Jahr vor der Publication von Riemann’s Nachlass erschienen ist. Auch erfuhr ich von Letzterem erst[^8], als meine Arbeit bereits in ihrer zweiten Fassung zum Druck übergehen war. Aber ich bin glücklich, mit Ihnen die Priorität der Entdeckung Riemann’s constatiren zu können. …

Sie haben in freundlicher Weise den Wunsch geäussert, Genaueres über die Prämissen meiner damaligen Arbeit zu erfahren. Die Anregung zum selbständigen Eindringen in die Potentialtheorie verdanke ich Herrn Helmholtz. Das in der Arbeit behandelte Problem, der ursprünglichen Auffassung nach der Potentialtheorie gehörig, und wesentliche Anschauungen meiner Arbeit sind aus mathematisch-physikalischen Autoren geschöpft. Ich nenne neben den Vorlesungen und Schriften von Herrn Helmholtz insbesondere ein mir gütig von Herrn O.E. Meyer geliehenes Heft noch nicht publicirter Vorlesungen von Herrn F. Neumann, dann ferner ein Buch über Elektrostatik von Herrn Kötteritzsch, etc. Die Durchführung der so gewonnenen Ideen wurde mir sodann wesentlich erleichtert durch Herrn Weierstrass’ Vorlesungen über Abel’sche Functionen, sowie besonders durch die von Herrn Schwarz publicirten Untersuchungen über das Abbildungsproblem einfach zusammenhängender Flächen. Mit Rücksicht auf die letzteren wurde auf den Rath meines hochverehrten Lehrers, Herrn Weierstrass, der ursprünglich überreichte Entwurf der Arbeit so abgehändert, dass sich dieselbe in beiden veröffentlichten Fassungen an die Untersuchungen von Herrn Schwarz anschliesst. …

This work of Schottky enjoyed early and great recognition among colleagues, and still today is frequently cited. The reasons of this success are multiple, but I cannot resist to quote first Le Vavasseur 1902 [@Le-Vavasseur_1902] \[since in Geneva there is a prominent artist bearing a similar name\], himself quoting Picard:

Dans le Tome II de son [*Traité d’Analyse*]{}, page 285, M. Émile Picard écrit: “Deux aires planes $A$ et $A_1$, limitées chacune par un même nombre de contours, ne peuvent pas, en général, être représentées d’une manière conforme l’une sur l’autre. L’étude approfondie de ce problème a été faite par M. Schottky dans un beau et important Mémoire.”

Plus loin même Tome, page 497, en note, M. Émile Picard écrit encore: “Nous avons déjà eu l’occasion de citer le beau travail de M. Schottky; c’est un Mémoire fondamental à plus d’un titre.”

The enthusiasm for Schottky’s work diffused from France to Italy, cf. especially Cecioni 1908 [@Cecioni_1908], who may be credited for the first rigorous proof of the parallel slit map. The reason of Schottky’s popularity is the quite amazing novelty of his work in prolongation of Riemann’ ideas—but so in retrospect only for Schottky was not directly influenced by Riemann. The methods range from potential-theoretic to algebraic functions, flourishing into an breathtaking variety of results. Beside the circle map for multiply-connected domains, it contains both what later will be known as the Kreisnormierungsprinzip (KNP), plus the parallel-slit mappings (PSM). —[*Warning.*]{} In fact I am not sure that it contains KNP, but could easily have on the basis of a naive parameter count. Also it is never clear if material was amputated from the first 1875 edition of Schottky’s Thesis. According to Klein’s quote (\[Klein-1923:quote:Riemann-1858\]), it seems however that the first Latin edition (1875) of Schottky’s Thesis contains the statement of the Kreisnormierung, yet “[*nur auf Grund einer Konstantenzählung*]{}”.— At any rate, it contains (explicitly or in embryo) virtually all of the varied canonical conformal maps which will be re-studied by Koebe during the period 1904–1930, trying even to extend the results to infinite connectivity. As is notorious, this ramifies to deep waters still not completely elucidated today, cf. He-Schramm 1993 [@He-Schramm_1993], which is still the best result reached so far on the Kreisnormierung problem. Schottky’s Thesis also contains the idea of symmetric reproduction of such a domain, where Klein identifies one of the first instance of automorphic functions. The name “Schottky uniformization” is still of widespread usage today (e.g. Bers, Maskit, etc.). The influence of Schottky’s work is also apparent in the jargon “Schottky differentials” widely used in several of Ahlfors’ papers, especially Ahlfors 1950 [@Ahlfors_1950]. (From the algebro-geometric viewpoint this probably just amounts to a real differential.) Last but not least, the Schwarz principle of symmetry (1869 [@Schwarz_1869-Ueber-einige-Abbildungsaufgaben]) \[which afterwards Klein liked to identify in Riemann’s Nachla[ß]{} [@Riemann_1857_Nachlass] already, as testimonies the many brackets added in his collected papers, e.g. Klein 1923 [@Klein-Werke-III_1923 p.631, line 3]\] enables one to form the so-called [*Schottky double*]{}. All this appears first in this single work of Schottky.

The admiration for Schottky’s Thesis propagated long through the ages, e.g.:

An understanding of all identities between domain functions may be obtained by sustained application of Schottky’s theory of multiply-connected domains \[15\](=1877). Schottky proved that there is a close relation between the mapping theory of these domains and the theory of closed Riemann surfaces; the identities among domain functions have their complete analogue in the theory of Abelian integrals and might be proved by means of the latter. \[…, and on p.214\]

[**Schottky functions and related classes.**]{} Schottky \[15\](=1877) was the first to consider the family $\frak R$ of all functions which are single-valued and meromorphic in $D$ \[a multiply-connected domain\] and have real boundary values on $C$ \[the full contour of $D$\]. He developed an interesting theory of conformal mapping of multiply-connected domains from the properties of this family and established by means of it the relation of this theory with the theory of closed Riemann surfaces. It is evident that functions $f(z)\in \frak R$ are very useful in the method of contour integration.

Schottky’s Thesis originated in the ambiguous context of physical intuition vs. Weierstra[ß]{}ian rigor. It is notorious that the ultimate redaction was a hard gestation process subjected to incessant revisions demanded by Weierstra[ß]{}. As we know (from Schottky himself (\[quote:Schottky-1882\]), plus the next two quotes by Klein) the first impulse was physically motivated (Helmholtz, F. Neumann, the father of C. Neumann, etc.), and then only lectures of Weierstra[ß]{} and papers of Schwarz came to influence the mathematical treatment. It is notorious that Schottky’s Dissertation writing must have been a very difficult gestation process through Weierstra[ß]{} supervision, who drifted the methodology towards that of Schwarz. For Klein this excessive Weierstrassization is regarded from a sceptical angle (cf. again the next two quotes).

It is a delicate question to wonder about the rigor reached in Schottky, despite its ultimate foundation over technology of Schwarz as a substitute to the Dirichlet principle. To ponder its ultimate rigor, it suffices to say that all of Schottky’s results where subsequently revisited, by the following workers:

$\bullet$ Koebe for KNP in a (torrential) series of paper spread from 1906 to 1922.

$\bullet$ Cecioni 1908 [@Cecioni_1908] for the PSM (=parallel-slit mapping); the latter even mark (discretely) the superiority of his proof by emphasizing that Schottky’s argument relies on a parameter count, whereas he proposes to prove PSM “[*direttamente”*]{} (cf. p.1). This technical “gap” was of course known to Klein, cf. the next Quote \[Klein-1923:quote:Riemann-1858\]. Also in Salvemini 1930 [@Salvemini_1930 p.3] (a student of Cecioni) the critique is made more explicit: “[*Questo risultato \[=PSM\] era stato enunciato dallo Schottky in base ad un computo di parametri, computo che non è poi esauriente.*]{}”

$\bullet$ Bieberbach 1925 [@Bieberbach_1925] for the circle mapping problem.

None of those writers attacks frontally the standards of rigor in Schottky (as based upon the complicated but solid foundations laid by Schwarz). Still, the technical complications was seen as a need to find simpler derivations of the geometrical results. After sufficiently time elapsed, the subsequent generation tends to ascribe the (rigorous) proof of Schottky’s result to this second wave of workers. E.g., Grunsky 1978 [@Grunsky_1978] ascribes Schottky’s circle maps to Bieberbach 1925 [@Bieberbach_1925] (cf. Quote \[quote:Grunsky-1978\]), and Bieberbach 1968 [@Bieberbach_1968-Das-Werk-Paul-Koebes] credits Koebe for the proof of KNP (in finite connectivity). All these redistributions are done without specific objections upon the original arguments Schottky’s. This is a usual loose process relegating methodologies just due to their cumbersomeness, as a sufficient reason for lack of rigor.

In contrast, even more contemporary workers still credits Schottky for the first proof of the Kreisnormierung result (cf. e.g., Schiffer-Hawley 1962 [@Schiffer-Hawley_1962 p.183]). So it is a subtle socio-cultural game to pinpoint precisely about which writer furnished the first acceptable proof.

Klein’s comments about Riemann-Schottky
---------------------------------------

In the third volume of his collected papers Klein makes several comments about Riemann and Schottky Thesis. He insists first on the physical motivations of Schottky, which were progressively “censured” under Weierstrass’ influence.

\[Klein-1923:quote:Schottky\]

Ich greife gern noch einmal auf die wiederholt genannte Arbeit Schottkys in Crelle Journal, Bd. 83 (1877) zurück, zumal ich weiter unten (S. 578/579) ohnehin ausfürlicher auf sie zurückkommen mu[ß]{}. Die gro[ß]{}e Ähnlichkeit der auf einen besonderen Fall bezüglichen Schottkyschen Untersuchungen mit den allgemeinen meiner Schrift war mir von vornherein aufgefallen. Ich schrieb also damals an Herrn Schottky und fragte ihn nach der Enstehung seiner Ideen. Hierauf antwortete her mir in einem Briefe von Mai 1882 (der in Bd. 20 der Math. Annalen abgedruckt wurde), da[ß]{} er in der Tat ursprünglich auch von der Bertrachtung der Strömungen einer inkompressiblen Flüssigkeit ausgegangen sei und diesen physikalischen Ausgangspunkt nur auf Rat von Weierstrass bei der Drucklegung durch die Bezugnahme auf Schwarz’ Untersuchungen über konforme Abbildung ersetz habe.

Then Klein recollects some more details in the following passage. This contains an anecdotic conflict (with Bieberbach 1925 [@Bieberbach_1925]=Quote \[quote:Bieberbach-1925\]) about the estimated date of Riemann’s Nachlass. More interestingly, Klein expresses the view that Schottky’s theorem (to the effect that a multiply-connected domain is conformal to a circle domain) may be seen as the planar case of Klein’s [*Rückkehrschnitttheorem*]{}, which in turn seems to be one of the weapon that Klein used in his early strategy toward uniformization (an approach not successfully completed until Brouwer-Koebe ca. 1911 [@Klein-Brouwer-Koebe_1912]).

\[Klein-1923:quote:Riemann-1858\]

Übrigens hat Riemann ja auch die andere Art automorpher Funktionen, die enstehen, indem man an einen von Vollkreisen begrenzten Bereich der Ebene an diesen Kreisen fortgesetzt symmetrisch reproduziert (Siehe das von H. Weber bearbeitete Fragment XXV in der ersten (1876 erschienenen) bzw. XXVI in der zweiten (1892 erschienenen) Auflage der Ges. math. Werke von Riemann.) Die Prüfung der Originalblätter hat ergeben, da[ß]{} Webers Mitteilungen den Vorbereitungen zu einer im Sommer 1858 gehaltenen Vorlesung entnommen sind. Und zwar geht Riemann dabei zunächst von der Aufgabe aus, für ein von mehreren Kugeln gebildetes Konduktorsystem das Gleichgewicht elektrostatischer Ladungen zu bestimmen. Hierfür war die Benutzung des Symmetrieprinzipes in den Arbeiten von W. Thompson vorgebildet, die als Briefe an Liouville in dessen Journal von 1845 an erschienen. Also auch hier sind die mathematischen Entwicklungen aus physikalischen Anregungen erwachsen.

Auf dieselben Funktionen ist dann unabhängig in seiner Berliner Dissertation 1875 Herr Schottky gekommen. Von seinem physikalischen Ausgangspunkte ist schon oben auf S.573, die Rede gewesen. Im übrigen sind die Schicksale der Schottkyschen Arbeit, wie sie sich nach persönlicher Mitteilung des Verfassers ergeben, so merkwürdig, da[ß]{} ich gern die Gelegenheit ergreife, sie hier mitzuteilen. Es erfolgten nach einander drei verschiedene Redaktionen:

a\) Eine lateinische Fassung, die nicht publiziert ist, sondern nur der Philosophischen Fakultät in Berlin vorgelegen hat,

b\) Eine deutsche Bearbeitung, welche 1875 in Berlin als Dissertation gedruckt wurde,

c\) Die umgearbeitete Darstellung in Crelles Journal, Bd. 83 (1877).

Bei Niederschrift von a) hat der Verfasser noch keine Fühlung mit Weierstrass gehabt, dafür aber ganz seiner freien Ideenbildung folgen können. Aus dem Gutachten, da[ß]{} Weierstrass über a) seinerzeit für die Fakultät abgegeben hat und von dem ich durch die Freundlichkeit von Herrn Schottky eine Abschrift vor Augen habe, scheint mit Gewi[ß]{}heit hervorzugehen, da[ß]{} Schottky hier, freilich nur auf Grund einer Konstantenzählung, das “Rückkehrschnitttheorem” für den besonderen, von ihm betrachteten Fall ausgeschprochen hat, d. h. die Möglichkeit, einen von $p+1$ regulären Randkurven begrenzten eben Bereich auf einen von $p+1$ Vollkreisen begrenzten Bereich konform abzubilden (also das Rückkehrschnitttheorem für den obersten orthosymmetrischen Fall, wie ich mich ausdrücke).

Die Redaktion b) ist dann durch eine erste Fühlungnahme mit Weierstrass bedingt. Bei der umfassenden Beherrschung ausgedehnter Teile der Mathematik und seiner stark ausgeprägten Persönlichkeit, die sich zu bestimmten Beweisgängen durchgearbeitet hatte, übte Weierstrass auf jüngere Forscher je nachdem einen au[ß]{}erordentlich fördernden, oder auch, wo ihm die Gedankengänge fremdartig waren, einen hemmenden Einflu[ß]{}. \[…\]. Schottky scheint ähnliche Erfahrungen gemacht zu haben, so da[ß]{} er in b) sich blo[ß]{} auf die Konstantenzählung beschränkt, ohne ihre Tragweite für das Fundamentaltheorem anzudeuten \[…\]. Die physikalische Ideenbildung aber, von der doch der Autor ausgegangen war, wird gänzlich ausgeschaltet und durch Zitate auf die das Existenzproblem der konformen Abbildungen betreffenden Arbeiten von Schwarz ersetzt.

In c) endlich ist auch noch besagte Konstantenzählung weggeblieben. \[\[\[Fu[ß]{}note: Dagegen hat Schottky in c) (S.330 daselbst), wiederum auf Grund blo[ß]{}er Konstantenzählung, den Satz ausgeschprochen, da[ß]{} sich jedes ebene, von $p+1$ Randkurven begrenzte, Gebiet umkehrbar eindeutig konform auf die Vollebene mit Ausnahme von $p+1$ geradlinigen, zur $x$-Achse parallelen Strecken abbilden lä[ß]{}t. Bereiche der letzteren Art spielen in der modernen Literatur unter dem Namen [*Schlitzbereiche*]{} bekanntlich eine wichtige Rolle.\]\]\] Statt dessen finden sich wertvolle, vorher nicht publizierte, Angaben über die verschiedenen Normalformen, die Weierstrass bei den Gebilden $p>2$ unterschied; \[…\]

Incidentally this [*Rückkehrschnitttheorem*]{}, may have some connection with the Ahlfors function albeit probably no direct link is evident, there is still some striking analogy developed in the next section.

A historical puzzle: why Klein missed the Ahlfors circle mapping?
-----------------------------------------------------------------

\[27.04.12\] After reading quite closely the above comments of Klein, plus having a vague idea of the content of Schottky’s Dissertation one is puzzled by how close Klein might have been (ca. 1882) to anticipate by circa 70 years the circle map of Ahlfors (1948–1950). Here is our reasoning.

First, Schottky’s Thesis (and in cryptical form already Riemann’s Nachlass) contains two striking results:

$\bullet$ the [*circle map*]{} (CM) of a (compact) multiply-connected domain to the disc, and beside

$\bullet$ what later came to be known (in Koebe’s era, cf. e.g., Koebe 1922 [@Koebe_1922]) as the [*Kreisnormierungsprinzip*]{} (KNP) to the effect that any such domain is conformally equivalent to a circular domain. (Recall from Klein’s quote (\[Klein-1923:quote:Riemann-1858\]) that this occurs only in the original Latin version of Schottky’s Thesis.)

Both results are natural extensions of RMT (=Riemann mapping theorem) either by allowing branched coverings or just by using faithful conformal diffeomorphisms (but then of course the target depends upon moduli).

Now loosely speaking one may consider both results (CM and KNP) as lying at the same order of difficulty (at least both are to be found in Schottky’s Thesis).

Next, Klein points out (cf. right above Quote \[Klein-1923:quote:Riemann-1858\]) that he was able in 1881–82 to prove an extension of (KNP) to positive genus $p>0$, which he calls (apparently with Fricke’s assistance—cf. Klein 1923 [@Klein-Werke-III_1923 p.623, footnote 4]) the [*Rückkehrschnitttheorem*]{} (RST). Klein was very proud of this result (cf. especially Klein 1923 [@Klein-Werke-III_1923 p.584], where this discovery is dated from September 1881 (Borkum)), comparing it (as a psychological experience) to Poincaré’s discovery of his general [*fonctions fuchsiennes*]{}.

Thus, there is an obvious commutative diagram (Fig.\[KNP-RST:fig\]), and whatsoever the actual meaning of Klein’s (RST) should be, there is only a single natural candidate to fill in the diagram at the (triple) question-marks, namely the Ahlfors circle map. This accentuates once more why Klein may have been a serious candidate to anticipate the Ahlfors circle map, at least without extremal interpretation.

-5pt0

Furthermore, in view of say Gabard 2006 [@Gabard_2006], the Ahlfors mapping amounts essentially to Jacobi’s inversion problem in the real case, and here again this was one of Klein’s major preoccupation (cf. e.g., Weichold’s Thesis 1883 [@Weichold_1883], Hurwitz’s work 1883 [@Hurwitz_1883], plus other sources, e.g. Klein 1892 [@Klein_1892_Realitaet]).

Of course, it would be an excellent project to try getting acquainted with Klein’s techniques so as to inspect if they lead to another elementary existence-proof of Ahlfors maps. Again it should be recalled that even if Klein himself was never able to complete his programme some helping hand from Brouwer-Koebe ultimately vindicated all of Klein’s intuitions.

Rückkehrschnittteorem (Klein 1881–82) {#sec:Ruckkehrschnittthm}
-------------------------------------

Klein found this theorem in 1881, and published it 1882 in [@Klein_1882_Ruckkehrschnitt]. From the start the paper confesses to use some irregular methods.

What does Klein in this paper? First he takes a closed Riemann surface of genus $p>1$ (w.l.o.g) and traces on it $p$ disjoint Rückkehrschnitten (retrosections) and asserts that the cutted Riemann surface may be mapped to a $2p$-ply connected domain on the sphere, whose corresponding boundaries $A_i'$, $A_i''$ are related by a linear substitution. He then uses these $p$ substitutions to reproduce the conformal mapping ad infinitum (a trick already present in Riemann’s Nachlass 1857 [@Riemann_1857_Nachlass]). He notes that the construction depends on the right number of free constants $3p$ compatible with Riemann’s moduli $3p-3$, thus yielding a sound evidence for some sort of uniformization. Of course it is not yet the standard uniformization as the reproduced domain filling more and more the sphere still avoids an infinite set (a Cantor set). In fact this construction gives an unramified infinite cover of the given closed Riemann surface by a subregion of the sphere (which is however not simply-connected).

Then he applies a similar method to the case of symmetric Riemann surfaces by using a symmetric system of retrosections while showing that the above construction may be done equivariantly. For instance in the simpler to visualize dividing case the above dissection process leads to a similar symmetric domain, symmetric with respect to the orthogonal symmetry of the sphere (whence the name [*orthosymmetric*]{}). In the nondividing case the structural symmetry is rather the diametral one (antipodal map), whence the name [*diasymmetric*]{}.

Logically it seems that Klein’s method depends on Schottky’s inasmuch as first doing the retrosections one is reduced to the schlichtartig case which turns out to be schlicht. (This result was extended by Koebe in 1908 [@Koebe_1908_UbaK3] to schlichtartig surfaces of infinite connectivity: schlichtartig implies schlicht!)

Clearly something remains to be understood on this RST, and our guess that it is sufficiently strong to imply Ahlfors theorem is quite disputable. At any rate Klein seems to have had a clear-cut conception of how his dichotomy ortho- vs. diasymmetric is reflected into the Riemann sphere with its two real structures (equatorial symmetry vs. antipody). However the issue that dividing curves are precisely those mapping to the equatorial sphere in a totally real fashion may have escaped his attention and does not seem to be logically reducible to his RST. Yet since RST is supposed to be the positive genus case of KNP (cf. Klein’s quote (\[Klein-1923:quote:Riemann-1858\])) it may be expected that one first establishes KNP and from here one deduces a circle map, much like Riemann was able to do for the zero genus case in his Nachlass [@Riemann_1857_Nachlass]. This suggests yet another strategy to approach Ahlfors theorem.

\[04.11.12\] A more naive idea could be to start from a bordered surface of type $(r,p)$, and make $p$ retrosections to get it planar (but with $r+2p$ contours). Then there is on the dissected surface a circle map of degree $r+2p$. Of course the map is a priori not assuming the same values on both ridges of the retrosections and even if we can arrange this, we would like those points to get mapped in the interior and not the boundary of the circle.

Grunsky’s bibliographical notes (Grunsky 1978)
----------------------------------------------

Let us now reproduce Grunsky’s historical comments (in his monumental book 1978 [@Grunsky_1978 p.198]) about circle maps. (Brackets are ours additions. We added author’s names in front of the bracket-references to improve readability, plus page numbers, and finally inserted the symbol $\bigstar$ when disagreeing with Grunsky’s cross-references.)

\[quote:Grunsky-1978\]

Theorem 4.1.1. goes back to Riemann 1857/58/76 [@Riemann_1857_Nachlass], who gave some hints for the proof if \[the domain\] $D$ is bounded by circles. The first proof is due to Bieberbach 1925 [@Bieberbach_1925], who used the Schottky-double and deep results in the theory of algebraic functions. Elementary proofs were given by Grunsky 1937 [@Grunsky_1937], 1941 [@Grunsky_1941_KA]; for 4.1.3. \[a sort of auxiliary lemma in linear algebra\] see Furtwängler 1936, Bourgin 1939. Related proofs in Akira Mori 1951 [@Mori_1951], Komatu 1953 [@Komatu_1953] (containing generalizations), Tsuji 1956 [@Tsuji_1956]; cf. Golusin 1952/57 [@Golusin_1952/57], Tsuji 1959 [@Tsuji_1959-BOOK/Chelsea1975]. A proof based on the method for Plateau’s problem: Courant 1937 [@Courant_1937] \[$\bigstar$ in Gabard’s opinion this paper does not reprove the circle mapping, but rather the mapping to a Kreisbereich, due to Schottky–Koebe, cf. p.709 and p.717, of [*loc.cit.*]{}\], generalized in Courant 1939 [@Courant_1939]; cf. Courant 1950 [@Courant_1950] \[especially p.183–187\]. Another proof, using, like Bieberbach 1925 [@Bieberbach_1925], the Schottky double in Wirtinger 1942 [@Wirtinger_1942]; cf. also Rodin-Sario 1968 [@Rodin-Sario_1968] \[where ???\]. Triply connected domains: Limaye 1973 [@Limaye_1973]. Representation of the mapping function (Ahlfors function, see 4.3.) by an orthonormal system in Meschkowski 1952 [@Meschkowski_1952], by the Bergman kernel in Nehari 1950 [@Nehari_1950]. Proofs using extremal properties in papers quoted in 4.3. and 6. \[More about this below (Quote \[quote:Grunsky-1978-B\]).\] An extension to certain domains of infinite connectivity in Röding 1975 [@Roeding_1975]. A more general type of image domain for doubly connected domains in Bieberbach 1957 [@Bieberbach_1957].

Some generalizations, based on ideas used in the aforementioned papers, mainly concerning Riemann surfaces in Nehari 1950 [@Nehari_1950], Tietz 1955 [@Tietz_1955] (cf. Köditz-Timmann [@Koeditz-Timmann_1975]), Mizumoto 1960 [@Mizumoto_1960], Timmann 1969 (Diss., Hannover) [@Timmann_1969], Röding 1972 (Diss., Würzburg) [@Roeding_1972], Röding 1977 [@Roeding_1977_mero]. Cf. Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] \[$\bigstar$ where?\], Carathéodory 1950 [@Caratheodory_1950_Buch_Funktionentheorie] \[$\bigstar$ where?\], Sario-Oikawa 1969 [@Sario-Oikawa_1969] \[$\bigstar$ where?\].

[**Comments (Gabard, Mai 2012):**]{} Alas, regarding the three last books no pagination is supplied by Grunsky, and as far as I browsed through them, I failed to locate any place where Ahlfors’ circle mapping is established anew.

Now we reproduce Grunsky 1978 [@Grunsky_1978 p.199]:

\[quote:Grunsky-1978-B\]

Theorem 4.3.1., a generalization of Schwarz’ lemma to multiply connected domains, is a special case of a more general theorem (individual bounds on each boundary component, prescribed zeros) proved by Grunsky in 1942 [@Grunsky_1942] (save for uniqueness, see Grunsky 1950 [@Grunsky_1950]). Cf. Hervé 1951 [@Herve_1951]. Another proof of 4.3.1. was given by Ahlfors in 1947 [@Ahlfors_1947], completed in Ahlfors 1950 [@Ahlfors_1950] (cf. Golusin 1952/57 [@Golusin_1952/57]) and the extremal function is called the “Ahlfors function”, a term frequently used in the broader sense of any function mapping \[the domain\] $D$ \[in a\] $(1,n)$ onto $U$ \[the unit disc\]; the result was carried on to characterization of the additional zeros of the extremal function. The method used by Ahlfors, Euler-Lagrange multipliers (also pointed out in Grunsky 1946 [@Grunsky_1946] and applied in Grunsky 1940 [@Grunsky_1940]) is likewise a basis for our §6. – For further proofs of our theorem see Nehari 1951 [@Nehari_1951] and Nehari 1952 [@Nehari_1952-BOOK pp.378 ff.], and some of the papers quoted for theorem 4.6.4. – Ahlfors function in a ring domain Kubo 1952 [@Kubo_1952]. – Applications of the Ahlfors function in Alenicyn 1956 [@Alenicyn_1959], 1961 [@Alenicyn_1961], (cf. Mitjuk 1965 [@Mitjuk_1965]).

Italian school: Cecioni 1908, Stella Li Chiavi 1932, Matildi 1945/48, Andreotti 1950 {#sec:Italian-school}
------------------------------------------------------------------------------------

Of course in the overall Grunsky’s comments and references are essentially sharp (especially a deep knowledge of Russian/Ukrainian works). Maybe only some contribution of the Italian school are ignored. (Those are however meticulously listed in the book Ahlfors-Sario 1960 [@Ahlfors-Sario_1960].)

For instance the simple continuity argument in the Harnack-maximal case based upon Riemann-Roch (without Roch) gives a simple proof in this case (compare e.g., Huisman 2001 [@Huisman_2001] or Gabard 2006 [@Gabard_2006]). This simple argument goes back to Enriques-Chisini seminal book 1915/18 [@Enriques-Chisini_1915-1918], and may have been implicit in Riemann’s original manuscript (not published), compare Bieberbach’s quote (\[quote:Bieberbach-1925\]).

Further, closely allied work is to be found in works of Cecioni 1908 [@Cecioni_1908], and his students: Salvemini 1930 [@Salvemini_1930], Stella Li Chiavi 1932 [@Stella-li-Chiavi_1932], etc.

Those works certainly deserve closer studying, but they do not seem to anticipate Ahlfors circle map. One notable exception is the article Matildi 1945/48 [@Matildi_1945/48] (discovered by the writer as late as \[13.07.12\]), where existence of circle maps for surfaces bounded by a single contour seems to be established. (This Italian work was known to Ahlfors (or Sario?) at least as late as 1960, again being quoted in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960].) Of course it would be interesting to see if Matildi’s method adapts to more contours, while trying to make (more) explicit the degree bound obtained by him. Andreotti 1950 [@Andreotti_1950] seems to go precisely in this sense by including several contours (alas the writer’s Italian declined fast enough to have failed understanding properly Andreotti’s achievements).

Is there any precursor to Ahlfors 1950? {#Sec:Precusors}
=======================================

What about Teichmüller 1941? {#sec:Teichmueller}
----------------------------

One can wonder about the content of Teichmüller’s Werke. Does it overlap with the Ahlfors function? While reading the long memoir of Teichmüller 1939 [@Teichmueller_1939] it transpires to anybody familiar with Klein’s work how strong the latter’s influence is; in particular Teichmüller gives a thorough account of the (now) so-called [*Klein surfaces*]{} (and their moduli). Of course such results were anticipated by Klein (at least at the heuristic level). Hence, it seems quite natural to wonder if Teichmüller anticipated the existence of Ahlfors function (for orientable membranes). Here is a report of those portions of Teichmüller’s works which looks closest to this goal, but it should still be debated how much of the Ahlfors circle maps was anticipated by Teichmüller.

The most relevant passage in Teichmüller’s writings seems to be the following extract of Teichmüller 1941 [@Teichmueller_1941] (reedited in [@Teichmueller_1982 p.554–5]):

\[quote:Teichmueller-1941\]

Wir beschäftigen uns nur mit [**orientierten endlichen Riemannschen Mannigfaltigkeiten.**]{} Diese können als Gebiete auf geschlossenen orientierten Riemannschen Flächen erklärt werden, die von endlich vielen geschlossenen, stückweise analytischen Kurven begrenzt werden. Sie sind entweder geschlossen, also selbst geschlossene orientierte Riemannsche Flächen, die man sich endlichvielblättrig über eine $z$-Kugel ausgebreitet vorstellen darf, oder berandet. Im letzteren Falle, kann man sie nach Klein durch konforme Abbildung auf folgende Normalform bringen: ein endlichvielblättriges Flächenstück über der oberen $z$-Halbebene mit endlich vielen Windungspunkten, das durch Spiegelung an der reellen Achse eine symmetrische geschlossene Riemannsche Fläche ergibt; \[…\]

(So lä[ß]{}t sich z.B. jedes Ringgebiet, d.h. jede schlichtartige endliche Riemannsche Mannigfaltigkeit mit zwei Randkurven, konform auf eine zweiblättrige Überlagerung der oberen Halbebene mit zwei Verzweigungspunkte abbilden.)

Unfortunately, no precise cross-reference to Klein is given and one needs to browse Klein’s works (with the option of some Göttingen Lectures Note 1891/92 [@Klein_1891--92_Vorlesung-Goettingen], [@Klein_1892_Vorlesung-Goettingen] not reproduced in Klein’s collected papers). This absence of precise location is quite annoying. A charitable excuse is the World War II context in which the paper was written: “[*Weil mir nur eine beschränkte Urlaubzeit zur Verfügung steht, kann ich vieles nicht begründen, sondern nur behaupten.*]{}” (compare [*loc.cit.*]{} [@Teichmueller_1982 p.554] 2nd parag.)

Detective work: Browsing Klein through the claim of Teichmüller
---------------------------------------------------------------

Regarding Teichmüller’s cryptical allusion to Klein (as discussed in the previous section) we have the following candidates in Klein’s works (none of which at the present stage of our historical search truly corroborates Teichmüller’s crediting):

\(1) Klein 1882 [@Klein_1882 p.75]=[@Klein-Werke-III_1923 p.567] where one reads:

**

Man hat also eine komplexe Funktion des Ortes, welche in symmetrisch gelegenen Punkten geiche reelle, aber entgegengesestzt gleich imaginäre Werte aufweist.

This looks quite close to the desired assignment, yet in reality only corresponds to the existence of a real morphism on any real curve; equivalently the existence for any (closed) symmetric Riemann surface of an equivariant holomorphic map to the sphere acted upon by the (usual) complex conjugation fixing an equator. Hence, in our opinion, this passage of Klein cannot be regarded as a genuine forerunner of the Ahlfors circle mapping.

\(2) Another place where Klein comes quite close to Teichmüller’s assertion occurs in the same 1882 booklet “[*Über Riemanns Theorie …*]{}”, where Klein computes the moduli of real algebraic curves—equivalently symmetric Riemann surfaces (cf. [@Klein-Werke-III_1923 p.568–9]):

\[quote:Klein-1882-moduli\]

Indem wir uns jetzt zu den [*symmetrischen*]{} Flächen wenden, haben wir noch eine kleine Zwischenbetrachtung zu machen. Zunächst ist ersichtlich, da[ß]{} zwei solche Flächen nur dann “symmetrisch” aufeinander bezogen werden können, wenn sie neben dem gleichen $p$ dieselbe Zahl $\pi$ der Übergangskurven \[=real “ovals”\] darbieten und überdies beide entweder der ersten Art oder der zweiten Art angehören. \[This is the dichotomy ortho- vs diasymmetric.\] Im übrigen wiederhole man speziell für die symmetrischen Flächen die Abzählungen des §13 betreffs der Zahl der in eindeutigen Funktionen enthaltenen Konstanten unter der Bedigung, da[ß]{} nur solche Funktionen in Betracht gezogen werden, welche an symmetrischen Stellen konjugiert imaginäre Werte aufweisen. Hiermit kombiniere man sodann nach dem Muster des §19 die Zahl solcher über der $Z$-Ebene konstuierbarer mehrblättrigen Flächen, welche in bezug auf die Achse der reellen Zahlen symmetrisch sind. \[…\] Die Sache ist dann so einfach, da[ß]{} ich sie nicht speziell durchzuführen brauche. Der Unterschied ist nur, da[ß]{} die in Betracht kommenden, früher unbeschränkten Konstanten nunmehr gezwungen sind, entweder [*einzeln reell*]{} oder [*paarweise konjugiert komplex*]{} zu sein. Infolgedessen reduzieren sich alle Willkürlichkeiten auf die Hälfte. Wir mögen folgenderma[ß]{}en sagen:

[*Zur Abbildbarkeit zweier symmetrischer Flächen $p>1$ aufeinander ist neben der Übereinstimmung in den Attributen das Bestehen von $(3p-3)$ Gleichungen zwischen den reellen Konstanten der Fläche erforderlich.*]{}

If this passage sounds a bit sketchy to the reader, we may refer to Klein’s subsequent lecture notes of 1892 [@Klein_1892_Vorlesung-Goettingen p.151–4], where full details are given.

The basic idea of this (Riemann-style) moduli count is to represent a given curve of genus $g$ as an $m$-sheeted cover of the line. If $m$ is large enough (so as to avoid exceptional cases of Riemann-Roch’s theorem), a group $g_m$ of $m$ points will move in a linear system of dimension $m-g$. To specify a map to ${\Bbb P}^1$ we may send the divisor $g_m=:D$ to $0$, say, and another $D'$ (linearly equivalent to the former) to $\infty$, leaving the possibility of a scaling factor. Thus the function depends on $2m-g+1$ constants. On the other hand by Riemann-Hurwitz such maps have $2m+2g-2$ branch points. Hence considering the totality of such covers modulo those yielding the same curve leaves $2m+2g-2-(2m-g+1)=3g-3$ essential constants. (cf. also Griffiths-Harris 1978 [@Griffiths-Harris_1978/94 p.256]).

\[15.12.12\] It is legitimate to wonder if this method (à la Riemann-Klein) is powerful enough to compute the gonality profile (cf. Definition \[gonality-profile:def\]).

Klein adapts this counting argument to the real case (again for full details we recommend Klein 1892 [@Klein_1892_Vorlesung-Goettingen p.151–4]). Doing so we may hope that he anticipates the Ahlfors mapping when the construction is particularized to the orthosymmetric case.

Since a totally real morphism lacks real ramification, we must prescribe imaginary conjugate branch points. However this necessary condition is not sufficient as shown by a quartic smoothing a visible conic plus an invisible one like $x^2+y^2=-1$ (alternatively consider the Fermat curve $x^4+y^4=1$ projected from the inside of the unique oval). In this case the projection from the interior of the oval yields a real map without real ramification, but not totally real.

We see no obvious link from Klein’s equivariant branched covers to the stronger assertion that fibres over real points consists only of real points, and consequently one of the orthosymmetric halves maps conformally to the upper half-plane (as Teichmüller credits to Klein).

Of course it is not impossible that a suitable complement to Klein’s method yields something like an Ahlfors mapping. By a continuity argument in Gabard 2006 [@Gabard_2006 Lemme 5.2], it would be enough to chose $g_m=:D$ as an [*unilateral*]{} divisor, i.e. one supported entirely by one half of the curve. Then we would be finished if the symmetric divisor $D^{\sigma}$ is linearly equivalent to $D$. But this condition is far from automatic and involves probably some lucky choice in the position of the initial divisor $D$.

Alternatively, one may try to specify the ramification and work out the Lüroth-Clebsch sort of argument to construct explicitly the finitely many conformal type of Riemann surfaces lying above the prescribed ramification. But the writer failed to draw any serious conclusion.

More is less: Teichmüller again (1939)
--------------------------------------

For those not overwhelmed by German prose, the following passage also bears some resemblances to the Ahlfors function:

Falls $\frak M$ eine orientierte und berandete Mannigfaltigkeit ist, braucht man $f$ nur auf $\frak M$ zu kennen, um $f$ auf $\frak F$ berechnen zu können. \[The latter is of course the doubled surface.\] $f$ mu[ß]{} dann auf den Randkurven von $\frak M$, die ja zu sich selbst punktweise konjugiert sind, reelle Werte haben. Umgekehrt ist eine Funktion der Fläche, die in unendlich vielen Randpunkten von $\frak M$ reell ist, eine Funktion von $\frak M$, denn sie stimmt mit der konjugierten in unendlich vielen Punkten überein und ist darum gleich ihrer konjugierten Funktion. Ja, wir können die Funktionen $f$ von $\frak M$ sogar ganz auf $\frak M$ charakterisieren:

Die Funktionen der orientierten berandeten endlichen Riemannschen Mannigfaltigkeit $\frak M$ sind genau die Funktionen $f$, die in $\frak M$ bis auf Pole regulär analytisch sind und die am Rande von $\frak M$ reell werden. D.h. die Punkte, wo die Funktion Werte eines abgeschlossenen Kreises der oberen oder der unteren Halbebene annimmt, sollen eine kompakte Menge im Innern von $\frak M$ bilden. In der Tat lassen sich diese Funktionen durch Spiegelung zu Funktionen von $\frak F$ machen, insbesondere sind sie auf den Randkurven von $\frak M$ stetig.

In this passage we note that just adding the single word “nur” in the third line of the 2nd parag. to read “die nur am Rande von $\frak M$ reell werden” would essentially lead to an anticipation of Ahlfors 1950. However taken literally this assertion of Teichmüller is weaker than Ahlfors’ and indeed the previous Quote \[quote:Teichmueller-1941\] is perhaps just a logical distortion (through hasty writing!) of the above more precise (but logically weaker) formulation. Under this hypothesis then we agree perfectly with Teichmüller 1941 (cf. again Quote \[quote:Teichmueller-1941\]) that this reality behaviour of functions was known to Klein.

The crucial distinction is between functions real on the boundary and those which are real only on the boundary. Now a priori a real function may be real on an interior point of the membrane, in which case the range (of the function) will not be contained in one of the half-plane, but overlap with both of them. In contrast a stronger reality behaviour arises when fibres of real points excludes imaginary conjugate points, in which case the range is contained in one of the half-plane, which is the context of Ahlfors’ circle mapping.

Courant 1937, 1939, 1950 {#sec:Courant}
------------------------

In the paper Courant 1939 [@Courant_1939], one detects another approach to the existence of circle maps via the methods of Plateau’s problem (at least so is claimed by Grunsky 1978, cf. Quote \[quote:Grunsky-1978\]). We cite some portion of Courant’s introduction:

\[quote:Courant-1939\]

The theory of Plateau’s and Douglas’ problem furnishes powerful tools for obtaining theorems on conformal mapping. Douglas emphasized (1931) that Riemann’s mapping theorem is a consequence of his solution of Plateau’s problem; then he treated doubly connected domains and in a recent paper (1939) multiply connected domains. With a different method I gave in a paper on Plateau’s problem (1937) a proof of the theorem that every $k$-fold connected domain can be mapped conformally on a plane domain bounded by $k$ circles. The same method can be applied to the proof of the parallel-slit theorem and, as will be shown in the thesis of Bella Manel, to mapping theorems for various other types of plane normal domains. It is the purpose of the present paper first to give a simplification of the method by utilizing an integral introduced by Riemann in his doctoral thesis, and secondly, to prove a mapping theorem of a different character referring to normal domains which are Riemann surfaces with several sheets. \[…\]

For the case $p=0$, the theorem was stated by Riemann, according to oral tradition. \[See Bieberbach 1925, where a proof is indicated; and Grunsky 1937, where another proof is given.\]

It should still be elucidated if this work by Courant (officially overlapping with Bieberbach-Grunsky) may also be connected to the Ahlfors circle mapping. This is still not completely clear to the writer. The topic is addressed again in Courant’s book of 1950, e.g., as follows:

[*Theorem 5.3:*]{} Every plane [\[footnote 12: As said before, in view of the general result of Chapter II the assumption that $G$ is a plane domain is not an essential restriction.\]]{} $k$-fold connected domain $G$ having no isolated boundary points can be mapped conformally onto a Riemann surface $B$ consisting of $k$ identical disks, e.g. interiors of unit circles, connected by branch points [\[footnote 13: The conformality of the mapping is of course interrupted at the branch points.\]]{} of total multiplicity $2k-2$. \[…\]

This somewhat loose footnote 12 of Courant may advance him as a forerunner of the Ahlfors circle map. Courant does not specify the degree derived by his method, but reading him literally one recovers (quite strikingly!) Ahlfors’ bound $r+2p$ (compare the following numerology):

The connectivity $k$ of a membrane of genus $p$ with $r$ contours is equal to $r+2p$ (each handles contributes 2 units to the connectivity). \[Alternatively, we may interpret the connectivity $k$ as $b_1+1$, where $b_1$ is the first Betti number. The Euler characteristic is $\chi=2-2p-r$, but also expressible as $\chi=1-b_1$ (since $b_2=0$). Back to the connectivity, we find $k=b_1+1=(1-\chi)+1=2-\chi=2-(2-2p-r)=2p+r$, as desired.\]

Adopting Courant’s branching multiplicity $b:=2k-2$, we compute the corresponding degree $d$. By Riemann-Hurwitz $\chi= d \cdot \chi (D^2)-b$, hence $d=\chi+b=(2-2p-r)+(2k-2)=2k-2p-r=2(r+2p)-2p-r=r+2p$. [*q.e.d.*]{}

This is pure numerology, without much control of the underlying geometry. More insight is suggested by Courant’s subsequent statement in [*loc.cit.*]{} [@Courant_1950 p.183–4, Thm5.3], which we reproduce:

Moreover, an arbitrarily fixed point $F_{\nu}$ on each boundary circle $\beta_{\nu}$ can be made to correspond to a fixed boundary point $P_\nu$ on the boundary continuum $\gamma_\nu$ of $G$, and the position of one simple branch point in $B$ may be prescribed. The class $\frak N$ of these domains depends on $3k-6$ real parameters: the $2k-3$ freely variable branch points represent $4k-6$ coordinates, while fixing the points $F_\nu$ reduces the number of parameters by $k$.

Extending this reasoning to (non-planar) membranes, we derive again Ahlfors’ bound, as follows:

We assume the membrane $F_{r,p}$ (of genus $p$ with $r$ contours) conformally mapped as a $d$-sheeted cover of the disc $D^2$ with $b$ branch points. As usual the Riemann-Hurwitz relation reads $\chi=d\cdot \chi(D^2)-b$. From the $b$ branch-points, one of them can be normalized to a definite position (through a conformal automorphism of the disc). Now the fibre over a boundary point of the disc gives $d$ points on $\partial F$. Those $d$ boundary points can be thought of as having a prescribed image. Thus the mapping itself is fully determined by $2(b-1)-d$ real constants.

On the other hand, we know since Klein 1882 (cf. our Quote \[quote:Klein-1882-moduli\]) that $F_{r,p}$ has $3g-3$ real moduli where $g$ is the genus of the double $2F$, i.e. $g=2p+(r-1)$. Positing the Ansatz that the family of $d$-sheeted covering surfaces has enough free-parameters to fill the full moduli space leads to the inequation $2(b-1)-d \ge 3g-3$. But $b=d-\chi$ and $2\chi=\chi(2F)=2-2g$. Hence $2(d-\chi-1)-d=d+(2g-2)-2\ge 3g-3$, i.e. $d\ge g+1$, which is Ahlfors’ bound $r+2p$.

Of course, this happy numerology (noticed by the writer the \[20.05.12\]) is no substitute to a serious proof of the Ahlfors circle map. However Courant formulates a variational problem à la Plateau-Douglas (or Dirichlet-Riemann-Hilbert) affording existence of a circle map (presumably with the same bound as predicted by Ahlfors as prompted by our heuristic count). Unfortunately, in Courant’s book the presentation is not directly adapted to the case of general membranes of positive genus ($p>0$), making the reading somewhat hard to digest. Hopefully someone will manage in the future to present a self-contained account based upon Courant’s method. (This project involves some hard analysis and will be deferred to a subsequent technical section. ABORTED: I had not the time/force to adapt Courant’s text to higher genera as suggested by his sloppy footnote 12.) Of course in view of Carathéodory’s philosophy (cf. Quote \[quote:Caratheodory-1928\]) one may wonder which of Courant’s vs. Ahlfors approach enjoys methodological superiority? Further remind that Ahlfors (1950 [@Ahlfors_1950 p.125–6]) has also an elementary argument for circle mapping involving no extremal problem.

Another puzzling feature of the above numerology is that it gives the impression that any $r+2p$ points prescribed on the contours may be mapped to a fixed point of the circle. Whether this is really true deserves to be investigated.

Trying to read Courant’s book 1950 [@Courant_1950] with the focus of the Ahlfors circle map is not an easy task (in our opinion). We may then hope that reading the original 1939 article [@Courant_1939] is easier due to its more restricted content. Let us write down its main statement:

\[quote:Courant\_1939:statement\]

We consider a Riemann surface on a $u,v$-plane consisting of the interior of $k$ unit circles which are connected in branch points of total multiplicity $2k-2$; to this surface we affix $p\ge 0$ full planes with two branch points each. Thus we define a class of domains $B$ with the boundary $b$ on the plane of $w=u+iv$.

Now our theorem is: Each $k$-fold connected domain $G$ in the $x,y$-plane with the boundary curves $g_1,g_2,\dots,g_k$ \[…\] can be mapped conformally on a domain $B$ of our class for any fixed $p$.

In this mapping the branch points on the full planes and one more branch point may be arbitrarily prescribed and, moreover, on each boundary circle $b_\nu$ of $B$ a fixed point may be made to correspond to a fixed point of $g_\nu$.

Personally, I find this statement hard-to-read for several reasons, I shall list subsequently. Moreover it is not clear if suitably interpreted, it really implies the Ahlfors circle mapping.

How to interpret this statement of Courant? Here are some critics probably due to the writer’s incompetence (rigid brain)! On the one hand, we have $B$, which moves in a class of domains. Perhaps those are Riemann surfaces? For instance the operation of affixing $p$ full planes may give a surface of genus $p$, at least this is what is suggested by a latter publication of Courant 1940 [@Courant_1940-Acta], whose relevant portion we quote again for definiteness:

On the basis of the previous results, the proof of the characteristic relation $\varphi(w)=0$ for the solution of the variational problem becomes very simple, if the underlying class of domains $B$ is chosen not as a domain in the plane but as a Riemann surface all of whose boundary lines are unit circles. This class is defined as follows:

We consider for the case of genus zero a $k$-fold connected domain $B$ formed by the discs of $k$ unit circles which are connected in branch points of the total multiplicity $2k-2$. For higher genus $p$, we obtain domains $B$ by affixing to the $k$-fold circular disc $p$ full planes each in $4$ branch points \[footnote 2: Each such full plane represents a “handle” and increases the genus by $1$.\].

Well, but then the domain $B$ of Quote \[quote:Courant\_1939:statement\] would have genus $p$. Then how is it possible for him to get mapped conformally (in a one-to-one fashion?) to the domain $G$, which seems to be planar since its connectivity is equal to the number of boundaries! Perhaps $G$ should be assumed to be $(k+2p)$-fold connected (or put more briefly $G$ should have genus $p$ and $k$ contours)?

If so then Courant gives a (conformal) one-to-one(?) map (=diffeomorphism) $G\to B$ onto a “normal” domain $B$. To make a link with Ahlfors, it would be desirable to know if $B$ maps to the disc even after the affixing of the $p$ full planes. (Incidentally, this operation is somewhat poorly defined, but perhaps better exposed in other publications, cf. e.g., Courant’s book 1950 [@Courant_1950 p.80 and ff.] or Courant 1949/52 [@Courant_1949-52:Book].)

Hence the crucial point would be to know if $B$ is a many-sheeted cover of the disc, and if yes: how many sheets are required? Very naively $k+p$ could suffice, in which case Courant would not only compete with Ahlfors 1950 [@Ahlfors_1950], but also with Gabard 2006 …(NB: This $(k+p)$-sheeted-ness occurs again in Courant 1940 [@Courant_1940-Acta p.78], and it would be of interest to decide if this constitutes an anticipation of Gabard 2006.)

If we push our misunderstanding of Courant to its ultimate limit, we may have the impression that what he do, is an attempt to mix the parallel-slit mapping he learned from Hilbert 1909 [@Hilbert_1909], with the Riemann-Schottky-Bieberbach-Grunsky theorem, but that the resulting surgery/transplantation does not lead to any really viable creature.

Of course, probably much of our misunderstanding is caused not merely from the difficult mathematics but also from a shift in language (plus perhaps some inaccuracies due to the torrential number of publications?), yet we may still hope that either an appropriate reading (or reorganization) of Courant’s thoughts may lead to an anticipation of the Ahlfors circle map. Hence, we encourage strongly any reader able to take the defense of Courant to publish an account in this direction. Finally, we cite another papers of Courant about conformal maps, which could be of some relevance:

$\bullet$ Courant 1937 [@Courant_1937], especially p.682, footnote 7, where we read: “[*If we assume the possibility of a conformal mapping on the unit circle for all surfaces admitted to competition \[…\]*]{}”. This could have some connection with Ahlfors circle maps, but probably does not. Later on, this article contains some conformal mapping theorems, which are only announced without proof. Perhaps, those could be of some relevance. Especially Fig.11, p.722, seems to be close to Klein’s Rückkehrschnitt-Theorem, and could eventually leads to a proof of Ahlfors? This paper also relates the ideas of J. Douglas about minimal surfaces (especially his extended version of the Plateau problem for surfaces of higher topological structures, where Douglas uses systematically Klein’s symmetric surfaces). One may therefore wonder if Ahlfors’ circle maps may somehow find application in this grandiose theory of minimal surfaces à la Plateau-Douglas-Radó-Courant, etc. As far as the writer knows no direct connection is presently available in print, despite the probable vicinity of both topics.

$\bullet$ Courant 1938 [@Courant_1938], especially p.522 “[*Every plane $k$-fold connected domain can be mapped conformally to a $k$-fold unit circle*]{}”. Hence the result—we are mostly interested in—occurs here already in 1938. In contrast to the 1939 version [@Courant_1939], here neither Riemann, nor Bieberbach 1925 [@Bieberbach_1925], not even Grunsky 1937 [@Grunsky_1937] are cited. Did Courant rediscovered the result independently?

$\bullet$ Finally we quote, Courant 1919 [@Courant_1919], where (under some influence of Hilbert 1909, and Koebe 1909) conformal mappings to “normal domains” are discussed for non-schlichtartig surfaces (of finite genus). This is also re-discussed in Courant’s book of 1950 [@Courant_1950].

Last but not least, it is perhaps relevant to remind that some doubts where expressed by Tromba 1983 [@Tromba_1983-PREPRINT] about the validity of Courant’s argumentation regarding higher genus cases of the Plateau-Douglas problem (compare also Jost 1985 [@Jost_1985]). It is not clear to the writer if Tromba’s objections compromise seriously the validity of Courant’s assertions (regarding higher genus conformal maps re-derived via the method of Plateau). This could be a another obstacle toward completing a Courant-style approach to the Ahlfors map.

Douglas 1931–36–39 {#sec:Douglas}
------------------

Having discussed (very coarsely) Courant, it would be unfair to neglect J. Douglas. His resolution of Plateau’s problem interacts strongly with conformal mapping, with the distinctive attitude (partially successful) of not getting subordinated to the latter. As already pointed out (in Courant’s Quote \[quote:Courant-1939\]), Douglas re-derived the (RMT) as the 2D-case of Plateau (cf. Douglas 1931 [@Douglas_1931-Solution p.268]). Subsequently, Douglas extended his Plateau solution to configurations of higher topological structure (cf. Douglas 1936 [@Douglas_1936-Some-new-results], 1939 [@Douglas_1939-min-surf], 1939 [@Douglas_1939-The-most-general]). Thus, it is nearly natural to ask if Douglas (himself, or at least his methods) may anticipate/recover the Ahlfors circle map? Ironically, Douglas’ work relied on Koebe’s, and interestingly took a systematic advantage of (Klein’s) symmetric Riemann surfaces (e.g., orthosymmetry). Without entering the details of all those exciting connections, we just refer to the cited original works, plus the account of Gray-Micallef 2008 [@Gray-Micallef_2008], of which we quote some extracts:

An unexpected bonus of Douglas’s method is a proof of the Riemann-Carathéodory-Osgood Theorem, which follows simply by taking $n=2$. \[…\] Douglas was rightly proud that his solution not only did not require any theorems from conformal mapping but that some such theorems could, in fact, be proved using his method.

However, Douglas did have to use Koebe’s theorem in order to establish that his solution had least area among discs spanning $\Gamma$. He had hoped to fix this blemish, but he never succeeded. That had to wait for contributions from Morrey \[1948\] and, more recently, from Hildebrandt and von der Mosel \[1999\]. \[…\]

Even before working out all the details for the disc case, Douglas was considering the Plateau problem for surfaces of higher connectivity and higher genus. \[…\] As early as 26 October 1929, Douglas announced that his methods could be extended to surfaces of arbitrary genus, orientable or not, with arbitrarily many boundary curves in a space of any dimension. He may well have had a programme at this early stage, but it is doubtful that he had complete proofs. Even when he did publish details in \[3\](=1939 [@Douglas_1939-min-surf]), the arguments are so cumbersome as to be unconvincing. One should remember that Teichmüller theory was still being worked out at that time and that the description of a Riemann surface as a branched cover of the sphere is not ideally suited for the calculation of the dependence of the $A$-functional on the conformal moduli of the surface. Courant’s treatment in \[7\](=1940 [@Courant_1940-Acta]) was more transparent but still awkward. The proper context in which to study minimal surfaces of higher connectivity and higher genus had to wait until the works of Sacks-Uhlenbeck \[19\](=1981), Schoen-Yau \[20\](=1979), Jost \[11\](=1985) and Tombi-Tromba \[21\](=1988). \[…\]

Finally, we mention the recent work of Hildebrandt-von der Mosel 2009 [@Hildebrandt-von-der-Mosel_2009], plus the survey Hildebrandt 2011 [@Hildebrandt_2011]. Here we learn, that Morrey 1966 [@Morrey_1966] was the first to re-prove Koebe’s KNP (=Kreisnormierungsprinzip) via Plateau, modulo a gap fixed by Jost 1985 [@Jost_1985]. The ultimate exposition of 2009 (of loc.cit. [@Hildebrandt-von-der-Mosel_2009]) is intended to be “[*possibly simpler and more direct*]{}” (loc.cit., 2009, p.137) and “[*are complete analogs of the approach of Douglas and Courant*]{}” (loc.cit., 2011, p.77).

As an agenda curiosity, the “Plateau-ization” of conformal mapping theorems does occur along diabolic chronological regularity. From Riemann 1851 [@Riemann_1851] to Douglas 1931 [@Douglas_1931-Solution], gives an elapsing period of 80 years. For circle maps, we have from Riemann 1858 to Courant 1939($-1$) also 8 decades, and from Koebe 1904 (announcement of KNP, in his Thesis talk, yet without convergence proof until 1907/08) to Jost 1984 [@Jost_1985] gives the same interval of time. Thus Ahlfors 1950 [@Ahlfors_1950] can safely wait up to 2030, before getting reproved via the method of Plateau?

Again, from our focused viewpoint, the critical question is whether within the problem of Plateau (à la Douglas–Radó–Courant, etc.) germinates an alternative proof of the Ahlfors mapping. As far as we know, the paper closest to this goal his Courant 1939 [@Courant_1939]. Yet, we cannot readily claim that it includes the result of Ahlfors 1950.

Cecioni and his students, esp. Matildi 1945/48, and Andreotti 1950
------------------------------------------------------------------

Among several interesting works of Cecioni and his students (cf. Sec.\[sec:Italian-school\]) we point out especially the article by Matildi 1948 [@Matildi_1945/48] (discovered by the writer as late as \[13.07.12\]). In it the existence of an (Ahlfors-type) circle map in the special case of surfaces with a single contour seems to be established via classical potential-theoretic tricks, plus at the end some algebraic geometry. This work was known to Ahlfors (or Sario?) at least as late as 1960, being quoted in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960]. It would be interesting to see if Matildi’s method adapts to an arbitrary number of contours, and also try to make (more!) explicit the degree bound obtained by him. In that case Matildi should be considered as a serious forerunner of Ahlfors 1950 [@Ahlfors_1950], at least at the qualitative level (no extremal problem). Perhaps, il professore Cecioni himself has—and may have—several works (some of which we could not consult as yet) coming quite close to the circle mapping thematic à la Ahlfors.

The idea that Matildi’s argument should extend easily to the case of several contours looks an accessible exercise. Andreotti 1950 [@Andreotti_1950] seems to go precisely in this sense.

A global picture (the kaleidoscope)
-----------------------------------

The place occupied by RMT (Riemann mapping theorem) is quite pivotal in conformal mapping with an organical explosion of results around it, like:

$\bullet$ KNP=Kreisnormierungsprinzip (implicit in Riemann 1857/8, Schottky 1875 (Latin version of his Thesis, cf. Klein’s Quote \[Klein-1923:quote:Riemann-1858\]), in full by Koebe 1905-10-20).

$\bullet$ RS=Riemann-Schottky mapping of a multiply-connected domain to the disc. (This is also known as the Bieberbach-Grunsky theorem, so RS$\approx$BG, if you want.)

$\bullet$ AM=the Ahlfors mapping (of a compact bordered surface to the disc).

$\bullet$ GKN=generalized Kreisnormierung (of a compact bordered surface to a circular domain inside a closed Riemann surface of constant curvature having the same genus $p$): apart from some anticipation for $p=1$ Strebel 1987 [@Strebel_1987] and Jost (unpublished), the full result is due to Haas 1984 [@Haas_1984] (existence), and Maskit 1989 [@Maskit_1989] (uniqueness). For an approach via circle packings, compare also He 1990 [@He_1990] and He-Schramm 1993 [@He-Schramm_1993].

$\bullet$ RST=Rückkehrschnitttheorem of Klein 1882 [@Klein_1882_Ruckkehrschnitt] is yet another form of generalized Kreisnormierung to positive genera, and for simplicity we identify it loosely to GKN. The first (rigorous) proof of RST is to be found in Koebe 1910 UAK2 [@Koebe_1910_UAK2], see also Bers 1975 [@Bers_1975] for a modern account via quasiconformal deformations.

Of course (at least modulo some sloppiness) we have universal implications (just by specializing the topological structure) like $$\textrm{AM}\Rightarrow \textrm{RS} \Rightarrow \textrm{RMT}
\Leftarrow \textrm{KNP} \Leftarrow \textrm{GKN}.$$ Besides it is desirable that GKN or RST$\Rightarrow$AM, at least this would resolve our big historical puzzle about Klein-Teichmüller as anticipating Ahlfors. This desideratum is a bit cavalier, yet akin to the implication KNP$\Rightarrow$RS, which is cryptical since Riemann’s Nachlass (1857 [@Riemann_1857_Nachlass]).

On the other hand there is a large panoply of methods including:

$\bullet$ algebraic functions (Abel 1826, Jacobi 1832, Riemann 1857, etc.),

$\bullet$ potentials (Dirichlet ca. 1840, Green 1828, Gauss 1839, Thomson 1848, etc.),

$\bullet$ iterative methods (Koebe, Carathéodory 1905–12),

$\bullet$ extremal problems (Fejér-Riesz 1922, Carathéodory 1928, Ostrowski 1929, etc.),

$\bullet$ orthogonal systems (Bergman kernel 1922, Szegö 1921)

$\bullet$ Plateau-Douglas functionals (Plateau 1849, Douglas 1930, Courant 1939 via Dirichlet resurrected),

$\bullet$ circle packings (originally in Koebe 1936, rediscovered by Andreev and Thurston 1985 with convergence proof by Rodin-Sullivan 1986),

$\bullet$ Ricci flow (Hamilton 1988 [@Hamilton_1988], which specialized to 2D enables one to recover the uniformization theorem); idem via Liouville’s equation (desideratum Schwarz, followed by Picard 1890–93, Poincaré 1899, Bieberbach 1916 [@Bieberbach_1916-Delta-u-und-die-automorphen-Funkt], etc., cf. Mazzeo-Taylor 2002 [@Mazzeo-Taylor_2002] for a modern account), and also e.g., Zhang [*et al.*]{} 2012 [@Zhang-et-al_2012], where a mixed Ricci flow/Koebe’s iteration is advocated.

Blending all these results with all those methods accessing them, we get the kaleidoscope depicted below (Fig.\[Kaleidoscope:fig\]) attempting to classify a body of results in a (more-or-less) systematic fashion. Black arrows stress out methods effective in solving a certain mapping problem, whose extremity points to the source (listed in our bibliography). Starting around RMT, arrows are propagated by translation to other locations (e.g., RS, or KNP). Arrows turn to white colored, if the corresponding method has not yet been applied to solve the relevant mapping problem. Of course several methods (like the balayage of Poincaré 1907, or some of Koebe’s method may be slightly outdated having few living practitioners). In contrast, Koebe’s iteration method is still quite popular due to its computational efficiency (see e.g., Zhang et al. 2012 [@Zhang-et-al_2012]), and presumably theoretically fruitful as well (where it is used in conjunction with the Ricci flow).

-55pt 0

Of course the picture is hard to make completely reliable, yet it may aid feeling the power (or popularity) of some methods (e.g, the extremal problem method seems to apply quite universally, except presently to GKN). On the other hand some recent methods like circle packings look very powerful, and may not have as yet explored their full range of applicability (e.g. regarding AM). As discussed in the previous section, we do not know if the Plateau method could crack the AM. Another powerful method is that of the Bergman kernel, which probably also leads to a derivation of the AM. When reading papers of the golden period (1948–1950, Bergman, Schiffer, Garabedian, etc.) this seems to be almost folklore, as well as in some papers of Bell (e.g. 2002 [@Bell_2002]). While spending some time reading precisely what is put on the paper, the writer rather developed the feeling that the positive genus case is never handled in full details. (As a general lamentation, it is an easy challenge to cite about 20 papers where results proved in the planar case are followed by the apocryphal allusion that the proof works through [*mutatis mutandis*]{} without planarity proviso.)

Digression on Bieberbach and Bergman {#Sec:Bieberbach-Bergman}
====================================

The Bergman kernel {#sec:Bergman}
------------------

Among the variety of methods mentioned in the previous section, one especially popular is the Bergman kernel function. This emerges in Bergman’s Thesis 1921/22 [@Bergman_1922]. The point of departure is an area minimal problem going back to Bieberbach 1914 [@Bieberbach_1914] capturing some salient geometric feature of the Riemann mapping. Interestingly, Bergman 1922 ( [@Bergman_1922 p.245]) confesses to be not able to reprove the RMT with this method:

In dem betrachteten Spezialfall (Minimalabbildung durch analytische Funktion) ist die erhaltene Minimalfunktion die Kreisabbildungsfunktion. Wie oben gezeigt, kann man die Existenz der ersteren unabhängig von dem Hauptsatze der Funktionentheorie beweisen; es besteht somit die Möglichkeit, den Hauptsatz auf diesem Wege von neuem zu beweisen, was mir aber bis jetzt nicht gelungen ist.

A similar lamentation is expressed by Bochner 1922 [@Bochner_1922 p.184]:

Aus der Möglichkeit der Kreisuniformisierung eines einfach zusammenhängenden Bereiches folgt aber, wie Bieberbach bemerkt hat (l.c.), da[ß]{} die Minimalabbildung mit eben der Kreisabbildung identisch ist, indes ist es mir nicht gelungen, aus der Minimalabbildung der Kreisuniformisierung aufs neue herzuleiten.

In a similar vein, some 3 decades later one among the prominent aficionados of the method wrote (source=Math.-Reviews for Lehto’s Thesis 1949 [@Lehto_1949]):

Despite its great intrinsic elegance and its adaptability for numerical computations, the theory of complex orthonormal functions (centering about the concept of the Bergman kernel function) had the drawback of being a mere representation theory; the fundamental existence theorems had to be borrowed from other fields. In $\S 4$ the author fills this gap in one important instance by giving an existence proof for the parallel-slit mappings (in the case of simply-connected domains this is identical with the Riemann mapping theorem \[provided the slit is extended to $\infty$\]) within the framework of the orthonormal function theory.

So somewhere in between 1922–1949 some technological turning point must have occurred amplifying dramatically the power of the Bergman kernel method. When and how did this occurred exactly? Probably through the Bergman–Schiffer collaboration in the 40’s, plus some fresh blood like Garabedian or Lehto. In several subsequent publications of Garabedian and Schiffer, it is emphasized that parallel-slit mappings are easier than circle maps (cf. Quotes \[quote:Garabedian-Schiffer\_1950\] and \[quote:Garabedian\_1949-52\]). However the Ahlfors circle mapping seems accessible to the Bergman-Szegö orthogonal system method as suggested in Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950], where only the planar case is handled in detail. Often in literature, yet not in the just cited paper, it is sloppily insinuated that a method implemented in the planar case extends to Riemann surfaces. A typical specimen is the earlier paper by Garabedian 1950 [@Garabedian_1950] claiming another proof of the full Ahlfors theorem by deploying a broad spectrum of techniques (yet not readily reducible to the Bergman kernel) ranging from Teichmüller 1939 [@Teichmueller_1939-Dreikreisesatzes], Grunsky 1941–42 [@Grunsky_1940], [@Grunsky_1942], Ahlfors 1947 [@Ahlfors_1947] and Schiffer’s inner variations.

Inspecting back the Bergman method itself, it is not hard to understand why it is most readily implementable in the planar case. It seems indeed to require a sort global ambient coordinate system. Let us look at the beautiful original paper Bergman 1922 [@Bergman_1922 p.240]. Here the key idea is a characterization of the Riemann function $w\colon B \to \Delta$ (of a \[simply-connected\] domain $B$ to the disc) as the one whose range $w(B)$ has smallest possible area amongst all functions $f\colon B \to {\Bbb C}$ constrained by $f'(0)=1$ (and $f(0)=0$ after harmless translation so that $0\in B$). The area swept out by $f$ is calculated by the integral $$\int \int_B f'(z) \overline{f'(z)} d\omega,$$ where $d \omega$ is the surface element in the $B$-plane, while the integrand $\vert f'(z)\vert^2$ measures the distortion effected at $z$. Following Bieberbach 1914 [@Bieberbach_1914] (who in turn seems inspired by Ritz 1908 [@Ritz_1908]), Bergman plugs in place of $f(z)$ a polynomial (recall the finitistic motto of Bloch “[*Nihil est infinito …*]{}”): $$w_n(z)=a_0+a_1z+\dots+a_n z^n,$$ with coefficients determined as to minimize the above integral under the constraint $w_n'(0)=1$. There is always a unique such polynomial, which is computed by usual methods (finite extremum-problem). The limiting function $\lim_{n\to
\infty} w_n$ gives—again Bieberbach is cited—the required mapping. The method is so simple and elegant that it is hardly conceived why it fails to reprove the RMT (which Bergman and others call the [*Hauptsatz der Theorie der konformen Abbildung*]{} (p.240)). The reason is however a quite simple vicious circle, namely that the above (Bieberbach) “areal” characterization of the Riemann function logically rests on RMT. Hence the minimum function (of Bergman) is eminently computable, but the resulting power series may not have a priori the required geometrical property of univalence and the right disc-range. I guessed the latter property follows from Bieberbach 1914 [@Bieberbach_1914], hence the real problem is univalence. However on \[13.06.12\], after reading Bergman 1947 [@Bergman_1947 p.32], the opposite looks true: namely univalence is easy but the disc-range issue is not. There are mentioned two contributions, one by Bergman 1932 [@Bergman_1932] and also Schiffer 1938 [@Schiffer_1938-CRAS-domaines-minima] where the desideratum (of reproving RMT) is established for starlike domains. So almost as importantly, this source (of 1947) points out that Bergman’s dream of 1922 (new proof of RMT via the area extremum problem) was not borne out until 1947, and therefore seems really to be credited to the newer generation like Garabedian and Lehto.

Generally speaking, extremum problems are often solvable (even uniquely soluble), but it is another piece of careful analysis to control precisely the geometric behavior of solutions, e.g. in the hope to re-crack RMT. Of course, the problem was ultimately solved, cf. e.g., Garabedian 1950 [@Garabedian_1950] or the already mentioned Thesis of Lehto 1949 [@Lehto_1949], which are the first completed Bergman-style approaches to RMT.

The point for re-exposing the hearth of the method is to emphasize the rôle of polynomials generated by $z^n$ as a preferred system of global functions on the domain $B$ out of which an ideal object is processed through an extremum procedure handled [*in finito*]{}. How can one adapt this on a Riemann surface where no global parameters are supplied a priori? This is a little puzzle to the writer \[06.06.12\], but the masters (Bergman, Garabedian, Bell, etc.) often claim the method to suit the broader context with minor changes. Compare, e.g., the following sources:

$\bullet$ Bergman 1950 [@Bergman_1950 p.24, Remark] justifies in this book extensibility to Riemann surfaces by referring to results of Sario 1949–50.

$\bullet$ Garabedian 1950 [@Garabedian_1950 p.361], where one reads “[*For the sake of a simple presentation of results we have merely stated the theorem for the case of schlicht domains of finite connectivity. However the theorem is true with only one change if $D$ is a Riemann surface [\[…\]]{}. The reader will easily verify that the proof which we shall give of the theorem carries over with minor changes to the more general situation.*]{}” If not pure bluff, it is sad that Garabedian did not write down the details at that time. If we believe in the unity of mathematics especially the algebro-geometric curves and analytic Riemann surfaces at the compact level, then the existential aspect of circle maps is frankly more trivial in the “schlicht” and even “schlichtartig” case, compare e.g., the argument in Gabard 2006 [@Gabard_2006] (reproduced below as Lemma \[Enriques-Chisini:lemma\]), which in substance is the one of Bieberbach 1925 [@Bieberbach_1925], Wirtinger 1942 [@Wirtinger_1942], but perhaps slightly streamlined by the mere usage of algebro-geometric language.

Minimizing the integral vs. maximizing the derivative (suction vs. injection), i.e. Bieberbach 1914-Bergman 1921/22 vs. Fejér-Riesz 1922, etc.
----------------------------------------------------------------------------------------------------------------------------------------------

Trying to avoid the vicissitudes of life concomitant with the Dirichlet principle, the early 1920’s imagined two methods of attack to the RMT via extremum problems. Given $B \ni
a$ a simply-connected domain in the complex plane ${\Bbb C}$, which is not the plane and therefore can easily be assumed to be bounded via a suitable transformation, RMT amounts to find a conformal map to the disc. The following (animalistic) acronyms are derived by contracting the contributors’ names:

$\bullet$ (BIBER)=(Bieberbach 1914 [@Bieberbach_1914] and Bergman\[n\] 1922 [@Bergman_1922]). \[Biber=German for “beaver” (=“castor” in French).\]

$\bigstar$ [*Amongst analytic functions $f\colon B
\to {\Bbb C}$ normed by $f(a)=0$ and $f'(a)=1$ minimize the integral $\int
\int_B \vert f'(z)\vert^2 d\omega$, where $d\omega$ is the surface element of the Euclidean metric.* ]{}

$\bullet$ (FROG)=(Fejér-Riesz 1922, Carathéodory 1928 [@Caratheodory_1928]$\leftrightarrow$Ostrowski 1929 [@Ostrowski_1929], and Grunsky 1940 [@Grunsky_1940], Ahlfors 1947 [@Ahlfors_1947] in the multiply-connected context)

$\bigstar$ [ *Amongst analytic functions $f\colon
B \to \Delta$(= unit disc) normed by $f(a)=0$ maximize the modulus $ \vert f'(a)\vert$.*]{}

As remembered in the previous section, the problem BIBER was not prompt in supplying an autonomous proof of RMT, while succeeding only in the late 1940’s (Garabedian’s or Lehto’s Thesis). Further this succeeded perhaps only under the proviso of smooth boundary (Jordan curve), cf. e.g. Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950 p.164]: “[*The most serious drawback in our method is, perhaps, that we must make assumptions upon the smoothness of the boundary of the domain we consider, so that the general case is reached only after a topological approximation argument is given.*]{}”.

In contrast FROG met earlier success (cf. e.g., Ostrowski 1928/29 [@Ostrowski_1929] and Carathéodory 1928 [@Caratheodory_1928]) streamlining previous work of Fejér-Riesz 1922 (published in Radó 1923 [@Rado_1923-Uber-konf-Abb]).

For extensions to multiple-connectivity, or even Riemann surfaces, we have the following contributions:

$\bullet$ FROG leads to the works of Grunsky 1940–42 [@Grunsky_1940; @Grunsky_1942] (schlicht domains of finite connectivity) and Ahlfors 1950 [@Ahlfors_1950] (non-planar compact bordered Riemann surfaces), where the derivative $f'(a)$ is computed w.r.t. any local chart. In fact Ahlfors rather considers the variant where given two points $a,b$ the modulus of $f(b)$ has to be maximized amongst functions with $f(a)=0$.

$\bullet$ BIBER is somewhat harder to formulate on a Riemann surface $F$ (taking the rôle of the domain $B$) as the magnitudes involved in the problem require something more than the Riemann surface structure. A Riemannian metric would make the problem meaningful, but which metric to choose? Of course there is the canonical conformal metric given by uniformization of the doubled membrane $F$. Of course we deviate slightly from a self-contained proof of RMT or Ahlfors (=AMT), but this is maybe not a dramatic concession.

Thus, even in its basic formulation, some ideas are required to set a perfect analogue of the problem BIBER for a (bordered) surface. If this can be done, it is likely (or desirable) that the extremal function (whose existence and uniqueness is derived by Hilbert’s spaces arguments) is a circle map, i.e. effects a conformal representation over the disc. (This is a priori not the unit disc, but renormalize so.) In the simply-connected case, both extremals of BIBER and FROG (denoted $\beta$ and $\alpha$ respectively) yield the one and the same object, namely the Riemann mapping $B\to\Delta$ (again after a harmless scaling of $\beta$, cf. Bergman 1950 [@Bergman_1950 p.24] for its exact value in terms of the Bergman kernel). Hence, it is plausible that the least area map for the surface $F$ coincides with the Ahlfors function. So this would be a sort of conformal identity, perhaps of some practical significance.

Of course, the primary interest would be to reobtain (via BIBER) a novel proof of Ahlfors 1950 [@Ahlfors_1950]. (This game may be already implicit in several works, as those of Bergman and Garabedian itemized in the previous section, but no pedestrian redaction is available in our opinion.) Yet, the real novelty would be the resulting “binocular view” of the one and same object (i.e., the Ahlfors extremal) through two different angles, yielding a sharper perception of it. Perhaps, this gives sharper differential-geometric insights about the Ahlfors map of a membrane, and may have some implications toward Gromov’s FAC(=filling area conjecture). Remember our naive conviction that this problem FAC should succumb just under the powerful methods of 2D-conformal geometry.

Bergman kernel on Riemann surfaces
----------------------------------

\[13.06.12\] Consulting other sources (e.g. Weill 1962 [@Weill_1962]), it seems that the theory of the Bergman kernel can be developed over any Riemann surface. The idea is to use the Hilbert space structure on the space of analytic differentials. A complete exposition is e.g., Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.302]. Whether or not this leads to another proof of Ahlfors circle maps is another question.

\[15.06.12\] Other references for the Bergman kernel on Riemann surfaces include Nagura 1951 [@Nagura_1951], and Nehari 1950 [@Nehari_1950] where the Ahlfors function is expressed in terms of the Bergman kernel.

\[25.06.12\] In fact the key observation is probably the [*conformal invariance*]{} of the integral involved in the minimum problem BIBER (of the previous section). Thus it may be hoped that this problem leads to an independent treatment of the Ahlfors mapping, treated from a Hilbert space \[of “areally” ([*aérolaire*]{}) square-integrable holomorphic functions\] viewpoint. This would give some culmination to the device of Bieberbach 1914 [@Bieberbach_1914].

So having in mind the possibility of extending the BIBER minimum area problem of the previous section to compact bordered Riemann surfaces (which looks reasonable in view of the conformal invariance of this area functional) we would like to reprove the existence of a circle map (à la Ahlfors 1950 [@Ahlfors_1950]).

Relevant literature on this problem (but from our naive viewpoint not completely satisfactory) includes in chronological order:

$\bullet$ Bieberbach 1914 [@Bieberbach_1914] (simply-connected schlicht case)

$\bullet$ Bergman 1950 [@Bergman_1950 p.24], where the fact that the range of the minimizing function is a circle is considered as well-known (with reference to Bieberbach’s Lehrbuch (1945 edition) [@Bieberbach_1945-Lehrbuch]). Later in Bergman’s book 1950 [@Bergman_1950 p.87] the circle map $B\to \Delta$ is recovered through the function $F(z,
\zeta)=\frac{\hat K(z, \bar \zeta)}{\hat L(z,  \zeta)}$ defined on p.86, but it is not clear if this function solves the least area problem. (Perhaps the connection is easy to do.)

$\bullet$ Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950 p.166–7] where the BIBER problem is again formulated, but somehow only in the purpose of showing existence of the reproducing kernel function, in the optic of re-deriving the PSM (parallel-slit maps). In particular one may wonder if it possible to show by a direct analysis if the minimum function is a circle map. Circle maps are reobtained later in the paper (p.182) however through a different procedure.

$\bullet$ Nehari’s book 1952 [@Nehari_1952-BOOK] where the BIBER minimum problem appears on p.362 (for multiply-connected domain only) and its relation to the Bergman kernel is made explicit in the subsequent pages (esp. p.368-9). However I do not think that the issue about the circle mapping property of the minimum function of BIBER is handled. Later in the book (p.378) the Ahlfors extremal function is treated, yet a priori there is no clear-cut identity between the Bieberbach and Ahlfors extremal function. Nehari’s book borrows a lot of ideas from other writers without referring to them, thus it is an easy task to observe strong overlap with the previous literature (e.g. Bergman 1950 and Garabedian-Schiffer 1950).

$\beta$ and $\alpha$ problems {#sec:beta-and-alpha-problems}
-----------------------------

\[27.06.12\] As already discussed in Section 7.8, there are essentially two problems BIBER and FROG amounting respectively to minimize an integral and to maximize a derivative. We may rebaptize them respectively the $\beta$-problem (for Bieberbach-Bergman) and the $\alpha$-problem for Ahlfors (albeit this should truly be Fejér-Riesz 1922, for historical sharpness).

For simplicity we restrict to domains, though the ultimate dream is to concoct didactic expositions pertaining to Riemann surfaces.

Regarding the $\beta$-problem (of minimizing the areal integral) it has a direct Hilbert-space interpretation (recall the affiliation Dirichlet-Riemann-Hilbert-(Schmidt)-Ritz-Bieberbach-Bergman), as finding the vector of minimal length on the hyperplane defined by the prescription $f'(t)=1$, where $t$ is some fixed point (previously denoted $a$). Such minimization traduces into orthogonality to this hyperplane, yielding the so-called [*reproducing property*]{} while permitting to identify the $\beta$-extremal with the Bergman kernel (function). For a detailed execution, cf. e.g. Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950 p.166–7] (henceforth abridged GS50).

Likewise the $\alpha$-problem received ultimately a similar treatment through Garabedian’s Thesis 1949 [@Garabedian_1949] (recast in the just cited Garabedian-Schiffer article), but the treatment is somewhat more involved appealing to the Szegö kernel instead, characterized via an orthogonalization taking place along the boundary of the domain (hence in substance the idea of length rather than area). It follows in particular an explicit formula for the derivative of the Ahlfors function $\vert
f'(t) \vert=2\pi k(t,t)$ in term of the Szegö kernel. (Garabedian’s work is such a tour de force that it was represented in virtually all major texts of that period, e.g. Bergman 1950 [@Bergman_1950] and Nehari 1952 [@Nehari_1950], plus also the paper GS50.)

Can we understand better the connection between both extremal problems? Our naive question is whether the $\beta$-map is a circle map. Remember that Bieberbach 1914 [@Bieberbach_1914] has an argument in the case where the domain is simply-connected (via his first Flächensatz saying that a map from the disc with normalized derivative expands the area of the disc unless it is the identity). Combining this with the Riemann mapping, Bieberbach argues that the $\beta$-map must be disc-ranged, for otherwise we could deflate the area by post-composing with the Riemann map, hence violating the minimum property.

Alas, it seems that this argument is hard (impossible?) to extend to the multiply-connected case. Thus it is puzzling to wonder if the $\beta$-map is a circle map. If it is the case, then we could inject the $\beta$-solution into the Ahlfors problem and compare them. In view of the explicit formula of Garabedian we can even try a direct comparison of the respective derivatives at $t$ and hope to find an equality in which case by uniqueness we would have $\beta=\alpha$ (modulo scaling), i.e. a perfect coincidence between the Bieberbach and Ahlfors functions.

Of course, ideally everything should be done geometrically from the extremal problem, without duelling with hard analysis. Recall that each problem has its allied reproducing kernel, which serves to express its solution. In particular we may hope to derive the circle mapping property of the $\beta$-function from the property of its allied (Bergman) kernel (cf. GS50, p.167). And if not, we may hope to connect the $\beta$ to the $\alpha$-map through a somewhat accidental identity between their kernels functions. As far as the writer knows this is not explicitly made, and perhaps wrong.

Let us emphasize a naive duality between the $\alpha$- and $\beta$-problem. The first amounts to a maximal pressurization (inflation) within a limited container (the unit disc), whereas $\beta$ is a free vacuum deflation leading ineluctably toward a big-crunch to a point (constant map) if there were not the initial explosion sustained by the derivative normalization $f'(t)=1$. Hence it is not so surprising that the Ahlfors map is a circle-map but the same issue for the Bieberbach least-area map seems more like an isoperimetric miracle.

We learned from Gaier’s 1978 survey [@Gaier_1978-JDMV p.34–35, §C] the following piece of information. Gaier’s article contains a proof of a striking fact due to Grötzsch 1931 (see also Gaier 1977 [@Gaier_1977-Roumaine], where the precise ref. is identified as Grötzsch 1931 [@Groetzsch_1931]) that a map (non-unique!) minimizing the area integral $\int\int \vert
f'(z) \vert^2 d \omega$ (à la Bieberbach 1914 [@Bieberbach_1914]–Bergman\[n\] 1922 [@Bergman_1922], but extended to the multiply-connected setting) under the schlichtness proviso (and the normalization $f(a)=0, f'(a)=1$) maps the domain upon a [*circular slit disc*]{} (with concentric circular slits centered at the origin). According to Gaier, Grötzsch’s paper contains no details outside the indication of using his [*Flächenstreifenmethode*]{} (striptease method). Gaier’s proof combines Carleman’s isoperimetric property of rings (relating the modulus to the area enclosed by the inner contour) with Bieberbach 1914 [@Bieberbach_1914] (first area theorem) to the effect that a schlicht normalized map from the disc inflates area, unless it is the identity. A natural question \[13.07.12\] is what happens if we relax schlichtness of the map? Do we recover an Ahlfors circle map?

As a historical curiosity, Gaier 1977 [@Gaier_1977-Roumaine] remarks that the above least-area problem for schlicht functions was reposed as a research problem as late as 1976 in the Durham meeting by Aharonov (compare for the exact ref. the Math. Review by Burbea of Gaier 1977 [@Gaier_1977-Roumaine]). It is apparently Kühnau (Grötzsch’s eminent student) who pointed Grötzsch’s priority in the reference just cited (Grötzsch 1931 [@Groetzsch_1931]). It should be remembered that several treatments existed in print (prior to Aharonov’s question), e.g. the one in Sario-Oikawa’s book of 1969 [@Sario-Oikawa_1969] (see pages as in MR of Gaier 1977 [@Gaier_1977-Roumaine]), which is inspired from Reich-Warschawski 1960 [@Reich-Warschawski_1960]. All these treatments are quite involved, and Gaier 1977 [@Gaier_1977-Roumaine] claims to simplify them.

A paper related to Gaier’s and to this circle of ideas—i.e. Bieberbach’s area minimization, yet, alas not exactly furnishing our naive desideratum—is Alenicyn 1981/82 [@Alenicyn_1981/82]: this gives the exact reference to the relevant work of Carleman 1918 [@Carleman_1918] as well as to that of Vo Dang Thao 1976 [@Vo-Dang-Thao_1976] (the latter being however slightly criticized for mistakenly assuming the schlichtness of some function).

Philosophically such Bieberbach-type area minimization problem amounts to a deflation as opposed to the inflation of Ahlfors-type problem maximizing the derivative. According to popular wisdom, both viewpoints could coincide since a semi-empty bottle is the same as a half-filled one. (This reminds the story of Ahlfors’ whiskey bottle used as a defense-weapon against an aggressor.)

\[17.07.12\] We can also switch completely of extremal problem by looking at an Ahlfors (for short $\alpha$-type) extremal (inflationist) problem of maximizing the derivative among schlicht functions. Given $D$ a multiply-connected domain marked interiorly at the point $a\in D$, find among all schlicht functions $f\colon D \to {\Bbb C}$ bounded-by-one $\vert f\vert
\le 1$ the one maximizing the modulus of the derivative $f'(a)$. It is reasonable to guess that “the” (unique?) extremal map will take $D$ upon the full circle with circular slits (schlichtness being only fulfilled on the interior). It seems that this behavior is the one described in Meschkowski 1953 [@Meschkowski_1953] (basing his analysis upon a distortion result of Rengel 1932 [@Rengel_1932-33]), and see also the treatment by Reich-Warschawski 1960 [@Reich-Warschawski_1960]. Added 27.07.12: Compare also Nehari 1953 [@Nehari_1953-Inequalities p.264–5], where another treatment of this problem is given, and credits is given to Grötzsch 1928 [@Groetzsch_1928] and Grunsky 1932 [@Grunsky_1932].

[**Optional digression:**]{} Asking schlichtness up to the boundary, we get maybe the Kreisnormierung of Koebe? This would be interesting since as pointed out in one of Meschkowski’s paper cited in the bibliography (locate where exactly!?, but anecdotic because cf. also Schiffer-Hawley 1962 [@Schiffer-Hawley_1962], Hejhal 1974 [@Hejhal_1974], etc.) there was in the 1950’s no clear-cut extremal problem leading to the Kreisnormierung (even in finite connectivity). Maybe the situation changed slightly after several works of Schiffer (and his collaborator Hawley) where some Fredholm eigenvalues came into the dance (compare several refs. cited below in the period 1959–1963).

At this stage combining the analysis of Gaier 1978 [@Gaier_1978-JDMV] for the $\beta$-problem and that of Meschkowski/Reich-Warschawski for the $\alpha$-problem (refs. as in the penultimate paragraph) we see a perfect duality between the behavior of the extremal [*schlicht*]{} functions (at least qualitatively since both mappings carry the domain upon the same canonical region of a circular slit disc). Maybe one can even identify both functions (after harmless scaling). Those works raise some hope that the schlicht-relaxed $\beta$-problem (area minimization à la Bieberbach) produces again the Ahlfors map (or at least enjoys the same property of being a circle map). As far as we know \[20.07.12\], there is no such published account corroborating this intuition. This would be highly desirable to complete the symmetry of the picture below (Fig.\[alpha:fig\]) summarizing our discussion.

\[22.07.12\] On reading Alenicyn 1981/81 [@Alenicyn_1981/81 p.202], 1981/82 [@Alenicyn_1981/82], where one is referred for the least-area problem back to Nehari’s book of 1952 [@Nehari_1952-BOOK], especially pp.340 (one can safely add p.341) and p.362. Nehari’s pages340–341 are perhaps not so relevant as it is merely a set of exercises. What is truly relevant is page 362, where the least area problem is posed and partially analyzed. In fact, this least area problem is handled earlier (with somewhat sharper information) in Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949 p.201] where the solution is represented as $M(z,a) \,
M'(a,a)^{-1}=:M^{\ast}(z,a)$, where $M(z,a)=[A(z,a)-B(z,a)]/2$ is a combination of $A,B$ the two canonical parallel slit maps of the domain $B$ upon horizontal (resp. vertical) slit domains taking $a$ to $\infty$ as a simple pole with residue $+1$ (compare p.200).

\[26.07.12\] In fact this solution is already announced in Grunsky’s Thesis 1932 [@Grunsky_1932 p.140]! As to the geometry of this map $M^{\ast}$, Garabedian-Schiffer (p.201) add the fact that it is at most $n$-valent ($n$ being the number of contours of the domain, equivalently, its connectivity). (This information is not to be found in Nehari 1952 [@Nehari_1952-BOOK].) Alas, Garabedian-Schiffer (1949 ) never seem to assert that the least-area map $M^{\ast}(z,a)$ is a circle map. On p.217, they show that any unitary function $E$ (=unit-circle map) may be expressed as a linear combination of the least-area maps $M(z, n_{\nu})$ centered at the $N$ zeros $n_{\nu}$ (‘Nullstellen’) of $E$ (assumed to be simple), compare Eq. (131) and (131’). Finally, on p.219 it is observed that the area of any such $E$, mapping the domain $D$ upon the unit-circle covered $N$ times, is exactly $N\cdot \pi$ (since area as to be counted with multiplicities). Of course, if our conjecture about the circle-mapping nature of least-area maps (there is one for each center $a$) is correct, then we could sharpen Garabedian-Schiffer’s assertion about the “at most $n$-valency” into an exact $n$-valency of those maps.

\[27.07.12\] It could be the case, that our conjecture about the circle mapping nature of the least area map is settled in Lehto’s Thesis 1949 [@Lehto_1949] (see especially p.41).

\[29.07.12\] However on consulting M. Maschler 1959 [@Maschler_1959] (esp. p.173) it seems to be asserted that the range of the least area maps are unknown for domains of connectivity higher than $2$.

\[26.07.12\] To our grand surprise, we notice that the least-area problem is handled in full generality (i.e., for compact bordered Riemann surfaces) in Schiffer-Spencer 1954 [@Schiffer-Spencer_1954 p.135]. However again (as in Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949]) it is not shown that the resulting extremal function is a circle map.

At this stage we see that there is a wide variety of extremal problems, and as a rough rule we may split the most common of them into the $\alpha$- and $\beta$-type (for Ahlfors and Bieberbach resp.) Each problem is hard to analyze precisely but there is a large body of wisdoms accumulated about them by the masters (Koebe, Carathéodory, Bieberbach, Grötzsch, Grunsky, Ahlfors, Schiffer, Garabedian, Golusin, etc.) Optionally by a nebulous bottle principle there may be a certain duality (even possibly an identity) between $\alpha$- and $\beta$-solutions. At least so is the case in the simplest simply-connected setting according to Bieberbach 1914 [@Bieberbach_1914], and apparently in the multi-connected setting we have at least coincidence of the range when considering the restricted schlicht problems. We may also speculate that a careful analysis of a suitable extremal problem may lead to a solution of the Gromov filling area conjecture.

Finally we mention a related extremal problem treated in Schiffer 1938 [@Schiffer_1938], namely that of minimizing the maximum modulus in the family of schlicht functions $f\colon B \to
{\Bbb C}$ normalized by $f(a)=0$ and $f'(a)=1$. The (or rather any) extremal is shown to map (conformally) the Bereich $B$ upon a circular slit disc.

Least area problem vs. least momentum
-------------------------------------

\[03.08.12\] The menagerie of extremal problems leading to the Riemann mapping can still be further enlarged. Each extremal problem exploits the ordered structure of the real line via some real-valued functional. One may incidentally get some feeling of regression about this massive usage of real numbers in complex geometry problems, but this is common and respectable practice since Dirichlet’s principle. Regarding the problem of circle maps [*per se*]{} it is not perfectly clear what is the [*ideally suited*]{} extremal problem (if any beside that of maximizing the derivative)? What is somehow missing is an extremal principle selecting the best extremal problem! The competitive nature of such extremal problems fascinated generations but requires strong classification aptitudes in view of the difficulty of each problem and the diversity of them.

First the [*least-area problem*]{} consists in minimizing the area of the range of an analytic function counted by multiplicity. This is measured by the functional $A[f]=\int
\int \vert f' \vert^2 d\omega$ (which seems much allied to the Dirichlet integral). (To extend the problem to Riemann surfaces one just needs to take notice of the conformal invariance of this integral upon conformal change of metrics.) To avoid the minimizers collapsing to the (uninteresting) constant functions, one imposes the side condition $f'(t)=1$ at some inner point $t$ of the domain $B$. The least-area map (which exists uniquely by Hilbert space theory) effects when $B$ is simply-connected nothing but than the Riemann mapping (Bieberbach 1914 [@Bieberbach_1914]). This viewpoint was widely pursued especially by Bergman, yielding in particular the concept of Minimalbereich. See for instance Bergman 1922 [@Bergman_1922], Bergman 1929 [@Bergman_1929-Hermite] where the concept seems to emerge, yet no precise definition. As noted in Maschler’s papers e.g. 1959 [@Maschler_1959] it seems that the nature of those minimal-domains was not completely elucidated in the late 1950’s. However, Maschler—extending a result of Schiffer 1938 [@Schiffer_1938-CRAS-domaines-minima]—observes that such minimal domains satisfy the mean property. Therefore on applying the result of Davis (as quoted in Aharonov-Shapiro 1976 [@Aharonov-Shapiro_1976]) characterizing the circle as the unique domain with a one-point quadrature identity (i.e. such that the mean value property holds for all harmonic functions) one may hope to infer our desideratum that the least area map has a range which is a disc.

As we remarked it is not very clear what the range of this map is, and we may speculate about it being a (multiply-covered) circle.

Another problem is that of the “least momentum” where one minimizes instead the integral $\int\int_B \vert f(z) \vert^2
d\omega$ (notice the suppression of the derivative) and again to avoid the trivial solution $f=0$ we impose $f'(t)=1$ at some point $t\in B$ of the domain. Another possible normalization is to ask $f(t)=1$, like in Fuchs 1945 [@Fuchs_1945/46]. Here again it seems reasonable to expect circularity of the range of the minimum mapping. The intuition being that the inertia-momentum of a rotating body gets minimized for a circular body (granting some atomical resistance avoiding a complete gravitational collapse of matter).

\[07.08.12\] Of course all those problems are super-classical, yet we still find it hard to delineate the relevant clear-cut results among the super-massive literature. Our naive intuition would be that such least-area (or momentum) map are closely allied to circle maps. However it is not sure that this is the pure truth for non-simply-connected domains (and a fortiori for bordered surfaces). As we already said the relevant sources includes for the area problem:

$\bullet$ Grunsky 1932 [@Grunsky_1932 p.140], alas no details, some more details in Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949] (but no assertion of circularity) only the Grunsky formula expressing the least-area map as combination of the two slit-maps.

$\bullet$ for the least momentum see many works of Bergman starting from his Thesis 1922 [@Bergman_1922].

Perhaps it should be observed that the least-momentum problem is perhaps somewhat less easily extensible to Riemann surfaces in view of the lack of conformal invariance of its functional.

Finally, we can mention Walsh’s 1935 survey (Mémorial [@Walsh_1935]) where all such problems are united under a generalized form where more points $z_1, \dots z_n$ are prescribed in the domain joint with some prescribed values $\gamma_1, \dots
\gamma_n$ and one is required to find the map minimizing the functional under the interpolating condition $f(z_i)=\gamma_i$. Alas, in Walsh’s survey attention is confined to the simply-connected case and the multi-connected variants where at that time not systematically understood.

A digression about Nehari’s paper of 1955 {#Nehari-digression:sec}
-----------------------------------------

In Nehari 1955 [@Nehari_1955], the author presents a nice application of Bieberbach’s 1925 [@Bieberbach_1925] existence theorem of a circle map for an $n$-ply connected domain upon the disc of degree $n$. Precisely Nehari deduces a bound on the number of linearly independent solutions to a certain extremal problem (akin to those treated by Szegö 1921 [@Szego_1921]). It seems plausible that this Nehari argument is sufficiently universal to extend directly to the more general setting of compact bordered Riemann surfaces (membranes for short) upon invoking Ahlfors 1950 [@Ahlfors_1950] instead of Bieberbach 1925 [@Bieberbach_1925]. As the argument uses only the circle mapping nature of the Ahlfors map, we may even appeal to Gabard 2006 [@Gabard_2006] to obtain a sharper bound. In reality what is truly relevant is the absolute invariant of the (separating) gonality à la Coppens 2011 [@Coppens_2011]. Let us try to explore this connection, albeit some details require to be better worked out in order to really understand this technique of Nehari.

We try first to go quickly to the hearth of Nehari’s ideas. The starting point is the following extremal problem formulated for $D$ a compact domain bounded by $n$ analytic curves (for simplicity) forming its complete boundary contour $C$. Further in the interior of $D$ a set $C_1$ consisting of a finite number of rectifiable Jordan arc and/or curves is given. \[Warning: in his paper [@Nehari_1955 p.29] Nehari writes “$C_1$ will stand for a subset of $C$”, which in our opinion is just a misprint! $C$ should be $D$!? Of course, our domain $D$ differs from Nehari’s as ours includes the contours.\] Let also $L^2=L^2(D)$ be the (Hilbert) space of analytic functions on $D$ with finite integral $\int_C \vert
f(z) \vert^2 ds<\infty$ where $ds$ is the (Euclidean) length element.

. Find the functions $f\in L^2$ minimizing the norm $\int_C \vert f(z) \vert^2 ds $ under the constraint $\int_{C_1} \vert f(z) \vert^2 ds=1$.

This problem suggests looking at the functional $$J(f)=\frac{\int_C \vert f(z) \vert^2 ds}{\int_{C_1} \vert f(z)
\vert^2 ds}$$ whose minimizers are (up to scaling) the solution of problem (P).

Next Nehari sets up a certain integral equation whose eigenspace attached to the lowest eigenvalue parametrize the extremals of (P). We skip the details, but the key issue is just the linearity of the set of solutions to Problem (P). With this at hand, we can plunge directly to the core of Nehari’s argument, namely the:

[(Nehari 1955 [@Nehari_1955 p.36])]{} Assuming (as above) the domain $D$ of connectivity $n$ (=number of contours), problem [(P)]{} admits at most $n$ linearly independent solutions.

Nehari’s argument splits in 4 short steps:

[**Step 1 (Bieberbach 1925)**]{} According to the latter ([@Bieberbach_1925]) there is a circle map $f\colon D \to
\overline{\Delta}=\{\vert z\vert \le 1\} $ of degree $n$. This means that $\vert f(z) \vert=1$ exactly on the contours (i.e. $f^{-1}(\partial\overline{\Delta}=S^1)=C$) and upon changing the origin to an unramified place we may assume that $f$ has exactly $n$ zeroes, say $z_1,\dots,z_n$.

[**Step 2 (Nehari’s trick in linear algebra)**]{} Assume by contradiction that (P) has $n+1$ linearly independent solutions $f_i$ ($i=1,\dots,n+1$). We consider the linear map $${\Bbb C}^{n+1} \to L^2 \to {\Bbb C}^n\,,$$ where the first arrow maps $(A_1,\dots, A_{n+1})\mapsto
\sum_{i=1}^{n+1} A_i f_i$ and the second is the evaluation $
\varphi\mapsto (\varphi(z_1),\dots, \varphi(z_n))$ at the zeroes of the (Bieberbach) function $f$. For dimensionality reasons, there is a non-zero vector $(A_i)$ in the kernel which creates the function $f_0:=\sum_{i=1}^{n+1}A_i f_i$ vanishing at all $z_i$, yet without being identically $0$ (the $f_i$ being linearly independent).

[**Step 3 (Nehari factorizes)**]{} The function $g$ defined by $g
\cdot f=f_0$ is regular in $D$ (since writing $g=f_0/f$ we see that the zeroes of $f$ are cancelled out by those of $f_0$ which by construction englobe those of $f$). Now using the property of the circle map $f$ we find the following strict inequality $$J(f_0)=\frac{\int_C \vert f_0(z) \vert^2 ds}{\int_{C_1} \vert
f_0(z) \vert^2 ds}=\frac{\int_C \vert g(z)\vert^2 \overbrace{
\vert f(z) \vert^2}^{=1} ds}{\int_{C_1} \vert g(z) \vert^2
\underbrace{\vert f(z) \vert^2}_{< 1} ds}>\frac{\int_C \vert
g(z) \vert^2 ds}{\int_{C_1} \vert g(z) \vert^2 ds}=J(g)\,.$$ (Moreover reading backwards the numerators we see that the norm of $g$ equals that of $f_0$ so that $g\in L^2$.) The just obtained inequation $J(g)<J(f_0)$ shows that $f_0$ fails to solve (P).

[**Step 4 (Using the linear structure)**]{} However the $f_i$ ($i=1,\dots,n+1$) solve (P), hence by virtue of the linear structure of the extremals to (P) \[which Nehari derives from an interpretation as the eigenspace attached to the lowest eigenvalue, but which perhaps may be derived more directly\] it follows that $f_0$ solves also (P) \[after scaling appropriately\], violating the conclusion of Step 3.

Albeit our presentation is not completely polished (and Nehari’s maybe not perfectly organized for the beginner), we see that the basic trick looks sufficiently universal, as to extend to the following context.

Instead of the finitely-connected domain $D$, we consider $F$ a compact bordered orientable Riemannian surface of genus $p$ and with $r$ contours. Now $ds$ denotes the induced length element attached to the (Riemannian) metric. As above, we specify a subset $C_1$ of the interior of $F$ consisting of a finite “drawing” of Jordan arcs and curves (perhaps they do not even need to be pairwise disjoint). Then we set up the extremal problem (P) in this context, and they above proof seems to work mutatis mutandis, except for trading Bieberbach 1925 [@Bieberbach_1925] by Ahlfors 1950 [@Ahlfors_1950] or Gabard 2006 [@Gabard_2006]. Precisely, we may consider a circle map $f\colon F \to \overline{\Delta}$ of least possible degree, say $\gamma$. By Gabard 2006 [@Gabard_2006] we know that $\gamma\le r+p$. So we arrive at the following statement:

Let $F$ be a membrane of genus $p$ with $r$ contours. Assume that $F$ has the gonality $\gamma$, i.e. the least degree of a circle map to the disc. (We know $\gamma\le r+p$) Then the extremal problem [(P)]{} admits at most $\gamma$ linearly independent solutions.

Ahlfors’ extremal problem
=========================

Ahlfors extremal problem (Grunsky 1940–42, Ahlfors 1947–50)
-----------------------------------------------------------

Ahlfors’ method involves solving the following extremal problem:

\[Ahlfors-extremal:thm\] [(Ahlfors 1950 [@Ahlfors_1950])]{} Given any compact bordered Riemann surface (membrane for short) and two interior points $a,b$, find among all (analytic) functions bounded-by-one taking $a$ to $0$ the one maximizing the modulus $\vert f(b) \vert$.

Such a function exists (normal families argument à la Vitali-Montel) and is unique up to a rotation (=multilication by an unimodular complex number $\omega=e^{i \theta}$). Hence it is unambiguously defined by the points $a,b$ if $f(b)$ is required to be positive real, and we denote $f_{a,b}$ the corresponding function.

Furthermore Ahlfors’ extremal function $f_{a,b}$ concretizes the given surface as a full-covering of the disc $\Delta$, of degree $$r\le \deg f_{a,b} \le r+2p, \label{Ahlfors:pinch}$$ where $r$ is the number of contours and $p$ the genus (of the given membrane).

It is nowadays quite customary—following (another) Russian school (Golusin, S.Ya. Havinson, etc.)—to call the extremal an [*Ahlfors function*]{}, albeit even Ahlfors seems to have been rather embarrassed by this probably unearned distinction (cf. his comments in Collected Papers [@Ahlfors_1982_Coll_papers p.438]). The same idea occurred somewhat earlier in works of Grunsky 1940–42 [@Grunsky_1940], [@Grunsky_1942], yet the latter confined attention to plane domains (as did Ahlfors 1947 [@Ahlfors_1947]). Being close colleagues—as materialized by their joint note (Ahlfors-Grunsky 1937 [@Ahlfors-Grunsky_1937]) about the best conjectural value for the [*Bloch constant*]{} (still open up to present days)—it is puzzling that both were not very aware of overlapping studies (admittedly imputable to the difficult World War II context).

Semi-fictional reconstruction of Ahlfors’ background (Fejér-Riesz 1922, Carathéodory 1928, Ostrowski 1929)
----------------------------------------------------------------------------------------------------------

Where does Ahlfors’ extremal problem come from? This is surely a non-trivial question yet let us attempt to give some elements of answers. The narrative is made more plausible by looking a bit around while trying to keep track of the historical continuity. We shall thus use several indirect sources, especially Remmert.

As notorious, the Dirichlet principle suffered ill-foundations during a long period of about 40 years (1860-1900). This was beneficial to Schwarz-Christoffel who developed some constructive methods for the RMT for polygons. Another trend involves directly rescuing the Dirichlet principle via the “alternierendes Verfahren” of Schwarz and the parallel work of C. Neumann. This influenced Picard’s [*méthodes des approximations successives*]{}, as well as Poincaré’s balayage.

Then came Hilbert’s breakthrough. Yet, alternative methods circumventing the intricacies of potential theory seemed worth attention. As reported in Remmert 1991 [@Remmert_1991], one can ascribe to Fejér-Riesz ca. 1921 (published by Radó 1923 [@Rado_1923-Uber-konf-Abb]) the first purely complex variables (potential-theoretic free) proof of the RMT by using the extremal problem of making the modulus of the derivative as large as it can be. Several technical simplifications were then obtained by Carathéodory 1928 [@Caratheodory_1928] and Ostrowski 1929 [@Ostrowski_1929] (independently). This leads in principle to the most elementary proof of the RMT. Extending this idea to multiply-connected domains (say first of finite connectivity) leads directly to the extremal problem considered by Grunsky 1940–42 [@Grunsky_1940], [@Grunsky_1942], and Ahlfors 1947 [@Ahlfors_1947], and Ahlfors 1950 [@Ahlfors_1950] when extended to Riemann surfaces.

In fact prior to Fejér-Riesz, it is fair to refer to Koebe’s (and Carathéodory’s) elementary proofs of the RMT, also via an extremal problem or at least iterative methods (compare e.g. Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950]).

As a matter of digression, it can be recalled that this extremal viewpoint leads as well to a proof of the uniformization theorem (without potential theory). Compare Carathéodory 1950 [@Caratheodory_1950], plus several papers by Grunsky (easily located in his collected papers).

Extremal problems and pure function-theoretic proofs of the RMT (Koebe, Carathéodory, Bieberbach)
-------------------------------------------------------------------------------------------------

The previous section is a bit caricatural and the real history is marvellously detailed in Gray 1994 [@Gray_1994]. Let us summarize the chronology of this period, in the center of which there is probably one of the main inspiring force toward the Ahlfors extremal function (namely the [*Schwarz lemma*]{} as Carathéodory christened it in 1912).

$\bullet$ Painlevé 1891 [@Painleve_1891]: boundary behavior of the Riemann mapping for a contour having an everywhere continuously varying tangent.

$\bullet$ Harnack 1887 [@Harnack_1887] provides a satisfactory proof for solving a suitable version of Dirichlet’s principle, and states what has become known as [*Harnack’s theorem*]{} on monotone limits of harmonic functions.

$\bullet$ Osgood 1900 [@Osgood_1900] applies Harnack’s theorem to draw the existence of a Green’s function for any simply-connected plane domain thereby resolving the Riemann mapping theorem (RMT). This dependance is eliminated in Koebe 1908 and Carathéodory 1912 (cf.items below), where Schwarz’s lemma is substituted.

$\bullet$ Poincaré 1907 [@Poincare_1907] (and independently Koebe 1907, cf. below) proves uniformization (rigourously). For this Poincaré combines his [*méthode de balayage*]{} (of 1890 [@Poincare_1890]) and simplifies it using Harnack’s theorem. From the Green’s function he deduces the conformal map of a Riemann surface (à la Weierstrass) to the disc, and uses earlier works of Osgood.

$\bullet$ Koebe 1907 also proves uniformization (UNI). In Koebe 1907c [@Koebe_1907_UbaK2] he compares his method to Poincaré’s. Like Poincaré he had relied on Schwarz’s method, but unlike him made a much more modest use of Harnack’s theorem. Koebe also insists upon his avoiding of the use of modular functions.

$\bullet$ Koebe 1908 [@Koebe_1908_UbaK3] supplies another proof (of UNI) avoiding completely Harnack’s theorem. \[Subsequently Koebe interacted widely with Fricke’s attempt to modernize the original [*continuity method*]{} of Klein-Poincaré, and showed how this could be rigorized overlapping thereby with simultaneous work by Brouwer. This interaction with Brouwer seems to have ended quite contentiously.\]

$\bullet$ Koebe 1909 [@Koebe_1909_UAK1], 1910 [@Koebe_1910_UAK2] proof of his [*Verzerrungssatz*]{} (distortion theorem). From it he derives, the first elementary proof of the (RMT) appealing to a long list series of name going back via Arzelà and Montel 1907 [@Montel_1907] to Ascoli 1883.

$\bullet$ Carathéodory 1912 [@Caratheodory_1912 p.109] notes that [*Schwarz’s lemma*]{} (which he was the first to call by this name, and which he locates in Schwarz’s Ges. Abh., vol. 2, p.109) can act as a substitute to Harnack’s theorem (upon which Osgood 1900 relied heavily). \[Interrupting the present narrative this will have to play a major rôle in Ahlfors’ extremal problem.\] Using the Schwarz’s lemma and Montel’s theorem, Carathéodory obtains the Riemann mapping using an exhaustion of the domain $G$ by subdomains $(G_n)$ each mapped via $f_n$ to the disc and studied under which condition on $G_n$ the $f_n$ converges to a function $f$ giving the Riemann mapping (again without potential theory).

$\bullet$ Carathéodory 1913a [@Caratheodory_1913a] proves Osgood’s conjecture, that the Riemann map extends to a homeomorphism of the boundary iff the boundary is a Jordan curve. In Carathéodory’s opinion this achievement is mostly a byproduct of Lebesgue’s far-reaching theory of integration (1902 [@Lebesgue_1902]), and the consequences drawn from it by Fatou 1906 [@Fatou_1906]. This reliance upon Lebesgue-Fatou was soon disputed by Koebe 1913 (cf. item below).

$\bullet$ Carathéodory 1913b [@Caratheodory_1913b] discusses the boundary behavior when the boundary curve is not a Jordan curve. This paper is oft regarded as inaugurating the concept of [*prime ends*]{} (although earlier origins are in the work of Osgood, and related ideas in Study-Blaschke 1912 [@Study-Blaschke_1912]).

$\bullet$ Koebe 1913 [@Koebe_1913] disputes the need for Lebesgue’s theory in Carathéodory’s treatment, showing how to generalize a theorem of Schwarz to the same effect. A similar result is claimed independently by Osgood-Taylor 1913 [@Osgood-Taylor_1913].

$\bullet$ Bieberbach 1913 [@Bieberbach_1913] wrote a short paper disputing the (in his opinion) excessive Carathéodory’s reliance on Schwarz’s lemma, proposing to use only Montel’s theorem. The next year Bieberbach 1914 [@Bieberbach_1914] invokes another extremum principle (area minimization of the range of the mapping suitably normalized) to simplify Carathéodory’s work. This freed the theory from any reliance upon Montel’s theorem (but uses instead ideas of Ritz).

$\bullet$ Back to Koebe, in 1912 [@Koebe_1912] could not resist after the stimulus aroused by Carathéodory’s work to go back to some old idea of his own ([*Quadratwurzeloperationen*]{}) to create his [*Schmiegungsverfahren*]{} (squeezing methods) for solving the Riemann mapping by the iterated taking of square roots. This presentation was entirely elementary.

$\bullet$ Carathéodory 1914 [@Caratheodory_1914] incorporated all these criticisms in his paper for the Schwarz Festschrift, which was to remain his final account until the newer methods of Perron were introduced. \[Here we may have also mentioned the argument of Fejér-Riesz 1921.\]

$\bullet$ Bieberbach 1915 in his pocket book Göschen [@Bieberbach_1915 p.95] also proposes to deal entirely within pure function theory, while rejecting the potential-theoretic approach (despite Hilbert’s work). This actually presents a version of Koebe’s Schmiegungsverfahren and concludes to the Riemann mapping theorem via Koebe’s Verzerrungssatz (seen as a preferred alternative over Schwarz’s lemma).

Interlude: Das Werk Paul Koebes
-------------------------------

In this section we digress slightly from our main path to look closer at the monumental works of Koebe. A useful guide is Bieberbach’s overview of Koebe’s work in 1968 [@Bieberbach_1968-Das-Werk-Paul-Koebes]. The main point of overlap of Koebe with our main theme (Ahlfors) lies in the Riemann-Schottky mapping (albeit for Koebe the mapping to a Kreisbereich is given full attention neglecting thereby the circle mapping). Of course, the other main aspect of Koebe’s life is the uniformization theorem of (Klein-Poincaré-Schwarz).

Again some chronology:

$\bullet$ Riemann 1857–58 [@Riemann_1857_Nachlass] and Schottky 1877 [@Schottky_1877] (maybe only in the 1875 Latin version?) proved that any $n$-ply connected domain maps conformally to a Kreisbereich (circular domain). \[Bieberbach and indeed Koebe 1910 [@Koebe_1910_JDMV] ascribe this to Riemann, albeit we are not sure to be in total agreement with this assertion.\]

$\bullet$ In Bieberbach’s opinion the above Riemann-Schottky Kreisbereich-mapping is first rigourously proved by Koebe in a series of four papers written in 1906, 1907, 1910, 1920 (which we attempt to summarize in more details):

\(1) Koebe 1906 [@Koebe_1906_JDMV]: this starts with a rigidity result for two Kreisbereiche as being conformal to each other only through linear transformations. The proof uses potential theory (and the Cauchy integral). In the case of a Kreisbereich with two contours there is a pencil of orthogonal circles whose cross-ratio of any member affords the conformal invariant of the ring. It follows that $(\varrho+1)$-ply-connected Kreisbereiche depend upon $3\varrho -3$ essential constants when $\varrho
\ge 2$, the same quantity as predicted by Schottky for general multiply-connected domains of the same connectivity. This yields some evidence for the possibility of mapping those to a Kreisbereich. Actually Koebe (p.150) reminds that the Kreisbereich mapping is (essentially) solved by Schottky and by Poincaré (referring loosely to the first volumes of Acta). \[In the next paper Koebe adopts a more critical position, and does not take this as granted.\] Next, he claims the result extends to schlichtartig surfaces. His argument amounts to fill the Riemann surface by discs, to get a closed surface of genus 0, and appeal to Schwarz 1870 [@Schwarz_1870] to map this to a sphere. Next, Koebe proposes to relax the schlichtartig character to formulate a similar result for positive genus. Again one fills the surface by discs to gain a closed surface of genus $p$. This can be mapped as a ramified cover of the sphere of degree $p+1$ (as well-known since Riemann, but for Koebe being Schwarz’s pupil Riemann is taboo and an ad hoc \[somewhat sketchy\] argument is supplied). At any rate the result is that any compact bordered Riemann surface of genus $p$ is conformally embeddable in a closed Riemann surface of genus $p$, hence representable as a $(p+1)$-sheeted cover of the sphere. Although this result concerns like Ahlfors 1950 [@Ahlfors_1950] compact bordered surfaces, it seems that this Koebe mapping lies not so deep as the image of the contours of the map are poorly controlled, in particular they need not coincide.

\(2) Koebe 1907 [@Koebe_1907_UrAK]: this starts by quoting again his rigidity result of the previous paper. Then more critically Koebe notices that the mapping of a planar $(\varrho+1)$-ply connected domain upon a Kreisbereich of the same connectivity is not so easily established, making abstraction of the Klein-Poincaré [*Kontinuitätsmethode*]{} ([*méthode de continuité*]{}) not yet effective in 1907. (This had to wait until the work of Brouwer and Koebe ca. 1911 [@Klein-Brouwer-Koebe_1912], and Koebe 1912 [@Koebe_1912_BdKm].) The rigidity result affords an essentially unique solution of the mapping problem. Then Koebe proceeds to show that a Kreisbereich mapping exists for triply-connected domains ($\varrho=2$), and generally if the domain is symmetric under complex conjugation provided the real axis cuts all contours. For triply connected domains, he takes the Schottky double, which conformally maps to a closed Riemann surface of genus $2$ (via massive quotations to Schwarz, Ges. Abh. II, S.133–143, S.144–171, S.175–210). As any curve of genus 2 this is hyperelliptic (canonical mapping via holomorphic 1-forms). As to the more general case, the problem involves cutting the domain along the real axis, yielding a simply-connected region. This is mapped conformally to the upper half-plane, and symmetrically reproduced. Then Green’s function is constructed via Harnack’s theorem (quotation to Harnack 1887 [@Harnack_1887], Poincaré 1883 [@Poincare_1883], Osgood 1900 [@Osgood_1900] and Johansson 1905).

\(3) Koebe 1910 [@Koebe_1910_JDMV]: the paper starts again with the objective to solve the [*Problem der konformen Abbildung eines $(p+1)$-fach zusammenhängenden Bereiches auf einen von $p+1$ Vollkreisen begrenzten Bereich*]{} (which he proposes to call [*Kreisbereich*]{} for short). Koebe recalls that the problem was first addressed by Schottky 1877 in his [*Doktordissertation*]{}, and earlier in Riemann’s Nachlass. He reminds from his first work \[item (1)\] that [*je zwei Kreisbereiche aufeinander nur durch lineare Funktionen konform abgebildet werden können*]{}. Then he repeats the two special cases he was able to solve previously, and now proposes to tackle the general case via two different methods (of his own): [*Überlagerungsfläche*]{} and [*iterierendes Verfahren*]{} \[cf. items (A) and (B) below\]. He proudly emphasizes that both methods have a larger applicability than to the present Kreisbereich problem, since their combination, allowed him to settle the whole series of classical mapping problems of Klein and Poincaré (1881–84) in their pioneering works on automorphic functions, and the allied uniformization. Hilbert’s 22th Problem (1900) is mentioned for reposing the uniformization question especially in connection to Poincaré 1883’s paper [@Poincare_1883]. Schwarz is again (justly) regarded as the father of the method [*der Überlagerungsfläche*]{}, which plays a key rôle in the newer developments in the automorphic theory, as exemplified through the work of Poincaré 1907 [@Poincare_1907] himself and Hilbert 1909 [@Hilbert_1909]. After these general remarks Koebe proceeds to prove the general Kreisbereich mapping. \[As warned in Bieberbach’s report, the present paper of Koebe does not contain full details, yet some lovely geometric ideas worth sketching. Complete details appear in the last contribution item (4), but then it is easy to get lost in technicalities.\]

\(A) Koebe assumes the contours of the domain $B$ to be analytic curves. Via some abstract [*Spiegelungsprozesses*]{} (ascribed to Schwarz) he constructs via symmetric reproduction of $B$ a schlichtartig Riemann surface $B^{(\infty)}$. (One must imagine $B$ glued with replicas thought of as the back-side of the domain.) Then he can apply his [*allgemeines Abbildungsprinzip*]{} to the effect that schlichartig implies schlicht (first established in Koebe 1908 [@Koebe_1908_UbaK3], with subsequent approaches by Hilbert 1909 [@Hilbert_1909] and in Courant’s Thesis 1910/12 [@Courant_1912]). The new schlicht domain $B^{(\infty)'}$ is tesselated by replicas of the conformal copy $B'$ of $B$. Hence $B'$ admits a complete infinite system of symmetric reproduction. This is enough (for Koebe) to characterize a Kreisbereich. (Here we may agree with Bieberbach’s diagnostic that Koebe’s exposition is sketchy, but details were supplied later in Koebe 1920 [@Koebe_1920].)

\(B) Then is exposed the promised [*iterierendes Verfahren*]{}. This is a beautiful device based upon successive applications of the RMT to circularize a specific contour and then reflecting by a [*Spiegelgung*]{} (inversion by reciprocal radii) the domain across this circularized contour. Koebe draws nice pictures (like below Fig.\[KoebeiV:fig\]) suggesting that this iteration scheme produces domains with [*sukzessive Steigerung der Spiegelungsfähigkeit des Bereichs*]{} whereupon it is made plausible that when repeated ad infinitum the resulting domain has an infinite aptitude of symmetric reproduction, hence must be a Kreisbereich. The convergence proof uses his [*Verzerrungssatz*]{} (distortion theorem).

\(4) Koebe 1920 [@Koebe_1920], where full details are supplied. $\bullet$ In parallel, Koebe concentrates his efforts on the uniformization problem starting with Koebe 1907 [@Koebe_1907_UrAK] devoted to the uniformization of real algebraic curves, yet the real technological breakthrough occurs in the next paper.

-35pt 0

$\bullet$ Koebe 1907 [@Koebe_1907_UbaK1] discovers a first version of his [*Verzerrungssatz*]{} (VZS), which turns out to be relevant both to the Riemann-Schottky Kreisbereich-mapping, as to uniformization. As forerunners of the (VZS) Bieberbach mentions the works of Landau, Schottky related to Picard’s theorem (1879 [@Picard_1879]). This Koebe’s paper also contains (what later came to be known) as the [*Viertelsatz*]{} to the effect that the range of any schlicht function on the unit disc normalized by $f(0)=0$ and $\vert f'(0) \vert=1$ contains a disc of some universal positive radius $\varrho$. The sharp value $\varrho=1/4$ is conjectured, but only established by Bieberbach 1915 [@Bieberbach_1915]. Armed with this Verzerrungssatz (yet without the precise bound) Koebe manages to prove uniformization. This represents a generalization of the RMT to simply-connected Riemann surfaces. Bieberbach recalls that according to oral tradition the trick of the universal covering surface is due to H.A. Schwarz (ca. 11. April 1882, as carefully reported in Klein’s Werke [@Klein-Werke-III_1923 p.584]).

$\bullet$ Simultaneously and independently Poincaré 1907 [@Poincare_1907] also proves the uniformization theorem via his [*méthode de balayage*]{}.

$\bullet$ Koebe 1907 [@Koebe_1907_UbaK2] inspects Poincaré’s proof and proposes a variant using Harnack’s theorem (in potential theory) circumventing thereby the Viertelsatz, as well as Poincaré’s balayage.

$\bullet$ The new ingredient (Verzerrungssatz of Koebe) turned out to act usefully in other uniformization problems envisioned by Klein (e.g., the [*Rückkehrschnitttheorem*]{}, etc.) In Koebe’s formulation this resulted to the conformal mapping of a schlichtartig Riemann surface to a schlicht domain of the Riemann sphere. This result appears in Koebe 1908 [@Koebe_1908_UbaK3]. Its proof uses beside the Verzerrungssatz a general convergence theorem (à la Montel-Vitali), which Koebe discovered independently \[according to Bieberbach\].

$\bullet$ Koebe 1909 [@Koebe_1909_UAK1] gives a sharper version of the [*Verzerrungssatz*]{} and applications to Klein’s general uniformization problem (via groups of linear transformations).

$\bullet$ Hilbert 1909 [@Hilbert_1909], using a variant of the Dirichlet principle, gives another method for the schlicht mapping of a schlichtartig surface (to the sphere), via a so-called parallel-slit mapping \[extending the Schottky-Cecioni result to infinite connectivity\].

$\bullet$ In response Koebe 1909 [@Koebe_1909_UbaK4], 1910 [@Koebe_1910_Hilbert] and independently Courant 1910/12 [@Courant_1912] proves anew the above Hilbert’s Ansatz about parallel-slit mappings.

$\bullet$ Already Schottky 1877 [@Schottky_1877] tried \[in Bieberbach’s opinion\] to prove the \[Riemannian\] theorem that every $n$-ply connected planar domain conformal-maps bijectively to a parallel Schlitzbereich. Hilbert’s new method proves this for arbitrary schlichartig Riemann surfaces. Koebe in the aforementioned two works, sharpens Hilbert’s theorem by noticing that the range of the mapping fill the full plane save a set of measure zero. At this occasion Koebe also formulates his [*Kreisnormierungsprinzip*]{}, still open today, despite the spectacular progress by He-Schramm 1993 [@He-Schramm_1993].

$\bullet$ Bieberbach emphasizes that the [*iterierendes Verfahren*]{} may really have first emerged through the Kreisbereich mapping problem. \[This conflicts slightly with Koebe’s claim that he employed it earlier for uniformization.\] At any rate Bieberbach writes “[*Solche iterierenden Verfahren entwickelt Koebe über Jahrzehnte hin immer weiter, bis alle Uniformisierungsprobleme algebraischer Gebilde dem iterierenden Verfahren zugänglich werden.*]{}”

$\bullet$ The proof of the (RMT) via repeated [*Quadratwurzelabbildungen*]{} itself constitutes an iterative method, which Koebe calls the [*Schmiegungsverfahren*]{}. Credit for this discovery is to be shared with Carathéodory.

$\bullet$ A rigorous foundation to the [*Kontinuitätsmethode*]{} of Klein-Poincaré is paid by Koebe much attention in a torrential series of paper starting with 1912 [@Koebe_1912_BdKm], 1912 [@Koebe_1912_BdKm2], 1914 [@Koebe_1914_UAK4], etc. Those works overlaps (and then may supplement) the works of Brouwer on the invariance of domain (and dimension), and its application to Riemann surfaces. The resulting priority question is very intricate. Even Klein in 1923 [@Klein-Werke-III_1923 p.734] writes: [*Die entscheidende Wendung trat aber erst 1911/12 durch das Einsetzen der Untersuchungen von Brouwer und Koebe ein. (Ich halte um so mehr an der alphabetischen Reihenfolge fest, als die gegenseitige Beziehung der beiden Forscher nicht ganz geklärt ist.)*]{} Soon afterwards Klein also cites footnote 2) in Brouwer 1919 [@Brouwer_1919], where Brouwer seems to revendicate some priority over Koebe, while reporting some falsification of his own (Gött. Nachr.) article via a citation to Koebe added after proof-reading.

Koebe and his relation to Klein or Ahlfors
------------------------------------------

In the overall Koebe’s monumental work is quite intricate with deep influences by methods of Schwarz (ca. 1870), results of Schottky (1875/77), visions of Klein and Poincaré (early 80’s), supplemented by methods of his own. The following chart (Fig.\[KoebeMap:fig\]) gives an Überblick maybe helping navigation through Koebe’s works and the logical links between his results.

-85pt 0

From our Ahlfors’ biased viewpoint several points are worth noticing:

\(1) Koebe frequently refers to Klein’s orthosymmetry for real algebraic curves. In view of the close connection between orthosymmetry and the Ahlfors circle mapping, it is tempting to wonder if Koebe was ever close to discover the Ahlfors circle mapping. Of course Koebe’s focus seems to have been more attracted by the uniformization problem (in particular for real algebraic curves), cf. Koebe 1907 [@Koebe_1907_UrAK]. However Klein’s orthosymmetry appears in many subsequent papers (e.g., 1919 [@Koebe_1919:47 p.29, p.35]), and we would not bet that one day someone discovers in Koebe some anticipation of the Ahlfors map (as it occurred say with the circles packing of Andreev–Thurston). If not directly, it could via the Rückkehrschnitttheorem of Klein (cf. Sec.\[sec:Ruckkehrschnittthm\]), which Koebe was the first to prove seriously (cf. Koebe 1910 UAK2 [@Koebe_1910_UAK2]). Hence schematically, there might exist a (harsh style) path like: $$\textrm{Koebe}\Rightarrow\textrm{Klein} \Rightarrow
\textrm{Teichm\"uller} \Rightarrow \textrm{Ahlfors}.$$

\(2) Koebe also notices at several places (e.g., 1907 UbaK1 [@Koebe_1907_UbaK1 p.199]) that the orthosymmetry concept for real algebraic curves extend to analytic real curves. One can then wonder if there is likewise a function theoretical characterization of orthosymmetry in terms of (totally real) mapping to the sphere. This would amount to say that any bordered surface is expressible as a total cover of the disc (taking boundary to boundary). Of course this might be a bit fantasist, but perhaps deserves to be analyzed more carefully. (Maybe this fails already for planar domains, cf. a work of Heins ca. 1954.)

Ahlfors’ background (Bergman 1941, Schiffer 1946, Schottky differentials)
-------------------------------------------------------------------------

Let us quote the introduction of Ahlfors 1950 [@Ahlfors_1950]:

\[Ahlfors-1950:quote\]

In the handling of the extremal problems we are in close contact with the methods of Bergman \[1941\]=[@Bergman_1941] and Schiffer \[1946\]=[@Schiffer_1946], which they have developed for plane regions. A convenient tool for applying these methods to regions on Riemann surfaces is found in the class of Schottky differentials, and it was the recognition that Bergman’s kernel-functions are in fact Schottky differentials that led us to undertake this study.

The second part of the paper (§§4–5) deals with an extremal problem that we have previously solved for plane regions. There are great simplification over my original proof for which I am partly indebted to my student P. Garabedian. An interesting point is that the extremal functions are again defined by means of Schottky differentials.

As a complement, we may reproduce a passage of Ahlfors’ comments in his collected papers [@Ahlfors_1982_Coll_papers p.438]:

\[Ahlfors-1982:quote\]

The purpose of \[36\](=Ahlfors 1950 [@Ahlfors_1950]) was to study open Riemann surfaces by solving extremal problems on compact subregions and passing to the limit as the subregions expand. The paper emphasizes the use of harmonic and analytic differentials in the language of differential forms. It is closely related to \[35\](=Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950]), but differs in two respects: (1) It deals with Riemann surfaces rather than plane regions and (2) the differentials play a greater rôle than the functions.

I regard \[36\] as one of my major papers. It was partly inspired by R. Nevanlinna, who together with P.J. Myrberg had initiated the classification theory of open Riemann surfaces, and partly by M. Schiffer (1943) and S. Bergman (1950), with whose work I had become acquainted shortly after the war[^9]. The paper also paved the way for my book on Riemann surfaces with L. Sario \[1960\], but it is probably more readable because of its more restricted contents.

I would also like to acknowledge that when writing this paper I made important use of an observation of P. Garabedian to the effect that the relevant extremal problems occur in pairs connected by a sort of duality. This is of course a classical phenomenon, but in the present connections it was sometimes not obvious how to formulate the dual problem.

The allied infinitesimal form of the extremal problem
-----------------------------------------------------

The input required to pose Ahlfors’ extremal problem (Theorem \[Ahlfors-extremal:thm\]), is a membrane with two interior marked points, denoted $a,b$. When the point $b$ converges to the point $a$ (becoming infinitely close to it), we may think of a unique point of multiplicity two. This limiting process mutates the extremum problem into:

Let $a$ be a single point in the membrane $W$. Among all functions $f$ analytic on $W$ with $\vert f \vert \le 1$ on $W$ it is required to find the one which makes the modulus of the derivative $f'(a)$ to a maximum. Here the derivative is computed w.r.t. any holomorphic chart. Its maximum value has no intrinsic meaning, yet the extremal function exists and is uniquely defined (up to a rotation) and denoted by $f_{a,a}=f_a$.

It seems to be folklore that such functions are also circle maps subjected to the same Ahlfors bound $\deg f_{a}\le r+2p$. Presumably a continuity argument reduces to the case of (bipolar) functions $f_{a,b}$, or maybe adapt the whole argument in Ahlfors 1950 [@Ahlfors_1950]. At any rate, the result is taken for granted in Yamada 1978 [@Yamada_1978], Gouma 1998 [@Gouma_1998]. This can maybe deduced as a special case of Jenkins-Suita 1979 [@Jenkins-Suita_1979].

Higher extremal problems=HEP$\approx$High energy physics, alias Pick-Nevanlinna interpolation
---------------------------------------------------------------------------------------------

What happens if we take more than two points? For instance three points $a,b,c$? Should we then maximize the area of the simplex spanned by the image points? If yes for which metric on the disc (Euclid vs. hyperbolic)? How does the problem reformulate when the 3 points coalesce at the subatomic level into a point affected by a multiplicity 3. Does the problem amount then to maximize the modulus of the first two derivatives?

Maybe this brings us in the realm of Pick-Nevanlinna interpolation, a theory initially developed in the disc. Compare e.g. Garabedian 1949 [@Garabedian_1949], Heins 1950 [@Heins_1950], Jenkins-Suita 1979 [@Jenkins-Suita_1979].

Perhaps for any (effective) divisor $D=d_1 p_1+\dots +d_n p_n$ interior to the membrane there is an extremal problem denoted $EP(D)$. Then how much of Ahlfors’ theory extends: existence, uniqueness and qualitative circle mapping nature of the function, and estimates over the degree of the extremals. In the classic theory where $\deg(D)=2$ we have $\deg f_{a,b}=r+2p$. Maybe in general denoting by $f_D$ the extremal function allied to the divisor $D$ we find $\deg f_{D}\le r+ \deg(D) p$. Compare Jenkins-Suita 1979 [@Jenkins-Suita_1979] for more serious answers. If we could find a divisor of degree one then this would recover Gabard’s bound $r+p$. Maybe not a divisor is required but an ordered collections of points, as in Ahlfors’ original problem where $a$ seems to have a preferred rôle over $b$, getting mapped to zero.

Such higher extremal problems depending upon a higher number of free parameters are probably more flexible in the sense that if the original $\deg(D)=2$ case of Ahlfors fails to realize the gonality, then maybe higher versions succeed. Perhaps there is even a universal quantum limit of such problem $EP\infty$ for a divisor of infinite degree, leading thereby to a branched (yet Randschlicht) version of the Bieberbach coefficient problem. This is to mean a version of the Ahlfors map where all derivatives are simultaneously maximized as a large convey? One can speculate about the existence of such an universal extremal problem whose solution would be a branched avatar (non schlicht) of the Koebe extremal function (involved in the Bieberbach-de Branges theorem). This would be for the given bordered surface the best circle mapping and arguably it ought to realize the gonality. \[05.11.12\] In the classic Bieberbach problem involving the disc the coefficients of schlicht power series are estimated by $\vert
a_n\vert\le n$ when $f'(0)=1$. If we replace the disc by a finite bordered surface $F$ we could expect that all maps $F\to \Delta\to
{\Bbb C}$ factorizing as a circle map (of minimal degree) followed by a schlicht map also admit universal estimates upon the coefficients w.r.t. to a chart. Perhaps the upper bounds sequence involved in Bieberbach-de Branges (regularly spaced integers $n$) has to be replaced by certain spectral eigenvalues of $F$ conceived as a vibrating membranes. So the problem is the following. Given a bordered surface $F$ marked interiorly at some point $a$. We look at all analytic maps $F\to {\Bbb C}$ with $f(a)=0$ and $f'(a)=1$ w.r.t. some chart. We develop $f$ in power series and expect some universal estimates on the coefficients at least when $f$ factorizes as a circle map of minimal degree followed by a schlicht map. The dream would be that there is a unique extremal function maximizing simultaneously all coefficients and this would be essentially the best possible Ahlfors map post-composed with the Koebe function.

Of course it may happen that all this generality is not necessary in case the basic Ahlfors map $f_{a,b}$ is already the most ergonomic object, in the sense of realizing the gonality for suitable centers $a,b$.

A more orthodox way to formulate higher versions of Ahlfors’ extremal problem involves the theory of Pick-Nevanlinna interpolation. Cf. for instance Jenkins-Suita 1979 [@Jenkins-Suita_1979]. The original theory being formulated in the disc $\Delta$, one may hope to lift things via an Ahlfors map but this probably leads nowhere. Genuine avatars of Ahlfors extremal problem are formulated by prescribing Taylor section (jets) at a given collection of points. Compare again Jenkins-Suita 1979 [@Jenkins-Suita_1979], building upon a paper of Heins 1975 [@Heins_1975]. In this extended context all features of the Ahlfors map persist: existence of an extremal (via normal families), uniqueness of the solution (Heins 1975), finite sheeted covering of the disc, and upper bound over the mapping degree. Again a crucial question is whether such problems always achieve the gonality.

Ahlfors’ proof {#Ahlfors:sec}
==============

\[January 2012\] This section is a superficial glimpse into Ahlfors’ original resolution of his extremal problem emphasizing that Ahlfors requires first the qualitative existence of a circle map. A more detailed analysis will be attempted later (Sec.\[Ahlfors-proof:sec\]).

Soft part of Ahlfors 1950: circle maps with $\le
r+2p=g+1$ sheets {#sec:Ahlfors-soft}
------------------------------------------------

When writing the paper Gabard 2006 [@Gabard_2006] (and a fortiori in my Thesis 2004 [@Gabard_2004]), I was very ignorant about the depth of Ahlfors’ paper (and the massive literature around it). To be honest I am still today quite ignorant having only a very fragmentary understanding of Ahlfors arguments. I take this opportunity, to rectify the arrogant claim (in [@Gabard_2006]) to the effect that a simplified proof of Ahlfors’ theorem is proposed. Of course, my paper only recovers the weaker assertion about existence of circle maps (in contradistinction to the deeper extremal problem analyzed by Ahlfors).

Furthermore even in the weaker circle maps context, I only realized recently \[January 2012\] that a much shorter portion of Ahlfors’ paper achieves this goal (cf. Ahlfors 1950 [@Ahlfors_1950 p.124–126]), even with the $r+2p=g+1$ bound on the degree. We reproduce the relevant extracts (p.124 and then p.126):

\[Ahlfors-1950-circle-map:quote\]

\[p.124\] It must first be proved that the class of functions with $F(a)=0$ and $\vert F \vert=1$ on $C$ \[=the boundary contours\] is not empty. In other words, we must show that $\overline{W}$ can be mapped onto a full covering surface of the unit circle.

\[p.126\] The function \[…\] maps $\overline{W}$ onto a covering surface of the unit circle\[=disk\], and a standard argument\[=just number conservation\] shows that every point is covered exactly $P+1$ times. \[$P$ is the genus of the double in Ahlfors’ notation\]

Thus, we have the following historical:

As early as Spring 1948, Ahlfors had an existence-proof of circle maps of degree $\le r+2p$.

This conjecture is supported by the remarks made in Nehari 1950 [@Nehari_1950] (cf. our Quote \[Nehari-1950:quote\]). In contrast, the issue that the same upper bound $r+2p$ holds true for Ahlfors extremals may have required Garabedian formulation of the dual extremal problem for differentials. This is somehow in line with Jenkins-Suita 1979 [@Jenkins-Suita_1979], who speak of the [*Garabedian bound*]{} following a coinage of Heins 1975 [@Heins_1975 p.4].

At any rate, it seems first crucial to understand the easy part of Ahlfors’ argument (existence of a circle map of degree $\le
r+2p$). Even here we failed as yet.

[**Anecdote (skip!)**]{} Ahlfors’ argument bears some vague resemblance with the argument exposed by myself in the RAAG-conference of 2001. Here the game was that (in view of Riemann without Roch) any group of $g+1$ points on the curve moves. The orthosymmetric curve in question is of course the Schottky double of the given bordered surface. If such points are chosen on the real locus we are forced in the non-Harnack-maximal case ($r<g+1$, $r\equiv g+1 \pmod 2$) to select two points on the same oval (pigeon hole principle). All the subtlety is to ensure that those points will circulate along the complex orientation (as the border of one half) without doing collision repulsing them in the imaginary locus, and thereby violating total reality. Using Abel’s theorem plus some incompressible fluid argument I tried to argue that this is always possible for a clever choice of (totally real) divisor. However the argument was slightly vicious, and it would require me too hard work to repair it. If I have enough energy I should try to write down this argument, while trying to analyze it properly.

In Ahlfors’ paper (1950 [@Ahlfors_1950]), one starts with a circle map of degree $\le r+2p$, and by a miraculous intervention of Garabedian the same bound turns out to be valid for all Ahlfors extremals. Let us refer to this vague principle as the Ahlfors-Garabedian divination (AGD). (Vagueness only alludes to my own poor understanding of their methods.)

Now in view of Gabard 2006 [@Gabard_2006], as well as the deeper investigation in Coppens 2011 [@Coppens_2011], we know that circle maps of lower degrees $\le r+p$ exist. Thus, granting the AGD-divination, we may expect to find Ahlfors extremals of correspondingly low degrees. Of course this amounts to take the best from two different worlds, and is extremely far from a serious argument. Hence a thorough study of Ahlfors 1950 [@Ahlfors_1950] perhaps suitably adapted (and augmented by other tricks) could lead to a confirmation of the naive Conjecture \[gonality:conj\]. Of course this is pure speculation, and arguably the emphasis could be a study circle maps [*per se*]{} without getting obnubilated by Ahlfors extremal problem.

Ahlfors hard extremal problem
-----------------------------

We have nothing to add for the moment, suffices to say that Lagrange multipliers play a crucial rôle (as in earlier work of Grunsky). Yet it would be nice to summarize the idea (and the logical structure):

\(1) [**Existence of extremals.**]{} Ahlfors first needs the existence of a circle map so as to arrange a nonempty set of competing functions (giving some ground under the foots to get started). Of course a function bounded-by-one would have been sufficient to get started, but Ahlfors achieves much more. Of course the normal families argument alone cannot supplant this preliminary study.

In papers subsequent to Ahlfors’, namely Read 1958 [@Read_1958_Acta] and Royden 1962 [@Royden_1962] existence is derived via more abstract functional-analysis (Hahn-Banach). More on this in Sec.\[Read-Royden:sec\].

Other treatments Heins 1950 [@Heins_1950] appeals to Martin’s theory and elementary convexity consideration, which expressed in more highbrow setting essentially amounts to Krein-Milman existence of extreme points in convex bodies (cf. esp. Forelli 1979 [@Forelli_1979] and the discussion in Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF]).

\(2) [**Uniqueness of the extremal.**]{} Looks easy (essentially like when defining something by a universal property in category theory). Universal properties of category theory are essentially akin to extremal problems in geometry. This is not completely true for some natural extremum problems admits several solutions). In the case at hand uniqueness is essentially a version of Schwarz’s lemma.

Other accounts of Ahlfors’ extremal problem
===========================================

Ahlfors’ paper of 1950 [@Ahlfors_1950] aroused quick interest among the conformal mappers community (Nehari 1950, Heins 1950, Garabedian 1949–50, Schiffer, etc.). Numerous papers seems to reprove Ahlfors’ theorem along (better?) routes (e.g., Read 1958, slightly optimized in Royden 1962). The latter article seems to be among the most popular revision of Ahlfors 1950 [@Ahlfors_1950], with identic results but possible simplifications in the proof. The present section tries to review those (second generation) contributions while providing link to subsequent critiques (e.g., Nehari 1950 is criticized by Tietz 1955, who in turn is attacked by Köditz-Timmann 1975).

Garabedian 1949, 1950
---------------------

Garabedian qualifies himself as a hard-worker, who could absorb simultaneously the influence of three giants: Ahlfors, Bergman and Schiffer. As a result, he seems to have exerted a notable influence over the final shape of Ahlfors 1950 [@Ahlfors_1950], and is even apparently able to reprove the full result of Ahlfors 1950 [@Ahlfors_1950] in the paper Garabedian 1950 [@Garabedian_1950 p.361]. (A little Riemann-Hurwitz computation is required to convince that Garabedian reobtains exactly the same degree $r+2p$ as Ahlfors.) The proof deploys a rich mixture of techniques (Teichmüller, Grunsky, Ahlfors, plus the variational method of Schiffer).

Another point worth noticing is the following issue oft emphasized by Garabedian [@Garabedian-Schiffer_1950 p.182]:

\[quote:Garabedian-Schiffer\_1950\]

Thus our procedure leads to the existence of the circle mapping $F(z)$ which is associated with Schwarz’s lemma. It is to be noted that the existence of this function lies somewhat deeper than the existence of the slit mappings $\varphi(w)$ and $\psi(w)$ in multiply-connected domains, and therefore it is not too surprising that the present section is more difficult that the preceding ones. Of course, for $n=1$, $F(z)$ is just the function found in the elementary Koebe proof of the Riemann mapping theorem.

Garabedian alone repeats a similar comment in Garabedian 1949 [@Garabedian_1949-52:Book p.207]:

\[quote:Garabedian\_1949-52\]

The conformal mappings which we obtain here are closely related to the generalization of Schwarz’s lemma to multiply connected domains in sharp form \[1, 7\] \[=resp. Ahlfors 1947 [@Ahlfors_1947], and Garabedian 1949, Duke Math. J.\], and their existence lies somewhat deeper than that of the more standard canonical maps in a multiply connected region.

Nehari 1950, Tietz 1955, Köditz-Timann 1975
-------------------------------------------

Regarding the first two mentioned papers (Nehari 1950 [@Nehari_1950], Tietz 1955 [@Tietz_1955]), I suggested in Gabard 2006 [@Gabard_2006 p.946], that those papers may have conjectured the improved control $r+p$ on the degree of circle maps. (When discovering the $r+p$ bound ca. 2001/02, I was not influenced by those papers which I located only later in 2005 while polishing the ultimate shape of Gabard 2006 [@Gabard_2006].) Nehari 1950 [@Nehari_1950] does not seem to give a new proof of circle maps (Ahlfors’ theorem), but inspired by it proposes to describe canonical slit maps (incidentally those for which Garabedian seems to have a lesser esteem, cf. Quotes \[quote:Garabedian-Schiffer\_1950\] and \[quote:Garabedian\_1949-52\]). Nehari also shows how to express the Ahlfors function in terms of the Bergman kernel function. (If I understand well the situation, this is just a representation theory yet not an alternative existence-proof.) Nehari’s paper shows that Ahlfors was in possession of the degree $r+2p$ as early as Spring 1948, at least for a circle map. It is a delicate question if the same bound for extremal maps requires Garabedian’s remark about the dual extremal problem. Heins’ paper 1975 [@Heins_1975] using the term “Garabedian’s bound” may suggest a positive answer. The reader is not well placed to guess the answer, but remember that the (published) proof in Ahlfors 1950 [@Ahlfors_1950] requires (and acknowledges) Garabedian’s dual problem. Let us quote the crucial extract of Nehari:

\[Nehari-1950:quote\]

It was recently shown by Ahlfors \[1\](=L. Ahlfors, Material presented in a colloquium lecture at Harvard University in Spring 1948.) that the well known canonical conformal mapping of a schlicht domain of connectivity $n$ onto an $n$-times covered circle \[5,7\] (=Bieberbach 1925 [@Bieberbach_1925], Grunsky 1937–41 [@Grunsky_1937], [@Grunsky_1941_KA]) can be generalized, in the case of an open Riemann surface, in the following manner: an open Riemann surface of genus $g$ which is bounded by $n$ closed curves can be mapped conformally onto a multiply-covered circle, the number of coverings not exceeding $n+2g$.

Soon afterwards, Tietz 1955 [@Tietz_1955 p.49] criticizes (slightly) some of Nehari’s asserted results:

\[Tietz-1955:quote\]

Bei der Herleitung seiner Schlitztheoreme kommt Herr Nehari ebenfalls auf diese Frage; sein Beweis für die genannte Vermutung ist jedoch unhaltbar.

Nimmt man jedoch diese Neharische Behauptung als richtig an, so hie[ß]{}e das, da[ß]{} $R$ aus $p+r$, und damit $R^2$ aus $2p+r=G+1$ Blättern bestünde; …

This seems to be a forerunner of the bound $r+p$ (by commutativity of addition!), at the conjectural level at least. \[Parenthetically, I do not understand Tietz’s claim about the sheet number of the double $R^2$. I believe that the degree keeps the same value $p+r$, as one has to double the map not just the space.\] Finally, Tietz concludes his paper [@Tietz_1955 p.49] as follows:

So Tietz does not seem to be able reprove a result as strong as the one of Ahlfors 1950. In fact, the situation looks even worse, since even Tietz’s weak version is questioned in the paper by Köditz-Tillmann 1975 [@Koeditz-Timmann_1975 p.157], as shown in the following extract (parenthetical reference are ours addition):

The extract is followed by a specific objection (not reproduced here). The article (of Köditz-Timmann 1975 [@Koeditz-Timmann_1975 Satz 3, p.159]) seems however to contain a proof of Ahlfors’ theorem based upon an “Approximationssatzes von Behnke u. Stein”, yet without any bound on the degree.

A propos Behnke-Stein 1947/49 [@Behnke-Stein_1947/49] (the famous paper going back to 1943), it contains the result that any open Riemann surface (arbitrary connectivity and genus) admits a non-constant analytic function.

Can one deduce this Behnke-Stein theorem by agglomerating Ahlfors extremals (or weaker circle maps) relative to compact subregions of an adequate exhaustion?

In this connection let us remember the paper by Nishino 1982 [@Nishino_1982], where Ahlfors is applied to prove existence of (non-constant) analytic functions on certain complex surfaces (four real dimensions). Since this Nishino paper employs Ahlfors bound $r+2p$, it would be nice to understand it thoroughly to see if some better constant leads to some sharpened result. (Alas it seems that a subsequent paper of Nihino ca. 1983 proves a stronger result (pertaining to arbitrary complex dimension) while eradicating apparently any logical dependence upon Ahlfors 1950. The MR-reviewer, M. Hervé, seems to have been a bit overwhelmed by the work.)

Heins 1950 {#sec:Heins}
----------

Heins being one of the most prolific and pleasant-to-read writers of the U.S. school (student of Walsh), it is not surprising to find several first classes contributions regarding our special Ahlfors map topic. Specifically, the paper Heins 1950 [@Heins_1950] reproves Ahlfors’ result in presumably its full strength (this even without quoting Ahlfors 1950 [@Ahlfors_1950] but the closely allied work Garabedian 1949 [@Garabedian_1949]). Remember that Ahlfors’ result was exposed at the Harvard seminar in Spring 1948 (cf. Nehari’s Quote \[Nehari-1950:quote\]), and must have widely circulated since then. Taking a closer look to Heins’ paper, it is at first sight not completely evident that a bound on the degree derives from his method but is quite likely to do since his quantity $m$ (number of generators of the fundamental group, cf. p.571) is easily recognized to be $2p+(r-1)$, where $p$ is the genus and $r$ the number of contours. Thus one certainly recovers exactly Ahlfors’ result with its bound. In some sense, Heins’ paper goes even deeper than Ahlfors by treating Pick-Nevanlinna interpolation.

Several subsequent works in Heins’ spirit (overlapping with Ahlfors theorem) are worth mentioning: Heins 1975 [@Heins_1975], Forelli 1979 [@Forelli_1979] and Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF].

Kuramochi 1952
--------------

The paper Kuramochi 1952 [@Kuramochi_1952] also seems to recover Ahlfors bound for circle maps using the extremal problem. This is maybe the sort of technical paper with too much [*notatio*]{} and not enough [*notio*]{}? (This is a joke of Hellmuth Kneser, compare his paper in JDMV.) Kuramochi’s work seems to be inspired by Ahlfors 1950 [@Ahlfors_1950] and by a 1951 paper by Nehari (confined to the planar case). Nehari offers a positive review (in MathReviews):

\[Nehari-Kuramochi:quote\]

[Generalizing a method developed by the reviewer for the case of plane domains \[Amer. J. Math. 73 (1951), 78–106\], the author discusses extremal problems for bounded analytic functions on open Riemann surfaces of positive genus. The procedure is illustrated by a detailed treatment of the case corresponding to the classical Schwarz lemma which had previously been discussed, by different methods, by L.V. Ahlfors \[1950\]. A complete characterization of the extremal function is obtained and Ahlfors’ positive differential is constructed.]{}

Read 1958, Royden 1962 (via Hahn-Banach) {#Read-Royden:sec}
----------------------------------------

We start with:

$\bullet$ Royden 1962 [@Royden_1962], where existence of a solution to Ahlfors’ extremal problem is achieved via conjunction of Hahn-Banach with Riesz’s representation theorem (circumventing thereby both Euler-Lagrange and normal families). Exploiting the duality pointed out by Garabedian (pair of extremal problems with a dualizing Schottky differential, i.e. one extending to the double), the control on the degree is achieved by the usual index formula $\deg(\vartheta)=2g-2$ (Poincaré 1881–85, but already in Riemann in the holomorphic case at hand). Ahlfors’ upper bound $\deg f_{a,b}\le r+2p$ follows. Royden’s paper is therefore quite remarkable for supplying alternatives arguments. It seems to have been inspired mostly by:

$\bullet$ Read 1958 (two papers [@Read_1958_Fenn], [@Read_1958_Acta]). Read is also a student of Ahlfors (as one may learn in Ahlfors 1958 [@Ahlfors_1958]) and already relies on Hahn-Banach to prove existence of an Ahlfors function (but, as Royden observes, does not take care of making the argument with the Schottky differential so as to bound the degree). The technique employed (by Read) to prove extremals is to relate the dual extremal problems (à la Garabedian-Ahlfors, 1949–1950) to conjugate extremum problems of the Lebesgue classes $L_p$ and $L_q$, where $p^{-1}+q^{-1}=1$, where one maximizes an $L_p$-norm versus vs. minimizing an $L_q$-norm. Such problems classically reduce to Hahn-Banach. For this reduction of Garabedian-Ahlfors to Hahn-Banach, Read employs a converse to Cauchy’s theorem (itself an application of Stokes) due to Rudin 1955 [@Rudin_1955-class-Hp] in the planar case. Methods of Rogosinski–Shapiro 1953 [@Rogosinski-Shapiro_1953] are another ingredient to the proof.

To summarize, the Read-Royden approach via Hahn-Banach (functional analysis, coinage of Hadamard) effects a little drift from the traditional Euler-Lagrange variational approach (used in Grunsky 1940–46 ([@Grunsky_1940], [@Grunsky_1946]), Ahlfors 1947 [@Ahlfors_1947], 1950 [@Ahlfors_1950]). As conceptually brilliant as it is, this new method does not lead to an improved degree control. The reason is quite simple, namely Ahlfors’ bound $r+2p$ is sharp within the extremal problem it solves (contribution of Yamada 1978 [@Yamada_1978] in the hyperelliptic case).

The game naturally splits in existence of extremals (either via Montel’s normal families or via Hahn-Banach) and then to analyze its geometric properties. Ahlfors’ 1950 treatment (apparently influenced by Garabedian’s dual extremal problem) supplies the trick to bound the degree via a Schottky differential, and Royden’s argument looks, in this second geometric step, virtually osculant to Ahlfors’ original.

Remember yet that Ahlfors’ original proof—presented in Spring 1948 at Harvard as reported in Nehari 1950 [@Nehari_1950 p.258, footnote]), and perhaps nearly similar with pages 124–126 of the published paper 1950 [@Ahlfors_1950]—manages without Garabedian’s influence to supply existence of circle maps of degree bounded by $r+2p$.

Existence of (inextremal) circle maps
=====================================

This section focuses on existence of circle maps on membranes (=finite bordered Riemann surfaces) without appeal to the extremal problem. In fact those are logically required (at least in Ahlfors’ account but not in Royden’s 1962 [@Royden_1962]) as a qualitative preparation to the analysis of the quantitative problem.

Ahlfors 1948/50, Garabedian 1949
--------------------------------

\[09.06.12\] We mean the papers Ahlfors 1950 [@Ahlfors_1950] and Garabedian 1949 [@Garabedian_1949]. The additional 1948 date is intended to reflect that Ahlfors lectured on this material somewhat earlier, as shown by Nehari’s Quote \[Nehari-1950:quote\]. Those writers address the deeper extremal problem $\max \vert f'(a)\vert$ amongst functions bounded-by-one $\vert f\vert\le 1$, however it seems that they are well-acquainted with topological methods (e.g., Garabedian 1949 [@Garabedian_1949] cites Alexandroff-Hopf’s classical 1935 treatise “[*Topologie*]{}”). Such a qualitative topological inspection seems a logical prerequisite to their treatments of the quantitative extremal problem. Prior to posing any extremal problem, it is vital to ensure non-emptiness of the set of permissible competing functions. Perhaps the following trivial remarks are worth doing. For domains bounded by $r$ Jordan curves, we have clearly some function bounded-by-one (take the identity map suitably scaled to shrink the domain inside the unit-disc). For a general compact bordered surface, it is less obvious that such functions exist at all. Of course one can take the Schottky double to apply Riemann’s existence theorem (of a morphism to ${\Bbb
P}^1$) and look at the image of the (compact) half. However the latter can still cover the full Riemann sphere, which is annoying for our purpose. \[05.11.12\] Using Klein’s work one can certainly find an equivariant map from the double to the sphere acted upon by orthosymmetry (standard complex conjugation), yet it may still be the case that the full sphere is covered by the half of the double. \[As a simple example we may take a conic $C_2$ with real points and project it from a real point $p$ outside of the unique oval. The corresponding map $C_2\to {\Bbb P}^1$ is equivariant and surjective when restricted to one half of the complex locus of the conic $C_2$. Indeed given a point of ${\Bbb P}^1$ is tantamount to give a line $L$ through the center of perspective $p$. This line $L$ cuts $C_2$ in two points (except for the two real tangents). If $L$ is a real line cutting the real locus $C_2({\Bbb R})$ we can take as antecedent a point on the border of the half Riemann surface. If $L$ does not cut $C_2({\Bbb R})$ its intersection with $C_2$ is a conjugate pairs of points one of them lying in the fixed half of $C_2$. Finally if $L$ is an imaginary line then its intersection with $C_2$ consists of two points distributed in both halves of $C_2$. Indeed $L$ can by continuity be degenerated to a real line $L_0$ missing the real locus of $C_2$ (recall that the pencil of line is just an equatorial sphere with equator corresponding to real lines) and since during the process no points of $L\cap C_2$ became real it follows that both $L$ and $L_0$ have the same distributional pattern when intersected with the conic.\]

Hence in general some preparatory qualitative “topological” investigation is required to see that the extremal problem is non-vacuously posed. Remember that Ahlfors directly attacks the existence of a circle map, where it may have been sufficient to prove existence of a function bounded-by-one. His argument is in part topological inasmuch as it involves annihilating the periods of the conjugate differential of a suitable harmonic function, but also contains a great deal of non-trivial analysis, plus basic principles of convex geometry. We shall try later to penetrate in more details in Ahlfors proof.

Regarding Garabedian 1949 [@Garabedian_1949], topological considerations also plays a vital rôle in conjunction with Abelian integrals, etc. We refer the reader to the original paper. In retrospect, it may just be too sad that this brilliant work was not directly written in the broader context of Riemann surfaces.

Mizumoto 1960
-------------

This is the paper Mizumoto 1960 [@Mizumoto_1960] (which I discovered only in March 2012), yet it looks quite original making use of a topological argument involving (Brouwer’s) topological degree of a continuous mapping. So it is spiritually close to Gabard 2006 [@Gabard_2006]. However Mizumoto [@Mizumoto_1960 p.63, Thm 1, with $N$ defined on p.58] only recovers the old bound of Ahlfors $r+2p$.

Gabard 2004–2006
----------------

The proof published in the writer’s Thesis 2004 [@Gabard_2004] is essentially the same as the one in 2006 [@Gabard_2006] (modulo slight modifications suggested by the referee, presumably J. Huisman). In fact J. Huisman already on the 2004 version supplied some corrections about naive little mistakes that I made (esp. a wrong statement of Abel’s theorem forgetting to ask both divisors to be of the same degree). Of course it is to be hoped that the new bound $\le r+p$ will stay correct in the long run. In case the result is true, it would be desirable if alternative more conventional analytic methods are able to reprove this bound $r+p$. Recent results of Coppens 2011 [@Coppens_2011] show the bound $r+p$ to be best-possible, at least for generic curves in the moduli space. Coppens’ work actually supplies a much sharper understanding of all intermediate gonalities (compare Sec.\[Coppens:subsec\] for more).

There is a little historical inaccuracy in Gabard 2006 [@Gabard_2006]. When writing the paper, I did not realized properly that Ahlfors has also a quite elementary proof of the existence of circle maps of degree $r+2p$. Alas, I still do not completely understand Ahlfors’ proof yet it is clear-cut that its elementary part does not use the extremal problem! Accordingly, the sentence in Gabard 2006, p.946 reading as follows is quite inaccurate: “[*[\[…\]]{} un résultat équivalent fut démontré par L.V. Ahlfors en 1950, qui déduit d’un problème d’extrémalisation la possibilité de représenter toute surface de Riemann à bord compacte comme revêtement holomorphe (ramifié) du disque.*]{}

Related results 
================

Some closely allied problems involves [*Parallelschlitzabbildung*]{} (parallel-slit mapping), the relationship with the Bergman kernel, etc. Although a bit outside our main theme of the Ahlfors map, the methods employed are quite similar and therefore a thorough knowledge of those proximate mapping problems can only reinforce the general understanding. In fact it is not to be excluded that the Kreisnormierung, or its positive genus case avatar, known as Klein’s Rückkehrschnitttheorem, is logically stronger than the Ahlfors mapping (but this is for the moment just a naked speculation).

Parallel slit mappings (Schottky 1877, Cecioni 1908, Hilbert 1909)
------------------------------------------------------------------

Those mappings (abridged PSM) involve several tentacles using varied technologies tabulated as follows:

$\bullet$ (Classical) Schottky 1877 [@Schottky_1877], Cecioni 1908 [@Cecioni_1908] (via methods of Schwarz, and Picard). Classically Schottky’s argument is criticized (by e.g. Klein, Cecioni, Salvemini, etc.) for depending only upon a constant count not fully sufficient to establish the mapping existence (this critique appears e.g., in Cecioni ) It is likely that subsequent rigorous continuity methods as developed by Brouwer upon topological ground can easily supplement Schottky’s heuristic argument (browse through Koebe’s works, etc.)\]

$\bullet$ (Dirichlet resurrected) Hilbert 1909 [@Hilbert_1909], Koebe 1910 [@Koebe_1910_Hilbert], Courant 1910/12 [@Courant_1912] (those writers extend the PSM to domains of infinite connectivity)

$\bullet$ (Extremal problem à la FROG Fejér-Riesz-Radó-(Carathéodory)-Ostrowski-\[Grunsky\]) de Possel 1931 [@de-Possel_1931], 1932 [@de-Possel_1932], Grötzsch, Rengel 1932/33 [@Rengel_1932-33], 1934 [@Rengel_1934],

$\bullet$ (Bergman kernel) Nehari 1949 [@Nehari_1949], Lehto 1949 [@Lehto_1949], Meschkowski 1951 [@Meschkowski_1951], etc.

A philosophical curiosity is that PSM is somewhat easier (according to specialists, cf. e.g. Garabedian’s Quote \[quote:Garabedian-Schiffer\_1950\] and Hejhal 1974 [@Hejhal_1974]) than the Kreisnormierung (KNP) (cf. next section), and this already in finite connectivity (cf. e.g. the very subtle approach to KNP imagined in Schiffer-Hawley 1962 [@Schiffer-Hawley_1962]). One may wonder about this sharp discrepancy of difficulty, since it is easily conceivable that for such canonical regions (bounded by elementary curves of the most elementary stock) one could easily pass from one normal-form to the other through explicit maps (at least in finite connectivity). \[Of course I do not claim that this is an easy game for me, but I suspect so for people like Schwarz-Christoffel or Schläfli it could be accessible. Of course there is maybe a difficulty in choosing the “accessory parameters” but this should be pulverizable through modern topological arguments à la Brouwer?\] Another striking asymmetry of the theory is that PSM hold true in infinite connectivity (since Hilbert 1909 [@Hilbert_1909] and the subsequent work of Koebe 1910 [@Koebe_1910_Hilbert]), whereas KNP is still wide open in infinite connectivity. (A very naive guess would be to deduce KNP$\infty$ from PSM$\infty$ through a continuity method for infinite dimensional manifolds. Maybe this suggests using Leray-Schauder theory as an infinite-dimensional avatar of the Brouwer degree?

Regarding PSM, lucid remarks are made in Burckel 1979 [@Burckel_1979 p.357–8], namely:

$\bullet$ the result in infinite connectivity is due to Hilbert, Koebe, Grötzsch, Rengel, de Possel (as we just said also),

$\bullet$ excellent book expositions are credited to Bieberbach 193?/67 [@Bieberbach_1967-BUCH-Einfuehrung-in-die-konf-Abb], Golusin 1952/57 [@Golusin_1952/57] and Nehari 1952 [@Nehari_1952-BOOK],

$\bullet$ de Possel’s proof in 1931 [@de-Possel_1931] (and the allied work by Rengel and Grötzsch) via an extremum problem is recognized as reminiscent of Fejér-Riesz’s proof of RMT. However at one point of the proof RMT is invoked. Later de Possel 1939 [@de-Possel_1939] found a (short) constructive way around this (see also Garabedian 1976 [@Garabedian_1976]).

$\bullet$ for an approach to PSM, and the other canonical regions (radial or circular slits), via the Dirichlet principle see Ahlfors 1966 [@Ahlfors_1966-BOOK-Cplx-Anal].

Kreisnormierungsprinzip (Riemann 1857, Schottky 1875/77, Koebe 1906-08-10-20-22, Denneberg 1932, Grötzsch 1935, Meschkowski 1951–52, Strebel 1951–53, Bers 1961, Sibner 1965–68, Morrey 1966, Haas 1984, He-Schramm 1993) {#sec:KNP}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

This (cavalier?) principle (abridged KNP) starts with the fact that a multiply-connected domain of finite connectivity maps conformally to a circular domain.

This was already implicit in Riemann’s Nachlass 1857/58/76 [@Riemann_1857_Nachlass] (according to Bieberbach 1968 [@Bieberbach_1968-Das-Werk-Paul-Koebes p.148–9] who apparently saw a copy of Riemann’s original manuscript, cf. our Quote \[quote:Bieberbach-1925\] reproducing Bieberbach 1925 [@Bieberbach_1925]; cf. also Koebe 1910 [@Koebe_1910_JDMV p.339]: “[*Den Hauptgegenstand dieser und des gegenwärtigen Vortrages bildet das Problem der konformen Abbildung eines $(p+1)$-fach zusammenhängenden Bereiches auf einen von $p+1$ Vollkreisen begrenzten Bereich, ein Problem, welches in der Literatur zuerst bei Schottky (Dissertation, Berlin 1875, umgearbeitet erschienen in Crelle 1877) in seiner bekannten Doktordissertation auftritt, jedoch früher bereits von Riemann in Betracht gezogen worden ist, wie aus seiner nachgelassenen Schriften hervorgeht.*]{}”).

The statement resurfaced more explicitly in Schottky’s Thesis 1875/77 [@Schottky_1877] (at least in the Latin 1875 version). The latter’s argument rests again only on a naive parameter count of moduli. Indeed, a circular domain with $r$ contours depends upon $3r$ free parameters to describe centers and radii of those $r$ circles, while removing the 6 (real) parameters involved in the automorphism group of the (Riemann) sphere, we get $3r-6$ essential constants. Invoking the (Schottky) double of the domain, whose genus is $g=r-1$, this number agrees with Riemann’s count of $3g-3$ moduli (where of course attention is restricted to “real” moduli). This adumbrates why circular domains are flexible enough to conformally represent any domain. Such naive counting arguments usually turn into rigorous proofs by appealing to some topological principles (like Brouwer’s invariance of the domain) vindicating the so-called continuity method. This sort of game occupy several of Koebe’s papers, who probably arranged this already; see also Grunsky 1978 [@Grunsky_1978] for an implementation of KNP in 12 pages (p.114–126).

Koebe devoted several papers to the KNP question in 1906 [@Koebe_1906_JDMV], 1907 [@Koebe_1907_JDMV], 1910 [@Koebe_1910_JDMV] (Überlagerunsfläche and iteration method), 1920 [@Koebe_1920].

As early as 1908 [@Koebe_1908_UbaK3], Koebe advanced conjecturally the validity of this principle for domains of infinite connectivity: an issue still undecided today (2012), but corroborated in He-Schramm 1993 [@He-Schramm_1993] in the case of countably many boundary components (via the method of circle packings). Most of the contributions (listed in our subtitle) are carefully referenced in He-Schramm’s paper just cited.

Other proofs of the basic (finitary) KNP result are obtained by:

$\bullet$ Courant 1950 [@Courant_1950] (via a Plateau-style approach) \[Micro-Warning: Hildebrandt-von der Mosel 2009 [@Hildebrandt-von-der-Mosel_2009] and also Hildebrandt 2011 [@Hildebrandt_2011] credit rather Morrey 1966 [@Morrey_1966] for the first rigorous proof, modulo yet another gap filled by Jost 1985 [@Jost_1985]\].

$\bullet$ Schiffer and Hawley in several papers: Schiffer 1959 [@Schiffer_1959] (via the Fredholm determinant) and Schiffer-Hawley 1962 [@Schiffer-Hawley_1962] (via an extremal problem of the Dirichlet type).

It is common folklore that the Kreisnormierung, like the uniformization and even the Ahlfors circle map belong to a somewhat deeper class of problems than the parallel-slit mapping succumbing quickly to elementary techniques of potential theory. (Compare Garabedian-Schiffer’s Quotes \[quote:Garabedian-Schiffer\_1950\] and \[quote:Garabedian\_1949-52\], and also Hejhal 1974 [@Hejhal_1974 p.19] who makes similar remarks, for instance “[*We remark that the Koebe \[circular\] mapping is similar to the universal covering map, in that neither an explicit formula nor an explicit differential equation is known for it.*]{}”)

Such higher stock problems make it challenging to ask whether KNP(finite) could not be handled via an extremal problem à la Ahlfors, or to be historically sharper in the spirit of FROG=Fejér-Riesz-(Carathéodory)-Ostrowski-Grunsky.

\[$\bigstar$ Warning the sequel looks attractive yet erroneous, cf. the next paragraph for a rectification $\heartsuit$\] Maybe the relevant extremal problem (under educated guess) is to maximize the modulus of the derivative at a fixed point $a$ of the domain amongst functions bounded-by-one (in modulus) while imposing schlichtness to the mappings (otherwise we recover Ahlfors’ many-sheeted discs). Intuitively, this maximum pressurization exerted at the point $a$ ensures surjectivity of the mapping while filling most of the container in which the function in constrained by the condition $\vert f \vert \le 1$, yet roundness of the residual set of the image looks less intuitive. Speculating further, this “Ahlfors-schlicht” extremum problem could crack the fully general KNP in infinite connectivity (KNP($\infty$)). However, it suffices to remind that several complications are reported for the usual Ahlfors function in infinite connectivity (existence easy and uniqueness due to Havinson 1961/64 [@Havinson_1961/64], Carleson 1960/67 [@Carleson_1967-book], see also Fisher 1969 [@Fisher_1969]) by subsequent investigators like Röding 1977 [@Roeding_1977_Ahlfors], Minda 1981 [@Minda_1981-image-Ahlfors-fct], Yamada 1983–92 [@Yamada_1983-rmk-image-Ahlfors-fct] [@Yamada_1992-Ahlfors-fct-on-Denjoy], where the Ahlfors extremal function ceases to be a circle map and start to omit values). It is therefore quite overoptimistic to hope an Ahlfors-type (=FROG) strategy toward the prestigious KNP($\infty$). \[05.11.12\] $\heartsuit$ [*Correction.*]{}—The beginning of the previous paragraph is quite erroneous since the analogue of the Ahlfors map under the schlichtness proviso (=injectivity) is known to take a multi-connected domain not on a Kreisbereich but on a circular-slit disc. This result is due to Grötzsch 1928 [@Groetzsch_1928], Grunsky 1932 [@Grunsky_1932], Nehari 1953 [@Nehari_1953-Inequalities p.264–5] (another proof while crediting the just two cited works by Grötzsch and Grunsky), Meschkowski 1953 [@Meschkowski_1953] and finally Reich-Warschawski 1960 [@Reich-Warschawski_1960]. (Those references were already listed in Sec.\[sec:beta-and-alpha-problems\].) It is yet to be observed that such circular-slit-disc ranged maps fail schlichtness up to the boundary, and one can legitimately speculate about a suitable extremal problem akin to Ahlfors’ establishing KNP (in finite connectivity at least).

Röding 1977 (still not read)
----------------------------

The paper Röding 1977 [@Roeding_1977_mero] is perhaps quite dangerous (for Gabard 2006 [@Gabard_2006]), yet I could not procure a copy as yet.

Behavior of the Ahlfors function in domains of infinite connectivity
--------------------------------------------------------------------

There is a series of works studying the behavior of the Ahlfors function for domains of infinite connectivity. Traditionally those works look more confined to the domain case.

The basic existence and uniqueness result are addressed by Havinson 1961/64 [@Havinson_1961/64], Carleson [@Carleson_1967-book] with simplifications in Fisher 1969 [@Fisher_1969]. In contrast to the finite case, the image of the Ahlfors function does not necessarily fill the full unit circle (=disc). We just list some main contributions:

Röding 1977 [@Roeding_1977_Ahlfors] (2 points are omitted), Minda 1981 [@Minda_1981-image-Ahlfors-fct] (fairly general discrete subset of omitted values), Yamada 1983 [@Yamada_1983-rmk-image-Ahlfors-fct] (omission of a fairly general set of logarithmic capacity zero), Yamada 1992 [@Yamada_1992-Ahlfors-fct-on-Denjoy] (characterization of omitted point-sets of the Ahlfors function in case of Denjoy domains).

The quest of best-possible bounds
=================================

The writers’s own contribution $r+p$ seems, at first glance, a dramatic improvement upon Ahlfors’ upper bound $r+2p$ (at least so sounded the diagnostic of the generous Zentralblatt reviewer of my article, namely Bujalance). In the long run it may be that Ahlfors’ extremals are always as good for suitable choices of points $a,b$, but only meagre evidence is presently available.

Distribution of Ahlfors’ degrees (Yamada 1978–2001, Gouma 1998) {#Yamada-Gouma:subsec}
---------------------------------------------------------------

The papers by Yamada 1978 [@Yamada_1978], 2001 [@Yamada_2001] and Gouma 1998 [@Gouma_1998] address the delicate question about the exact values realized as degrees of Ahlfors functions. Ahlfors’ pinching $r\le \deg(f_{a,b})\le r+2p$ collapses for planar surfaces ($p=0$) to an equality, and the question is trivially settled in this case.

Yamada and Gouma rather consider the infinitesimal form of the problem where just a single interior point $a$ is prescribed while maximizing $\vert f'(a) \vert$. They obtain spectacular complete results for membranes having a hyperelliptic double ([*hyperelliptic membranes*]{}), yet without being planar ($p=0$) in which case we are in the trivial range already discussed.

For a hyperelliptic membrane, the followings hold true:

[(1) (Yamada 1978)]{} The ponctual Ahlfors function $f_a$ has degree $g+1$ at the fixed points of the hyperelliptic involution (so-called Weierstrass points).

[(2) (Gouma 1998)]{} The degree of $f_a$ can only assume values $2$ or $g+1$.

[(3) (Yamada 2001)]{} The case of degree $2$ is always realized at suitable points.

\[05.11.12\] Gouma’s result shows large discrepancy between degrees taken by Ahlfors extremals and those of general circle maps. Of course the latter are more flexible with a specimen of degree 2 (just quotient by the hyperelliptic involution), whence circle maps exist in all even degrees (post-compose with a power map $z\mapsto z^k$).

Those works promise a grandiose link between Ahlfors and the classic tradition of Weierstrass points, which probably also regulate the degree of Ahlfors maps for general (non-hyperelliptic) surfaces.

Separating gonality (Coppens 2011) {#Coppens:subsec}
----------------------------------

In another direction of dramatic depth, Marc Coppens 2011 [@Coppens_2011] is able to show sharpness of the bound $r+p$ claimed in Gabard 2006 [@Gabard_2006]. Actually, Coppens establishes the more spectacular realizability of all intermediate values for the gonality. Even if Coppens’ result looks at first sight subsumed to that of Gabard, it is in reality logically independent, so that a possible misfortune of Gabard’s result should not necessarily affect the truth of Coppens’ one. To be more specific, we introduce the following definition:

\[gonality:def\] The gonality (denoted $\gamma$) of a membrane (i.e. a compact bordered Riemann surface) is the least degree of a full (or total) covering map to the disc.

\[05.11.12\] A full (or total) covering map can be defined just as non-constant analytic map taking boundary to boundary. Then it makes good sense to Schottky-double the map and classic theory ensures the local power-map $z\mapsto z^k$ character of analytic functions, whence the branched cover nature of the map, in particular its surjectivity (via a clopen argument). The jargon “total” is borrowed from Stoilow 1938 [@Stoilow_1938-Lecons] and quite compatible with the “total reality” jargon (of Geyer-Martens 1977 [@Geyer-Martens_1977]) incarnating the algebro-geometric pendant of Ahlfors circle maps.

It is easy to show that a total map lacks ramification along the boundary. (Possible argument: Else it behaves locally like $z\mapsto z^2$ near a boundary uniformizer, but then the half-space is wrapped to a full domain expanding outside the permissible range of the map.)

In particular such a total map induces a usual (unramified) cover of the circle $\partial W \to
\partial D=S^1$, whereupon the trivial lower bound $r
\le \gamma$ follows, where $r$ is the number of boundary contours of the membrane $W$. On the other hand Gabard’s main result in 2006 [@Gabard_2006] asserts the upper bound $\gamma \le r+p$, where $p$ is the genus of $W$. Coppens’s striking result states:

[(Coppens 2011)]{} Practically, all intermediate values of the gonality compatible with the pinching $r\le\gamma\le r+p$ are realized as the gonality of a suitable membrane of topological type $(r,p)$. More accurately, there is a single trivial exception when $r=1$ and $p>0$, in which case the value $\gamma=1$ must be excluded.

Taking $\gamma=r+p$ supplies sharpness of Gabard’s upper bound. On the other hand, Coppens’ theorem tightens considerably Ahlfors’ squeezing $$r\le \deg f_{a,b} \le r+2p$$ into $$r\le\gamma\le \deg f_{a,b}\le r+2p,$$ yielding a notable contribution to Yamada-Gouma’s general question on the distribution of Ahlfors degrees (cf. previous section). Of course the contraction becomes most stringent when the gonality $\gamma$ attains its maximum value (i.e., $\gamma =r+p$ if thrusting Gabard), as it does for generic membranes in the moduli space ${\cal M}_{r,p}$ (parameterizing isomorphism classes of bordered Riemann surfaces).

Of course the moduli space stratifies through the gonalities. Imitating Riemann’s original count in our context it should be possible to predict dimensions of the varied strata. Such a deeper investigation looks desirable to complement the theory of Ahlfors circle maps. More on this in Sec.\[sec:gonality-sequence\].

\[05.11.12\] We are presently not aware of any total-bordered avatar of the simple Riemann-type counting argument, so efficient for closed surfaces in predicting correctly the gonality $[\frac{g+3}{2}]$ as well as the dimensions of moduli strata of lower gonalities. It is suspected that this asymmetry is inherent to the boundary behavior of total maps which causes certain difficulties. Of course the difficulty is somewhat akin to the intricacies arising when doing real instead of complex algebraic geometry. Yet the problem is certainly not insurmountable.

Naive question: Ahlfors degree vs. the gonality
-----------------------------------------------

All information mentioned so far is summarized in the string of estimates: $$r\le \gamma \le \begin{Bmatrix} \le\deg f_{a,b} \\ \le r+p
\end{Bmatrix} \le r+2p.$$ An obvious question is whether inequality $\gamma \le \deg
f_{a,b}$ is best-possible:

Is Ahlfors extremal problem flexible enough that each membrane has an Ahlfors map $f_{a,b}$ of degree as low as the gonality $\gamma$ for suitable centers?

Yamada’s deep result (2001 [@Yamada_2001]) positively answers the case of hyperelliptic membranes (those which are 2-gonal $\gamma=2$).

Other sources (Fay 1973, Černe-Forstnerič 2002)
-----------------------------------------------

In Fay 1973 [@Fay_1973 p.116], one reads the following assertion:

\[Cerne-Forstneric-2002:quote\]

It has been proved in \[3, p.126\](=Ahlfors 1950 [@Ahlfors_1950]) that there are always unitary functions with exactly $g+1$ zeroes [*all*]{} in $R$; and when $R$ is a planar domain, it is shown in Prop.6.16 that $S_{0,\dots,0}\cap
\Sigma_a$ is empty for $a\in R$ and that the unitary functions holomorphic on $R$ with $g+1$ zeroes are parametrized by the torus $S_0$.”

A similar comment is to be found in Černe-Forstnerič 2002 [@Cerne-Forstneric_2002 p.686]

\[Cerne-Forstneric-2002:quote\]

It is proved in Ahlfors 1950 [@Ahlfors_1950 pp.124–126] that on every bordered Riemann surface of genus $p$ with $r$ boundary components there is an inner function with multiplicity $2p+r$ (although the so-called Ahlfors functions may have smaller multiplicity).

Actually, it seems that Ahlfors’ proof shows even the slightly stronger fact that each integer $d\ge r+2p$ do arise as the degree of a circle map.

It is not perfectly clear to the writer how this claim must be interpreted: either as an exact degree $2p+r$ or as $\le 2p+r$.

On page 684, Rudin 1969 [@Rudin_1969] is quoted. Also on page 693 we find an interesting stability of inner functions of degrees $\ge r+2p-1$.

Applications of the Ahlfors mapping {#Sec:Applications-of-the-Ahlf-map}
===================================

This section lists some of the known applications of Ahlfors maps. Those applications either require the extremal property or merely conformality and the essentially topological feature of circle maps.

Gleichgewicht der Electricität (Riemann 1857)
---------------------------------------------

This source (Riemann 1857/58/76 [@Riemann_1857_Nachlass]) is the very origin of all our story. Alas the physical applications Riemann had in mind were apparently only partially reproduced in H. Weber’s reconstruction of the original manuscript. Can someone imagine what Riemann had exactly in mind (eventually on the basis of the original manuscript, which must still be dormant somewhere in Göttingen)?

Here are some well-known remarks concerning this posthumous fragment; compare the “original” (as edited by H. Weber and reproduced in part below) as well as the remarks in Bieberbach 1925 [@Bieberbach_1925 p.9, §7].

Interestingly, Riemann starts with the first boundary value problem for plane domains, and actually uses the conformal circle map to solve it, whereas the reverse engineering may look more natural in view of his Dirichlet principle philosophy. Strikingly, Riemann anticipates both the Schwarz symmetry/reflection principle (Schwarz 1869 [@Schwarz_1869-Ueber-einige-Abbildungsaufgaben p.106]) as well as the Schottky double (Schottky 1875–77 [@Schottky_1877]). Typical to Riemann, an equality sign is virtually put between potential theory and algebraic functions: the Green theorem is used and Abelian integrals (of the third species) and their periods (Periodicitätsmoduln) enter the scene. \[05.11.12\] Recall also that Bieberbach 1968 [@Bieberbach_1968-Das-Werk-Paul-Koebes] asserts that Riemann’s work also contains a trace of the Kreisnormierung, and so does earlier Koebe 1910 [@Koebe_1910_JDMV]. Besides, Bieberbach 1925 [@Bieberbach_1925] (cf. Quote \[quote:Bieberbach-1925\]) gives full credit to Riemann for the proof of circle maps in the planar case (both via potential theory and algebraic functions) emphasizing that Weber’s account is not completely faithful of the original manuscript. In contrast when based only on Weber’s account, reviewers of Riemann’s work tend to be more minimalist. E.g., Grunsky 1978 [@Grunsky_1978 p.198] writes: “[*Theorem 4.1.1. \[i.e. full covers of the disc for multi-connected domains\] goes back to Riemann, [\[423\]]{}, who gave some hints for the proof when $D$ is bounded by circles. The first proof is due to Bieberbach [\[88\](=1925 [@Bieberbach_1925])]{}, who used the Schottky-double and deep results in the theory of algebraic functions. Elementary proofs were given by Grunsky [\[195\](=1937–41); \[…\]]{}* ]{}”

\[quote:Riemann\]

Das Problem, die Vertheilung der statischen Electricität oder der Temperatur im stationären Zustand in unendlichen cylindrischen Leitern mit parallelen Erzeugenden zu bestimmen, vorausgesetzt, dass im ersteren Fall die vertheilenden Kräfte, im letzteren die Temperaturen der Oberfächen constant sind längs geraden Linien, die zu den Erzeugenden parallel sind, ist gelöst, so bald eine Lösung der folgenden mathematischen Aufgabe gefunden ist:

In einer ebenen, zusammenhängenden, einfach ausgebreiteten, aber von beliebigen Curven begrenzten Fläche $S$ eine Funktion $u$ der rechtwinkligen Coordinaten $x,y$ so zu bestimmen, dass sie im Innern der Fläche $S$ der Differentialgleichung genügt: $$\frac{\partial^2 u}{\partial x^2} +\frac{\partial^2
u}{\partial y^2}=0$$ und an den Grenzen beliebige vorgeschriebene Werthe annimmt. \[So this is ‘just’ the first boundary value problem, alias Dirichlet problem.\]

Diese Aufgabe lässt sich zunächst auf eine einfachere zurückführen:

Man bestimme eine Function $\zeta=\xi + \eta i$ des complexen Arguments $z=x+iy$, welche an sämmtlichen Grenzcurven von $S$ nur reell ist, in je einem Punkt einer jeden dieser Grenzcurven unendlich von der ersten Ordnung wird, übrigens aber in der ganzen Fläche $S$ endlich und stetig bleibt. Es lässt sich von dieser Function leicht zeigen, dass sie jeden beliebigen reellen Werth auf jeder der Grenzcurven ein und nur einmal annimmt, und dass sie im Innern der Fläche $S$ jeden complexen Werth mit positiv imaginärem Theil $n$mal annimmt, wenn $n$ die Anzahl der Grenzcurven von $S$ ist, vorausgesetzt, dass bei einem positiven Umgang um eine der Grenzcurven $\zeta$ von $-\infty$ bis $+\infty$ geht. Durch diese Function erhält man auf der obern Hälfte der Ebene, welche die complexe Variable $\zeta$ repräsentirt, eine $n$fach ausgebreitete Fläche $T$, welche ein conformes Abbild der Fläche $S$ liefert, und welche durch die Linien begrenzt ist, die in den $n$ Blättern mit der reellen Axe zusammenfallen. Da die Fläche $S$ und $T$ gleich[^10] vielfach zusammenhängend sein müssen, nämlich $n$-fach, so hat $T$ in seinem Innern $2n-2$ einfache Verzweigungspunkte (vgl. Theorie der Abelschen Functionen, Art.7, S.113) und unsere Aufgabe ist zurückgeführt auf die folgende:

Eine wie $T$ verzweigte Function des complexen Arguments $\zeta$ zu finden, deren reeller Theil $u$ im Innern von $T$ stetig ist und an den $n$ Begrenzungslinien beliebige vorgeschriebene Werthe hat.

Kennt man nun eine wie $T$ verzweigte Function $\widetilde{\omega}=h+ig$ von $\zeta$, welche in einem beliebigen Punkt $\varepsilon$ im Innern von $T$ logarithmisch unendlich ist, deren imaginärer Theil $ig$ ausser in $\varepsilon$ in $T$ stetig ist und an der Grenze von $T$ verschwindet, so hat man nach dem Greenschen Satze (Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse Art.10. S.18f.): $$u_{\varepsilon}=-\frac{1}{2\pi} \int u \frac{\partial
g}{\partial \eta} d \xi\,,$$ wo die Integration über die $n$ Begrenzungslinien von $T$ erstreckt ist.

Die Function $g$ aber lässt sich auf folgende Art bestimmen. Man setze die Fläche $T$ über die ganze Ebene $\zeta$ fort, indem man auf der unteren Hälfte (wo $\zeta$ einen negativ imaginären Theil besitzt) das Spiegelbild der oberen Hälfte hinzufügt. Dadurch erhält man eine die ganze Ebene $\zeta$ $n$fach bedeckende Fläche, welche $4n-4$ einfache Verzweigungspunkte besitzt und welche sonach zu einer Klasse algebraischer Functionen gehört, für welche die Zahl $p=n-1$ ist. (Theorie der Abel’schen Functionen Art.7 und 12, S.113, 119.)

Die Function $ig$ ist nun der imaginäre Theil eines Integrals dritter Gattung, dessen Unstetigkeitspunkte in dem Punkt $\varepsilon$ und in dem dazu conjugirten $\varepsilon'$ liegen, und dessen Periodicitätsmoduln sämmtlich reell sind. Eine solche Function ist bis auf eine additive Constante völlig bestimmt und unsere Aufgabe ist somit gelöst, sobald es gelungen ist, die Function $\zeta$ von $z$ zu finden.

Wir werden diese letztere Aufgabe unter der Voraussetzung weiter behandeln, dass die Begrenzung von $S$ aus $n$ Kreisen gebildet ist. Es können dabei entweder sämmtliche Kreise ausser einander liegen, so dass sich die Fläche $S$ ins Unendliche erstreckt, oder es kann ein Kreis alle übrigen einschliessen, wobei $S$ endlich bleibt. Der eine Fall kann durch Abbildung mittelst reciproker Radien leicht auf den andern zurückgeführt werden.

Ist die Function $\zeta$ von $z$ in $S$ bestimmt, so lässt sich dieselbe über die Begrenzung von $S$ stetig fortsetzen, dadurch dass man zu jedem Punkt von $S$ in Bezug auf jeden der Grenzkreise den harmonischen Pol nimmt und in diesem der Function $\zeta$ den conjugirt imaginären Werth ertheilt. Dadurch wird das Gebiet $S$ für die Function $\zeta$ erweitert, seine Begrenzung besteht aber wieder aus Kreisen, mit denen man ebenso verfahren kann, und diese Operation lässt sich ins Unendliche fortsetzen, wodurch das Gebiet der Function $\zeta $ mehr und mehr über die ganze $z$-Ebene ausgedehnt wird. \[…\]

This last paragraph is the one where Klein identifies (by Riemann) early examples of “automorphic functions” (compare Quote \[Klein-1923:quote:Riemann-1858\]).

Painlevé’s problem (Painlevé 1888, Denjoy 1909, Besicovitch, Ahlfors 1947, Ahlfors-Beurling 1950, Vitushkin, Melnikov, Garnett, Marshall, Jones, Tolsa 2003)
------------------------------------------------------------------------------------------------------------------------------------------------------------

This connection is first explored in Ahlfors 1947 [@Ahlfors_1947]. The point of departure is usually identified (modulo notorious sloppiness on finding the modern formulation) in Painlevé’s Thesis 1888 [@Painleve_1888] concerned with generalizations of Riemann’s removable singularity theorem: [*when do all bounded analytic functions defined in the vicinity of a compactum extend across the compactum?*]{} Riemann’s theorem settles removability of singletons.

A necessary and sufficient condition for removability is the vanishing of a certain numerical invariant directly attached to the Ahlfors function, the so-called [*analytic capacity*]{}. This is nothing but the maximum possible distortion $\vert
f'(\infty)\vert$ measured at infinity among all analytic functions defined on the complement of the compactum and bounded-by-one there. This characterization (due to Ahlfors 1947 [@Ahlfors_1947]) is not regarded as a satisfactory answer to Painlevé problem requiring a purely geometric (quasi-optical) recognition procedure of removable sets. If the compact set lies on a rectifiable curve of the plane, removability is tantamount to zero length (Denjoy’s conjecture 1909 [@Denjoy_1909-Painleve/Sur-les-fct-anal-unif-a-sing-discontinues], initially a theorem which turned out to be “gapped”, but confirmed via Calderón 1977 [@Calderon_1977] in Marshall [@Marshall_1978?]). In the general case, Vitushkin 1967 [@Vitushkin_1967] proposed a characterization via “invisible sets” (due to Besicovitch in the 1930’s), i.e. those sets having orthogonal projections of zero Lebesgue measure along almost all directions. Verdera 2004 [@Verdera_2004] explains brilliantly a metaphor with ghost objects virtually impossible to photography. Alas, Vitushkin’s expectation turned out to be not entirely correct, cf. Jones-Murai 1988 [@Jones-Murai_1988] for a counterexample, yet it gives already an approximate idea of the whole problem. For instance, the prototypical example is the [*one-quarter Cantor set*]{}: a unit-square subdivided in $4\times
4=16$ congruent subsquares whose only 4 extreme “corner-squares” are kept, with this operation iterated ad infinitum (Fig.\[Cantor:fig\]). The resulting Cantor set turns out to be removable (Garnett 1970 [@Garnett_1970]), but has positive (Hausdorff) length since its projection on the line of slope $1/2$ fills a whole interval (again Fig.\[Cantor:fig\]). Here the 1/2-slope photography of the set is Lebesgue massive (hence “visible”), yet most other projections give sets of zero measures, in accordance with the removability of the set. Of course Garnett argues differently using in particular the classic analytic theory of Ahlfors-Garabedian.

The Painlevé problem engaged many investigators (Painlevé 1888 [@Painleve_1888], Denjoy 1909 [@Denjoy_1909-Painleve/Sur-les-fct-anal-unif-a-sing-discontinues], Urysohn, Besicovitch 1930’s, Ahlfors 1947 [@Ahlfors_1947], Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950], Vitushkin 1958 [@Vitushkin_1958], 1967 [@Vitushkin_1967], Garnett 1970 [@Garnett_1970], Melnikov 1967 [@Melnikov_1967], 1995 [@Melnikov_1995], Calderón 1977 [@Calderon_1977], 1978 [@Calderon_1978-ICM], G. David 1998 [@David_1998], and many others up to its ultimate solution in Tolsa 2003 [@Tolsa_2003]. This tour de force blends a vast array of technologies (Melnikov’s Menger curvature, stopping processes à la Carleson already involved in the corona, etc.)

As a naive question how much of this theory extends to Riemann surfaces, using say Ahlfors 1950 [@Ahlfors_1950] instead of Ahlfors 1947 [@Ahlfors_1947]. \[06.11.12\] Of course it may be argued that most compactums of interest are phagocytable in a chart or a schlichtartig region hence planar via Koebe’s theorem. However it may seem that non-planar compactums exist as well on Riemann surfaces? What is the simplest example if any? Of course I certainly miss(ed) something trivial. A naive example is to take the $1/4$-Cantor set and project it down to the torus ${\Bbb
R}^2/{\Bbb Z}^2$, but of course the latter set may be planarized again (via suitable retrosections).

Classical literature discussing Painlevé’s problem includes:

$\bullet$ Ahlfors 1947 [@Ahlfors_1947]

$\bullet$ Garabedian 1949 [@Garabedian_1949]

Type problem (Kusunoki 1952)
----------------------------

In a 1952 paper [@Kusunoki_1952], Kusunoki found a clever application of the Ahlfors function to the type of open Riemann surfaces. Beware that the type of open Riemann surfaces is here understood in the analytic sense due the Finnish school (Myrberg 1933 [@Myrberg_1933], Nevanlinna 1941 [@Nevanlinna_1941]) of having a Nullrand.

More precisely, Nevanlinna 1941 () introduced a notion of surfaces with [*null-boundary*]{} (Nullrand). This amounts to exhaust the open Riemann surface by compact subregions $F_n$, while solving via Dirichlet (rescued by Schwarz, Hilbert, etc.) the boundary problem $\omega_n$ equal $0$ on $\Gamma_0=\partial
F_0$ and equal to $1$ on $\Gamma_n=\partial F_n$. As the subregions $F_n$ expand to infinity, two scenarios are possible:

$\bullet$ either the $\omega_n$ converges to $0$, or

$\bullet$ the sequence $\omega_n$ converges to a positive harmonic function, $\omega$.

In the first case, the open Riemann surface $F$ is said to have null-boundary, and in the second case to have positive boundary.

Null-boundary is equivalent to having no Green’s function, or a recurrent Brownian motion. More relevant to Kusunoki’s work is Nevanlinna’s equivalent formulation in terms of the convergence to $0$ of the Dirichlet integral $d_n=D[\omega_n]$.

Now, Kusunoki proves the following estimate (yielding a null-boundary criterion in case the right hand-side explodes to infinity):

[(Kusunoki, 1952)]{} $$\frac{1}{2\pi \lambda_n} \log{\frac{1}{\bar{r}_n}} \le
\frac{1}{d_n},$$ where $\lambda_n\le r_n+2p_n$ is the degree of an Ahlfors function $f_n\colon F_n\to D$ and $\bar{r}_n$ the maximum value of $f_n$ achieved on the Anfangsbereich $F_0$ of the exhaustion.

Kusunoki’s argument does not seem to use in any fundamental way the extremal property of the Ahlfors function. Thus perhaps any circle map (of possibly lower degree, e.g. $\le r_n+p_n$ via Gabard 2006 [@Gabard_2006]) accomplishes the job as well. This option is also corroborated by the fact that Kusunoki also appeals to Bieberbach 1925 [@Bieberbach_1925] where no extremal property is put in the forefront. Accordingly there is some hope to derive a sharper Kusunoki’s estimate. Alas the magnitudes $r_n$ change during the process so the net bonus is hard to quantify.

Carathéodory metric (Carathéodory 1926, Grunsky 1940, etc.)
-----------------------------------------------------------

Cf. for instance Grunsky 1940 [@Grunsky_1940 p.232, §3], Burbea 1977 [@Burbea_1977-Caratheodory].

Corona (Carleson 1962, Alling 1964, Stout 1964, Hara-Nakai 1985) 
-----------------------------------------------------------------

In Alling 1964 [@Alling_1964], the explicit degree bound $r+2p$ of the Ahlfors map is [*not*]{} employed. In fact any “innocent” circle map (of finite degree and not necessarily solving Ahlfors’ extremal problem) suffices to transplant the truth of Carleson’ corona theorem (1962 [@Carleson_1962]) from the disc to any finite bordered Riemann surface. Assuming that Ahlfors circle mapping theorem is really involved to prove, or speculating on a very apocalyptic earthquake destroying simultaneously all the ca. 13 proofs presently available, it is still true that the Alling/Stout extension of the corona persists all such crashes. Recall that Köditz-Timmann 1975 [@Koeditz-Timmann_1975] prove existence of a circle map (via a Behnke-Stein approximation theorem) without any control on the mapping degree. This weak form of Ahlfors is enough to complete Alling’s proof.

In contrast, Hara-Nakai 1985 [@Hara-Nakai_1985] exploit fully Ahlfors bound $r+2p$ for the finer [*corona problem with bound*]{}. The obvious problem is whether one can produce better corona bounds using circle maps of lowered degrees (e.g. those in Gabard 2006 [@Gabard_2006]). What probably plagues the game is that even in the disc case sharp estimation of the best corona constant is still an open difficult matter. Cf. e.g. Treil 2002 [@Treil_2002], where the best upper estimate of Uchiyama 1980 (Preprint) is supplemented by a lower bound improving one of Tolokonnikov 1981.

Literature includes:

$\bullet$ For the disc: Carleson 1962 [@Carleson_1962], Hörmander, Gamelin 1980 [@Gamelin_1980-Wolff's-proof] (Wolff’s proof), Garnett’s book 1981 [@Garnett_1981-BOOK], etc.

$\bullet$ For bordered surfaces: Alling 1964 [@Alling_1964], Hara-Nakai 1985 [@Hara-Nakai_1985], Oh 2008 [@Oh_2008].

Quadrature domains (Aharonov-Shapiro 1976, Sakai 1982, Gustaffson 1983, Bell 2004, Yakubovich 2006) 
----------------------------------------------------------------------------------------------------

This is another discipline bearing deep connections with the Ahlfors function. For instance Aharonov-Shapiro 1976 [@Aharonov-Shapiro_1976] prove that Ahlfors maps associated to quadrature domains are algebraic. Combining this with works by Gustafsson 1983 [@Gustafsson_1983-Quadrature], Bell 2005 [@Bell_2005-Quadrature-domains] arrives at the striking conclusion: “[It is proved that quadrature domains are ubiquitous in a very strong sense in the realm of smoothly bounded multiply connected domains in the plane. In fact they are so dense that one might as well assume that any given smooth domain one is dealing with is a quadrature domain, and this allows access to a host of strong conditions on the classical kernel functions associated to the domain.]{}”

Compare also Yakubovich 2006 [@Yakubovich_2006], and the references therein.

Wilson’s optical recognition of dividing curves (Gabard 2004)
-------------------------------------------------------------

\[30.12.12\] Another highbrow (yet poorly explored) application of Ahlfors theorem was sketched in Gabard’s Thesis (2004 [@Gabard_2004 p.7]). This was an answer to Wilson’s question (1978 [@Wilson_1978 p.67]) on deciding the dividing character of a plane curve by sole inspection of its real locus. Here again Ahlfors theorem affords an answer: a real curve is dividing iff it admits a total pencil (with possibly imaginary conjugate basepoints). Yet it must be admitted that the answer, albeit perfectly geometric, has probably little algorithmic value unless complemented by further insights.

Steklov eigenvalues (Fraser-Schoen 2010, Girouard-Polterovich 2012)
-------------------------------------------------------------------

Compare the paper by Fraser-Schoen 2010/11 [@Fraser-Schoen_2011] where, for the first time, the Ahlfors map is applied to spectral theory (the first Steklov eingenvalue). Of course the basic trick of conformal transplantation is akin to the closed case (Yang-Yau 1980 [@Yang-Yau_1980]), yet in the bordered case it seems that the Ahlfors map respects precisely what should be, when it comes to take care of the Neumann boundary condition. In this respect the Fraser-Schoen contribution looks extremely original.

Building upon a paper of Payne-Polya-Schiffer, Girouard-Polterovich 2012 [@Girouard-Polterovich_2012] are able to extend the (Fraser-Schoen) estimate to higher eigenvalues.

Other (Dirichlet-Neumann) eigenvalues (Gabard 2011)
---------------------------------------------------

Inspired by Fraser-Schoen exciting paper, I also tried the game with the modest arXiv note Gabard 2011 [@Gabard_2011], where the second inequality of Hersch 1970 [@Hersch_1970] is adapted to configurations of higher topological structure. Note that the other two remaining inequalities of Hersch are probably likewise extensible (involving the quadrant and octant of a sphere).

Klein’s intuition (Klein 1876, Marin 1988, Viro 2013, Gabard 2013)
------------------------------------------------------------------

Another little application of the Ahlfors map can be given to Klein’s intuition that a orthosymmetric (i.e. dividing or of type I) curve in the plane cannot acquire a solitary double point by progressive variation of its coefficients. This goes back to Klein 1876, and was probably justified by several workers though in a somewhat different shape from this original statement (e.g. Marin 1988 [@Marin_1988], and based upon him Viro 1986/86 [@Viro_1986/86-Progress]). For a clear-cut arguement using the deep Ahlfors theorem, cf. our Lemma \[Klein-via-Ahlfors(Viro-Gabard):lem\] below, which was essentially suggested to me by Viro (though in the modern Marin-Viro formulation differing somewhat from Klein’s original assertion). Our lemma below (\[Klein-via-Ahlfors(Viro-Gabard):lem\]) is however exactly Klein’s assertion, though proved by the device of Ahlfors maps. Even if Teichmüller 1941 [@Teichmueller_1941] should be right by ascribing to Klein the existence of Ahlfors circle maps, it is quite unlikely that Klein disposed of this as early as 1876 (the critical range being rather ca. 1882 right before the psychological collapse of Klein due to overwork). Accordingly our (Viro inspired) proof of Klein’s assertion via Ahlfors might be a bit too eclectic, yet it is quite hygienical while requiring little topological concentration.

Eclectic applications of the Ahlfors map {#Sec:Virtual-applications-Ahlf-map}
========================================

Those are only oneiric applications of Ahlfors maps, i.e. topics bearing only vague analogy to our main focus.

Filling area conjecture (Loewner 1949, Pu 1952, Gromov 1983)
------------------------------------------------------------

This was already discussed in the Introduction. One may wonder whether the FAC is also meaningful (and true) for non-orientable membranes. It seems so, imagine, e.g, a hemisphere surmounted by a microscopic cross-cap over a “glass-of-wine shaped” protuberance at the north pole (Fig.\[Wineglass:fig\]). This membranes satisfy FAC, for it effects no shortening of the intrinsic distance of the circle while having an area slightly larger than that of the hemisphere. One (possibly more accessible) question is whether the FAC holds true for membrane having the topological structure of a Möbius band (equivalently a disc with a single cross-cap). This case in view of simplicity of the topological structure is perhaps already known, or at least accessible via the traditional methods of Loewner-Pu, etc. (Alas, I am not aware of a specific reference.)

Another option is to generalize Gromov’s problem to membranes filling several contours (as suggested by J. Huisman ca. Sept. 2011). Arguably, a disjoint union of hemispheres is the best filling, at least when the contours are completely insulated (at infinite distance). Perhaps specifying some finite distance-functions $\rho_{i,j}$ between each pair of circles one can expect a least-area connected filling (without shortenings), but I have presently no clear view on how to pose properly such generalized problems.

Open Riemann surfaces embed in ${\Bbb C}^2$ (Narasimhan, Gromov, Slovenian school, etc.) {#Open-RS-embed-in-C2:sec}
----------------------------------------------------------------------------------------

The Slovenian school of complex geometry (Černe, Forstnerič, Globevnik, etc.) are also frequently employing the Ahlfors map. One among the most notorious elusive open problem (Narasimhan, Gromov, Forstnerič, Wold, etc.) is:

\[Narasimhan:conj\] Any open Riemann surface embeds properly in ${\Bbb C}^2$ (equivalently such that the image is a closed set).

In Forstnerič-Wold 2009 [@Forstneric-Wold_2009], the full problem (or at least the case of interiors of finite Riemann surfaces) is reduced to the following finitary version, seemingly much more accessible, yet apparently still out of reach:

[(FW2009)]{} \[FW2009:conj\] Each compact bordered Riemann surface $F$ embeds holomorphically in the plane ${\Bbb C}^2$.

\[06.11.12 (based on ideas of ca. Sept. 2011)\] Such an embedding is possible whenever the corresponding real curve $C$ (namely the Schottky double of $F$) admits a [*totally real pencil of lines*]{}. This is for instance the case for Klein’s Gürtelkurve (any real plane quartic with 2 nested ovals). Fig.\[Coppens:fig\] below provides plenty of other baby examples (alas most of them being only immersed). Indeed in this situation (total pencil of lines) the corresponding projection is totally real and the allied morphism $C\to {\Bbb P}^1$ induces a continuous map between the imaginary loci, i.e. $C({\Bbb
C})-C({\Bbb R})\to {\Bbb P}^1({\Bbb C})-{\Bbb P}^1({\Bbb R})$. It follows that an imaginary line of the pencil cuts the curve $C$ [*unilaterally*]{} (i.e. only along one half of the orthosymmetric Riemann surface). Removing such an imaginary line from ${\Bbb P}^2({\Bbb C})$ leaves a replica of ${\Bbb C}^2$ containing entirely the original bordered surface $F$. This simple method fails miles-away from the full Forstnerič-Wold desideratum. Indeed Ahlfors theorem (1950 [@Ahlfors_1950]) only implies existence of a totally real pencil but a priori involving auxiliary curves of order higher than one. On the other hand when starting from the abstract bordered surface (and its double) we may have first a projective model in ${\Bbb P}^3$, which projected down to the plane ${\Bbb P}^2$ may develop singularities. Hence the model in question is only immersed in general. Our naive approach only helps grasping the notorious difficulty of the question, yet still permits to settle a limited collection of special cases. Actually the method, requiring a totally real pencil of lines, applies only to real dividing smooth curves of order $m=2k$ having a deep nest of profundity $k$ (that is, higher order avatars of the Gürtelkurve).

Another classical idea was to exhibit the required embedding $F\hookrightarrow {\Bbb C}^2$ via a suitable pair of Ahlfors circle maps (not necessarily extremals). This works in special cases, e.g. hyperelliptic configurations; see Černe-Forstnerič 2002 [@Cerne-Forstneric_2002], and also the related paper Rudin 1969 [@Rudin_1969]. Maybe more sophisticated variants of Ahlfors maps arising in the broader Pick-Nevanlinna context could do the job, but this looks extremely delicate.

Another natural strategy is to embed one representant in each topological type (this is actually possible by Černe-Forstnerič 2002 [@Cerne-Forstneric_2002 Theorem 1.1]), while trying to use a continuity argument inside Teichmüller (moduli) space as suggested in Forstnerič-Wold 2009 [@Forstneric-Wold_2009].

Naive approaches to the Forstnerič-Wold question
------------------------------------------------

This section tries (unsuccessfully) to connect some highbrow geometry on the isometric resp. conformal embedding problem with the FW-desideratum of the previous section. Available are some rather formidable weapons cooked resp. by Nash-Kuiper-Gromov and Teichmüller-Garsia-Rüedy-Ko, which alas lack some rigid analytic character upon the image model as to assess anything like the FW-conjecture. Of course the conformal embedding technique is most likely to pierce the hearth of the FW-problem, yet the merely smooth character of the conformal model hinders realizability as a holomorphic curve. However it is still conceivable that a suitable tour de force, somewhat akin to Garsia’s (1962/63 [@Garsia_1962/63-algebraic-surfaces]) conformal realizability as a real algebraic surface in $E^3$, is able to unlock the secret of the FW-problem.

\[06.11.12\] Another little puzzle is whether there is a connection with (Gromov’s and probably others) question as to whether [*any Riemannian surface embeds isometrically in Euclidean $4$-space $E^4$*]{}. (Compare Gromov 1999 [@Gromov_1999] delightful preprint “Spaces and questions”, note yet the article Gromov-Rohlin 1969 [@Gromov-Rohlin_1969] where the (real) projective plane with its round “elliptic” geometry is shown to lack such an embedding.) Thus orientability is required. Via a bordered version, we can probably embed our Riemann surface $F$ (equipped with a conformal Riemannian metric) in $E^4$ isometrically hence conformally. Via hasty thinking, the FW-desideratum (\[FW2009:conj\]) follows, but alas it does not due to the lacking rigid analytic nature of the image-model.

\[16.11.12\] A more rigid constant curvature version of Gromov’s isometric embedding conjecture would be actually sufficient:

\[Space-form-embedding:conj\] [(Space-forms embedding)]{} Any orientable bordered Riemannian surface of constant Gaussian curvature $K\equiv -1$ (and totally geodesic boundary) isometrically embeds in $E^4$.

When combined with the uniformization theorem, one should be able to deduce the FW2009 conjecture (but again this is illusory unless one is able to ensure complex analyticity of the image).

The space-form embedding [(\[Space-form-embedding:conj\])]{} implies the [FW2009]{} conjecture (and perhaps the full proper embedding problem [(\[Narasimhan:conj\])]{} via an exhaustion trick).

Given the bordered Riemann surface, we take its double $2F$, which is acted upon by a canonic involution $\sigma$. On this closed Riemann surface, there is by the uniformization theorem (Poincaré–Koebe 1907 [@Poincare_1907], [@Koebe_1907_UbaK1]) a conformal hyperbolic metric (whenever $\chi (F)=\frac{1}{2} \chi(2F)<0$) and the involution $\sigma$ becomes isometric. (This equivariant uniformization is due to Koebe 1907 [@Koebe_1907_UrAK], cf. also Jost 1985 [@Jost_1985] for another approach via Plateau.) It follows that the boundary of $\partial F$ are geodesics. On applying the space-form embedding (\[Space-form-embedding:conj\]) to $F$ we get the FW2009 desideratum.

Naively it seems that bordered hyperbolic space-forms already embed isometrically in $E^3$, cf. Fig.\[SpaceForms:fig\] for some qualitative pictures. Of course finding a hyperbolic model for a membrane of type $(r,p)=(1,1)$ is more tricky to visualize.

However on tessellating the hyperbolic pants one would (under suitable junctures) get probably trouble with the Cebyshev-Hilbert obstruction to realizing the hyperbolic geometry in 3-space. So maybe one must still accept variable (negative) curvature.

More flexible and suited to the problem at hand is the theory of Teichmüller-Loewner-Garsia-Rüedy realizing in the vicinity of any smoothly embedded closed surface in $E^3$ any conformal type of Riemann surface having the same topology via normal deformations. In particular:

\[Garsia’s-thm\] [(Garsia 1961 [@Garsia_1961])]{} Any closed Riemann surface embeds conformally in Euclidean $3$-space $E^3$. (The image model can also be made real-algebraic by techniques à la Nash, etc., cf. [Garsia 1962/63 [@Garsia_1962/63-algebraic-surfaces]]{}.)

(Rüedy 1971 [@Ruedy_1971] extended the result to open Riemann surfaces, and may also have contributed to the embedded version of Garsia, if the latter only showed an immersion, at least so is claimed in Ko 1993 [@Ko_1993-finite-type], yet not so in the papers by Rüedy.)

A direct bordered version of Garsia’s result is the following:

\[Conformal-embeddings:conj\] [(Conformal embeddings)]{} Any compact bordered Riemann surface embeds conformally in $E^3$.

This conjecture would still be sufficient to answer the FW2009 conjecture. In fact this last conjecture (\[Conformal-embeddings:conj\]) may appear as a direct consequence of Garsia’s theorem upon taking the Schottky double:

Any compact bordered Riemann surface embeds conformally in $E^3$.

Let $F$ be the given surface. Take its (Schottky) double, to get the closed Riemann surface $2F$. By Garsia’s theorem (\[Garsia’s-thm\]) $2F$ is conformally diffeomorphic to a classical surface in $E^3$. We conclude by restricting this embedding to the original half of the orthosymmetric Riemann surface $2F$.

The Garsia-Rüedy theorem climaxes the Riemann-Prym-Klein conception of the Riemann surface seen as a classic (Euclid-Gauss) differential-geometric curved surface in $3$-space (compare the introduction of Klein 1882 [@Klein_1882], equivalently Quote \[quote:Klein-Prym\]).

Next there is a series of papers by Ko starting with his Thesis in 1989 where the Garsia-Rüedy conformal embedding is extended by trading ambient 3-space by an arbitrary preassigned Riemannian manifold of dim $\ge 3$. Specifically, he obtains the following results:

[(Ko 1989, 1991, 1999, 2001)]{} Given any ambient orientable Riemannian manifold $\frak M$ of dimension $\ge 3$, then any Riemann surface $F$ embeds conformally in $\frak M$ provided:

[(1)]{} $F$ is compact(=closed) [(Ko’s Thesis 1989 [@Ko_1989-compact] reissued as Ko 2001 [@Ko_2001])]{}.

[(2)]{} $F$ has finite topological type, i.e. $\pi_1$ is of finite generation or equivalently homeomorphic to a finitely many punctured closed surface [(Ko 1993 [@Ko_1993-finite-type])]{};

[(3)]{} [nothing!]{}, i.e. $F$ is a completely arbitrary open surface [(Ko 1999 [@Ko_1999-open])]{}.

Specializing (1) to $\frak M=E^4$ seems to approach the desideratum of FW2009. However there is a serious plague, for when applied to a closed surface we get a conformal embedding in ${\Bbb
R}^4={\Bbb C}^2$, while complex-analyticity of the image is inhibited by the lack of non-constant bounded analytic functions on closed Riemann surfaces. Again the whole point is that the conformal model (of the Garsia-Rüedy-Ko=GRK theory) are only smooth $C^{\infty}$-surfaces and not holomorphic curves in ${\Bbb
C}^2$. While it is impossible to ensure complex-analyticity in the closed case, there is no evident obstruction in the (compact) bordered realm, except that the maximum modulus of any linear projection on a complex line must take its maximum modulus on the boundary.

\[27.12.12\] Assuming $F$ holomorphically embedded in ${\Bbb C}^2$, we get a family of holomorphic maps $\pi_t\colon F\to {\Bbb C}_t$ parameterized by the Riemann sphere of all (complex) lines through the origin. Since $F$ is compact bordered each such holomorphic mapping has compact range with maximum modulus reached on a boundary point. Of course the image is a priori not a disc (in which case we would have a circle map), but some more complicated shadow of the Riemann surface $F$. Concentrating much on this geometry it may be hoped that for some $F$ (alas not all remind the half of the Gürtelkurve) some obstruction is detected and FW is false. Alternatively effecting a linear projection on the Riemann sphere of all lines through a point $p=(x,y)\in{\Bbb C}^2$ gives the family of projections $\lambda_p\colon F \to {\Bbb
P}^1_p$, where the latter symbol is the pencil of lines through $p$. Using translation in ${\Bbb C}^2$ all such pencils identify to ${\Bbb P}^1_0$, where $0=(0,0)$ is the origin, and we get $\Lambda_p\colon F \to {\Bbb P}^1_0={\Bbb P}^1({\Bbb C})$ a holomorphic family parameterized by $p\in{\Bbb C}^2$. Again some obstruction could occur here, but it is hard to capture. Last one could look at the tangent line assignment yielding a sort of Gauss mapping $F\to {\Bbb P}^1({\Bbb C})$ and explore its geometry in the hope of detecting some fine geometric obstruction. All this is merely canary singing without tangible grounding.

In the other optimistic direction it can be hoped that a much boosted version of the Garsia-Rüedy-Ko theory proves the FW-conjecture.

\[19.12.12\] First we know (from Černe-Forstnerič 2002 [@Cerne-Forstneric_2002]) that any topological type of finite bordered surface contains a representative holomorphically embedded in ${\Bbb C}^2$. Applying the high-dimensional version of Garsia (due to Ko 1989 [@Ko_1989-compact], plus subsequent articles) we can realize all Riemann surfaces within a normal tubular neighborhood via an (infimal) normal variation. This is akin to a cellulite bubbling, alas destroying a priori the holomorphic character of the initial model. However it is not to be excluded that better controlled vibrations of the pudding permit to explore the full moduli space. This would assess the full Forstnerič-Wold conjecture. Of course what we are saying here is nothing new that was not already said in FW2009, and one requires serious new idea to make progresses.

A naive idea would be to take a holomorphic tube around the bordered surface $F\subset {\Bbb C^2}$, i.e. a neighborhood $N$ of $F$ together with a framing, i.e. a biholomorphic trivialization $N \to F\times \Delta$ where $\Delta$ is the unit disc. (I hope that a such exists but I am not sure!) One may also suppose with a compact tube involving the closed disc $\overline{\Delta}$, so $t\colon \overline N \to F\times \overline \Delta$. Via this trivialization any holomorphic normal variation amounts to a circle map, provided the amplitude of the variation is maximum along the boundary. Conversely given a circle map $f\colon F \to \overline \Delta$ we construct a holomorphic deformation by considering the image under the map $$F \buildrel{id\times f}\over{\longrightarrow} F \times \overline
\Delta \buildrel{t^{-1}}\over{\to} \overline  N.$$ However the image curve is biholomorphic to the original $F$ and our variation is trivial. Perhaps one should use quasi-conformal avatars of circle maps (QCM for short) to perform a genuine variation of the complex structure. Such maps clearly exist, and we get so perhaps a tangible strategy toward Forstnerič-Wold 2009. Note however that the normal variation effected by a QCM destroys the analytic character of the image. On the other hand it is not essential to work with circle maps to get normal deformations so perhaps there is some freedom to be gained here. As another vague idea on how to construct the required trivialization of the normal bundle (or thickening $N$) one could imagine that it is fixed in the smooth category and then sliced by the normal 2-planes orthogonal to $F$. Each slice would be essentially a simply-connected domain and one would construct the trivialization by a version of the Riemann mapping theorem with parameters. This looks dubious but maybe leads somewhere? It is likely now that the trivialization $t$ is not holomorphic globally but only so in restriction to each slice (=fibre of the normal bundle). This is good for varying moduli, but of course disrupting the holomorphic character of the deformation.

Let us try to summarize a naive strategy toward FW2009:

[*Step 1*]{}.—Fix a semi-holomorphic trivialization $t$ of the normal neighborhood of $F$ in ${\Bbb C}^2$. (Seems accessible via a Riemann mapping theorem with parameters).

[*Step 2*]{}.—Among all normal deformations (cellulite bubbling) described by $C^\infty$ maps $f$ to the disc $\overline \Delta$ those inducing “rigid” analytic curves under the above displayed map.

[*Step 3*]{}.—By an avatar of the Teichmüller-Garsia-Rüedy technique try to calculate the moduli of the resulting deformations. Ko’s theorem ensures that all moduli are realized via soft $C^{\infty}$ deformations, but what happens when we restrict to the class of rigid perturbations?

Pick-Nevanlinna interpolation
-----------------------------

Compare the paper Jenkins-Suita 1979 [@Jenkins-Suita_1979].

Klein-Rohlin maximality conjecture(s) (Gabard 2013)
---------------------------------------------------

\[11.01.13\] The first paper were the notion of dividing curves appeared (namely Klein 1876 [@Klein_1876]) is concluded by some cryptical allusions which Klein might have derived from experimental data or by a theoretical argument involving his deep geometric intuition of Riemann surfaces.

Those intuitions were nearly forgotten for ca. 102 years until Rohlin picked them up again in his seminal paper 1978 [@Rohlin_1978] enriching the complete solution ca. 1969 by Gudkov’s of Hilbert’s 16th problem for sextics by the data of complex characteristics, i.e. Klein’s types I/II (erster und zweiter Art). This Rohlin achievement was in good part made possible by Arnold 1971 [@Arnold_1971/72] breakthrough of filling the half of an $M$-curves (or more generally) one of type I by discs bounding the ovals. Once this is in place Klein’s assertion or intuition found quite spectacular evidence and were somehow distorted by Rohlin in a related but somewhat more Hilbertian and grandiose conjecture: “a real scheme is of type I iff it is maximal”. One of the application was apparently destroyed in Shustin 1985/85 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin], yet the direct half “type I$\Rightarrow$maximal” could still be true.

A somewhat elusive desideratum would be that this reputed difficult conjecture of Rohlin follows from Ahlfors theorem. More about this in Sec.\[Klein-Rohlin-conj:sec\] below.

Starting from zero knowledge {#Sec:Starting-from-zero}
============================

As yet the text was mostly historiographical, but from now on our intention is to elevate to the higher sphere of complete mathematical arguments. (Of course the title of this section is borrowed from a joke of academician V.I. Arnold.)

The Harnack-maximal case (Enriques-Chisini 1915, Bieberbach 1925, Wirtinger 1942, Huisman 2001)
-----------------------------------------------------------------------------------------------

The theorem of Ahlfors (existence of circle maps) is easier in the planar case (and due in this case to Riemann-Schottky-Bieberbach-Grunsky, etc.). Using the corresponding Schottky double which is a real curve (of Harnack-maximal type), the assertion follows quite immediately from Riemann-Roch (Riemann’s inequality) via a simple continuity argument. This argument is implicit in Enriques-Chisini 1915 [@Enriques-Chisini_1915-1918] (perhaps even in Riemann 1857/58 manuscript [@Riemann_1857]), and was then rediscovered by many peoples including Bieberbach 1925 [@Bieberbach_1925], Wirtinger 1942 [@Wirtinger_1942], Johannes Huisman 1999/01 [@Huisman_2001], and myself Gabard 2006 [@Gabard_2006]. The nomenclature [*Bieberbach-Grunsky theorem*]{} used say by much of the Japanese school (e.g. A. Mori 1951 [@Mori_1951], etc.) is thus slightly in jeopardy.

\[Enriques-Chisini:lemma\] [(Riemann 1857/8, Schottky 1875/77 ($\pm$), Enriques-Chisini 1915 ($\pm$), Bieberbach 1925, Wirtinger 1942, etc.)]{} Any planar bordered Riemann surface with $r$ contours has a circle map of degree $r$. Moreover the fibre over a boundary point may be prescribed as any collection of points having one point on each contour.

Double the surface to get a closed one of genus $g=r-1$. On the corresponding Harnack-maximal curve (i.e. $r=g+1$), pick one point $p_i$ on each oval to get a divisor $D_0$ of degree $g+1$. Riemann’s inequality states $\dim \vert D \vert
\ge d-g$, where $\vert D \vert$ is the complete linear system spanned by the divisor $D$ and $d$ is its degree. (This is Riemann-Roch without Roch, and follows easily from Abel’s theorem.) It follows that the divisor $D_0=\sum_{i=1}^{g+1} p_i $, moves in its linear equivalence class. We may thus choose in the linear system $\vert D_0 \vert$ a line (classically denoted) $g^1_d$, consisting of groups of $d=g+1$ points. Subtracting eventual basepoints, this $g^1_\delta$ ($\delta\le d$) induces a totally real morphism to ${\Bbb P}^1$, since by continuity the points $p_i$ cannot escape their respective ovals. Indeed looking at Fig.\[Enriques:fig\] while imagining one point evading the real locus $C({\Bbb R})$ another one must instantaneously jump to locate himself symmetrically w.r.t. the (Galois-Klein) symmetry $\sigma$ induced by complex conjugation. Since a totally real morphism has degree $\ge r$, the final degree $\delta$ must be $g+1=r$.

-35pt 0

Gabard’s argument: circle maps of $\deg\le r+p$ {#sec:Sketch-of-Gabard}
-----------------------------------------------

The basic principle used in Gabard 2006 [@Gabard_2006] to prove existence of circle maps is some topological stability of the embedding of a closed Riemann surface into its Jacobian via the Abel map, which is quite insensitive to variations of the complex structure. This is how we derived universal existence theorem valid for all Riemann surfaces with upper control on the degree of such maps.

We suspect that the same method (suitably adapted to closed surfaces) enables one to recover the Riemann-Meis bound $[\frac{g+3}{2}]$ for the minimal sheet number concretizing a genus $g$ curve as a branched cover of the line ${\Bbb P}^1({\Bbb
C})$ (cf. Riemann 1857 [@Riemann_1857] and Meis 1960 [@Meis_1960]). Yet we failed presently to write down the details.

\[22.10.12\] Let us sketch rapidly the argument in Gabard 2006 [@Gabard_2006], to which we refer for more details.

\[Gabard:thm\] Any bordered surface $\overline{F}=\overline{F}_{r,p}$ of type $(r,p)$ supports a circle map of degree $\le r+p$.

Using the Schottky double $C=2 \overline{F}$, it is enough locating an unilateral divisor $D$ (i.e. one supported in the interior denoted $F$) linearly equivalent to its conjugate $D^{\sigma}$. By a simple continuity argument the pencil spanned by the pair $D, D^{\sigma}$ is totally real, hence induces a circle map; compare Lemme 5.2 in Gabard 2006 [@Gabard_2006] which we reproduce now:

The morphism induced by a pencil spanned by an unilateral pair of linearly equivalent divisors $D, D^{\sigma}$ is totally real.

Consider $g^1_d$ the linear series spanned by $D,
D^{\sigma}$. It is readily verified to be real. To check total reality imagine $D$ degenerating toward a point $D_0$ on the equator $g^1_d({\Bbb R})$ of the pencil $g^1_d$ (cf. Fig.\[Continuity-Gab:fig\]).

-5pt0

As long as $D$ stays imaginary it cannot acquire a real point (else as the morphism induced by $g^1_d$ is real it would have a real image). Therefore $D$ is so-to-speak magnetically confined to the original half, hence itself unilateral. Yet when $D$ becomes real it corresponds to a symmetric divisor (invariant under the involution $\sigma$), which must be the limit of unilateral divisors. The only possibility is for $D_0$ to be totally real. Since in a sphere, any point of the equator is accessible from the north pole, it follows that $D_0$ is always totally real. This completes the proof.

The task is thus reduced to exhibit an unilateral divisor such that $D\sim D^{\sigma}$ (linear equivalence on the curve $C$). Using Abel’s map $\alpha\colon C \to J$ to the Jacobian (variety) this amounts to say that $\alpha(D)$ is a real point of the Jacobian. Looking in the quotient $J/J({\Bbb R})$ this amounts to express zero as a sum of unilateral points. Taking any point $x_d$ in $F$, we search points $x_i\in F$ so that $$x_1+\dots +x_{d-1}=-x_d.$$ To solve this equation we use a principle of topological irrigation (subsumable to Brouwer’s theory of the mapping degree), but whose essence lies in the periodic behavior of the Abel map. Specifically, we know that $\alpha$ induces an isomorphism on the first homology. In a similar way (cf. Fig.\[Orthosym-basis:fig\]), the $r-1$ semi-cycles $\beta_1^+,
\dots, \beta_{r-1}^+$ (linking one contour to the others) and the $2p$ cycles $\widetilde{\alpha}_1, \dots,
\widetilde{\alpha}_p,\widetilde{\beta}_1, \dots,
\widetilde{\beta}_p$ winding around the $p$ handles form a basis of the first homology of the quotient $T^g:=J/J({\Bbb R})$, a $g$-dimensional (real) torus. Note that the extremity of the semi-cycles $\beta_i^+$ are pasted together when passing to the quotient.

-5pt0

The irrigation principle says that if we have $g$ cycles representing a basis of the 1st-homology of a $g$-dimensional torus $T^g$ then any point of the torus is expressible as the sum of at most $g$ points situated on the given cycles. Applying this, we can solve the above equation for $d-1\le (r-1)+2p$, i.e. $d\le
r+2p$ recovering Ahlfors bound. Since we are presently unable to reprove Ahlfors theorem via his original argument, let us state this as an independent theorem. (Note at the didactic level that our proof merely use Abel 1826 [@Abel_1826], and perhaps some Riemann in as much as we use that a curve of genus $g$ supports $g$ many holomorphic $1$-forms involved in the definition of the Abel map, yet nothing like say Green 1828 [@Green_1828] which is pivotal in Ahlfors’ implementation, although this is only stressed subconsciously.)

[(Ahlfors 1950, via Gabard’s method)]{} \[Ahlfors-via-Gabard:thm\] Any bordered surface $\overline{F}=\overline{F}_{r,p}$ of type $(r,p)$ supports a circle map of degree $\le r+2p$.

Now our points $x_i$ are situated on curves traced in advance around the handles. This constraint is not inherent to our problem, where only unilaterality is required. Thus the points enjoy more freedom and this is how we discovered (ca. 2002) the possibility of improving Ahlfors. More formally, we can imagine instead of the two cycles $\widetilde{\alpha}_i,\widetilde{\beta}_i$ winding around a handle a 2-cycle $\widetilde{\alpha}_i \star\widetilde{\beta}_i$ having the shape of a 2-torus. The latter torus is not traced on our surface $F$, but a vanishing cycle operation makes the torus visible. This torus is interpreted as a cycle with stronger irrigating power. Summarizing, we have in the quotient $T^g$ the $(r-1)$ semi-cycles and $p$ many 2-tori of stronger irrigating power. An (evident) variant of the irrigation principle gives solubility of the above equation for $d-1\le (r-1)+p$, i.e. $d\le
r+p$ (Gabard’s bound).

[*Warning.*]{}—\[06.11.12\] Presenting the full details in some less intuitive manner occupies the last 7 pages of Gabard 2006 (). It is hoped that the $r+p$ result is correct, but it should not be excluded that something wrong happened (or at least that the proof is not convincing enough). Thus more investigations require to be made to assess or disprove the above theorem. Of course the first part, where only Ahlfors’ bound $r+2p$ is recovered, seems less subjected to “corrosion”, because the irrigating cycles are readily traced on the bordered surface (without appeal to vanishing cycles, homologies, etc.).

Assigning zeroes and the gonality sequence {#sec:gonality-sequence}
------------------------------------------

\[22.10.12\] Here we explore some little new ideas inspired by the irrigating method discussed in the previous section. Alas, details are a bit messy (mostly due to severe degradations of the little I knew about algebraic curves). Most propositions of this section suffer the plague of hypothetical character. We hope that, despite vagueness of conclusions, the thematic addressed is worth clarifying. A general question of some interest is that of calculating for a given bordered surface the list of all integers arising as degrees of circle maps tolerated by the given surface. We call this invariant the [*gonality sequence*]{}. Another noteworthy issue is that apparently Ahlfors’ upper bound $r+2p$ is always effectively realized, in sharp contrast to Gabard’s one $r+p$ which can fail to be.

In the above argument (proof of (\[Gabard:thm\])) we may replace the point $x_d\in F$ by a collection of $k$ points say $z_1,
\dots, z_k\in F$. By the irrigation principle it is still possible to solve the following equation in the group $T^g=J/J({\Bbb R})$ $$x_1+\dots +x_{d-1}=-(z_1+ \dots+ z_k)$$ for $d-1\le (r-1)+p$. Alas, if the divisor $z_1+ \dots+ z_k$ is linearly equivalent to its conjugate the right hand side vanishes in $T^g$, and all $x_i$ could lye on the boundary of the semi-cycle (violating the unilaterality condition). However, in this case there is a circle map of degree $k$ exactly given by the divisor $D=\sum_{i=1}^k z_i$. Thus, we can still conclude the following:

(Circle maps with assigned zeroes) Given any collection $z_i$ of $k$ points in a bordered surface $\overline{F}$ of type $(r,p)$ there is a circle map of degree $\le (r-1)+p+k$ vanishing on the assigned points $z_i$.

It must just be observed that the pencil through $D,D^{\sigma}$, where $D=x_1+\dots +x_{d-1}+(z_1+ \dots+ z_k)$ is basepoint free due to the unilaterality of this divisor. (This holds true even if some of the $x_i$ or $z_i$ come to coincide.)

It seems even that there exists circles maps of any degree $d\ge
r+p$, but I am not sure about this point. Checking the truth of this requires the assertion that any point in the torus is expressible as the sum of the exact number of cycles available in the irrigating system. At first glance, this looks untrue in the trivial irrigating system for the flat 2-torus ${\Bbb R}^2/{\Bbb
Z}^2$ consisting of the 2 factors. Yet the origin may be redundantly expressed as sum of two points. Idem for a point on the vertical axis, there is an expression as that point plus the origin. So maybe it works. The general (hypothetical) statement would be:

(Hypothetical lemma=Sharp irrigation principle) Given cycles $\gamma_1, \dots, \gamma_k$ of dimensions (say one and two, yet this is certainly not essential) in a $g$-torus $T^g$ such that their Pontrjagin product $\gamma_1 \star \dots \star
\gamma_k$ represents the fundamental class of $T^g$. Then any point of $T^g$ is expressible as the sum of $k$ points $x_i$, one situated on each $\gamma_i$. (Some $x_i$ may coincide.)

Granting this we seem to get a sharpener version of the previous proposition.

(Very hypothetical!!!) \[hypothetical:prop\] Given any collection $z_i$ of $k$ points in a bordered surface $F$ of type $(r,p)$ there is a circle map of degree exactly $(r-1)+p+k$ vanishing on the assigned points $z_i$. In particular there exists circles maps of any degree $d\ge r+p$.

In fact the real problem is that our irrigating system involves the $r$ semi-cycles on $F$ (which close up into $J/J({\Bbb R})$). If the sum involves points located on the boundary of those semi-cycles, then those points must be discarded to ensure unilaterality of the divisor. Thus our method gives only an upper bound on the degree of the final map, but never an exact control.

Basic examples show that special Riemann surfaces may well admit circle maps of degree $d<r+p$ (cf. e.g. Fig.\[Chambery:fig\]).

The [*gonality*]{} $\gamma=\gamma(F)$ of a compact bordered Riemann surface $F$ is the least possible degree of a circle map tolerated by $F$.

Evidently $r\le \gamma\le r+p$ (the second estimate being Gabard’s claim). One can ask if each value $d\ge \gamma$ above the gonality occurs as the degree of a circle map. Alas, the above irrigation technique fails close to imply this. Our guess is that the response is in the negative, that is, there may be “gaps” in the sequence of all circle mapping degrees.

Thus to detect a gap it is natural to look among “special” surfaces of small gonality in comparison to its generic value $r+p$. A rapid glance at the combinatorics of Fig.\[Coppens:fig\] (below) helps us identify the simplest such example as a hyperelliptic surface with $(r,p)=(2,1)$. Then $\gamma=2<r+p=3$. Borrowing an idea of Klein, we can think of the corresponding real curve as a doubled conic. This occurs actually via the so-called canonical mapping (of algebraic geometry) which fails injectiveness for hyperelliptic curves. (Note: we switch constantly from bordered surfaces to real dividing curves, committing oft slight abuses of language.) Klein regards this doubled conic as a degeneration of the general Gürtelkurve (with two nested ovals) when both of them come to coalesce. This projective model of the hyperelliptic surface suggests that when projected from the doubled curve it has degree 2, but if the center of projection moves in the inside of the conic then the projection acquires degree 4 suddenly, without visiting the value 3. However substituting to the bordered surface this double conic is a bit fraudulent, e.g. because the latter is reducible and correspond rather to a disconnected Riemann surface. Also the doubled conic looks 2-gonal in $\infty^1$ ways whereas the original surface is uniquely 2-gonal. Thus another more reliable argument requires to be given. (This must probably be akin to the lemma proving uniqueness of the hyperelliptic involution.)

($\bigstar$) If I remember well there is a lemma saying that any basepoint free pencil $g^1_d$ on a hyperelliptic curve is composed with the hyperelliptic involution $g^1_2$. In more concrete words, any morphism to the line factors through the hyperelliptic projection, and so has even degree. If this is correct, Prop.\[hypothetical:prop\] is corrupted since the gonality sequence is exactly the set of even integers $2{\Bbb N}$. This remark would equally apply to any hyperelliptic membrane with $r=1$ or $2$, $p$ arbitrary.

However this conclusion conflicts with the Černe-Forstnerič claim (cf. 2002 [@Cerne-Forstneric_2002]) that Ahlfors proved any surface to exhibit a circle map of degree $r+2p$ exactly (take $r$ odd equal to 1). Of course the mistake is mine and to be found in the parag. ($\bigstar$) right above, as shown by the following example.

(Convention) Below and in the sequel, we shall often say just total morphism instead of totally real morphism.

[**Example 1.**]{} Consider a quartic with one node (so of genus $g=2$). This is hyperelliptic (alias 2-gonal) when projected from the node. However the curve also admits maps to the line of degree 3 (projection from a smooth point). Manufacturing a real picture gives the picture nicknamed 112 on Fig.\[F112:fig\] deduced via sense-preserving smoothings of both ellipses (ensuring the dividing character of the resulting curve by Fiedler 1981 [@Fiedler_1981]). The dashed circle indicates the node left unsmoothed. To avoid any mysticism, our nicknaming coding consists in writing the 3 invariants $r,p, \gamma$ as the string $rp\gamma$.

-5pt0

Picture 112 shows total morphisms (i.e. with totally real fibers over real points) of degree 2 (projection from the node), of degree 3 (projection from the inner loop) and of degree 4 (projection from inside the inner loop). One would like to know if $5$ is also the degree of a circle map, etc. We believe the answer to be positive: Ahlfors exhibits circle maps of degree exactly $r+2p$, and of all higher values too (as follows from his convexity argument). Hence the gonality sequence seems to be $\gamma=2=r+p, 3=r+2p, \dots$, where the dots mean all subsequent gonalities do occur after Ahlfors bound. This being a bit notationally messy, we introduce a graphical punching-card system on the figure, where the gonality sequence 234…is decorated by a triangle for $r+p$, an underlining of Ahlfors’ bound $r+2p$ (after which the gonality sequence is full), and the arrow indicating the least position from whereon the sequence is full. The given example does [*not*]{} confirm our initial guess about gaps in the gonality sequence, so let us examine another example.

[**Example 2.**]{} Consider a hyperelliptic model of type $(r,p)=(2,1)$. Then the genus $g$ of the double is $g=(r-1)+2p=1+2=3$. This prompts looking at plane smooth quartics having the right genus $3$, but alas the wrong gonality 3 (not 2). Thus we move to quintics (“virtual” smooth genus 6) and to lower down to $g=3$ we introduce one triple point (counting like 3 double points since perturbing slightly 3 coincident/concurrent lines creates 3 ordinary nodes). This gives the correct gonality $5-3=2$. Doing a real picture one may draw picture 212 on Fig.\[F112:fig\]. (Keep in mind the orientation-consistent smoothing ensuring the dividing=orthosymmetric character of the curve). It has $r=2$, and $p=\frac{g-(r-1)}{2}=1$. Notice total maps of degrees 2 (projection from the “tri-node”=triple point), degree 4 (projection from the inner circuit) and degree 5 (projections from the inside of this inner circuit). Yet we missed degree 3. Over the complexes such a curve is not 3-gonal (because it is 2-gonal from the tri-node and 4-gonal when projected from a smooth point). Consequently, the allied bordered surface has circle maps of degrees $2,4,5$ but not $3$, which is missing. Hence this example probably corrupts our naive Prop.\[hypothetical:prop\]. Also, Gabard’s bound $r+p$ needs [*not*]{} to be exactly the degree of a circle map. Further this example shows the gonality sequence to be gapped in general.

Now one general question is to wonder what can be said about the following invariant.

\[def:gonality-sequence\] The [*gonality sequence*]{} $\Lambda=\Lambda(F)$ consists of the ordered list $\gamma<\gamma_1<\gamma_2<\dots$ of all integers occurring as degrees of circle maps tolerated by the given bordered Riemann surface $F$.

Fragmentary information includes the following facts, gathered as a theorem. (To nuance reliability of the varied constituents we assign them some percentages of truth likelihood, with frankly Schopenhauerian scepticism!)

For any bordered Riemann surface with topological invariant $(r,p)$ (viz. number of contours $r$ and genus $p$) and gonality $\gamma$ (i.e. the least degree of a circle map), the following estimates hold good ([*en principe[^11]*]{}):

[\[100%\] $\bullet$ (T) (Trivial)]{} $r\le\gamma$.

[\[99%\] $\bullet$ (KTA)]{} $\Lambda$ is nonempty or equivalently $\gamma<\infty$ is finite [(Ahlfors 1950 [@Ahlfors_1950], or Teichmüller 1941 [@Teichmueller_1941] crediting Klein for the result; cf. also Köditz-Timmann 1975 [@Koeditz-Timmann_1975] for a proof via Behnke-Stein)]{}.

[\[100%\] $\bullet$ (Semigroup property)]{} the set $\Lambda$ is “multiplicative”, i.e. whenever it contains an element $\lambda
\in \Lambda$ it contains all integral multiples $k \lambda$. (This follows by composing the corresponding circle map by a power map $z\mapsto z^k$ from the disc to itself.) In particular [(KTA)]{} implies that $\Lambda$ is always infinite.

[\[98%\] $\bullet$ (A50) (Ahlfors 1950)]{} $\gamma\le r+2p$.

[\[75%\] $\bullet$ (G06) (Gabard 2006)]{} $\gamma\le r+p$.

[\[79%\] $\bullet$ (C11) (Coppens 2011)]{} $\gamma $ takes all intermediate values $r \le\gamma\le r+p$ (if $r=1$ the lower bound $r$ must be modified as $2$, excepted when $p=0$).

[\[97%\] $\bullet$ (AFCF) (Ahlfors 1950 [@Ahlfors_1950 p.126], adhered to in Fay 1973 [@Fay_1973 p.116] and Černe-Forstnerič 2002 [@Cerne-Forstneric_2002])]{} Ahlfors bound $r+2p\in \Lambda$ always belongs the gonality sequence; and so do all higher values.

The last assertion follows from Ahlfors proof (1950 [@Ahlfors_1950 pp.124–126]) where the origin is expressed as convex sum of points lying on a collection of circuits in ${\Bbb R}^g$. This is always feasible for $g+1=r+2p$ points, and a fortiori for more points. We shall try to digest Ahlfors argument in subsequent sections.

In contrast to (AFCF), Example 2 (=212 on Fig.\[F112:fig\]) above shows (or at least indicates strongly) that Gabard’s bound $r+p$ is not necessarily in the gonality sequence.

Further evaluations of the gonality sequence are tabulated on Fig.\[Coppens:fig\] as bold fonts. As before, the underlined number is Ahlfors (universal) bound $r+2p$, after which all gonalities are realized. The position pointed onto, by a triangle, is Gabard’s bound $r+p$. The little arrow is a pointer indicating the lowest integer after which the gonality sequence is full.

At an early stage of the tabulation, it seemed realist to advance the following.

\[conj:full-above-Gabard\] (Naive, destroyed by Example $4$) Strictly above $r+p$ each gonality occurs.

This is pure guessing, but if true it would considerably lower Ahlfors’ universal lower bound $r+2p$ for “fullness”. The next example still supports the guess, but the next Example 4 ought to violate it.

Consider, within the topological type $(r,p)=(1,2)$ where $g=(r-1)+2p=4$, a hyperelliptic model ($\gamma=2$). Looking at quintics (virtual genus 6) requires 2 nodes to correct the genus, but then the (complex) gonality is still 3 (and not 2 as we would like). The trick is (like in Example 2) to increase further the degree to permit a high order singularity lowering drastically the gonality. So we move to sextics (virtual genus 10) with a 4-node (counting for 6 ordinary nodes) decreasing correctly the genus to 4. As initial configuration we consider 3 coincident lines plus a conic through the coincidence and another line (in general position). An appropriate smoothing generates picture 122 on Fig.\[F122:fig\] with $r=1$ (all real circuits being connected through $\infty$). The gonality sequence seems to be $2, 5, 6,
\dots$. However 4 must be added to the list (being a multiple of 2). Hence the true sequence is $2,4,5,6, \dots$. Gabard’s bound is $r+p=3$, and strictly above it all values are realized (Ahlfors bound is $r+2p=5$).

-10pt0

\[23.10.12\]—[*Vague philosophy.*]{} An interesting feature of this example (122 on Fig.\[F122:fig\]) is that when gonality is very low in comparison to topological complexity, the Riemann surface, after having dispensed much energy to reach such a low gonality, seems falling into some dormant state without creating many new gonalities (missing the value 3). Perhaps this is a general phenomenon prompted by a principle of energy conservation.

\[28.12.12\]—[*Warning.*]{} On looking carefully at picture 122 above (Fig.\[F122:fig\]) it is seen that as the center of perspective is dragged from the $4$-node toward the two red loops or even their insides we may loose total reality for some lines of the pencil become tangent to the circuit somewhere (cf. dashed lines on picture 122) so that a suitable perturbation let disappear two intersections in the imaginary locus. Accordingly, it is not even evident form the picture that degrees $5, 6$ occur as degrees of total maps.

We now consider, in the topological type $(r,p)=(2,2)$ for which $g=(r-1)+2p=5$, again a hyperelliptic model. Looking at sextics with smooth genus $10$, we must use a correction by 5 (alas not a triangular number as those involved in multiple points). Thus we move to septics (order $m=7$) of smooth genus $\tilde{g}=\frac{(m-1)(m-2)}{2}=15$, and a 5-node (counting for $1+2+3+4=10$ ordinary nodes) effects the desired correction upon the genus. Smoothing a suitable configuration gives picture 222 on Fig.\[F122:fig\] with $r=2$ (two real circuits red and green colored). The gonality sequence includes the values $2,4,6=r+2p, \dots$. Six being Ahlfors bound the sequence is full from there on. When projected from the 5-node the degree is 2. Dragging the center of perspective along one of the two red loops gives total maps of degrees $7-1=6$ ([*warning:*]{} this is not even true, cf. again the dashed line on the picture!). The value 4 is not visible on the projective model, yet arises by the semigroup property. Studying the gonality over the complexes, it seems evident that 3 and 5 are not even complex gonalities, and we should be able to conclude that $2,4,6,\dots$ is the exact gonality sequence. (Here the “dots” refer again to the issue that all higher values belong to the gonality list, according to Ahlfors.) But then our conjecture \[conj:full-above-Gabard\] is violated (as $5$ does not belong the list). Incidentally, this example shows sharpness of Ahlfors bound $r+2p$ as the place from where the sequence is full.

Those examples can be iterated for higher values of the invariant $(r,p)$ while staying in the hyperelliptic realm. The arithmetical issue is the possibility to compensate the genus by a high-order singularity. We obtain for $(r,p)=(1,3)$, hence $g=6$, the figure 132 on Fig.\[F122:fig\], an octic (smooth genus 21) with a 6-node (counting for $\frac{6\cdot
5}{2}=15$ ordinary nodes) hence lowering down the genus to $6$. The gonality sequence is $2,4,6,7=r+2p, \dots$. Similarly, for $(r,p)=(2,3)$, $g=7$. Browsing through increasing degrees the genus are $10, 15, 21, 28, \dots$, whereas the nodes give the list $1,3,6,10,15,21, \dots$. The right pair is thus $28-21=7$. So we take a 9-tic (smooth genus $\tilde
g=\frac{(9-1)(8-1)}{2}=\frac{56}{2}=28$) with a 7-node. We construct easily picture 232 on Fig.\[F122:fig\], a curve whose gonality sequence is $2,4,6,8=r+2p, \dots$. (Note that in this case Ahlfors bound is sharp for the fullness of the sequence, but it was not in the previous example. It may again be observed that in the first example the $r+p$ bound occurs as a gonality, but it does not in the second example.)

The real outcome of these constructions is that for (certain, all?) hyperelliptic curves we can be totally explicit about the gonality sequence. Iterating ad infinitum we have:

For any topological type $(r,p)$ there is a surface of hyperelliptic type $(r,p)$ (with $r=1$ or $2$) whose gonality sequence $\Lambda$ is known explicitly. Namely,

$\bullet$ if $r=1$, then $\Lambda=\{2,4,6, \dots, 2p, r+2p, \dots
\}$, where the first “dots” runs through even values and the second means fullness after Ahlfors bound $r+2p$.

$\bullet$ if $r=2$, then $\Lambda=\{2,4,6, \dots, r+2p, \dots \}$, where the first “dots” runs through even values and the second means fullness after Ahlfors bound $r+2p$.

The natural question is of course to know if this spectrum distribution is specific to our models or generally valid for all hyperelliptic surfaces. (This looks likely, we think, maybe just by counting moduli.) Of course it is evident by the semigroup property that the gonality sequence contains the value listed, and is full after Ahlfors bound, yet the assertion that it reduces to this requires some argument.

A conjecture about fullness
---------------------------

\[23.10.12\] At this stage the situation is admittedly a bit messy. We try to clarify it by bringing into the picture the [*fullness invariant*]{} $\varphi$, that is the least integer from whereon the gonality sequence is full. (On the pictures discussed this is nothing but the little arrow used previously.) We have the string of inequalities: $$r\le \gamma \le \begin{Bmatrix}  \buildrel{{\rm Ga}}\over{\le} r+p \le \\
\hskip4pt \le \hskip8pt \varphi \hskip8pt \buildrel{{\rm
Ah}}\over{\le}
\end{Bmatrix} \le r+2p.$$ Is any comparison possible between $r+p$ and $\varphi$? On example 212 of Fig.\[Coppens:fig\] $r+p=3$ beats the fullness $\varphi=4$. Many examples on Fig.\[Coppens:fig\] do satisfy $r+p\le \varphi$, but there is also several counter indicators, e.g. pictures 313, 414 or 223.

The following is a trivial consequence of inequation $\gamma\le
\varphi$:

Fullness below Gabard’s bound (i.e. $\varphi < r+p$) implies low-gonality (i.e. $\gamma < r+p$).

The converse fails, see pictures 212 or 222.

On the pictures of Fig.\[Coppens:fig\] the fullness $\varphi$ is indicated by a little upward arrow. Examining examples on this figure it seems that when the surface has generic gonality (i.e. $\gamma=r+p$) then its fullness coincides with the gonality (i.e. $\varphi=\gamma$). It would be interesting to know if a general theorem hides behind this experimental observation.

(Pressing up and down: fullness conjecture) \[conj:fullness\] If $\gamma=r+p$, then $\varphi=\gamma$. In other words if $\gamma$ achieves maximum value (granting the truth of Gabard’s bound!) then $\varphi$ collapses to its minimum value (namely $\gamma$). In particular the gonality sequence of a generic surface would be perfectly explicit, as being full from $r+p$. This would also show that generically Ahlfors bound $r+2p$ for fullness can be drastically lowered.

It seems plausible that an adaptation of Gabard 2006 could prove this conjecture. (Ahlfors’ original proof can also be useful.) The idea would be that in the irrigation method the equation $x_1+\dots +x_{d-1}=-x_{d}$ which is soluble for $d\le (r-1)+p$ points is, by the assumption made on $\gamma$, not soluble for fewer points. One would then like to extend this “exact solubility” to the equation $x_1+\dots+
x_{d-1}=-(z_1+\dots+z_k)$, where the $z_i$ is a collection of points assigned in $F$. A vague idea is then that if some $x_i$ (or their lifts to $\overline{F}$ belong to the border) then upon dragging the $z_i$ we may hope to displace them to avoid this circumstance (incompatible with unilaterality). This would construct an unilateral divisor of any assigned degree $\ge r+p$, producing in turn circle maps of all such degrees. The conjecture would follow.

Another basic phenomenon is that even when two surfaces have the same invariants $(r,p)$ and the same gonality $\gamma$ their gonality sequences may differ. (See for such an example both pictures 324 on Fig.\[F324:fig\]). Interestingly the left figure 324 is 4-gonal in $\infty^1$ ways (projection from the inner oval), whereas its companion 324bis, is 4-gonal in only 4 ways (projection from the nodes). Again some conservation law seems involved for all the energy absorbed by the many pencils of degree 4 living on the first model seems to provoke the missing of pencils of degree 6. It could be imagined that the right curve is totally real when swept out by the pencil of conics through the 4 nodes of degree $2\cdot 6 -4 \cdot 2=12-8=4$, but it fails to be total for the circular conics through the 4 nodes certainly misses the outer oval.

-5pt0

To investigate the fullness conjecture (\[conj:fullness\]) further, we test curves of higher topological structure.

$\bullet$ For $(r,p)=(3,2)$, we seek a surface with maximum gonality $\gamma=r+p=5$. If we imagine this gonality arising via linear projection it is natural to look at a sextic having a deep nest. The virtual genus is then $10$, but we want genus $g=(r-1)+2p=2+4=6$. Thus we introduce 4 nodal singularities. To keep the gonality maximum those nodes must not be accessible from the inner oval, and consequently we distribute the dashed circles (indicating unsmoothed nodes) in the “periphery”. We thus obtain curve nicknamed 325 (on Fig.\[Generic:fig\]). It has $\gamma=5$ and the gonality sequence is $5,6,7=r+2p,\dots$. In fact $\gamma$ could be $<5$ via some nonlinear pencil harder to visualize. A pencil of conics with 4 basepoints matching with the $4$ nodes creates a series of degree $2\cdot 6-4\cdot 2=12-8=4$, more economical than our $5$. However looking at picture 325, the special conic consisting of two horizontal lines fails to intersect the inner oval. Thus this pencil is not total, and we safely conclude that $\gamma=5$, exactly. In particular, the fullness conjecture (\[conj:fullness\]) is verified on this example.

-5pt0

$\bullet$ For $(r,p)=(4,2)$, we seek a surface with maximum gonality $\gamma=r+p=6$. Imagine again this gonality arising via linear projection, we consider a septic with a deep nest. The virtual genus is then $15$, but we want $g=(r-1)+2p=3+4=7$, hence we conserve 8 nodal singularities. We obtain so the curve labelled 426 on Fig.\[Generic:fig\]. It has $\gamma=6$ and the gonality sequence is $6,7,8=r+2p,\dots$. The fullness conjecture (\[conj:fullness\]) seems verified on this example. Warning \[25.10.12\]: now the claim $\gamma=6$ is possibly an optical illusion, for a pencil of cubics with basepoints assigned on the nodes has degree $3\cdot 7-8\cdot 2=21-16=5<r+p$. If the latter is total then $\gamma\le 5$, violating our claim $\gamma=6$. Of course tracing pencil of cubics is not an easy game. Experience tell us that total pencils arise when basepoints are deeply rooted inside the deepest ovals. In the case at hand (curve 426), this feature is not fulfilled. The 8 basepoints of the cubics pencil lye outside the inner oval, yet, it could be that the 9th basepoint falls (by a lucky stroke) inside this oval. In fact, it is enough to observe that the cubic, consisting of the ellipse through the 6 points lying highest on figure 426, plus the line through the remaining 2 points (lying lowest on the same figure), fails to cut the inner oval. This gives evidence that our pencil of cubics is not total. We conclude $\gamma=6$, exactly. In fact one must check that the 8 assigned basepoints impose independent conditions on cubics, and so our pencil is forced to contain the special reducible cubic just described. Independence is checked by the usual stratification method, where one imposes more and more conditions while verifying that each extra condition drops dimension by checking that the corresponding inclusion is strict. The method seems to apply to our situation, and we conclude $\gamma=6$ (with reasonable self-confidence). Of course another detail that must be taken care is our somewhat tacit supposition that the gonality (or the gonality sequence) do not depend tremendously on the choice of smoothing. The classical method of small perturbation (Brusotti, etc.) asserts existence of a curve effecting the assigned smoothings, but there is an infinitude of choices for the coefficients. A priori the fine gonality invariants are sensitive to the choice effected. Remember that Brusotti’s method relies on the fact that the initial curve (thought of as a point in the discriminant hypersurface) has a neighborhood consisting of several “falde analytice”, i.e. a divisor with normal crossings each branch of which corresponding to preserving a certain node. This explains the liberal way to smooth away nodes of our initial configurations. Yet more hazardous is the claim of a smoothing conserving the exact location of all nodes. This remark hinders slightly the previous argument made on figure 426.

$\bullet$ We next test the invariant $(r,p)=(5,2)$, and within it seek again a representative of maximum gonality $\gamma=r+p=7$. Using the same device as above, we are inclined to look at an octic (order $m=8$) with an interior oval kept protected from intrusion of singularities. The smooth genus is then $\tilde
g=\frac{7\cdot 6}{2}=21$, but need be lowered down to $g=(r-1)+2p=8$. We thus consider a distribution of 13 nodes distant from the inner oval to produce the curve nicknamed 337 (on Fig.\[Coppens:fig\], see also Fig.\[Gabard:fig\] for a larger depiction). This curve has $r=3$ (not 5 as desired!). This means that I am a bad experimentalist, but the curve 337 is worth looking at closer. Since $g=8$ by construction, and $r=3$ we have $p=3$ (recall $p=\frac{g-(r-1)}{2}$). When projected from a point on the inner oval the curve is 7-gonal. This degree is [*larger*]{} than Gabard’s bound $r+p=6$! The example seems to violate Gabard’s bound $r+p$.

[*Summary.*]{}—While testing the fullness conjecture, we rather arrived to a counterexample to Gabard’s estimate $\gamma\le r+p$. We thus switch slightly of game, but try to keep in mind the fullness problem for later.

Potential counterexamples to Gabard 2006 ($\gamma\le r+p$)
----------------------------------------------------------

\[24.10.12\] The curve just discussed (337) seems a potential violation of the theorem $\gamma\le r+p$ asserted in Gabard 2006 [@Gabard_2006]. Can we solve this paradoxical situation? Either Gabard’s bound $r+p$ is false or something wrong happened. A possible explanation is that we were too cavalier when claiming $\gamma=7$; in fact the total lines pencil on curve 337 just shows $\gamma\le 7$. A priori there might be optical illusion about evaluating gonality. For instance sweeping our octic with 13 nodes by a pencil of cubics with 9 basepoints located on the nodes gives a linear series of degree $3\cdot 8-9\cdot 2=24-18=6$, rescuing the $r+p=6$ bound. Of course, it is another story to convince that such a map can be chosen total. Thus 337 still represents a severe aggression against $\gamma\le r+p$.

Similar counterexamples (be they illusory or real) can be manufactured in lower topological complexity. Starting with a configuration of 3 conics, we conserve the deep nest, but keep the maximum number of singularities in the periphery so as to lower the genus as much as possible. Keeping 7 nodes unsmoothed, but smooth away all others crossings in a sense-preserving fashion (to ensure the dividing character of the curve), we obtain the curve 215 with $r=2$ (on Fig.\[Gabard:fig\]). Its genus is $g=10-7=3$. Thus the genus of the half (complex locus split by the real one) is $p=\frac{g-(r-1)}{2}=\frac{3-1}{2}=1$. Projecting from the interior oval gives $\infty^{1}$ total maps of degree $5$, and the hasty guess is that $\gamma=5$. Since $r+p=3$, Gabard’s bound $\gamma\le r+p$ looks again corrupted.

-25pt 0

However the curve at hand (215) having $g=3$ (and being dividing), elementary knowledge of Klein’s theory prompts that the canonical map $C\to {\Bbb P}^{g-1}$, here ${\Bbb P}^2$, will exhibit the curve as a “Gürtelkurve”, i.e. a quartic with two nested ovals. Then the gonality is reevaluated as $\gamma=3$, and Gabard’s bound is vindicated again (by the rating agency!).

Another way to argue, would be to take a pencil of cubics with 7 basepoints assigned on the 7 nodes and another basepoint on the curve. The degree is then $3\cdot 6- 7\cdot 2- 1\cdot
1=18-14-1=3$. The bound $r+p$ looks rescued again. Yet some hard work is required to check total reality of a suitable pencil. Perhaps there is some conceptual argument, else one really requires tracing carefully the pencil after an educated guess of where to place the extra assigned basepoint.

It is even possible to construct a quintic with “visual” gonality exceeding $r+p$. The cooking recipe is the same as above. Start from a configuration of 2 conics and one line, keep the inner oval while maximizing the number of peripheral singularities. It results picture 214 on Fig.\[Gabard:fig\]. We see $r=2$ real circuits. The genus is $g=6-3=3$ (3 nodes must be subtracted), and thus $p=1$. The naive gonality seems to be $4$, exceeding (hence violating) Gabard’s bound $r+p=3$. Again to resolve the paradox one can either argue via the canonical map carrying the curve to a Gürtelkurve, or find a total linear series of lower degree. Here this would involve a pencil of conics through the 3 nodes plus one assigned basepoint inside the deep oval. The resulting series has degree $2\cdot 5 - 3\cdot 2- 1\cdot
1=10-6-1=3$, in agreement with the $r+p$ bound.

A drawback of figure 214 is that the 3 remaining nodes are nearly collinear, rendering nearly impossible the depiction of the conics pencil. (In reality the 3 nodes are not aligned, else the line through them cuts the quintic in 6 points.) It is convenient to consider rather a related quintic $214bis$ on Fig.\[Pencil:fig\], where the line has penetrated the inner oval (yet without destroying it). All invariants $r,p$ (as well as the naive gonality) keep the same value as on the previous example 214. The new curve makes it easier to trace a total series cut out by a pencil of conics, where the extra basepoint has been chosen most symmetrically. Each member of it has beside the 4 assigned basepoints (counting for $3\cdot 2+ 1\cdot 1=7$ intersections) 3 moving points which are permanently all real, as follows (only?) through patient inspection of the picture.

-45pt 0 -35pt0

Fig.\[Pencil:fig\] attempts to show various members of the conics pencil, as well as the 3 mobile points of the series. Those sections are depicted by the same letters e.g. 1,1,1 corresponds to the section by the ellipse invariant under symmetry about the vertical axis. Ultimately, the whole figure has to be extended by this symmetry, but this is better done mentally for not surcharging the figure. The dynamics (circulation) is quite tricky to understand, but the motion looks much accelerated (hence hard-to-follow) when the (red) curve crosses the basepoints of the pencil. This is a bit if the particle motion would be much accelerated by a gravitational black hole. Once the picture is carefully analyzed, it is evident that all 3 mobile points stay permanently real. Thus our quintic curve has gonality $\gamma\le
3$. (Gabard’s bound is rescued on this simple example.)

Perhaps similar miracles (via high-order pencils hard-to-visualize) produce for all other pseudo-counterexamples to $\gamma \le r+p$. Yet this probably requires considerable work even just for the previous curves (of Fig.\[Gabard:fig\]). In full generality, some divine act of faith is required to imbue with chimeric respect the last vestiges of truth imputable to Gabard’s result. Note that pencil of cubics are required for the examples (Fig.\[Gabard:fig\]): even our sextic with 7 nodes achieves, via conics, only gonality $2\cdot 6- 4\cdot 2=4$, not as economic as the $r+p$ bound.

[*Summary.*]{}—Two scenarios are possible: either Gabard’s bound $\gamma\le r+p$ is false (which is not quite improbable as its proof is intricate and there is an infinite menagerie of potential counterexamples), or it is true in which case it might be proved extrinsically by a highbrow extension of the last example described (Fig.\[Pencil:fig\]). This brings us to the next section, which albeit not very tangible in our fingers is perhaps technically implementable (at least at the level of Ahlfors bound $\gamma \le r+2p$).

Brill-Noether-type (extrinsic) approach to Ahlfors via total reality {#sec:Brill-Noether-approach-to-Ahlfors}
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\[26.10.12\] Neutralizing all virtual counterexamples (of the previous section) to $\gamma\le r+p$ amounts a sort of high-powered Brill-Noether theory for totally real pencils able to reprove Ahlfors theorem $\gamma \le r+2p$ (and then optionally to corroborate Gabard’s $\gamma\le r+p$) in a purely synthetic way. This section touches superficially this grandiose programme, we are quite unable to complete.

Let us be more explicit. Any smooth projective curve (or, what is the same, closed Riemann surface) embeds in ${\Bbb P}^3$. A generic projection will realize the curve (like a knot projection) as a nodal model in the plane ${\Bbb P}^2$ having at worst ordinary double points. Specializing to real (orthosymmetric) curves we get a model in the plane, on which one can hope to first prove existence of a total pencil while evaluating the least degree of such a pencil. This should amount considering adjoint curves passing through the nodes so as to lower most the degree. The procedure would be as follows.

Let $F$ be a bordered Riemann surface of invariant $(r,p)$. We consider its Schottky double $C=2F$, interpreted as a real orthosymmetric curve of genus $g=(r-1)+2p$ with $r$ real circuits. Using a generic immersion in the plane gives a model $\Gamma_m$ of the curve $C_g$ of order $m$ having $r$ real circuits, and a certain number of nodes $\delta$. For simplicity assume the nodes to be simple, though the more general situation must perhaps not be excluded. We have of course $g=\frac{(m-1)(m-2)}{2}-\delta$. Let $\Delta$ be the divisor of double points of $\Gamma_m$. (Those can occur in conjugate pairs under complex conjugation.) Consider in the complete linear system $\vert kH\vert:=\vert {\cal O}_{\Bbb
P^2}(k)\vert$ of all curves of degree $k$, a linear pencil $L$ of curves passing through the nodes $\Delta$ of $\Gamma$ (adjunction condition). The resulting series has degree $\le k\cdot m- 2\cdot
\delta$. In fact a better control must be possible. First $k$ has to be chosen large enough so that the adjunction condition is possible at all. Since $\dim \vert k H\vert=\binom{k+2}{2}-1$, the integer $k$ is chosen as the least integer such that this dimension exceeds $\delta$. Then we may have some excess permitting to assign other (simple) basepoints.

Let us be even more explicit (we work first over the complex, for simplicity). So assume given $C$ a curve of genus $g$. We look first at the canonical embedding $\varphi\colon C \to {\Bbb
P}^{g-1}$. The image curve has degree $2g-2$. We manufacture a plane model via successive projections from points chosen on the curve. This lowers the degree by one unit after each projection. We arrive ultimately at a nodal model $\Gamma_m\in {\Bbb P}^2$ of degree $m=(2g-2)-[(g-1)-2]=g+1$. Experimental study or an inspired guess suggests considering adjoint curves of degree $k=m-3$. This value is calibrated so that our $k$-tics have enough free parameters to visit all $\delta$ nodes of $\Gamma$. Indeed $$\begin{aligned}
\dim \vert k H\vert
=\binom{k+2}{2}-1=\frac{(k+2)(k+1)}{2}-1&=\frac{(m-1)(m-2)}{2}-1\cr
&\ge\frac{(m-1)(m-2)}{2}-g=\delta.\end{aligned}$$ We look at all curves of degree $k$ going through the nodes $\Delta$ of $\Gamma$. Denote $\frak d=\vert kH(-\Delta)\vert$ the corresponding linear system, and let $\varepsilon$ be its dimension. Obviously $$\varepsilon \ge \dim\vert kH\vert-\delta.$$ (In fact since nodes of an $m$-tic impose independent conditions upon adjoint curves of degree $m-3$ this is an equality. But we do not this deep fact essentially equivalent to Riemann-Roch.) Both displayed formulas show that $\varepsilon\ge 0$, and we may thus impose to our $k$-tics to pass through $\varepsilon-1$ extra points, while still moving inside a linear system of dimension $\ge 1$ (a so-called pencil). This gives a pencil $L\subset \frak
d$ of degree $$\begin{aligned}
&\le k\cdot m- 2\delta -1 \cdot (\varepsilon-1)\cr
&\le k\cdot m- 2\delta-\Bigl[\binom{k+2}{2}-1-\delta\Bigr]+1\cr
&= k\cdot m- \delta-\binom{k+2}{2}+2\cr
&= k\cdot m+ g-\binom{m-1}{2}-\binom{k+2}{2}+2 \quad \textrm{[now
recall $m=g+1$]}\cr
&= (m-3) m+ (m-1)-2\binom{m-1}{2}+2\cr
&= (m-3) m+ (m-1)-(m-1)(m-2)+2\cr
&= (m-3) m+ (m-1)\underbrace{[1-(m-2)]}_{-(m-3)}+2\cr
&= (m-3) [m-(m-1)]+2\cr
&= m-1=g.
$$ This proves that any curve of genus $g$ admits a pencil of degree $\le g$, which made basepoint-free induces a map of, eventually, lower degree. (Of course our assertion fails when $g=1$, but true otherwise granting some knowledge.) This “degree $g$” bound is a bit sharper than the usual degree $g+1$ prompted by Riemann(-Roch)’s inequality, but much weaker than the Riemann-Meis bound $[\frac{g+3}{2}]$ for the complex gonality. A natural wish is obtaining the Riemann-Meis bound via the above strategy, hoping that special configurations of $\varepsilon-1$ points on the curve impose less conditions than expected, leaving some free room for additional constraints lowering further the degree. This is essentially what Riemann was able to do (at least heuristically) via transcendental methods, and (exactly) what Brill-Noether’s theory is about at the pure algebro-geometric level. Recall, yet, that both works apparently fail satisfying modern standards, cf. e.g. Kleiman-Laksov 1972 [@Kleiman-Laksov_1972] and H.H. Martens 1967 [@Martens_Henrik_1967], where the problem was not yet solved apart via Meis’ analytic (Teichmüller-style) approach.

At this stage, starts the difficulties. The big programme would be to adapt the above trick to real orthosymmetric curves, in order to tackle Ahlfors theorem. The latter prompts the bound $g+1$ rather, but this little discrepancy should not discourage us. So in some vague sense a “real” Brill-Noether theory is required, combining probably also principles occurring in Harnack’s proof (1876 [@Harnack_1876]) of the after him named inequality.

From the real locus $\Gamma({\Bbb R})$ one shall identify deep nests, and it is favorable to choose them as the extra basepoints to ensure total reality of the pencil we are trying to construct. Then there is also a foliation on the projective plane induced by the members of the pencil. Inside each oval, the foliation must exhibit singularities (otherwise total reality is violated). In fact total reality imposes the foliation to be transverse to the real circuits. Hence if there is no singularity we would have a foliation of the disc which is impossible. Perhaps Poincaré’s index formula is also required. To be brief there is some little hope that a very careful analysis of the geometry establishes existence of a total pencil of degree $g+1=r+2p$, recovering so Ahlfors result.

This would be pure geometry (or the allied devil of algebra) without intrusion of either potential theory, neither transcendental Abelian integrals, nor even topological principles. Perhaps only elementary topological tricks are required to ensure total reality by gaining extra intersections via a continuity argument akin to Harnack’s. This offers maybe another approach to Ahlfors, yet it requires some deep patience. It looks perhaps somewhat cavernous as (extrinsic) plane curves with singularities are just a “Plato cavern”-style shadow of the full Riemannian universe.

If this dream of a synthetic proof of Ahlfors theorem is possible, then it would be nice (if possible) to boost the method at the deeper level of special groups of points to gain the sharper Riemann-Brill-Noether-Meis sharp control upon the gonality, whose real orthosymmetric pendant is expected to be the $r+p$ bound (of Gabard).

Last, I know (only through cross-citations) the work of Chaudary 1995 [@Chaudary_1995-Thesis] where a real Brill-Noether theory is developed. This probably helps clarifying the above ideas.

[*Philosophical remark.*]{}—Everybody experimented difficulties when playing with extrinsic models of Riemann surfaces. A typical instance occurs with Harnack’s inequality $r\le g+1$, whose extrinsic proof (Harnack 1876 [@Harnack_1876]) is pretty more intricate than Klein’s intrinsic version (same year 1876 [@Klein_1876]) based on Riemann’s conception of the genus. By analogy, one can predict that any synthetic programme toward Ahlfors will ineluctably share some unpleasant features of Harnack’s proof. The substance of the latter is a spontaneous creation of additional intersection points forced by topological reasons, leading to an excess violating Bézout. Arguments similar to Harnack’s might be required to ensure total reality of a well chosen pencil. Instead of being obnubilated by real loci (of both the curve and the plane), it is sometimes fruitful to move in the “complex domain” to understand better reality. A typical example is Lemme 5.2. (in Gabard 2006 [@Gabard_2006]) about unilateral divisors linearly equivalent to their conjugates. This was one of the key in my approach to Ahlfors maps. Perhaps this lemma is also relevant to the problem at hand ensuring total reality quite automatically. In the series of adjoint curves $\frak d$, one then imposes passing not through deeply nested ovals, but rather through imaginary points all located on the same half. The difficulty is of course showing existence of such a curve intersecting the fixed one only along one half (unilaterality condition), except eventually for some assigned basepoints (either real or imaginary conjugate).

Extrinsic significance of Ahlfors theorem
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\[07.11.12\] Another (less retrograde) desideratum is to explicit the extrinsic significance of Ahlfors theorem for real algebraic (immersed) plane curves. We touched this already in the Slovenian section \[Open-RS-embed-in-C2:sec\] but now a sharper idea is explored. The point is delicate to make precise and already quite implicit in my Thesis (2004 [@Gabard_2004], especially p.7 second “bullet”) plus of course in Rohlin 1978 [@Rohlin_1978] (albeit the latter may never have been aware of Ahlfors theorem). Today I discovered a certain complement which is perhaps worth presenting.

First Ahlfors theorem traduces in the following.

Any real orthosymmetric (=dividing) algebraic curve admits a totally real morphism to the line.

The half of the dividing curve is a bordered surface. By Ahlfors 1950, the latter tolerates a circle map, which Schottky-doubled gives the required total map. For another proof cf. e.g. the first half of Gabard 2006 [@Gabard_2006].

This pertains to abstract curves (equivalently Riemann surfaces) but it acquires some extra flavor when the curve becomes concrete. Of course the ontological problem of concreteness is that there are plenty of ways for an abstract object to become concrete. Thus concreteness is oft the opposite extreme of canonicalness. Arguably, there is perhaps still a preferred “Plato cavern” namely the projective plane which can be used as an ambient space where to trace all Riemann surfaces provided we accept nodal singularities. Concretely this is done via generic projections from a higher projective space (${\Bbb P}^3$ actually suffices to embed any abstract curve), and then projecting down to the plane ${\Bbb P}^2$ gives a nodal model. All this being pure synthetic geometry it transpierces matters regarding fields of definition (A. Weil’s jargon) and so adapts to the reality setting. As yet just trivialities, but now we aim interpreting synthetically the (non-trivial) Ahlfors theorem.

Starting from a real dividing curve in some projective space, suitable projections exhibit a birational model, $C$, in the plane as a nodal curve. Existence of a total morphism traduces into that of a total pencil, i.e. one all of whose member cut only real points on the curve $C$, at least as soon as they are mobile. A priori basepoints may include conjugate pairs of points. (A simple example arises when we look at the pencil of circles through 2 points. Recall that circles always pass to the so-called cyclic points at $\infty$, but this is just an affine conception).

In extrinsic terms, Ahlfors theorem takes essentially the following form.

[(IAS=Immersed Ahlfors via Kurvenscharen)]{} Given a dividing (real algebraic) curve $C$ immersed nodally in the plane ${\Bbb P}^2$. There is a totally real pencil of (auxiliary) curves of some order $k$, all of whose members cut on $C$ solely real points plus eventually imaginary conjugate pairs of basepoints.

This reduces to the basic theorem that any abstract morphism of algebraic geometry admits a concrete description in terms of ambient linear systems when the abstract object is projectively concretized. In substance this is just the spirit of Riemann (algebraic curves=Riemann surfaces) but extended to the realm of morphisms. So the required theorem is just basic algebraic geometry but I forgot all the foundations. Historically add to Riemann, certainly Cayley-Bachach, Brill-Noether, (Klein?), all the Italians, and finally Weil, Grothendieck, plus of course many others.

Now the new observation \[07.11.12\] is that we may always assume $k=1$ (in the theorem IAS) up to changing of birational nodal model. The idea is that we may first reembed the curve $C$ via the complete linear system of all curves of degree $k$ (alias Veronese embedding) in some higher space ${\Bbb P}^N$, where $N=\binom{k+2}{2}-1$. Then the image curve $C'$ is (totally) swept out by a pencil of hyperplanes corresponding to the original total pencil $L$ of $k$-tics in the plane ($k$-tics=curves of degree $k$). If we project from the base locus of the hyperplane pencil which is a linear variety of codimension 2 we arrive down again in ${\Bbb P}^2$, but now with a new model total under a pencil of lines. It seems to me that this trick works and we get the:

[(IAP=Immersed Ahlfors via lines pencils)]{} Given an abstract dividing (real algebraic) curve, there is always a nodal(ly immersed) model in the plane ${\Bbb P}^2$ which is total under a pencil of lines.

This permits to remove one of the obstruction in our discussion of the Forstnerič-Wold problem (already touched in Sec.\[Open-RS-embed-in-C2:sec\]). We now deduce the stronger assertion:

Any finite bordered Riemann surface immerses in ${\Bbb C}^2$.

Let $F$ be the bordered surface, and $C:=2F$ be its Schottky double which is real orthosymmetric. By the theorem (IAP) we find a nodal model in the plane ${\Bbb P}^2$ total under a pencil of lines. The pencil being real its unique basepoint $p$ is forced to be real. Since the allied morphism (projection) is total the fibre of an imaginary point is an unilateral divisor, i.e. confined to one half of the curve. This means that all imaginary lines through the basepoint cuts unilaterally the curve. It suffices thus to remove (from ${\Bbb P}^2({\Bbb C})$) an imaginary line through $p$ to obtain an immersed replica of $F$ in ${\Bbb C^2}$. Note that if $p$ lies on the (nodal) curve then only the open half (interior of $F$) is so embedded, but we can probably arrange this by displacing slightly the center of perspective $p$ outside the curve while conserving total reality. The net bonus is that the whole bordered surface (boundary included) is in ${\Bbb C}^2$.

Of course this is still millions of lightyears away from Forstnerič-Wold postulated embedding (for all finite bordered surfaces), yet represents already a nice application of Ahlfors. Of course the corollary is also the special (finitary) case of the famous Gunning-Narasimhan theorem (1967 [@Gunning-Narasimhan_1967]), immersing any open Riemann surface in ${\Bbb C}^2$. Maybe their immersions are proper also, whereas ours are not. Maybe the Fatou-Bieberbach trick arranges this issue always, cf. e.g. Forstnerič-Wold 2009 [@Forstneric-Wold_2009]. Anyway using the quantitative form of Ahlfors (not used as yet) one can go perhaps further, maybe saying things on the degree of the model. Note also that the viewpoint of nodal model of orthosymmetric curves affords another numerical invariant, namely:

[(quite implicit in Matildi 1945/48 [@Matildi_1945/48])]{} Given an abstract dividing real curve $C$. The least order $\delta$ of a nodal birational model of $C$ is termed (by us) the nodality of the curve $C$. Via Schottky-doubling this invariant also makes sense for finite bordered Riemann surfaces.

Projecting down to ${\Bbb P}^2$ the canonical model in ${\Bbb
P}^{g-1}$ of a curve of genus $g$, we get a nodal model of degree $g+1=(2g-2)-[(g-1)-2]$ (each projection from a point on the curve decreases the degree by one unit). Hence $\delta \le g+1$.

If the theorem (IAP) is correct, one could also try to define the linear gonality of a bordered surface (or the allied orthosymmetric double) as the least degree of a nodal plane model totally real under a pencil of lines. This gives perhaps yet another invariant $\lambda$, which seems to satisfy $\gamma\le
\lambda+1$.

Another dream of longstanding (Gabard’s Thesis 2004 [@Gabard_2004]) is whether Ahlfors’ theorem implies Rohlin’s inequality $r\ge m/2$ for a smooth dividing curve of order $m$. If such a curve $C=C_m$ is total under a pencil of lines, then sweeping out the curve by the pencil gives collections of $m$ real points. When rotating the line around the basepoint, those $m$ points never enter in collision (else smoothness is violated), nor do they disappear in the imaginary locus (else total reality is violated). After a 180 degree rotation already, the line returns to its initial position while the group of $m$ points recover its initial position giving raise to a monodromy permutation. Total reality forces each circuit of the curve $C$ to be transverse to the foliation underlying the pencil of lines. It follows that the monodromy transformation is an involution (order 2) and we deduce:

[(Rohlin essentially)]{} Let $C_m$ be a smooth real curve of order $m$ totally real under a pencil of line. Then the real locus $C_m({\Bbb R})$ consists of a deep nest of $m/2$ ovals when $m$ is even, and if $m$ is odd there is as usual one pseudoline and ovals distribute in a nest of depth $(m-1)/2$.

In particular Rohlin’s inequality $r\ge m/2$ follows in this special case where total reality is given by a pencil of lines. The general case of Rohlin still appeals to some formidable work, but perhaps may be derived via a linear pencil on a nodal model. Alas we are unable to complete this project.

Let us however try to be more explicit. Given a smooth dividing $C_m$. Let $L$ be a total pencil of $k$-tics given by Ahlfors (theorem IAS). Then one can either try to study directly the corresponding foliation appealing to Poincaré’s index formula, and hope to mimic the above argument. Alternatively one can try to use the reembedding trick, where we use another model total under a pencil of lines. Now on the new nodal model of degree say $\lambda$, we apply the same sweeping procedure. We see on one initial line $L_0$ (assumed generic, i.e. avoiding the nodes) $\lambda$ points all real. When rotating by a half-twist the line we see groups of $\lambda$ points which now may cross themselves, but one can still assign a monodromy permutation. Naively any point finishes its trajectory on the other side of the basepoint (alas this makes no sense since a projective real line is a circle not disconnected by a puncture). The number of real circuits $r$ of the curve $C$ is equal to the number of cycles of the monodromy permutation, but a priori the latter number can be very low since crossings are permitted. (Imagine e.g. a spiral which after several growing revolution times closes up to form a single circuit.) Note that we do not yet exploited the smoothness hypothesis of the original model $C_m$. A naive way to exploit this is via the complex gonality $\gamma_{\Bbb C}$. We have indeed $m-1=\gamma_{\Bbb C}\le \gamma$ ($C$ being smooth). On the other hand $\gamma\le \lambda-1$. Hence $m\le \lambda$. This is interesting yet certainly not enough to conclude Rohlin’s inequality. So we give up the question for the moment.

The gonality spectrum
---------------------

An idea perhaps worth exploring is to enrich the gonality sequence (Definition \[def:gonality-sequence\]) into what could be called the [*gonality spectrum*]{}. This would just be the former weighted by the dimension of the space of all circle maps having the prescribed degree.

As we already observed earlier (hyperelliptic examples) it seems that when a surface has a very low gonality then it “somnolates” without creating new gonalities. Thus more generally, the intuition behind this spectrum invariant would be a conservation law somewhat akin to Gauss-Bonnet: whatever the Riemannian incarnation of a topological surface the curvatura integra keeps constant value equal to the Euler characteristic ($ \int_F K
d\omega= 2\pi \chi(F)$).

Of course experiments requires to be made (using e.g. the specimens on Fig.\[Coppens:fig\]). Alas I had not presently the time to do serious investigations about this spectrum. It seems also expectable that from a certain range on, the spectrum is independent from the conformal structure. (At least so is the case for the gonality sequence which is always full after $r+2p$.)

Of course some convention is required, probably consider only maps up to automorphisms of the disc.

Example the only example where the spectrum is very easy to describe is the disc: in this case the $\gamma$-sequence is full starting from 1, and there is essentially only one map of degree one (the Riemann map). Given any unilateral group $D$ of $d$ points in the disc, thought of as the north hemisphere of the Riemann sphere the pencil through $D$ and its complex conjugate $D^{\sigma}$ induces a totally real map. (cf. Lemme 5.2 in Gabard 2006 [@Gabard_2006]). Conversely, given the map its fibre over 0 gives an unilateral divisor, which up to a range automorphism may be assumed to contain 0. Normalizing by a rotation there are thus the map depends upon $2d-3$ real constants. (Make this more precise…). Such maps are (in the complex function literature) often called finite Blaschke products.

Once the setting is well understood, this gonality spectrum encodes valuable information upon all circle maps. Of course one perhaps still want to know more; e.g. to understand the incidence relation among the varied maps, especially how high-degree maps may degenerate to lower degree ones. Fig.\[Coppens:fig\] shows some interesting examples. Considering e.g. picture 313 we see that both maps of degree 3 are limit of maps of degree 4 (actually can be connected by such), and both of them are also limits of maps of degree 5.

Looking at picture 112 (again on Fig.\[Coppens:fig\]) we see that the unique (total) map of degree 2 is also the limit of maps of degrees 3 and 4. The gonality sequence $2,3,4,\dots$ can be enriched by weighting by dimensions to get $2_0,3_1, 4_2, \dots$. Beware that probably there are other maps of degree 4 than those visible on the picture as linear projection, namely the unique 2-gonal map post-composed by circle maps of degree 2 from the disc to itself. Our guess is that such Blaschke maps may degenerate to their originator (the hyperelliptic projection) but not to maps of degree 3.

More lowbrow counterexamples to $\gamma\le r+p$
-----------------------------------------------

\[27.10.12\] We now pursue the project of multiplying and diminishing further the order of virtual counterexamples to Gabard’s estimate $\gamma\le r+p$ (cf. Fig.\[Gabard:fig\] and Fig.\[Pencil:fig\]). There we found curves (via an uniform recipe) seemingly violating the gonality upper bound $r+p$. The simplest example had order 5, but it is easy to get examples of order 4. The game is again to depict total pencils vindicating Gabard’s bound. Albeit very modest corroboration of the bound, we found instructive to visualize the corresponding total pencils.

First remind the general recipe: to manufacture an (at least virtual) counterexample to $\gamma\le r+p$, we leave tranquil the inner oval but maximize the number of singularities, so as to lower the genus $g=(r-1)+2p$, and hence $(r,p)$. Having left quiet the inner oval the virtual gonality via linear projection is one less than the degree, but $r+p$ may go lower down this value.

We first consider a configuration of order 5 consisting of 2 conics plus one line, see picture 304 below (Fig.\[F304:fig\]). Smoothing it as dictated by orientations while keeping unsmoothed the dashed circles gives a curve with $r=3$ real circuits of genus $g=6-4=2$. Hence $p=\frac{g-(r-1)}{2}=0$. The virtual gonality is $\gamma^\ast=4$ (projection from the inner oval). This seems to violate $\gamma\le r+p=3$. Looking at the pencil of conics through the 4 nodes gives a series of degree $2\cdot 5-4\cdot 2=10-8=2$. This violates the trivial bound $r\le \gamma$, but of course this pencil is not total: e.g. the conic consisting of the 2 horizontal (or better oblique) lines misses the inner oval. Assigning instead one of the 4 basepoints on the inner oval gives a pencil of degree $3$, which is claimed to be total.

Totality of the morphism requires examining (patiently) that each conic of the pencil cuts only real points on the quintic $C_5$. This is depicted on the large part of Fig.\[F304:fig\], where each triad of moving points of the series are labelled by triples $1,1,1$, then $2,2,2$, etc. Let us start from the conic consisting of the oblique line through $1,1$, plus the horizontal line. The latter cut the red pseudoline at infinity. This pair of lines deforms to a hyperbola cutting the triad $2,2,2$. This hyperbola is in turn pinched toward a pair of lines cutting the group $3,3,3$, etc, up to $7,7,7$. From here on, things becomes harder to visualize. (Alas our picture is not optimally designed.) The conic of the pencil now becomes very close to the primitive conic involved in the generation of the quintic $C_5$ via small perturbation. The net effect is that points on the green branch nearly “osculated” by the primitive ellipse are (violently) accelerated (like in CERN’s particles accelerator). At this stage it is quite delicate to make a consistent picture, but total reality seems to work: all particles stay real during the motion without disappearing as ghost in the imaginary locus (as conjugate pairs of points under Galois).

We promised a similar example of degree 4; this will be pictured later (Sec.\[sec:degree-four\]), being now sidetracked to another topic which looks more exciting.

Some crazy ideas about gravitation and unification of forces {#sec:gravitation}
============================================================

From gravitation to electrodynamics {#sec:electrodynamics}
-----------------------------------

Now we arrive at the following crazy interpretation (discovered the 27.10.12 at ca. 13h58). It would be nice if there is some relation of the Ahlfors maps with periodic solutions to the $n$ body problem in gravitation (celestial mechanics). The 4 basepoints of Fig.\[F304:fig\] may be thought of as supermassive black-holes, so massive that there is no interaction between them (imagine purely static objects lying in different sheets of the multiverse). Dually, the moving points of the linear system are imagined as massless microparticles (electrons, or better photons). There is also no gravitational interaction between them. Thus the sole interactions reigning are those between black holes and photons. It is also imagined that a photon can traverse a black-hole (without captivation).

As a wild speculation, the trajectories described by the 3 photons on Fig.\[F304:fig\] may satisfy exactly Newton’s law of gravitation. In particular the full trajectory would be the real locus of an algebraic curve! This would of course be a wide extension of Kepler’s law (on the rôle of conic sections in the simplest case of one sun and one planet).

If this is true we see a deep connection between Klein’s orthosymmetric curves, Ahlfors maps of conformal geometry and the totally real circulations positing periodic stable motions along circuits of an orthosymmetric curve. Exaggerating a bit this should explain the ultimate constitution of matter (and its relative stability) not via knots (as Lord Kelvin desired via Helmholtz vortices) but via bordered Riemann surfaces (probably quite ubiquitous already in the so-called string theory).

Note that our basic experiment (with Fig.\[F304:fig\]) is—as far as speed of motion is concerned—quite in line with this interpretation.

Let us look at one of the simplest example of orthosymmetric curve, namely the (Zeuthen-Klein) Gürtelkurve (aka [*courbe annulaire*]{}). This is a quartic with two nested ovals arising by smoothing two transverse ellipses having 4 intersections. The picture is given below (Fig.\[FGuert:fig\]). One can convincingly argue that the shapes of trajectories (especially the outer oval) are unlikely to be gravitational orbits. It seems that some hidden force repulses the particles (labelled 1 on the figure). Invoking some other (electric) force effecting repulsion between particles, then the trajectories of the Gürtelkurve look again physically tolerable. Thus the “physical” model should include two types of interactions: gravitational and electromagnetic.

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Of course one can drag the position of the sun while still having a totally real pencil. This gives the next figure (Fig.\[FGuert2:fig\]). Note that we did not changed the curve, yet it is still plausible that for suitable initial conditions (velocity vectors) the orbits of our 4 bodies follows exactly the same quartic curve.

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We arrive at the following metatheorem \[14h57\]:

[(Kepler generalized?)]{} \[metatheorem:thm\] Given any orthosymmetric real (algebraic) curve embedded (or immersed) in the Euclid plane ${\Bbb R^2}$ and a totally real pencil (existence ensured by Ahlfors theorem). There exists initial conditions (velocity vectors) such that the trajectories of particles obeying the inverse square law of Newtonian attraction resp. Coulombian repulsion match exactly the real circuits (“ovals”) of the given real algebraic curve. Further the dynamics (speed of motion) is dictated by the pencil. In particular there is plenty of periodic solutions to the $n$-body problem, essentially one for each such curve.

How to prove this? Philosophically, algebraicity might be not so surprising: recall Laplace’s potential-theoretic interpretation of Newton, and from Laplace there is just one step to Riemann, hence to Klein. The miracle should be essentially akin to Riemann’s existence theorem prompting any closed Riemann surface (an a priori completely fluid object) to rigidify canonically as an algebraic curve. Even if true the metatheorem is quite modest because in practice (meteorites, apocalyptic black holes scenarios, etc.) one is given the initial conditions and the goal is to predict the future evolution of the system. Here in contrast, we know in advance the trajectories (hence the destiny) while claiming existence of initial conditions compatible with the orbital structure. Generally, integrating the differential equations governing some motion, we meet a highly complex dynamical system subjected to the paradigm of chaotic determinism à la Poincaré. Note that a Euclidean model of the projective curve is required to give sense to Newton’s inverse square law.

Several questions naturally occurs assuming the truth of the metatheorem. The theorem affords plenty of periodic motions. Essentially we obtain as many periodic motions as there are real orthosymmetric curves. Even more than that, one requires an Ahlfors circle map (equivalently a totally real morphism à la Klein-Teichmüller). A first naive question is: do this recipe exhausts all periodic motions? Certainly not, try Euler and Lagrange’s periodic motions. Roughly all algebraic motions are periodic, but the converse has no chance to be true.

Observationally, Fig.\[FGuert2:fig\] looks anomalous because the series 1,2,3,4 closest to the sun looks much slowed down, whereas we are accustomed (Kepler) to rapid motions near a massive star. One requires perhaps a third type of interaction, say the [*strong interaction*]{}, to explain this. Namely both particles the proton and the electron are of a dualistic nature, hence they tend to “love” themselves like partners staying close together over a long period of time. This third force would have the net effect of diminishing the real speed by a factor proportional to the (squared?) distance separating the bodies. What is then the fourth force, alias [*weak interaction*]{} in contemporary physics? Maybe none is required in our model? Perhaps dually, particles of the same nature (namely electrons) dislike themselves like competitors and the [*weak force*]{} just produces some acceleration of the motion when they are in close vicinity. Visually this behavior is perhaps observed near the groups labelled 2,3 on the top part of Fig.\[FGuert2:fig\].

We have now a model with 4 fundamental forces. One must of course still define time. This would, on our example, just be the angular parameter of the pencil. Presumably the metatheorem should take into account these two extra forces, becoming somewhat sophisticated, yet probably still completely deterministic and hopefully reasonably easy to integrate. The miracle would be that it admits dividing (=orthosymmetric) real curves as periodic orbits. Of course to relativize, one can do similar games with real diasymmetric curves, but then there is no total reality prompted by Ahlfors theorem and particles sometimes disappear in the imaginary locus. We leave to the reader’s imagination appropriate physical interpretations (ghost particles, anti-matter, etc.)

Perhaps there is a more elementary way to explain slowness of the motion near the star (without appealing to the exotic forces at the subatomic level). Recall Kepler’s law in the elliptic case, that identic sectorial areas are swept out during the same amount of time. This suggests that the time parameter is not the angular parameter but the areal one. Of course one gets other troubles since the distant electron is supposed to move synchronously with the one closer to the proton (cf. Fig.\[FGuert:fig\]).

Some little objections
----------------------

\[28.10.12\] Another objection to our metatheorem (\[metatheorem:thm\]) is the following one. Assume the given orthosymmetric curve to be of the simplest stock, namely a line swept out by a total pencil of lines. Then one must assume that there is no forces between the two bodies to explain the rectilinear motion.

A more serious objection arises when $C$ is an ellipse swept out by a total pencil of lines through the middle of both foci. If all (four) fundamental forces involved satisfy the inverse square law, then so does the resulting force. Hence all interactions reduce to a single one which is attractive (to get an elliptic trajectory). However according to Kepler the orbit must be an ellipse with the sun located at one of the foci. Hence our geometric model where the basepoint of the total pencil lies at the center of the ellipse is not physically relevant.

This example suggests that the metatheorem requires corrections. Maybe one is only given in advance the orthosymmetric curve but not the total pencil, while the metatheorem states existence of a pencil physically observable. For an ellipse we would only be allowed to take pencil of lines through one of both foci; if the ellipse degenerates to a circle only the center would be permissible. \[30.12.12\] At this stage it might be relevant to remind that there is a vast theory of foci for high-order algebraic curves, due it seems to Plücker first and then Siebeck 1864 [@Siebeck_1864], etc. cf. e.g. Casas-Alvero 2013 [@Casas-Alvero_2013].

Of course this Kepler obstruction should not preclude physical systems obeying more complicated interactions laws with say several fundamental forces, maybe not all subsumed to the inverse square law. Such could validate exotic orbital structures, e.g. an ellipse with a sun at its center, as physically reasonable.

\[28.12.12\] Another possible objection comes from the following curve Fig.\[F324bis:fig\]. This possesses a total pencil of lines, yet some particles do not repulse, rather crossing themselves unsensitive of each other. On relabelling the particles one can posit a repulsion acting rather as a bounce of billiard balls (elastic shock). Another explanation could involve some quantumchromodynamics like assigning spins to the electrons neutralizing interaction between some of them.

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Back to insignificant geometry
------------------------------

Let us now leave such complex modelling question, to contemplate more complicated systems arising from other curves than the Gürtelkurve, especially some of higher order. First staying of order 4 there is, dual to the Gürtelkurve, the curve arising by reversing orientation of one of the ellipses (cf. arrows on Fig.\[F4oval:fig\]). This gives a quartic with 4 ovals when smoothing compatibly with the prescribed orientations. A total pencil arises from all conics through 4 basepoints distributed inside the ovals (Fig.\[F4oval:fig\]).

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Initially the point $a_1$ animated by a suitable horizontal velocity vector is mostly subjected to the attraction of the nearby star (=upper basepoints of the conics pencil). If $a_1$ and this star were to be alone in the universe, $a_1$’s orbit would be close to the dashed ellipse of “vertical eccentricity”, provided the upper star coincides with the focus of this ellipse. Yet in reality, as the body $a_1$ arrives near position $d_1$ and meanwhile body $a_8$ reached position $d_8$, electric repulsion is becoming predominant causing a (finally violent) deviation from the elliptic trajectory.

Instead of appealing to gravitation one can just imagine the basepoints (alias “stars” previously) as positively charged protons, the whole system reducing to an electrodynamical one obeying only Coulomb’s law of attraction resp. repulsion. The fixed protons would however not repulse, maintaining their fixed positions due to some nuclear cohesion (strong/weak forces).

It is easy to produce examples of higher topological complexity via curves of higher orders. Instead of starting with two ellipses, take three of them and smooth the configuration in a sense-preserving way to get Fig.\[Fsextic:fig\].

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Reversing orientation of one of the ellipses (say that with horizontal major axis) gives the more interesting Fig.\[Fsext2:fig\] requiring a pencil of conics to exhibit total reality.

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Again we use the same labelling as before, namely the first (cyan=pale blue colored) conic consisting of the vertical and horizontal lines cuts on the sextic $C_6$ the group of points labelled $a_1,\dots, a_{12}$, all of them being real. Moving clockwise from the top, a subsequent conic (blue colored) cuts the series denoted $b_i$, etc. One checks easily all conics of the pencil to cut only real points on the $C_6$. Looking at the corresponding dynamical process, we note that $a_1$ is first repulsed against $a_2$ being rejected as far as $c_1$, then attraction of the 4 protons (mostly the North and East one) track back the orbit to position $d_1$ where a repulsion against $d_{12}$ takes place, deflecting again the orbit along the way of the North proton, but then vicinity of $d_2$ causes another repulsion towards $e_1$ and $f_1$, which is finally gently repulsed by $a_{11}$. etc. The sequel of the story reproduces symmetrically.

It is now fairly evident how to construct similar dynamical systems of ever increasing complexity. It may be observed that the totally real map induced by the pencil gives a circle map of degree 12. Now the topological invariants are $r=5$ and $g=10$. Hence the half-genus is $p=\frac{g-(r-1)}{2}=3$. Hence this map has degree exceeding Ahlfors bound $r+2p=11(=g+1)$. However a parietal degeneration of the 4 basepoints against the ovals immediately enclosing them (cf. “squigarrows” on Fig.\[Fsext2:fig\]) exhibits a total map of degree $2\cdot
6-4\cdot 1=12-4=8$. This is actually in accordance with the $r+p$ bound predicted in Gabard 2006 [@Gabard_2006].

It is tempting to consider the (mildly singular) foliation induced by the pencil (of conics). It seems clear from the picture that there is a relation between the sum of Poincaré indices extended to the interior of an oval and the number of points circulating on the oval. Observe also that the foliation is transverse to the boundary of the disc bounding the oval. This property is general and follows at once from the fact that totally real maps lack real ramification points. Using Ahlfors total reality paradigm combined maybe with Poincaré’s index formula we suspect that some old (and perhaps new?) information on the topology of real plane (dividing) curves can be re-derived. In particular we suspect that it must be possible to recover Rohlin’s inequality. This states $r\ge m/2$, i.e. any smooth dividing plane curve of order $m$ has at least so many circuits as the half value of its order. This is a fantastic project, but we leave it aside for now. \[vague details p.32 of hand-notes\].

\[08.11.12\] Another highbrow (yet poorly explored) application of Ahlfors theorem was sketched in Gabard’s Thesis (2004 [@Gabard_2004 p.7]). This was an answer to Wilson’s question (1978 [@Wilson_1978 p.67]) on deciding the dividing character of a plane curve by sole inspection of its real locus. Here again Ahlfors theorem affords an answer: a real curve is dividing iff it admits a total pencil (with possibly imaginary conjugate basepoints). Yet it must be admitted that the answer, albeit perfectly geometric, has probably little algorithmic value unless complemented by further insights. Of course another question is to decide the dividing character from the sole data of a ternary form (homogeneous polynomial in 3 variables with real coefficients). The simplest case of Wilson’s question is that of a deep nest, i.e. a smooth curve $C_m$ of say even degree $m=2k$ with a completely nested collection of $k$ ovals. Then linear projection from a point on the deepest oval is total of degree $m-1$. Since the complex gonality is also $m-1$, we deduce that the gonality $\gamma$ is also $m-1$. On the other hand the topological invariants are $r=k$ and $g=\frac{(m-1)(m-2)}{2}$. Hence in this case Ahlfors bound $r+2p=g+1$ is strongly beaten by the gonality $\gamma=m-1<\!\!<g+1=[1+2+3+\dots+(m-2)]+1$. Gabard’s bound $r+p$ is also much greater than the exact $\gamma=m-1$; indeed $r+p$ is nothing but the mean value of $r$ and $g+1$ and in the case at hand the former is $m/2$ but the latter is quadratic in $m$.

\[29.10.12\] We consider next an octic (Fig.\[Foctic:fig\]) arising from a sense-preserving perturbation of 4 ellipses rotated by 45 degrees. Of course if all ellipses are oriented clockwise we get a nest of depth 4 and accordingly a total pencil of lines through the innermost oval. Here instead, we reverse some orientations to create 16 ovals and no nesting (cf. black curve on Fig.\[Foctic:fig\]). The theorem of Ahlfors predicts existence of a total pencil. The general principle is to impose basepoints inside the deepest ovals, hence the desired pencil must have degree 4. At this stage depiction can be a fairly difficult artform (reminiscent of gothical “rosaces”= rosewindows). Our trick was to use a ground ellipse of pretty large eccentricity so that oblique line of (angular) slope different from $\pi/4$ (the green and lilac colored ones) also passes through the deep nests. Of course this trick is not supposed to affect the generality of the method (i.e. Ahlfors theorem) but just intended to simplify the artwork!

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As to the arithmetics, recall that (plane) quartics depends upon $\binom{4+2}{2}-1$ parameters (coefficients counting), hence one is free to assign 13 basepoints. On the other hand, our dividing octic $C_8$ has genus $g=\frac{(m-1)(m-2)}{2}=\frac{7\cdot
6}{2}=21$ and $r=16$ ovals, thus the genus of the half (semi Riemann surface) is $p=\frac{g-(r-1)}{2}=3$. Imagine now that among all 16 basepoints of the pencil 13 moves against the ovals, then a series of (reduced) degree $4\cdot 8- 13\cdot 1=32-13=19$ is obtained. This matches with the $r+p$ bound on the degree of circle maps predicted in Gabard 2006 [@Gabard_2006]. Geometrically it is pleasant to observe that certain members of the pencil are Gürtelkurven (see the lilac-colored curve). Those are not connected. Hence total reality of a pencil is not necessarily allied to connectedness of the auxiliary curves. For the fun of depiction, one can increase the number of curves of the pencil while sweeping out more and more of the full color spectrum, creating a sort of rainbow effect (cf. Fig.\[Foctic2:fig\]).

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At this stage one gets the impression that the theory (or rather the pictures) works only for highly symmetric patterns. However the strength of Ahlfors result lies in its universal validness for all curves regardless of symmetry. This imbues some suitable respect plus a certain feeling of vertigo about the whole Ahlfors result.

Of course there is another possible orthosymmetric smoothing of our configuration of 4 ellipses. This is given by reversing one of the orientations of the ellipses, and we obtain the black-traced curve on Fig.\[Foctic3:fig\]. This times there is only 4 deep nests and a pencil of conics suffices to exhibit total reality.

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As to arithmetic matters, this octic has still $g=21$ but now only $r=6$ ovals. Hence the semi-genus $p=\frac{g-(r-1)}{2}=8$. Dragging the 4 basepoints against the deep ovals gives a total map of degree $2\cdot 8-4\cdot 1=16-4=12$. This is more economical that the $r+p$ bound, here equal to $14$.

Finally there is yet another smoothing of our 4 ellipses producing Fig.\[Foctic4:fig\] with 4 nests of depth 2. A pencil of conics suffices to show total reality.

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Regarding the topological invariants we have $r=8$, hence $p=7$. As before there is a total map of degree 12 (via parietal degeneration), which is better that Gabard’s bound $r+p=15$. Naively this relative improvement over the previous example (in comparison to the $r+p$ bound) could be explainable by the higher symmetry of the new curve probably reflecting a further particularization of the “moduli”. (Recall that if one believes Gabard 2006 [@Gabard_2006] and especially Coppens 2011 [@Coppens_2011] a bordered surface of type $(r,p)$ has generically gonality $\gamma =r+p$.)

Note yet that our total pencil of conics persists for any octic with 4 nests of depth 2, hence the symmetry of the pattern can be greatly damaged by large deformation of the coefficients without affecting the (estimated) gonality. So we certainly have the:

Any octic curve with $4$ nests of depth $2$ has gonality $\gamma\le 12$ (and presumably not lower, yet this remains to be elucidated).

Having clearly exhausted the smoothing options of our 4 ellipses, one is somehow disappointed that pencils of cubics were not yet required. Looking on p.7 of my Thesis [@Gabard_2004] I rediscover a simple such example involving only a sextic. Let me reproduce this with the rainbow technology. We start now from a configuration of 3 ellipses one of which is a circle and get Fig.\[Fcubic:fig\].

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The sextic has $g=10$ and $r=9$ (hence pre-maximal amond dividing curves), and thus $p=1$. Cubics depends on $\binom{3+2}{2}-1=10-1=9$ parameters, hence 8 basepoints may be freely assigned. Pushing them along ovals gives a total map of degree $3\cdot 6-8\cdot 1=10$. This matches with Gabard’s bound $r+p$, hence the curve should be considered has having general moduli. Of course if the smoothing is done very symmetrically and if moreover we play with the radius of the initial circle, we can perhaps arrange that all 9 basepoints lands on the sextic curve in which case the gonality would lover to 9 the minimum value (recall $r\le \gamma$).

Starting from the above sextic, one can perform a large deformation of the coefficients staying inside the space of all smooth sextic curves. The real locus picture may then undergo drastic change of shape yet its topological type keeps unaltered and so in particular the orthosymmetric character. It is not clear anymore that our simple minded pencil of cubics (spanned by 2 pairs of 3 lines) suffices to exhibit total reality. This amounts essentially to the claim that for any 8 basepoints distributed among the ovals then the ninth basepoint luckily falls into the remaining one. This luckiness phenomenon becomes even more hazardous when it comes to vindicate Gabard’s bound by a synthetical procedure. The latter seems equivalent to the claim that given any such curve (orthosymmetric with 9 non-nested ovals it is always possible to choose 8 points one on each oval) so that the pencil through them creates an extra basepoint inside the remaining oval. This lucky-stroke phenomenon should perhaps be further explored either as an application of the $r+p$ bound or as a way to disprove it.

\[08.11.12\] Let us fail to be more specific as follows. Remember first that a real sextic curve with 9 unnested ovals needs not to be dividing, cf. e.g. Gabard’s Thesis 2004 [@Gabard_2004] p.8, but this is of course well-known since at least the Rohlin-Fiedler era, e.g. Rohlin 1978 [@Rohlin_1978]. Second it is not even clear a priori that the conditions “dividing plus 9 unnested ovals” specifies a unique rigid-isotopy type of curves, i.e. a unique chamber in the space of all smooth sextics. This is a projective space of dimension $\binom{6+2}{2}-1=28-1=27$ parcelled into chambers by the discriminant hypersurface of degree $3(m-1)^2=3
\cdot 5^2=75$. (Inserted \[24.01.13\]: However this is true by a deep result of Nikulin 1979 [@Nikulin_1979/80].)

(very hypothetical!!) Any dividing sextic with $9$ unnested ovals admits a total pencil of cubics with $8$ basepoints on the sextic and the $9$th basepoint inside the remaining oval.

(pseudo-proof!) Since the curve is dividing we know by Ahlfors that there is a total pencil. We have very poor control on the degree of the curves of the pencil. We only know Ahlfors bound $r+2p=g+1=11$, Gabard’s one $r+p=10$ and the complex gonality $\gamma_{\Bbb C}=5$ which is completely useless. Stronger information comes from the trivial bound $r\le \gamma$. So the gonality $\gamma$ is fairly well squeezed as $9=r\le \gamma \le
r+p=10$. A priori a least degree total map could be given by a pencil of quartics. Then the degree could be as low as $4\cdot
6-16=24-16=8$; for quintics as low as $5 \cdot 6-25=5$; for sextics as low as $6\cdot 6-36=0$; septics $7\cdot 6-49=-7$; $k$-tics $k\cdot 6- k^2$ highly negative! Hence we have virtually no control on the degree of (members of) a total pencil, despite the bounds on the degree of the abstract total map. Let us thus shamefully postulate that the pencil in question can be chosen among cubics. For foliated reasons it is clear that the nine basepoints (elliptic points or “foyers” of Poincaré index $+1$) must be surjectively distributed among the 9 ovals. Indeed the total pencil is transverse to the real circuits and the disc bounding an oval cannot be foliated transversely (Euler-Poincaré obstruction). Hence we have the:

All basepoints of a total cubics pencil on a smooth sextic with $9$ unnested ovals are real, distinct, and surjectively(=equitably) distributed between the $9$ ovals (either in their insides or their periphery).

Applying the parietal degeneration trick we can take any 8 of the basepoints and drag them to the ovals. During the process we get new pencils (of possibly jumping dimension?) while the 9th basepoint could a priori escape its enclosing oval. The difficulty looks so insurmountable that we have to abort the project.

In fact the following principle is worth noticing. It gives a basic lower bound on the degree of total pencils, yet as we saw the real difficulty is rather upper bounds! As a matter of annoying nomenclature crash, note that the degree of the pencil is not that of the allied map but that of its constituting curves, so we should perhaps rather speak of the order of a (total) pencil.

[(Poincaré-style lower bound on the order of total pencils)]{}\[Poincare-lower-bound\] Given a (smooth) (dividing) plane curve with a total pencil of $k$-tics with $D$ many deepest ovals (i.e. the minimal elements of the nesting ordered structure). Then $D\le k^2$ or $k\ge \sqrt{D}$.

Each deep oval must enclose at least one singularity of the foliation. Remember that the latter is transverse to the curve by total reality. Poincaré’s index formula (1882/85) says that the sum of all indices equates the Euler characteristic. Applied to the disc bounding a deepest oval this forces the latter to encloses at least one singularity of index +1. Warning: one must explain why the disc could not be foliated by say two singularity of index $1/2$, so-called thorn singularities. The pencil has at most $k^2$ singularities of the foyer type (index=+1) materialized by the basepoints. Thus $D\le k^2$. Indeed for each deepest oval chose one foyer inside it. We get a map from the set of deepest oval to that of basepoints, which is injective since the deepest ovals are disjoint at least for a smooth curve. Try to clarify if smoothness is really required as a hypothesis!

\[30.10.12\] Let us look at another intriguing example. Start again with 2 ellipses invariant under rotation by 90 degrees, and add a concentric circle as the dashed one on Fig.\[Fcubic:fig\], but shrink its radius slightly beyond the critical radius where the circle passes through the 4 intersections of the 2 ellipses. Smoothing this configuration along our choice of arrows gives Fig.\[FcubicA:fig\]: a sextic with $r=9$ ovals one of them enclosing all others.

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The picture has the annoying property that ovals are pretty small, challenging a bit the visual perception of homo habilis. Since the curve is dividing, Ahlfors theorem predicts the existence of a total map. It is evident that no pencil of lines, nor of conics, is total. (This is either optically clear or deduced from Poincaré’s bound $k\ge \sqrt{D}=\sqrt{8}=2.828\dots$, i.e. Lemma \[Poincare-lower-bound\]). The 8 deep ovals prompts seeking among pencil of cubics. Of course we may just assign 8 basepoints inside those deep ovals and hope for total reality. Yet to manufacture a concrete picture it is natural to assign basepoints in the most symmetric way. Once this is done one try to identify special singular curves passing through the 8 points. We find 4 degenerate cubics consisting of a line plus a conic (cf. colored curves on Fig.\[FcubicA:fig\]). Once those are detected it is an easy matter to interpolate between them (by continuity) to trace a qualitative picture of the pencil (Fig.\[Fcubic3:fig\]).

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This archipelago sextic $C_6$ has $g=10$ as usual, and $r=9$, thus $p=1$. The total pencil can be lowered to degree $3\cdot 6- 8\cdot
1=10$, as predicted by the $r+p$ bound.

\[08.11.12\] Again several questions poses themselves naturally. (The sequel uses some jargon of Rohlin 1978 [@Rohlin_1978], for instance the [*real scheme*]{} of a smooth plane real curve is the isotopy class of the embedding of its real locus in the real projective plane):

\(1) Is any sextic $C_6$ belonging to the real scheme of the archipelago (i.e. 8 unnested ovals altogether surrounded by an outer oval) of dividing type? The answer is probably known to Rohlin and his students, especially if there is a nondividing counterexample? \[Update 24.01.13: yes there is one and this was well-known at least since Rohlin 1978 [@Rohlin_1978], yet his article was far from explicit when it comes to constructions. Personally I understood this point only after reading Marin 1979 [@Marin_1979], compare our Fig.\[GudHilbMarin:fig\] much below (virtually copied from Marin), which is a (clever) variant of Hilbert’s method of vibration.\]

Rohlin distinguishes real schemes as definite or indefinite depending on whether all its representatives belong to the same type or not, w.r.t. Klein’s dichotomy (ortho- vs. diasymmetric). (cf. Rohlin 1978 [@Rohlin_1978])

\(2) What is the exact gonalities occurring in this archipelago scheme (of course restricting attention to dividing models in case the scheme is indefinite)? If we believe in Gabard’s bound $\gamma\le r+p$, we have $9=r\le \gamma \le r+p=10$.

Perhaps answers are to be searched along the following direction. Maybe it is true that for any 8 basepoints (injectively) distributed in the 8 deepest ovals the corresponding cubics pencil is total. On counting intersections, we get roughly $8\cdot 2=16$ many coming from the 8 deep ovals and the outer oval should also contributes for 2 intersections. This is at least evident if the real part of the cubics are connected since the real circuit of each such cubic has to go at “infinity” (in the sense of moving outside the outer oval, for otherwise it would be contractible inside the bounding disc of the latter, whereas we know the cubic circuit to realize an “odd” nontrivial class in the fundamental group $\pi_1({\Bbb R}P^2)$ or just the allied homology). On the other hand, the cubic circuit must also visit the 8 assigned basepoint inside the outer oval, and so is forced to intercepts the latter. We arrive at a total of 18 real intersections, the maximum permissible by Bézout ($3\cdot 6=18$). Total reality would follow.

I remind vaguely of a standard result claiming that for a generic collection of 8 points there is a pencil of rational (hence connected) cubic interpolating them. (Cf. e.g. Kharlamov-Degtyarev survey ca. 2002). Now if all this is true, the archipelago scheme is dividing, and any such curve admits plenty of total cubics pencil of degree $3\cdot 6- 8\cdot 1=10$ (essentially one for each selection of 8 points on the deep ovals). It seems however hard to lower the gonality $\gamma$ up to the absolute minimum $r=9$, but I know no argument.

Total reality in the Harnack-maximal case {#sec:Total-reality-Harnack-max-case}
-----------------------------------------

\[08.03.13\] Much of this section is by now much illuminated by Le Touzé’s observation in Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics], where it is remarked that a very simple prescription of basepoints ensure total reality of a pencil of cubics on an $M$-quintic.

\[ca. 31.10.12\] Quite paradoxically it is much harder to depict total pencils on Harnack-maximal curves, alias $M$-curves (in Russia since Petrovskii 1938 [@Petrowsky_1938], cf. Gudkov 1974 [@Gudkov_1974/74 p.18]), especially when the order is $m\ge 5$. (For lower orders $m\le 4$ everything is essentially trivial: since $m=4$ just requires a pencil of conics passing through the 4 ovals of the quartic (with $g=3$).) Recall indeed that Ahlfors theorem is much easier in the planar case $p=0$, where it goes back to Bieberbach-Grunsky, if not earlier. Logically the argument simplifies much via Riemann-Roch and the absence of collision, cf. e.g. Gabard 2006 [@Gabard_2006 Prop.4.1] or Lemma \[Enriques-Chisini:lemma\] above in this text.

Shamefully, the following section climaxes the poor level of organization of the present text. Of course the game is quite outside the main stream of our subject (Ahlfors theorem), yet we think that some phenomena require to be clarified. In particular we were not able to make any reliable picture of a total pencil on a Harnack-maximal (smooth) plane curve of order $m\ge 5$. After some three days of pictorial tergiversation we found a sort of weak obstruction to manufacturing such pictures involving a basic type of pencil spanned by two special cubics. This obstruction is described at the end of the section, which otherwise reduces to a messy gallery of failing attempts of the desired easy depiction! Yet the abstract theorem of Bieberbach-Grunsky implies the existence of total pencil but they probably involve delicate-to-visualize pencil of cubics (in the quintic case). We would like to challenge gifted amateurs to picture them appropriately.

Let us first recall the construction of such $M$-curves due to Harnack (in the variant of Hilbert). We start with degree 5. Consider as primitive configuration an ellipse $E_2$ plus a line $L_1$. Take further 3 parallel lines $l_1,l_2,l_3$. There is some psychological difficulties to know if we should first smooth $E_2
\cup L_1$ and then perturb along $l_1\cup l_2\cup l_3$ or if we can directly perturb $E_2 \cup L_1$ without taking care of smoothing. Let us adopt the shorter route (actually so do Hilbert) by putting directly $C_3=(E_2\cup L_1)+\varepsilon
\vartheta_3$. This cubic (in black thick stroke) oscillates across the ellipse $E_2$ meeting it in the maximum number of 6 real points. Next smoothing their union (=product) $C_3 \cup E_2$ we get the (red-colored) quintic $C_5$ realizing the maximum number $r=7(=g+1)$ of ovals (one of them being in fact a pseudoline i.e. a Jordan curve in ${\Bbb R}P^2$ not bounding a disc).

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\[31.10.12\] Now the (perpetual) game is to find a total pencil on this dividing curve $C_5$ (recall that Harnack-maximal curves are always dividing). As usual the recipe is to distribute imposed basepoints $p_1, \dots, p_6$ in the deepest ovals. Those are fixed once for all and marked by black points on Fig.\[Harna1:fig\]. Since there are 6 ovals, pencil of lines or conics are not flexible enough to reveal the total reality of our $C_5$. We thus have to look among pencils of cubics. In view of the (vertical) symmetry of the curve $C_5$ it is natural to seek a symmetric pencil. We shall define them by specifying two of its members. A first vertically symmetric cubic through the 6 basepoints is the union of the 3 cyan-colored lines. This special (cyan) cubic $C_3$ cuts our quintic $C_5$ twice along each oval and once on the pseudoline, hence in $12+1=13$ points. Those are at finite distance but looking at infinity both horizontal cyan lines cuts the pseudoline branch of $C_5$ in two extra points, yielding a total of 15 point, the maximum possible (all of them being real). Beside, we consider another vertically symmetric cubic, namely the red-colored cubic $R_3$ consisting of the red ellipse through 5 points $p_i$ plus the red horizontal line (denoted $C$) through the remaining $p_i$. We can now consider the corresponding pencil spanned by the cyan and red cubics (equation $\lambda C_3+ \mu
R_3=0$). Unfortunately, the red cubic cuts $C_5$ along $2\cdot
6=12$ points on the ovals and only once at infinity. Indeed the pseudoline branch of $C_5$ is asymptotic to the line $D$ which in transverse to the red line $C$. Hence the intersection $R_3\cap
C_5$ is not totally real. Of course this defect does not prevent us from tracing the corresponding (non-total) pencil.

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[*Note.*]{}—A pencil of cubic may be defined by assigning 8 basepoints. By letting degenerate those against the 6 ovals (or the pseudoline) we get a series of degree $3\cdot 5 - 8\cdot
1=15-8=7$ as predicted by Bieberbach-Grunsky (cf. e.g. Lemma \[Enriques-Chisini:lemma\]). But it is far from evident to ensure total reality. Of course a coarse calculation would stipulate that the 6 ovals contributes for $2\cdot 6=12$ many intersections and imposing 2 extra basepoints on the pseudoline gives 2 additional intersection, totalizing 14 many hence the last man surviving is forced to be real as well. This argument certainly holds good if we know that all cubics of the pencil are connected but a priori a cubic may well have an oval which could be nested in one of the tiny ovals of our sextic. If so is the case then this one cubic’s oval only visits one of the 8 basepoints, without spontaneous creation of intersection on one oval of the quintic $C_5$. Maybe this scenario is quite improbable but I missed some argument.

A modest improvement over our previous attempt is to take a red-colored cubic satisfying total reality. This is given by changing the red-colored ellipse by taking the one passing through the 5 “highest” (relatively to our figure Fig.\[Harna2:fig\]) black-colored basepoints. Symmetry forces us then to take an additional red-colored line passing through the “lowest” basepoint. We obtain the following Fig.\[Harna2:fig\]. Alas it is not evident that total reality is satisfied.

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A third option is to change the cyan configuration of 3 lines and we get the following Fig.\[Harna3:fig\], which alas again seems to fail total reality.

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\[01.11.12\] Of course we would like ultimately to extend the game to sextic. Let us first reproduce a picture in Hilbert 1909 [@Hilbert_1909-Ueber-die-Gestalt-sextic]. The idea is again that a union of two ellipses is vibrated into a quartic $C_4$ oscillating across one of the ellipse $E_2$ (which is a circle on Fig.\[Hilb1:fig\], left), and next $E_2\cup C_4$ is smoothed to a sextic with 11 ovals (compare Fig.\[Hilb1:fig\], right).

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Again the challenge would be to trace a total pencil of curves on this $C_6$. We have 10 deep ovals, thus pencils of cubics look overwhelmed already with their only 8 assignable basepoints (and maximally 9 of them). Quartics have $\binom{4+2}{2}-1=15-1=14$ free parameters hence we can impose 13 basepoints. Choosing them in the deep ovals and doing a parietal degeneration gives a series of degree $4\cdot 6- 13 \cdot 1=24-13=11$. This matches with the Bieberbach-Grunsky bound, however it is far from evident that total reality is ensured.

In general if $C_m$ is a Harnack-maximal curve of order $m$, the previous examples (with $m=5,6$) suggest to consider auxiliary curves of degree $m-2$ forming a space of dimension $\binom{(m-2)+2}{2}-1=\binom{m}{2}-1$ and thus assigning $\binom{m}{2}-2$ basepoints will define a pencil. By parietal degeneration the resulting series has degree $(m-2)m-[\binom{m}{2}-2]$, and this is easily calculated as being equal to $$\begin{aligned}
(m-2)m-[\binom{m}{2}-2]&=(m-2)m-\frac{m(m-1)}{2}+2\cr
&=\frac{1}{2}[2(m-2)m-m(m-1)+2]+1=\frac{1}{2}[m^2-3m+2]+1\cr
&=\frac{(m-1)(m-2)}{2}+1=g+1, \end{aligned}$$ where $g$ is the genus. This again agrees with the Bieberbach-Grunsky theorem, but of course does not reprove it, be it just for the simple reason that smooth plane curves have specialized moduli among all curve sof the same genus. Still it would be exciting to manufacture tangible pictures of such total pencils in the planar case.

Now let us try again to do better pictures of the $M$-quintic. Any such $M$-quintic has 6 ovals and one pseudoline. By Bézout no three ovals can be aligned (otherwise 6 intersection with a line). Thus the six ovals are somehow distributed along a configuration resembling a hexagon. This raises some hope to draw reasonable pencil of cubics spanned by two configurations of 3 lines according to one of the following patterns (left of Fig.\[Hilb2:fig\]). This suggested to draw another model whose 6 ovals are nearly situated like a regular hexagon. Using cyclotomy, we get quickly the right part of Fig.\[Hilb2:fig\].

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A little piece of comment on the last Fig.\[Hilb2:fig\]: of course we started with a circle divided primarily in 6 equal parts, and have chosen the 3 horizontal lines as passing through the cyclotomic points. Those three lines are those used for the Harnack-Hilbert vibration trick, and the rest of the picture should be self-explanatory. Alas the bottom portion is quite difficult to observe. Yet a clear-cut portrait of Lars Valerian clearly emerges: the bottom oval is the mouth, then just above two big eyes “with an air of determination”, as well as some hairs emanating from the beret. In fact the portrait looks more like an alien, but the resemblance with Lars is much more flagrant when the circle is depicted as a “vertically oblong” ellipse. \[I apologize for adding some extra prose as otherwise the figures desynchronize from the text.\]

Now we consider the following pencil spanned by the cyan and red collections of lines (Fig.\[Hilb3:fig\]). Alas it fails to be totally real, for it contains the green cubic cutting only 13 real points on the quintic $C_5$. Of course the advantage of our pencil is that it is simple to draw, yet its disadvantage is that it has only 6 among the 8 assignable points located on the quintic. Somehow one should try to conciliate both properties.

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Testing the other configuration (of 2 pairs of 3 lines through the hexagon) one gets Fig.\[Hilb4:fig\]. The situation is not much improved. Now the 3 additional basepoints (intersection of pairs of parallel lines) are ejected at infinity but are not lying on the (black-colored) quintic curve $C_5$ whose pseudoline is asymptotic to the horizontal line. The corresponding pencil of cubics (spanned by the cyan and red colored lines) is probably not total, for it should contain a nearly circular ellipse through the hexagon plus the line at infinity, and the aggregated corresponding cubic seems to cut the $C_5$ only along 12, plus one at infinity, so a total of only 13 real points!?

-0pt

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One can also make the following picture Fig.\[Hilb5:fig\], where the 3 additional basepoints are marked by circles, one of them lying, alas, quite outside the range of the picture. A possible, yet delicate, desideratum would be to distort the configuration (pair of 3 lines arrangements) so that 2 of those circled basepoints lands on the quintic $C_5$. Then we would get a good candidate for an easy to depict total pencil of cubics on our quintic. Evidently this desideratum is probably impossible to arrange (a so-called “Irrweg”).

-0pt

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Maybe another arrangement worth looking at is the following Fig.\[Hilb6:fig\]. Now among the 3 extra basepoints at least one (that one corresponding to the intersection of both horizontal lines) is located on the quintic $C_5$ (at infinity). Hence 13 points are ensured to be real for all members of the pencil. It is easily checked that both fundamental curves of the pencil (cyan and red cubics) cut the $C_5$ in a totally real fashion (15 real points). For symmetry reasons (along the axe at 120 degrees) the nearly circular ellipse through the 6 points at finite distance plus the line at angle 120 degrees belongs to the pencil, but alas its intersection with the $C_5$ it hard to understand. Note by the way that the hexagonal configuration of 6 points is slightly perturbed thus there is no perfectly well defined such ellipse. At this stage the whole exercise is akin to a dolorous acupuncture session. Note that our symmetry deduced member of the pencil has the wrong behavior through the basepoints at infinity, hence the right curve belonging to the pencil includes rather the line at infinity (or at least a slight perturbation thereof). Thus we count 12 intersections with the oval coming from the nearly circular circuit, and just one intersection at infinity. This underscored total of 13 seems to indicate that this pencil again fails total reality.

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Albeit our exposition is not from the best stock, we hope at least to have demonstrated that the synthetic construction of total pencils on $M$-curves is not an easy matter. Of course it is not improbable that I missed something fairly easy!

[**Isoperimetric digression.**]{} During the session I wondered if the following problem makes sense. One of the notorious difficulty when trying to do real pictures of algebraic curves is that some ovals tend to be microscopic (especially for Harnack-maximal curves). Is there some optimal curve best suited for depiction? Admittedly the problem makes sense only for Euclidean affine models as opposed to projective curves (which could be pictured on the sphere up to a double cover). One could for instance ask the curves to enclose maximum area for a given length of the circuits. (Of course this makes sense only for curves of even degrees, except if we neglect the pseudoline.) This would be a sort of isoperimetric problem for curves competing among algebraic ones (of some fixed degree). Of course for degree two the isoperimetric solution is the circle. What about degree 4? A candidate is perhaps the Fermat curve $x^4+y^4=1$ whose real picture is somewhere between a circle $x^2+y^2=1$ and a square $x^{\infty}+y^{\infty}=1$. Of course one could argue that the optimal quartic is just a circle counted by multiplicity 2, but then the length of the circuit has to be counted twice. We have no certitude that our problem is well posed, nor that it is truly interesting. The naive scenario would be that the optimum is always the Fermat curves of higher even orders, yet what about $M$-curves? Maybe we need to restrict the problem to them, and ask for the best Euclidean realization of an $M$-curve? So for instance what is the best $M$-quartic? The best $M$-quintic? Does it looks like Ahlfors’ portrait (on Fig.\[Hilb2:fig\])?

Let us a last time return to our main problem of tracing a totally real pencil for an $M$-quintic. Remember once more that theoretical existence is ensured by the baby case (Bieberbach-Grunsky) of Ahlfors theorem on circle maps. Our dream would be that for such a quintic there is a simple-to-draw pencil generated by 2 configurations of 3 lines. Psychologically it is helpful to reverse the viewpoint. Instead of starting from the quintic $M$-curve $C_5$ and trying hard to depict the pencil, we shall start from the pencil and try to construct a curve tailored to it.

So we consider the pencil generated by 2 systems of parallel lines (colored cyan and red) with 9 basepoints (multicolored intersections) and try to build around this perfectly explicit pencil (cf. the previous Fig.\[Fcubic:fig\] Fig.\[Hilb6b:fig\]b below) a quintic having the following schematic picture (Fig.\[Hilb6b:fig\]a). This is to mean that each of the 6 ovals encloses one of the 9 basepoints, with the Bézout restriction that no aligned triad are enclosed (else 6 intersections in $C_5$ with a line) and further the pseudoline passes through 2 other basepoints. If such a “real scheme” (Rohlin’s jargon) exists then each curve of the pencil will cut on the $C_5$ a total of 15 real points. Indeed the 6 ovals contribute each for twice (now Fig.\[Hilb6b:fig\]b ensures connectedness of all cubics forming the pencil!) and the pseudoline for 2, hence a total of 14 and the last one is forced to be real as well (for algebraic “Galois theoretic” reasons).

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So exhibiting this scheme would complete our goal. Note the absence of Bézout-type obstruction to the posited real scheme (Fig.\[Hilb6b:fig\]a). Yet maybe there is deeper topological obstructions involving say the foliation underlying the pencil. In fact the argument is more modest. The two basepoints connected by the pseudoline are separated by the green ellipse. So the arc joining them (choose one!) is forced to have an extra intersection with the green ellipse (on Fig.\[Hilb6b:fig\]b). Topology forces the creation of a second intersection (intuitively the pseudoline once trapped in the green ellipse has to escape it). Thus we arrive at a total of $12$ (6 ovals), plus the $2$ assigned basepoints on the pseudo-line and plus the 2 extra-points just created. This gives 16 intersections between $C_5$ and the green cubic (enough to overwhelm Bézout). This prohibits the desired scheme.

Another (a priori) tangible real scheme is the one depicted on Fig.\[Hilb6b:fig\]c. Then it seems that arguing with the lilac conic we may repeat something like the previous argument. More precisely, if the pseudoline never penetrates inside the lilac ellipse $L_2$ then it has to be tangent to it at the 2 assigned basepoints but this gives already 4 extra-points which added to the 12 God-given produce an excess $16>15$! Thus we may assume the pseudoline $P$ to penetrate in the lilac ellipse (total of 13 intersection). Then several cases may occur. If $P$ tries to evade from the lilac ellipse $L_2$ then we have $14$ intersections, yet it must still pass to the second basepoint and (being now outside the $L_2$) this creates at least 2 intersections (counted by multiplicity). So eventually the pseudoline $P$ is forced to reach the other basepoint while staying inside the lilac $L_2$, and hence to cut the lilac axis of this ellipse. The latter axis being contained in the inside of the green ellipse, we get again 4 extra intersections with the green cubic (beside the 12 arising de facto from the ovals); too much for Bézout.

All this (if correct?, and suitably simplified!) should prove the following:

It is impossible to sweep out in a totally real fashion an $M$-quintic via a basic pencil of cubics spanned by two arrangements of parallel lines.

If true and suitably generalized to other configurations (see $\bigstar$ right below) this explains perhaps why we had so much trouble to make an appropriate depiction of the desired pencil. Again totally real pencils exist in abstracto hence in concreto, yet are probably of a somewhat more elaborated vintage.

\[02.11.12\] $\bigstar$ For instance it should be noticed that there is another possible scheme (distribution of 6 ovals) satisfying the “no-three-in-line” condition prompted by Bézout. This is depicted on Fig.\[Hilb6b:fig\]d which is admissible provided the horizontal diagonal is not aligned. Hence the real picture looks rather like Fig.\[Hilb6b:fig\]e. Of course it would be too cavalier to claim that the previous obstruction to the case at hand as the ellipses were destroyed during the process.

We leave the problem in this very unsatisfactory state of affairs, but let us perhaps try to motivate why the explicit depiction project could be fruitful!

From the viewpoint of gravitational systems (cf. the previous Sec.\[sec:electrodynamics\]) the interest of $M$-curves is that they express in some sense the most complex orbital structure permissible for a given genus (at least the maximum number of real circuits). Hence if Metatheorem \[metatheorem:thm\] is reliable such $M$-curves should display some remarkable motions. The intricacy of the trajectories is already suggested by Hilbert’s $M$-sextic on Fig.\[Hilb1:fig\]. However until the total pencil (of Bieberbach-Grunsky-Ahlfors) is not made explicit the dynamics of the electrons is imbued by mystery and darkness. Remind from Bieberbach-Grunsky (=Lemma \[Enriques-Chisini:lemma\]) that Hilbert’s $M$-sextic is not only static object but one animated by a circulation (total pencil) having one electron on each oval. We can from the static picture vaguely try to guess where repulsions take places and arrive at something like Fig.\[Hilb7:fig\].

0 -10pt0

On Fig.\[Hilb7:fig\], italics numbers enumerate ovals while roman numbers indicates positions at various times $1,2,3,4$. Note that our Harnack-maximal curve being dividing, it has a complex orientation (as the border of one half). This orientation agrees with that inherited from the smoothing. Further it has to be respected by the circulation due to the holomorphic character of the (Bieberbach-Grunsky) circle map. Having this is mind it is straightforward to make the picture above (Fig.\[Hilb7:fig\]) using the rule that whenever a repulsion is observed then electrons must be in close vicinity and thus any pair of points minimizing the distance between two neighboring ovals must be synchronized, hence labelled by the same time unit. In contrast when two close ovals do not repulse them (like ovals [*1*]{} and [*10*]{}) then they must be anti-synchronized in the sense that both particles do not visit the contiguity zone at the same moment. For instance there is also a repulsion between electrons on ovals [*1*]{} and [*11*]{} at time 1. So far so good. However on completing the picture one sees between ovals [*6*]{} and [*11*]{} some anomalous (asynchronic) repulsion. Maybe one can explain this via distant repulsion involving other particles of the system (especially the electron on oval [*10*]{}).

All this is very informal and saliently illustrates the sort of obscurantism caused by a lack of explicit knowledge of the total pencil. This perhaps motivates once more to complete the programme of the present section (construction of total pencils in Harnack-maximal cases). Ultimately one could dream of a computer program showing in real time the circulation of electrons prompted by the Bieberbach-Grunsky Kreisabbildung(en) along an Hilbert $M$-sextic.

Let us finally observe that there are other $M$-sextics (Harnack’s, Hilbert’s and even Gudkov’s). Basically the one we depicted (Hilbert’s) is gained by smoothing the configuration $E_2\cdot C_4=0$ consisting of an ellipse $E_2$ (circle on the picture) and an $M$-quartic $C_4$ one of whose oval oscillates across the ellipse $E_2$. It may be noticed that the oscillating oval lies mostly inside the ellipse (cf. the left-top part of Fig.\[Hilb8:fig\]). \[This schematic—yet Bézout compatible—style of depiction is borrowed from Gudkov 1974 [@Gudkov_1974/74 p.20].\] One can reverse this situation, by putting the vibrating oval outside the ground ellipse to get another $M$-sextic (cf. the right-top part of Fig.\[Hilb8:fig\]). A concrete construction this is achieved on the bottom part of Fig.\[Hilb8:fig\]).

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This curve has one “big” oval enclosing nine “small” ovals and the other lies outside. Of course if our metatheorem (\[metatheorem:thm\]) is plausible then it is challenging to interpret the dynamics especially the orbit along this long oval enclosing all others but one. Of course this would essentially boils down to visualize a total pencil for this $C_6$.

Finally let us make a little remark. We see that there must a deep reaching connection between Ahlfors theory of circle maps and the extrinsic geometry of real dividing curves, the link being given by the notion of total pencil. Another basic application of total pencils could arise in curve plotting problems. Assume given an algebraic equation $f(x,y)=0$ and a machine supposed to make a plot of the real locus. Suppose e.g. that the polynomial has degree 5, defines a smooth curve and that we have already traced within reasonable accuracy 2 ovals and a pseudoline and finally that both ovals are nested. Then the theory of total maps (but in fact Bézout suffices) ensures that the real locus has already been exhausted and we may stop the “root finding” algorithm. Of course the story becomes even more grandiose on appealing to Newton-Cayley iteration method and the allied fractals appearing as attracting basins. Likewise if an octic has 4 nests of depth 2 its real locus has already been exhausted (compare Fig.\[Foctic4:fig\]). Indeed in that case the pencil of conics through the 4 deeply nested ovals imposed to pass through another hypothetical point would create an excess of $8\cdot 2+1=17>16$ intersection points.

A baby pseudo-counterexample in degree 4 {#sec:degree-four}
----------------------------------------

We now give an example in degree 4. The recipe is always is the same and we get the example 102 below (Fig.\[F102:fig\]). It has $g=0$, $r=1$, thus $p=0$. At first glance the visual gonality as measured via a pencil of lines is $\gamma^{\ast}=2$ (projection from one of the nodes). This seems of course to violate Gabard’s bound $\gamma\le r+p$. However using a pencil of conics passing through the 3 nodes plus the point (labelled $8$ on the figure) gives a total pencil of the right degree. Of course the example is a paroxysm of triviality, yet it is still a nice case to visualize the fairly complex dynamics of total pencils. The forward semi-orbit of the series is depicted by points $1,2,\dots, 8$ after which the motion reproduces symmetrically.

Another example arises when we keep less singularities unsmoothed. We obtain so a linkage of “heartsuits” (cf. the middle picture 202 on Fig.\[F102:fig\]). Now $r=2$, $g=3-2=1$, and so $p=0$. A linear projection from one of the 2 nodes suffices to exhibit total reality, and so the gonality is $\gamma=2$. One can still trace pencils of conics through the nodes plus 2 extra points on the curve to get series of degree $2\cdot 4- 2\cdot 2-2\cdot
1=8-4-2=2$. Those gives more maps realizing the gonality. Of course one can also materialize such a curve as a smooth plane cubic, in which case we also see $\infty^1$ total pencils induced by linear projection from the unique oval. (Projecting from the pseudo-line, the oval of the cubic has some “apparent contour” and total reality fails.) One can also get the bottom picture 202 on Fig.\[F102:fig\], which has the same invariants.

Low-degree circle maps in all topological types by Harnack-maximal reduction {#sec:Chambery}
----------------------------------------------------------------------------

\[Source=Gabard 2005, Chambéry talk (unpublished as yet)\] Once Ahlfors theorem is known in the simple Harnack-maximal case (cf. Lemma \[Enriques-Chisini:lemma\]) one can easily exhibit in any topological type some very special surfaces (in Euclid’s 3-space) admitting a circle map to the disc having very low degree. Of course this is far remote from reassessing the full Ahlfors theorem, yet it is an interesting construction, which perhaps could lead to a general proof when combined with some Teichmüller theory. But this is only a vague project we shall not be able to pursue further.

Let us start with a membrane in Euclidean 3-space (endowed with the conformal structure induced by the Euclidean metric). Suppose the surface invariant under a symmetry of order two (cf. Fig.\[Chambery:fig\]). The key feature of this figure is that the axis of rotation “perforates” each “hole” of the pretzel. Hence, when taking the quotient all handles are killed, and we get a proper(=total) morphism to a schlichtartig configuration (i.e. of genus $p=0$). This in turn admits a circle map of degree equal to the number of contours (by the Bieberbach-Grunsky theorem=Lemma \[Enriques-Chisini:lemma\]).

The composed mapping gives a circle map of degree $2\cdot
\frac{r}{2}=r$ when $r$ is even, and of degree $2\cdot
\frac{r+1}{2}=r+1$ if $r$ is odd. (Compare again Fig.\[Chambery:fig\].)

Of course this has little weight in comparison to the general theorem of Ahlfors (1950 [@Ahlfors_1950]), yet it is a simple example showing that the degree of circle maps can be fairly lower than the degrees $r+2p$ or even $r+p$.

Experimental evidence for Coppens’ gonality
-------------------------------------------

\[24.03.12\]/\[19.10.12\] In this section we discuss Coppens result (2011 [@Coppens_2011]) on the realizability of all gonalities compatible with the $r+p$ bound (Gabard 2006 [@Gabard_2006]) on the degree of an Ahlfors circle map. Our superficial approach will not recover Coppens full result, yet is worth presenting for it enhances the depth of Coppens’ result. Looking at explicit projective models of Riemann surfaces always makes Riemann-type existence theorems (like Ahlfors maps) look quite formidable jewels (not to say miracles) when looked at experimentally through the Plato cavern of extrinsic algebraic geometry. The game is also pleasant because sometimes one gets the impression that Gabard’s bound $r+p$ looks blatantly violated. Also interesting is the issue that such basic experimental studies (akin to the CERN particles collider at a modest scale) are quite useful for understanding the failure of connectivity of the space of minimal circle maps (those of lowest possible degree). Further experiments should contribute to add some valuable insights over Ahlfors’ theory. (A. Einstein puts it as follows: “Any knowledge of the world starts and ends with experiments.”)

Coppens’ result is the following. To stay closer to Ahlfors’ viewpoint, we paraphrase it in the language of [*compact bordered Riemann surfaces*]{} (abridged membranes) instead of that of real dividing curves. Albeit most of our examples are derived via algebraic geometry, we will never have to write down any (boring) equation due to the graphical flexibility of plane curves à la Brusotti/Klein/Plücker (reverse historical order). So we are drifted to a sort of synthetic geometry.

[(Coppens 2011 [@Coppens_2011])]{} Given any two integers $r\ge 1$ and $p\ge 0$, and any integer $\gamma$ satisfying $\max\{2, r \}\le \gamma \le r+p$, there is membrane $F_{r,p}$ with $r$ contours of genus $p$ whose gonality is the assigned value $\gamma$.

Recall that the gonality of the membrane is understood as the least degree of a circle map from the given membrane (to the disc).

$\bullet$ For $(r,p)=(1,0)$, the statement becomes vacuous, but of course we can alter the range of permissible values as $r\le
\gamma\le r+p$.

$\bullet$ When $p=0$, $\gamma$ can take only the value $r$ and the latter is realized via the Bieberbach-Grunsky theorem (Lemma \[Enriques-Chisini:lemma\]).

$\bullet$ For $(r,p)=(1,1)$, the double has genus $g=(r-1)+2p=2$ hence is hyperelliptic. This actually proves the existence of a circle map of degree 2 ($=r+p$) in accordance with Gabard’s bound $r+p$. Coppens’s realizability theorem is trivially verified in this case for $\gamma$ can only assume value 2.

$\bullet$ For $(r,p)=(2,1)$, the range of $\gamma$ is $2\le \gamma
\le 3$. The value $\gamma=2$ is realized by a hyperelliptic model. The value $\gamma=3$ is obtained by considering a smooth quartic $C_4$ with two nested ovals while projecting it from a point on the innermost oval. This gives a [*totally real*]{} morphism of degree $3$. Total reality means that fibers above real points consists entirely of real points. We use also the abridged jargon [*total map*]{} which is quite in line with terminology used by Stoïlow 1938 [@Stoilow_1938-Lecons] or Ahlfors-Sario 1960 [@Ahlfors-Sario_1960], who use “complete coverings”.

-75pt 0

$\bullet$ For $(r,p)=(3,1)$, the genus of the double is $g=(r-1)+2p=2+2=4$. This is not the genus $g=\frac{(m-1)(m-2)}{2}$ of a smooth plane curve of order $m$ which belongs to the list $0,1,3,6,10,
\dots$ of triangular numbers, yet suggests looking at a quintic $C_5$ with two nodes. We thus consider a configuration of two conics plus a line and smooth it out in a orientation preserving way (so as to ensure the dividing character of the curve by a result of Fiedler 1981 [@Fiedler_1981]). We obtain so the curve depicted on Fig.\[Coppens:fig\] bearing the nickname 313. This actually encodes the value of the invariant $(r,p,
\gamma)$ written as the string $rp\gamma$, yet a priori the gonality $\gamma$ is not known and its value must be justified. On that figure 313 the dashed circles indicate those crossings that were [*not*]{} smoothed. The half of this curve is a bordered surface of type $(r,p)=(3,1)$, since $p=\frac{g-(r-1)}{2}$. It remains to evaluate its gonality. The idea is always to look at the curve from the innermost oval. In the case at hand, we project the curve from one of the two nodes to get a total morphism of degree $5-2=3$. Since $r=3$ is a lower bound on the gonality $\gamma$, it follows that $\gamma=3$, exactly. Note that this example seems to [*answer in the negative our question about the connectivity of the space parameterizing minimal circle maps*]{}. Further one can drag one point to the other while travelling only through [*total*]{} maps of degrees 4 (namely projections from points located in the intersection of the interiors of the blue resp. red ovals). \[09.11.12\]—[*Warning.*]{} Remember that a similar picture (Fig.\[F102:fig\], right-middle part) gave an example where the curve looked 2-gonal in only 2 ways, but another model of the curve (as a plane cubic) prompted the same gonality in $\infty^1$ fashions. So some deeper argument is required either to assess (or disprove) the italicized assertion.

Next, still for the same topological invariants $(r,p)=(3,1)$, we would like to find a membrane of gonality $\gamma=4$. This may be obtained from the same initial arrangement while moving the location of the dashed circles (of inert crossings) to get picture labelled 314 on Fig.\[Coppens:fig\]. The corresponding quintic projected from a point situated on the inner (blue-colored) oval has $\gamma\le 4$. Over the complexes, this quintic has gonality $\gamma_{\Bbb C}=3$ (projection from one of the nodes) and this is the only way for the curve to be trigonal. Yet over the real picture (our 314) none of these (trigonal) projections is total (since the inner oval has an apparent contour, i.e. some tangent to it passes through the node). It follows that $\gamma=4$, exactly.

$\bullet$ Let us next examine $(r,p)=(4,1)$. Then $g=(r-1)+2p=5$, so we look at quintics with one node. To create as many ovals, it proves convenient to reverse the orientation of one of the conics. We obtain so the figure coded 415. After noting that $r=4$, we project the curve via a pencil of conics assigned to pass through 4 points chosen in the innermost ovals (asterisks on the figure). Letting those 4 points degenerate against the ovals while exploiting the possibility of pushing one of them toward the node (so as to lower by 2 units the degree) we find $\gamma\le 2\cdot 5- 3\cdot 1
-1\cdot 2=10-3-2=5$. Over the complexes, the curve at hand (uninodal quintic) is trigonal only when seen from its unique node and 4-gonal only when projected from a smooth point. Inspection of the figure shows that none of these maps is total. It follows that $\gamma=5$ exactly.

It remains to find an example with $\gamma=4$. For this we just drag below the dashed circle (cf. label 414 on Fig.\[Coppens:fig\]), do the prescribed smoothing (always in the orientation consistent way). The resulting curve has $r=4$ (as it should). The novel feature is that the node is now accessible from 2 basepoints of the pencil of conics assigned in the deep ovals. This permits a lowering of the degree to $\gamma\le 2\cdot
5- 2\cdot 1 -2\cdot 2=10-2-4=4$. Remarking that the unique morphism of lower degree 3 (linear projection from the node) is not total we deduce that $\gamma=4$ exactly. The other morphisms of degrees 4 (namely projections from real points on the curve) obviously fails to be total, thus we infer that the curve (or the allied membrane) is uniquely minimal (i.e. there is a unique circle map of minimum degree).

Before embarking on larger values of the invariants $(r,p)$, we make a general remark, related to the previous Sec.\[sec:Chambery\]. There a suitable membrane in 3-space invariant under rotation by $\pi=180^{0}$ with a totally vertical array of handles (cf. Fig.\[Chambery:fig\]) showed the following:

[(Barbecue/Bratwurst principle)]{}\[Barbecue:lem\] $\bullet$ If $r$ is even, there is for any value of $p$ a membrane of type $(r,p)$ admitting a circle map of degree $r$ (the minimum possible value), whose gonality is therefore $\gamma=r$ exactly.

$\bullet$ If $r$ is odd ($p$ arbitrary), there is a membrane of type $(r,p)$ admitting a circle map of degree $r+1$, whose gonality $\gamma$ is therefore $r\le \gamma\le r+1$. (Alas, the exact value remains a bit undetermined!)

This lemma fills quickly several positions of our Fig.\[Coppens:fig\], namely those marked by a square. In the special case $r=1$ (belonging to the indefinite odd case), we can get rid off the annoying indetermination, because as soon as $p\ge
1$ the minimal value $r$ of the range $r\le \gamma\le r+1$ cannot be attained. Corresponding invariants are reported by rhombuses (squares rotated by $\pi/4$) on Fig.\[Coppens:fig\].

$\bullet$ Next we study $(r,p)=(5,1)$. Then $g=(r-1)+2p=6$, prompting to look at smooth quintics (without nodes). Consider the curve denoted 516 on Fig.\[Coppens:fig\], which has $r=5$. When projected via a pencil of conics through the assigned 4 basepoints (depicted by asterisks on the figure) and letting them degenerate toward the ovals gives a total map of degree $2\cdot 5-
4\cdot 1 =10-4=6$. Hence $\gamma \le 6$. Morphisms of lower degrees exist in degree 4 (linear projection from a point situated on the curve), and degree 5 (projection from points outside the curve). Clearly none of these maps is total, so that $\gamma=6$ exactly. Of course the minimal degree maps considered are plenty (no uniqueness), yet their parameter space is connected.

Next we require a specimen with $\gamma=5$. It seems evident that we have exhausted the patience of quintics (at least for the given arrangement), hence let us move to sextics of genus 10 (when non-singular). To get the right genus $g=6$, we have to conserve 4 nodes. Starting from a configuration of 3 conics suitably oriented and smoothed we obtain the figure denoted 515 with $r=5$ (still on Fig.\[Coppens:fig\]). Using a pencil of conics with 4 assigned basepoints (asterisks on the figure) gives a (probably total) map of degree $2\cdot 6-1\cdot 2-3\cdot 1=12-2-3=7$. This agrees with Ahlfors bound $r+2p$, but seems to challenge Gabard’s bound $r+p=6$. Maybe a pencil of cubics is required instead. Such a cubics pencil has 9 basepoints but only 8 of them may be assigned. Hence creating some 4 new basepoints (denoted by bold letters [**1,2,3,4**]{} on the figure) and letting them degenerate to the ovals or better the nodes (when some are accessible) gives a map of degree $3\cdot 6-3\cdot 1- 5\cdot 2=18-3-10=5$, rescuing Gabard’s $r+p=6$ and also giving the desired gonality $\gamma=5$. Admittedly this example is quite complex and perhaps not the best suited to illustrate Coppens’ gonality result. Its interest is still that it seems to corrupt Gabard’s bound $r+p$, and the latter can only be rescued by appealing to fairly sophisticated pencils. Of course it could be the case a priori that our curve (515) admits a pencil of conics of lower degree than 7, but under the totality condition basepoints must be distributed in the deep ovals by a Poincaré index argument (cf. Lemma \[Poincare-lower-bound\]). This impedes a lowering of the degree via a more massive degeneration of the base-locus to the nodes of picture 515 on Fig.\[Coppens:fig\]. Admittedly the predicted total pencil of cubics ought to be described more carefully.

[*Summary of the situation.*]{}—Of course one should still work out the higher values of $r$ while keeping $p=1$. As you notice our method is far from systematic. (All the difficulties encountered so far already enhance the power of Coppens’ result.)

$\bullet$ Then one must also handle higher values of $p$, starting with $(r,p)=(1,2)$. The case $\gamma=2$ is easy (via the barbecue construction, Lemma \[Barbecue:lem\]). For $\gamma=3$ we can imagine a surface with 3-fold rotational symmetry (cf. picture 123X on Fig.\[Coppens:fig\]). For it $\gamma\le 3$, but how to show equality? Alternatively, one may consider an algebraic model. Since $g=(r-1)+2p=4$, we look among quintics with 2 nodes. A suitable smoothing gives figure named 123, with $r=1$ (one circuit). Linear projection for the “inner” node gives a total map of degree $1\cdot 5-1\cdot 2=5-2=3$, so $\gamma\le 3$. But the complex gonality of such a quintic is $\gamma_{\Bbb C}=3$. Since $\gamma_{\Bbb C}\le \gamma$ it follows that $\gamma=3$ exactly.

$\bullet$ Let us next explore $(r,p)=(2,2)$. Then $g=5$. A surface with $\gamma=2$ is easily found (barbecue rotational symmetry). To realize the other gonalities we look among quintics with one node. We first meet figure 223, which has a total morphism of degree 3 (projection from the node). Hence $\gamma\le 3$ which is in fact an equality, since 3 is also the complex gonality of an uninodal quintic. To get a curve with $\gamma=4$ we just drag the unsmoothed singularity to get figure 224. Projection from the node is not total anymore, but a total map arises when projecting from the (green) oval giving rise to degree $5-1=4$. Since such a quintic is uniquely trigonal (via projection from the unique node, which which failed to be total), we infer that $\gamma=4$, exactly. Coppens’s theorem is verified for this topological type.

[**Premature conclusion/State of the art.**]{} It is clear that one can continue the game to tackle higher and higher values of the invariants. Instead of looking solely in ${\Bbb P}^2$ it is also pleasant to trace curves in ${\Bbb P}^1\times{\Bbb P}^1$, albeit ${\Bbb P}^2$ is a universal receptacle (any Riemann surfaces nodally immerses in the projective plane). However it is clear that our naive approach is quite time consuming and as yet we did not deciphered a combinatorial pattern permitting to boost the speed of the procedure to the level of an inductive process. (Curves or Riemann surfaces of higher topological structures are like [*homo sapiens*]{}, the result of a long, intricate morphogenesis.) Coppens proved the full result in one stroke by somehow penetrating the genetic code governing the evolution of all species.

Minimal sheet number of a genus $g$ curve as a cover of the line {#Minimal-sheet:sec}
----------------------------------------------------------------

It is classical (since Riemann 1857 [@Riemann_1857 §5, p.122–123]) that a general curve of genus $g$ is expressible as a branched cover of the sphere ${\Bbb P}^1$ of degree the least integer $\ge \frac{g}{2}+1$ (equivalently of degree $[\frac{g+3}{2}]$). \[Indeed if $g$ is even $g=2k$ the first value is $k+1$ and $[\frac{g+3}{2}]=[(2k+3)/2]=(2k+2)/2=k+1$; if $g=2k+1$ is odd then the first value is $ g/2+1$ which rounded from above gives $(2k+2)/2+1=k+2$, and $[\frac{g+3}{2}]=[(2k+4)/2]=k+2$.\]

Riemann’s truly remarkable argument (involving Abelian integrals) is beautifully cryptical (I should still study it properly). It is not clear (to me) if it includes the stronger assertion that [*any*]{} curve of genus $g$ admits a sphere-map of degree $\le
[\frac{g+3}{2}]$. At any rate, all modern specialists agree that the first acceptable proof of this pièce de résistance is Meis’ account (1960 [@Meis_1960]). (Meanwhile the algebro-geometric community devised several alternative approaches.)

Another allied (but different?) argument is the one to be found in Klein’s lectures 1892 [@Klein_1891--92_Vorlesung-Goettingen p.98–99], cf. also Griffiths-Harris 1978 [@Griffiths-Harris_1978/94 p.261].

The latter’s argument works as follows. Assume there is a $d$-sheeted map $C_g \to {\Bbb P}^1\approx S^2$ of a genus $g$ surface to the Riemann sphere. Then Euler characteristics are related by $\chi(C_g)= d \chi(S^2)-b$, where $b$ is the number of branch points. This gives $b$ ramified positions, whose locations determine the overlying Riemann surface up to finitely many ambiguities. So the $d$-sheeted surface depends upon $b-3$ essential parameters (after substraction of the 3 arising from the linear transformations on ${\Bbb P}^1$). This quantity has to be $\ge 3g-3$ the number of moduli of genus $g$ curves. This implies $b\ge 3g$, i.e. $2d-\chi(C_g)\ge 3g$, or $2d\ge 3g+(2-2g)=g+2$. [*q.e.d.*]{}

So far as we know, a similar computation as never been written down for the case of a bordered Riemann surface expressed as a $d$-sheeted cover of the disc (i.e., the context of Ahlfors circle maps). The reason is probably quite mysterious, yet also quite simply that the naive parameter count seems to lead nowhere.

Let us attempt the naive computation. Suppose $F_{r,p}\to D^2$ to be a membrane of genus $p$ with $r$ contours expressed as a $d$-sheeted cover of the disc. Euler characteristics are related by $\chi(F)=d\chi(D^2)-b$, where $b$ is the number of branch points. The group of conformal automorphisms of the disc as (real) dimension $3$. Hence our $d$-sheeted surface depends upon $2b-3$ real constants, whereas the membrane $F$ itself depends on $3g-3$ real constants, where $g$ is the genus of the double (cf. Klein 1882 [@Klein_1882]). The Ansatz $2b-3\ge 3g-3$ gives $2b\ge
3g$, and since $b=d-\chi$ and $g=(r-1)+2p$, this gives $2d \ge
3g+2\chi=3[(r-1)+2p]+2(2-2p-r)=r+2p+1$, equivalently $d\ge
(r+1)/2+p$.

This beats the value $r+p$ predicted in Gabard 2006 [@Gabard_2006], but looks blatantly overoptimistic. For instance taking $p=0$, gives the degree $\frac{r+1}{2}$ violating the absolute lower bound $r$ on the degree of a circle map. The provisory conclusion is that the naive parameters count leads nowhere in the bordered case. Does somebody know an explanation?

\[21.10.12\] A crude attempt of explanation is that the above count merely uses the Euler characteristic which is a complete topological invariant only for closed surfaces, but not for bordered ones. (Since $\chi(F_{r,p})=2-2p-r$, trading one handle against two contours leaves $\chi$ unchanged.) Of course the above counting uses also $g$ (the genus of the double $2F$) but the latter is also uniquely defined by $\chi(F)$, via the relation $2-2g=\chi(2F)=2\chi(F)$. Thus it is maybe not so surprising that Riemann(-Hurwitz)’s count predicts correctly the gonality of closed Riemann surfaces but fails seriously to do so in the bordered case. It could be challenging to find a moduli count existence-proof of Ahlfors circle maps supplemented probably by an adequate continuity method. For an (unsuccessful) attempt cf. Sec.\[Hurwitz-type\].

A very naive (numerological) parade is to introduce a new bound $\nu:=\max\{p+ \frac{r+1}{2}, r \}$ between the one found above and $r$ the absolute minimum of a total morphism. However a simple example probably shows this to be overoptimistic as well. Consider the plane quintic $C_5$ derived via a sense preserving smoothing of the depicted configurations of $2$ conics and a line (cf. Fig.\[quintic:fig\]).

Its genus is $g=\frac{(m-1)(m-2)}{2}=\frac{4\cdot3}{2}=6$, and we see $r=5$ real circuits. The relation $g=(r-1)+2p$ gives $p=1$ (genus of the half). Hence the new bound is $p+
\frac{r+1}{2}=1+3=4$, but $r=5$ so $\nu=\max=5$. However the membrane (corresponding to one half of the dividing curve $C_5$) cannot be represented with 5 sheets over the disc. Indeed a morphism of degree 5 from $C_5$ to the line ${\Bbb P}^1$ can arise through linear projection of the quintic $C_5$ from a point not on the curve (else degree $4$), but no such projection is totally real (compare central part of Fig.\[quintic:fig\], or argue via the Poincaré index, cf. Lemma \[Poincare-lower-bound\]).

\[09.11.12\] [*WARNING about the underlined “only”.*]{}—This argument looks at first sight quite convincing, yet it appears to be insufficient, and possibly the assertion itself on the gonality $\gamma(C_5)=6$ is erroneous. First, a total morphism of degree $r+p=5+1=6$ (as predicted in Gabard 2006 [@Gabard_2006]) should exist. This is corroborated by taking a pencil of conics through 4 points inside the 4 ovals of the above depicted $C_5$ (cf. Fig.\[FGuerN:fig\], left part) and letting them degenerate against the ovals, giving a total map of degree $2 \cdot 5-4=6$. This tell us only $\gamma \le 6$. A priori, it could be the case that higher order pencils access the low degree $5$, and with some good-fortune do it in a totally real way. In that case the gonality lowers down to $\gamma=5$ (the minimum permissible as $r=5$). Let us quickly discuss how this could happen, at least over the complexes. A priori pencils of cubics may have degrees as low as $3\cdot 5-3^2=6$ (hence not violating the previous token); quartics as low as $4\cdot 5-
4^2=4$, but quartics have dimension $\binom{4+2}{2}-1=14$ so that in reality only 13 basepoints may be assigned freely, hence the right value is $4\cdot 5- 13=7$; for quintics this is as low as $5\cdot 5- 5^2=0$ (yet all values $<4$ violates already the complex gonality of a smooth quintic, cf. e.g. Arbarello et al. 1985 [@Arbarello-Cornalba-Griffiths-Harris_1985-BOOK p.56, Exercise 18]). In fact the dimension of quintics is $\binom{5+2}{2}-1=20$ and thus the minimum degree is $5\cdot 5- 19=6$. For sextics the degree is as low as $6\cdot 5- 6^2=-6$, but since the sextics dimension is $\binom{6+2}{2}-1=27$, the real minimum degree is $30-26=4$ (and this beats linear projections from outside the curve). Recall incidentally that this is the value of the universal Riemann-Meis bound $[\frac{g+3}{2}]=[9/2]=4$, which was already attained by linear projections from the curve but nobody will exclude a priori a second return. Actually all 26 assigned basepoints fails to impose independent conditions on sextics, because our quintic $C_5$ aggregated to any line is a sextic meeting the requirement and varying among $\infty^2$ parameters (and not just the expected $\infty^1$ pencil). Thus we seem to fail getting a genuine pencil, but contrast this with the just remembered Riemann-Meis gonality. The situation is quite more tricky than initially expected. Another torpedo against the naive belief that a smooth $C_5$ has only $\infty^1$ series of type $g^1_5$ is the existence theorem of Brill-Noether-Kempf-Kleiman-Laksov theory (cf. e.g. Arbarello et al. [@Arbarello-Cornalba-Griffiths-Harris_1985-BOOK p.206]). The latter states the following.

Let $C$ be a (complex) curve of genus $g$. Every component of the variety $G^r_d$ parameterizing all linear series $g^r_d$ of dimension $r$ and degree $d$ has dimension at least equal to the so-called [*Brill-Noether number*]{} $\rho$, symbolically: $$\dim_{\ast} G^r_d\ge \rho:=g-(r+1)(g-d+r).$$ In particular when the latter number $\rho$ is $\ge 0$ the variety $G^r_d$ is nonempty.

In the case at hand it follows that $\dim_{\ast} G^1_5 \ge
6-(1+1)(6-5+2)=6-2\cdot2=6-4=2$. Hence there are other pencils of degree 5 than those readily visualized on the projective realization! This shows how vicious the Plato cavern is! Of course our appeal to the above general theorem, is a violation against the principle of do-it-yourself-ness, since low genus cases are in best treated by hand (cf. Arbarello et cie [@Arbarello-Cornalba-Griffiths-Harris_1985-BOOK p.209–211] for a possible treatment, alas not perfectly self-contained).

The following summarizes the swampy situation (while trying to extend the generality):

[(To be clarified with percentages of truth)]{}

$\bullet$ \[100 %\] Any smooth real quintics $C_5$ with $r=5$ (hence $4$ ovals and one pseudoline) is unnested (otherwise the line through the nest plus another oval gives 6 intersections, corrupting Bézout).

$\bullet$ \[80 %\] Furthermore taking a pencil of conics through the $4$ nests gives a total pencil (why exactly? clear on the Fig.ref[FGuerN:fig]{}(left part) but why in general?).

$\bullet$ \[79 %\] Assuming the previous point, the gonality is $\gamma\le 2\cdot5-4\cdot 1= 6$ (in accordance with Gabard’s bound $r+p$, but it is preferable to mistrust this!).

$\bullet$ \[100 %=0 %\] Alas it is not clear a priori that pencils of orders $\ge 6$ do not induce total pencils of possibly lower degree $=5$. (Recall that $r=5$ is an absolute lower bound for total maps!)

The only delicate point is the assertion about total reality of the pencil of conics. In fact this is fairly evident on the figure above, but in the general case I see no reason.

\[10.11.12\] In the light of the Kempf-Kleiman-Laksov existence theorem of special divisors (ESD) in the case of complex curves one may wonder about its relativization in the Ahlfors context of total maps. The point is of course that for $g^1_d$’s the existence theorem (ESD) boils down to the Riemann-Meis bound $\gamma_{\Bbb C}\le [\frac{g+3}{2}]$ for the gonality of complex curves. (Plug $d\ge g/2+1$ in the Brill-Noether number $\rho$ and notice its non-negativity.) Since Ahlfors 1950 $\gamma\le r+2p$ or maybe Gabard 2006 $\gamma \le r+p$ is to be considered as the genuine bordered (or orthosymmetric) avatar of the Riemann-Meis theorem one can dream of an orthosymmetric(=dividing) version of the whole special of divisor theory. It is not clear how to extend total reality for higher series $g^r_d$ which are not pencils $g^1_d$. Of course one can ask that all real members are totally real but this seems too restrictive. Is there any example at all? Perhaps not for simple dimension reason. For $g^2_d$’s this would amount to a plane model of the curve cut by all real lines in real points only. This looks overambitious by just perturbing a tangent at a non-inflection point outside the sense of curvature.

At any rate the theory surely works for pencils and the bonus is that we have a certain variety akin to $G^1_d$ parameterizing all total pencils of degree $d$ on a given dividing curve. How to denote it? I never understood for what the “$g$ or $G$” of resp. $g^1_d$ or $G^1_d$ is standing? (Candidates: groups of points, Gerade, Gebilde, Grassmann, ?) Improvising notation, we define $T^1_d$ the variety of total linear series of degree $d$ on a given dividing curve. We dream about repeating all the phenomenology of the classic theory, cf. e.g. p.203 of ACGH 1985 [@Arbarello-Cornalba-Griffiths-Harris_1985-BOOK]:

“[*A genus $g$ curve depends on $3g-3$ parameters, describing the so-called moduli. Our goal is to describe how the projective realizations of a curve vary with its moduli, and what it means to say that a curve is general or special. Accordingly, we would like to know, what linear series can we expect to find on a general curve and what the subvarieties of the moduli space corresponding to curves possessing a series of specified type look like. \[…\] A natural question is, how can we tell one curve from another by looking at these configurations \[$G^r_d$\], or more precisely, what do these look like in general, and how—and where—can they degenerate?*]{}”

For our “totality” varieties $T^1_d$ of total pencils we would gather them into a “telescope” $T^1:=\cup_{d=1}^{\infty} T^1_d$ naturally embedded in $C^{(\infty)}$, the infinite symmetric power of the (dividing) curve $C$. We have the degree function $\deg\colon C^{(\infty)}\to {\Bbb N}$, and the image of $T^1$ is nothing but than the gonality sequence $\Lambda$ (Definition \[def:gonality-sequence\]), whose least member is the (separating) gonality $\gamma$ (of Coppens). One would like to understand how total pencils may degenerate to lower degrees w.r.t the natural topology induced by $C^{(\infty)}$. We probably get a sort of telescope with high strata attached to lower dimensional ones (like in a CW-complex) and the game would be to understand the geometry or combinatoric of this tower. Understanding all this is arguably the most refined form of Ahlfors theorem one could desire. One would then like to know not only the gonality spectrum telling one the dimension of each strata $T^1_d$, but also know how they can degenerate to lower strata. Degeneration could still be encoded combinatorially in a simplicial-complex $\Lambda^{\ast}$ with vertices $\Lambda$ (gonality sequence). Two vertices $d_1<d_2\in \Lambda$ are linked by an edge if a total $g^1_{d_2}$ can degenerate to a $g^1_{d_1}$. More generally $d_1<d_2<\dots<d_{k+1}\in \Lambda$ form a $k$-simplex whenever each integer of the sequence admits a representative $g^1_d$ degenerating to its immediate predecessor, hence to all predecessors.

Working out this explicitly looks tedious already for simple example. For the Gürtelkurve (any smooth quartic $C_4$ with 2 nested ovals) the variety $T^1_3$ is a circle and $T^1_4$ is a 2-cell attached to the former in a natural way. Of course when a total $g^1_4$ degenerates to a total $g^1_3$ it acquires a basepoint, which as to be deleted (particle destruction). Total $g^1_d$ will ultimately be denoted as $t^1_d$’s. In view of the Brill-Noether theorem (ESD) the variety $G^1_4$ has dimension $\ge
\rho=3-2(3-4+1)=3$ and so we have a priori more than the $\infty^2$ evident total pencils $t^1_4$ arising via projection from the inner oval. For instance pencils of conics may have degree as low as $2\cdot 4- 4 \cdot 1=8-4=4$. Can they be total? I would have guessed not, but it seems that they can. Compare Fig.\[FGuerN:fig\] below.

It would be desirable if some continuity principle can ensure total reality, e.g. if the 4 basepoints are distributed both inside and outside the nested resp. unnested oval. Then like a salesman traveller, the conic has to visit all 4 basepoints and thus creates at least $8$ real intersections. Our picture would just be the limiting position of such a bipartite pencil, and the variety $T^1_4$ would be $\infty^4$, a much larger dimension than initially expected. Further if 3, among the 4 basepoints, become collinear then it may be argued that the conics pencil specializes to one of lines (after removing the static line). All this remains to be better analyzed.

Heuristic moduli count to justify Ahlfors or Gabard (Huisman 2001)
------------------------------------------------------------------

It is still plausible that one may gain some evidence in favor of the Ahlfors circle map (either with Ahlfors $r+2p$ or preferably the improved Gabard’s bound $r+p$) by arguing via a moduli count. (The reader reminds to have discussed orally this option with Natanzon and Huisman in Rennes in Summer 2001, resp. December 2001.) I do not know if it is possible to supply a better count than the unrealistic one of the previous section.

\[14.10.12\] In fact at a time when I only conjectured the bound $r+p$, Huisman (December 2001 or 2002?) reacted instantaneously with a parameter count giving some evidence to the conjecture. Let me reproduce this faithfully from hand written notes.

We adopt the viewpoint of dividing real (algebraic) curves. So let $C$ be a such with $r\ge 1$ ovals and of genus $g$. I mentioned to Johannes Huisman the intuition that there is a totally real morphism $C\to {\Bbb P}^1$ (i.e. inverse image of real locus contained in the real locus) whose degree is the barycenter of $r$ and $g+1$, that is $\frac{r+(g+1)}{2}$. (The heuristic reason behind this 2001 intuition are given in Gabard 2006 [@Gabard_2006], and in its most primitive form in the previous Section\[sec:Sketch-of-Gabard\].) “Let us count parameters!”. Thus spoke Huisman, like Zarathustra.

First the Riemann-Hurwitz relation written for the Euler characteristic is $\chi(C)= d \cdot \chi({\Bbb P}^1)-b $, where $d$ is the degree and $b$ the number of branch points (with multiplicity). Now we count real moduli. The ramification divisor of any totally real morphism actually lies in the imaginary locus of the sphere (not on the equator), but is of course symmetric w.r.t. the involution. Hence we may imagine the $b/2$ branch points prescribed only in the north hemisphere, thus depending on $2\cdot (b/2)=b$ real constants. The curve itself depends on $b-3$ moduli (subtract the dimension of the automorphism group of ${\Bbb
P}^1$ defined over ${\Bbb R}$), that is $$\begin{aligned}
b-3= d \cdot \chi({\Bbb P}^1)-\chi(C)&=\frac{r+g+1}{2} \cdot
2-\chi(C)-3\cr
&=(r+g+1)-(2-2g)-3=3g-4+r\ge 3g-3.\end{aligned}$$ This prompts enough free parameters to sweep out the full moduli space. Of course this does not reprove the existence of circle maps of the prescribed degree, yet give some evidence to the assertion.

\[15.10.12\] A notable defect of this Huisman count is that it is a posteriori, giving no hint why the degree value should be given by our Ansatz. It is thus preferable to make the same computation in a more organical way. As above the curve $C$ depends on $b-3$ real moduli, and we demand $b-3\ge 3g-3$. This gives $d\cdot
\chi(S^2)-\chi(C)\ge 3g$, i.e. $2d\ge 3g+(2-2g)=g+2$, or $d\ge
g/2+1$.

Two remarks are in order. The above is exactly the same heuristic calculation as the that (going back to Riemann) to be found in Griffiths-Harris for the complex gonality of a curves, and which we remembered before. (The least integer $d\ge (g+2)/2$ is $[\frac{g+3}{2}]$, obvious for $g$ even and also obvious when $g$ is odd.) Hence in substance this modification of Huisman’s count truly just assert that Gabard’s bound is compatible with the gonality of the underlying complex curve, yet does not predict the bound $(r+g+1)/2$.

Perhaps there is a better way to count, compare the section devoted to Courant (Sec.\[sec:Courant\]).

Since Riemann 1857 we know that a complex curve of genus $g$ depends on $3g-3$ moduli. For real curves the same assertion holds true by virtue of Klein 1882 [@Klein_1882]. (The modernized treatment is of course due to Teichmüller 1939 [@Teichmueller_1939].) In view of this we may formulate the Ansatz of an (Ahlfors) circle map to the disc and try to compute the minimal number of sheet and the allied ramification points required to supply the branched Riemann surface with enough free parameters so as to paint out the full moduli space.

I tried recently to redo this computation, but was not very successful. So we leave this as an easy project to be clarified at the occasion.

Other application of the irrigation method (Riemann 1857, Brill-Noether 1874, Klein, etc.)
------------------------------------------------------------------------------------------

The method used in Gabard 2006 [@Gabard_2006] is primarily based upon an irrigation principle in a torus, which in turn is logically reducible to the surjectivity criterion via the (Brouwer) topological degree of a mapping to a manifold.

Via this method we obtained (in ) the existence of an (Ahlfors) circle map of degree $\le r+p$. As pointed out there, the method also supplies a purely topological proof of Jacobi inversion theorem, to the effect that the Abel-Jacobi mapping from the symmetric powers $C^{(d)}$ of a complex curve to its Jacobian is surjective as soon as dimension permits (that is for $d\ge g$).

Of course the complex (or closed) avatar of the Ahlfors mapping is just the mapping of a closed genus $g$ surface as a branched cover of the sphere. In this situation it is classically known since Riemann 1857 [@Riemann_1857] and Brill-Noether 1874 [@Brill-Noether_1874] (but disputed by the modern writers) that the most economical sheet number required is $[\frac{g+3}{2}]$.

Contributions on this problem is vast (and according to the modern consensus first [*rigorously*]{} proved in Meis 1960 [@Meis_1960] for linear series of dimension one, whereas some classic references includes the more case of arbitrary dimensional series, esp. Brill-Noether and Severi)

$\bullet$ Riemann 1857 (Theorie der Abel’schen Functionen) [@Riemann_1857 §4],

$\bullet$ Brill-Noether 1874 [@Brill-Noether_1874] (working with plane curves with singularities, so a pure algebraization of Riemann’s theory if one does not fell claustrophobic in the Plato cavern.)

$\bullet$ Klein’s lectures of 1891 [@Klein_1891--92_Vorlesung-Goettingen p.99] (based on Abelian integrals and Riemann-Roch, essentially akin to Riemann’s original derivation)

$\bullet$ Hensel-Landsberg 1902 [@Hensel-Landsberg_1902 Lecture 31] (probably quite similar to Brill-Noether or inspired by Dedekind-Weber)

$\bullet$ Severi 1921 [@Severi_1921-Vorlesungen-u-alg.-Geom-BUCH Anhang G]

Then the modern era begins with:

$\bullet$ Meis 1960 [@Meis_1960] (Teichmüller theoretic) \[alas, this monograph is notoriously difficult to obtain\] $\bullet$ H.H. Martens 1967 [@Martens_Henrik_1967] (no proof, but a remarkable study of the geometry assuming non-emptiness)

$\bullet$ Kempf 1971 [@Kempf_1971] the first existence proof (simultaneous with the next contributors) of special divisors in general case (linear series of arbitrary dimension, extending thereby the pencil case first established by Meis 1960)

$\bullet$ Kleiman-Laksov 1972–74 [@Kleiman-Laksov_1972] [@Kleiman-Laksov_1974] (using resp. Schubert calculus, plus Poincaré’s formula and resp. singularity theory à la Thom, Porteous)

$\bullet$ Gunning 1972 [@Gunning_1972] using MacDonald computation of the homology of the symmetric power of the curve

$\bullet$ Griffiths-Harris 1978 [@Griffiths-Harris_1978/94 p.261], where the heuristic count à la Riemann-Klein is reproduced; and latter a rigorous argument (p.358) is supplied (along the line of Kempf’s Thesis ca. 1970).

In view of the interest aroused by this Riemann-Meis bound, and the apparent difficulty to prove it (appealing to a variety of ingenious devices), it seems reasonable to wonder if there is not a much simpler argument based upon the same “irrigation method” as the one used by the writer in relation with the Ahlfors map. This would merely use simple homology theory and the allied surjectivity criterion in term of the Brouwer degree. Heuristically, this amounts to see the genus $g$ pretzel inside its Jacobian and let it homologically degenerate over a bouquet (wedge) of $g$ $2$-tori irrigating the Jacobian. Thus it seems evident that with roughly $g/2$ points we may find a pair of (effective) divisor of that degree collapsing to the same point of the Jacobian. This pair of disjoint divisors serves to define the desired morphism to ${\Bbb P}^1$. The writer as yet did not found the energy to write down the details, but is quite confident that the strategy is worth paying attention. Of course it could be the case that this merely boils down (up to phraseological details) to the already implemented attack of Gunning 1972 [@Gunning_1972]. (Shamefully, I did not yet had the time to consult this properly.) Thus, it seems rather obvious that this result of Riemann-Brill-Noether-Meis as well (at least the existence of a mapping with such a degree) permits a proof via the irrigation method employed in Gabard 2006 [@Gabard_2006]. Of course “irrigation” would not establish the sharpness of Meis’ bound (which is another question), but could predict its value as universal upper-bound upon the gonality.

Another application: Complex manifolds homeomorphic to tori
-----------------------------------------------------------

This section deviates from the mainbody of the text, but serves to illustrate another spinoff of the irrigation method. The writer wondered about the following naive question (ca. 2001/2?). Assume given a complex (analytic) manifold (arbitrary dimension), and suppose also the underlying manifold to be homeomorphic to a torus. [*Must such a manifold be biholomorphic to a complex torus, i.e. ${\Bbb C}^n$ modulo a lattice?*]{} The answer is easy in dimension one (Abel essentially). In general the answer is negative, by virtue of a construction of Blanchard (Thesis ca. 1955) closely allied to the Penrose twistor. Basically there is over $S^2$ a certain bundle parametrizing quaternionic structures, and taking a fiber product with an elliptic curve yields on the torus $T^6$ (of 6 real dimensions) a complex structure which turns out to be not Kähler. This answers negatively the question when the complex dimension is 3. (For more details cf. also work by Sommese (ca. 1978), etc.)

All this is rather exotic complex geometry, but one may wonder if the assertion becomes true under the Kähler assumption. Then Hodge theory applies, and we dispose of a bona fide analog of the Abel mapping (sometimes called the Albanese mapping). The latter is also a map to a complex torus (called Albanese) and using the irrigation principle it is easy to show that $\alpha$ induces an isomorphism on the top-dimensional homology. First, it induces an isomorphism on the $H_1$, but the latter elevates up to the top-dimension since tori have a total homology $H_{\ast}$ modelled upon the exterior algebra over the $H_1$. By the Brouwer degree argument (irrigation intuitively), it follows that $\alpha$ is surjective. Then one can show that it is injective as well (I have forgotten the exact argument, but essentially if Albanese collapse a submanifold then like by Abel it collapses linear varieties which are simply-connected projective spaces, hence liftable to the universal cover of the Albanese torus).

Any torus shaped Kähler manifold is biholomorphic to its Albanese torus.

Of course this is surely well-known, but we just wanted to remember this as another high dimensional—but baby—application of the irrigation principle. Further Kodaira’s classification of (complex analytic) surfaces plus a deformation argument of Andreotti-Grauert (which I learned from R. Narasimhan) implied also a positive answer to the basic question in (complex) dimension 2. But I take refuge in my failing memory, and to not remember the exact details. Thus in principle, Blanchard’s 3-dimensional counterexample is sharp.

Invisible real curves (Witt 1934, Geyer 1964, Martens 1978) {#sec:Witt}
-----------------------------------------------------------

Ahlfors’ theorem bears some analogy with Witt’s theorem (1934 [@Witt_1934]) stating that a (smooth) real curve without real points admits a morphism (defined over the reals ${\Bbb R}$) to the invisible real line (materialized by the conic $x_0^2+x_1^2+x_2^2=0$). The analogy is again that when there is no topological obstruction, then a geometric mapping exists.

Subsequent works along Witt’s direction are due to:

$\bullet$ Geyer 1964/67 [@Geyer_1964-67] (alternative proof of Weichold, and Witt via Galois cohomology and Hilbert’s Satz 90);

$\bullet$ Martens 1978 [@Martens_1978], where the precise bound on the degree of the Witt mapping has been determined.

Philosophically, it seems challenging to examine if such strongly algebraic techniques (Riemann-Roch algebraized à la Hensel–Landsberg 1902, Artin, etc.) are susceptible to crack as well the Ahlfors mapping? Geyer, Martens or others are perhaps able to address this challenge? (So far as we know, no such account exist in print.)

Martens’ statement (quantitative version of Witt) is the following.

Given a closed non-orientable Klein surface with algebraic genus $g$ (i.e. the genus of the orientable double cover[^12]) there is a morphism to the projective plane of degree $\le
g+1$. Moreover this is the best we can hope for, i.e. for each $g$ there is a Klein surface not expressible with fewer sheets.

Perhaps the first portion of the statement is already in Witt 1934 [@Witt_1934]. Of course this can—via the Schottky-Klein Verdoppelung—also be stated in term of symmetric Riemann surfaces (equivalently real algebraic curves) as follows:

Given a symmetric Riemann surface of genus $g$ without fixed point, there is an equivariant conformal mapping to the diasymmetric sphere of degree $g+1$. Moreover the bound is sharp.

This formulation of Martens’s result also appears in Ross 1997 [@Ross_1997 p.3097], who supplies additional comments which are quite in accordance with our own sentiments, especially the issue that the short argument by Li-Yau 1982 [@Li-Yau_1982 p.272] does not appear as very convincing. Moreover Ross supplies some attractive differential geometric applications of this Witt-Martens mapping theorem, e.g. to the effect that the totally geodesic ${\Bbb R}P^2$ is the only stable minimal surface in ${\Bbb R}P^3$.

The three mapping theorems (Riemann 1857, Ahlfors 1950, Witt 1934)
------------------------------------------------------------------

From the conformal viewpoint we have thus three basic mapping theorems enabling a gravitational collapse of all compact surfaces to their simplest representatives (the sphere, the disc or the projective plane) depending on whether the original surface is:

$\bullet$ closed orientable (Riemann 1857 [@Riemann_1857]);

$\bullet$ compact bordered orientable (Ahlfors 1950 [@Ahlfors_1950]);

$\bullet$ closed non-orientable (Witt 1934 [@Witt_1934]).

None of those results tells what to do with a compact bordered non-orientable surface (whose simplest specimen is the Möbius band/strip). The latter does not carry positive curvature, which implies finiteness of the fundamental group for complete metrics (else punctured sphere). Alternatively the orientable double cover of Möbius is the torus, which has already moduli. Hence it is quite clear that the above three theorems form an exhaustive list of truths positing a fundamental trichotomy (of definitive crystallized shape). The motto “Alle guten Dinge sind drei”, is quite ubiquitous in life and mathematics! It is reasonable to expect that each of those mappings will pursue to find valuable applications in the future, yet much work remain to be done as to the stratification of the moduli space induced by the degree of such representations, etc.

For each of these 3 concretization problems one is interested in the exact determination of the lowest possible sheet number. In principle the answer is already known as follows:

For all $3$ types of conformal mapping to elementary surfaces of positive Euler characteristics $\chi >0$ (including $\chi (S^2)=2,
\chi (\Bbb R P^2)=1, \chi (\Delta=D^2)=1$) the sharp universal bound on the degree of such representation is known. More precisely,

$\bullet$ $[\frac{g+3}{2}]$ always concretizes closed genus $g$ surfaces expressed as cover of the sphere [(Riemann, Meis 1960 [@Meis_1960])]{}, and the bound is sharp [(again Meis 1960 [@Meis_1960])]{}.

$\bullet$ $g+1$ always concretizes non-orientable closed surface of algebraic genus $g$ (i.e. genus of the orientation double cover) expresses as cover of the projective plane [(Witt 1934 [@Witt_1934])]{}, and the bound is sharp [(Martens 1978 [@Martens_1978])]{}.

$\bullet$ $r+p$ always concretizes bordered orientable surfaces with $r$ contours and $p$ handles as (full or total) cover of the disc [(Gabard 2006 [@Gabard_2006])]{}, and the bound is sharp [(Coppens 2011 [@Coppens_2011])]{}.

Adhering to Klein’s viewpoint of symmetric surfaces, one can always interpret such objects as real curves of some genus $g$ (the first class is an exception except if one tolerates disconnected surfaces). In the third bordered case $g=(r-1)+2p$. The $r+p$ bound can be rewritten as $\frac{r+(g+1)}{2}$. If $r$ is lowest, i.e. $r=1$, this is statistically equal to $g/2$, as so is the first Riemann-Meis bound. In contrast the Witt-Martens bound looks much higher. Of course if $r=g+1$ is highest (Harnack-maximality) then $r+p=r+0=g+1$, agreing with Witt-Martens’s bound. In the overall it may be argued that both Martens’ and Gabard’s bound are fairly less economical that Riemann-Meis’, and that this is due to the equivariance or even total reality of the corresponding maps. On the other hand Ahlfors bound $r+2p=g+1$ looks much more compatible with Martens’ and if one is sceptical about Gabard’s version one could imagine that Ahlfors is asymptotically sharp for large values of the invariants. This scenario remains hypothetically possible in case we are unable to reassess through other mean Gabard’s $r+p$ or able to disprove its validity.

The following tabulation summarizes the key contributions:

\(1) Riemann 1857: any (or at least the general) closed Riemann(ian) surface maps conformally to the sphere with $\le
[\frac{g+3}{2}]$ sheets, where $g$ is the genus. It is not clear-cut if Riemann showed sharpness of the bound.

Related works includes (in chronological order):

$\bullet$ Brill-Noether 1874 [@Brill-Noether_1874];

$\bullet$ Klein 1891 [@Klein_1891--92_Vorlesung-Goettingen p.99];

$\bullet$ Severi 1921 [@Severi_1921-Vorlesungen-u-alg.-Geom-BUCH];

$\bullet$ B. Segre 1928 [@Segre_1928];

$\bullet$ Meis 1960 [@Meis_1960];

$\bullet$ Kempf 1971 [@Kempf_1971] and Kleiman-Laksov 1972–74 [@Kleiman-Laksov_1972] [@Kleiman-Laksov_1974];

$\bullet$ Gunning 1972 [@Gunning_1972];

$\bullet$ Griffiths-Harris 1978 [@Griffiths-Harris_1978/94 p.261];

$\bullet$ Arbarello-Cornalba 1981 [@Arbarello-Cornalba_1981].

This sharp bound $[\frac{g+3}{2}]$ as applied to spectral theory is observed in El Soufi-Ilias 1983/84 [@El-Soufi-Ilias_1983/84] (Yang-Yau 1980 [@Yang-Yau_1980] contented themselves with the weaker value $g+1$.) An interesting aspect of the Italian works is that they not only focus on the gonality upper bound, but also compute the dimensions of the lower dimensional strata for a prescribed gonality. Of course, the answer is the expected one (as easily predicted by Riemann-Hurwitz). \[The above Italian works, especially Segre has however a little objection to the simplicity of the exercise.\] We point out this is issue as it could be interesting to make a similar count for the Ahlfors circle map (bordered case). This topic will be briefly addressed in the next Sec.\[sec:profile-histogram\].

\(2) Ahlfors 1950 [@Ahlfors_1950]: any compact bordered Riemann surface maps conformally to the disc with $\le r+2p$ sheets (where as usual $r$ is the number of boundary contours and $p$ the genus). This bound is not sharp (at least for low values of the invariants $(r,p)$, e.g. for the Gürtelkurve type $(r,p)=(2,1)$). Modulo a mistake by the writer (in Gabard 2006 [@Gabard_2006]), Ahlfors bound can be improved as $\le r+p$. The latter is in turn sharp according to Coppens 2011 [@Coppens_2011].

\(3) Witt 1934 [@Witt_1934]: any closed non-orientable surface maps conformally to the projective plane ${\Bbb R}P^2$. Witt does not specify a bound (?), or maybe he does but sharpness was obtained by Martens 1978 [@Martens_1978].

Witt’s result received, arguably, only sporadic spectral applications, except in the article Li-Yau 1982 [@Li-Yau_1982], which however does not quote Witt, but whose authors were apparently able to reprove the result by their own \[compare their argument on p.272\]. (As already mentioned, Ross 1997 [@Ross_1997] does not seem to be convinced by the Li-Yau argument.)

Of course all this “diaporama” is the direct heritage of Riemann (plus maybe indirectly some Abel!) the first result being often called Riemann’s existence theorem. The 2 avatars of Ahlfors and Witt are akin to the absolute case of Riemann, via the trick of the Schottky-Klein double (or Verdoppelung as Teichmüller calls it) but then some equivariance or total reality is required, acting as a sort of boundary condition explaining probably why those versions took longer to emerge. Of course such equivariance/or boundary behaviors just hide a reality condition encoded in the field of definition of the allied Riemann surfaces. All this is best summarized diagrammatically:

$$\begin{aligned}
i(p^*,{\cal F}^*)=1-\frac{c_*-c_*'}{4}&=1-\frac{2c-2c'}{4}\cr
&=2\bigl( 1-\frac{c-c'}{4}\bigr)-1
=2j(p, {\cal F})-1.\end{aligned}$$

The gonality profile, moduli strata and the Ahlfors space {#sec:profile-histogram}
---------------------------------------------------------

\[10.11.12\] Heuristically (cf. e.g. Segre 1928 [@Segre_1928]) one can count the dimension of closed Riemann surfaces expressible as coverings of degree $d$ of the sphere as follows. By Riemann-Hurwitz $\chi(C_g)=d \chi(S^2)-b$. Hence there are $b-3$ free parameters, that is $$b-3=(2d-\chi)-3=2d-(2-2g)-3=2d+2g-5.$$ In particular the strata of given gonality $\gamma=d$ increases its dimension along a very simple arithmetic progression (as a function of $d$) until the full moduli space is exhausted for $d$ the least integer $\ge g/2+1$ (Riemann-Meis bound). The smallest strata is the [*hyperelliptic locus*]{} ($d=2$) of dimension $4+2g-5=2g-1=(2g+2)-3$, in accordance with the $2g+2$ ramification points visible as invariant points of an half twist acting upon a purely vertical pretzel in 3-space. I do not know if such a regularity occurs for bordered surfaces. Coppens’s theorem states another form of regularity, namely full realizability of all intermediated gonalites, but it does not pertain to the dimensions of the corresponding moduli strata.

On behalf of Coppens’s theorem the situation could be as follows. For a given topological type $(r,p)$, Coppens tells us that all intermediate $r\le \gamma \le r+p$ are realized. So we have $p+1$ possible gonalities, the largest of which $\gamma=r+p$ fills the full moduli space of real dimension $3g-3$ (Klein’s count conjointly with Gabard’s bound). As usual $g=(r-1)+2p$, so $3g-3=6p+3r-6$. If we knew the number of moduli of the minimal strata $\gamma=r$ we could try a linear interpolation as a possible scenario for the dimensions increments of the gonality strata. Naively our rotationally invariant picture (Fig.\[Chambery:fig\]) could act as a bordered substitute to the hyperelliptic closed case (at least for $r$ even). If so is the case can we count its moduli? Everything would be determined by the quotient planar surface with $r/2=r'$ contours. This planar surface (whose double has genus $g'=r'-1$) depends on $3g'-3$ moduli. This expressed in terms of $r$, gives he following $3g'-3=3r'-6=3/2 \cdot r-6$. This a candidate for the dimension of the lowest strata. Looking for a progression in $p$ steps toward the maximum value, we consider the difference $[6p+3r-6]-[3/2
\cdot r-6]=6p+3/2\cdot r=1/2[12p+3r]$, which is not easily divided by $p$. $\bullet$ In fact we have looked at the quotient but barely omitted the branched locus. Taking this into account we get rather a dependance on $3g'-3+2(2p+2)$ (real) moduli for the lowest strata. Expressing this in terms of $(r,p)$, gives $3/2
\cdot r+4p-2$. Hence the difference of the top and lowest strata would be $2p+3/2\cdot r-1$, which is alas still not nicely divisible by $p$. $\bullet$ Another idea is just to use maps from $F_{r,p}$ to the disc of minimum degree $r$. Then we have $\chi(F)=r \chi(\Delta)-b$. Hence there are $2b-3$ free real parameters. Expressed in terms of $(r,p)$, this is $2b-3=2(r-\chi)-3=2(r-(2-2p-r))-3=4r+4p-7$. Hence the difference between the top dimensional and the lowest dimensional strata is $\delta=(6p+3r-6)-(4r+4p-7)=2p-r+1$, which is not even positive in general. It looks again dubious to divide this in $p$ equal parts as suggests Coppens result. Again this just confirms what we already noticed (earlier in the text) that the Riemann-Hurwitz count looks seriously jeopardized in the bordered case, at least as long as we apply it so naively as we do.

One can reverse the game: instead of speculating on the size of the lowest strata we can speculate on the increment as being by 2 real units (like in the complex case) and draw the dimension $\lambda$ of the lowest strata. This would give $\lambda=\dim
({\rm top \;strata})-p\cdot 2=(6p+3r-6)-2p=4p+3r-6$. Testing this on the type of the Gürtelkurve $(r,p)=(2,1)$ gives $\lambda=4+6-6=4$, whereas the hyperelliptic model depends on $2g+2-3=2\cdot3+2-3=5$ real parameters. Hence the later has codimension 1 in the full moduli of the Gürtekurve type, which as dimension $3g-3=3\cdot 3-3=9-3=6$. This motivates modifying the increment to one of only 1 unit. This leads to the following Ansatz: $\lambda=5p+3r-6$. This gives for $(r,p)=(2,1)$, $\lambda=5+6-6=5$ the correct number. But if we look at the type $(r,p)=(2,2)$ we get $\lambda =10+6-6=10$; but on the other hand the hyperelliptic models have $2g+2-3=2\cdot 5+2-3=9$ moduli conflicting the new Ansatz for $\lambda$.

Of course the real scenario about the increments might be pretty more complicated than the linear progression observed in the complex case (corresponding to closed Riemann surface). Another more neutral way to look at the question is as follows. Given is $(r,p)$ a pair of integers. Allied to this there is a moduli space ${\cal M}_{r,p}$ of all bordered (Riemann) surfaces of type $(r,p)$. Its dimension is $3g-3$ (Klein 1882 [@Klein_1882]), where $g$ is the genus of the double. We imagine the range of all possible gonalities $r\le\gamma\le r+p$ as a horizontal array of entries above each of which is reported the dimension of the moduli space of curve having gonality $\le
\gamma$. This is depicted as a vertical bar. At first, only the top dimension attached to $\gamma=r+p$ is known as $3g-3$. By Coppens we know that there will be $p$ descents of this highest bar to the lower gonalities between $r$ and $r+p$. Pause at this stage to notice that assigned to the sole data $(r,p)$ there is assigned unambiguously such a histogram of gonalities (cf. Fig.\[Histogr:fig\]).

One special case in which we can hope to be more explicit regarding the lowest strata is when $r$ equals 1 or 2. In this case we know that the moduli space contains hyperelliptic membranes. Assuming $p$ large enough ($p\ge 1$) the lowest gonality is $\gamma=2$. It is tautological that the hyperelliptic locus has this gonality, and conversely. So we control explicitly the dimension of the lowest strata. We find $(2g+2)-3$ real constants. Thus the dimension difference $\delta$ of the top and lowest strata is $\delta=3g-3-[(2g+2)-3]=g-2$. This rewritten in terms of $(r,p)$ is also $g-2=(r-1)+2p-2=r+2p-3$.

$\bullet$ If $r=1$, this gives $\delta=2p-2=2(p-1)$. Positing linearity of the increment, this ought to be divided in $p-1$ equals parts (since $r=1$ itself is not a gonality when $p\ge 1$), and we get exactly a progression by 2 units. Hence under the Ansatz of linearity the histogram would be completely known.

$\bullet$ If $r=2$, this gives $\delta=2p-1$. Assuming linearity of the increment, this ought to be divided in $p$ equals parts, and we get something like a progression by 2 units. However the non-divisibility implies that in this case it is impossible to have a linear progression of the histogram. Hence some jumps must occurs.

So in these cases there is some hope to be completely explicit about the histogram attached to $(r,p)$. It would essentially suffices to decide where occur some irregular jumps.

Let us formalize a bit. Given a pair of integers $(r,p)$, we have a moduli space ${\cal M}:={\cal M}_{r,p}$ of all bordered Riemann surfaces of type $(r,p)$. (To allege notation with omit the indices $(r,p)$, as the topology is fixed once for all.) Its dimension is invariably $3g-3$, where $g=(r-1)+2p$ is the genus of the double.

\[gonality-profile:def\] [Inside the full moduli space ${\cal M}:={\cal M}_{r,p}$, consider the sublocus $M_d$ of all surfaces with gonality $\gamma\le d$, and let $\mu_d=\dim M_d$ be its dimension. The histogram we were speaking about is essentially the function $d\mapsto \mu_d$, which we call the [*gonality profile*]{}.]{}

It is evidently monotone but a priori not strictly. Misinterpreting Coppens’s result one would guess strict monotony, but Coppens states only that each gonality is exactly realized, hence in symbols that $M_d-M_{d-1}$ is non-void (at least for $d$ in the range $[r,r+p]$). Thus a priori it could be the case that when incrementing the parameter $d$ we get new surfaces but their variety is not of larger dimension. Of course this scenario may look a bit unlikely due to the algebro-geometric character of the whole topic, but I do not know an argument. The domain of our function $d\mapsto \mu_d$ is the set of all integers but the interesting range is $[r, r+p]$ at least taking Gabard for granted. The latter amounts to say that $\mu_{r+p}=3g-3$.

Now if $r=1$ or $2$, then the moduli space ${\cal M}={\cal
M}_{r,p}$ contains hyperelliptic representatives, and the latter exhaust the locus $M_2$. We calculate easily $\mu_2=(2g+2)-3=2g-1$ and deduced the difference $\delta=\mu_{r+p}-\mu_2=(3g-3)-(2g-1)=g-2$. From here we inferred that:

$\bullet$ when $r=1$ (and $p\ge 1$) then $r$ itself is not a gonality and so there is really only $p-1$ descents. Since $\delta=g-2=r+2p-3=2p-2=2(p-1)$, we can divide (without rest) this by the number of $p-1$ descents, to get a statistical increment of 2 units. If one believes in the linearity regularity then the histogram would be completely known in that case.

$\bullet$ when $r=2$ then $r$ is a gonality, and we have exactly $p$ admissible descents along the range $[r,r+p]$. Now $\delta=g-2=r+2p-3=2p-1$, which is not divisible by $p$. We infer an obstruction to the scenario of linearly evolving histogram. (In other words the function is not linear on the segment $[r, r+p]$.) Perhaps it is just doing a gentle seesaw at some early place?

At this stage we may have exhausted all what can be said on trivial arithmetical grounds. Going further probably requires some geometric impetus, like looking at explicit models (extending the hyperelliptic case). So one needs probably to describe large families of $d$-gonal surfaces for $d\ge 2$. If a general result describing the gonality profile $d\mapsto \mu_d$ looks out of reach, one can start examining low values of $(r,p)$ to explore the situation.

\[11.11.12\] [*Examples.*]{}—$\bullet$ E.g. for $(r,p)=(2,1)$ (thus $g=3$) (the Gürtelkurve type) then the profile is completely known, namely $\mu_2=5$ (hyperelliptic locus of dimension $(2g+2)-3$) and $\mu_3=6$ (equal to $3g-3$). Of course Gabard’s $\gamma\le r+p$ follows in this case via the canonical embedding realizing the curve as a Gürtelkurve in ${\Bbb P}^2$.

$\bullet$ For $(r,p)=(2,2)$ (thus $g=5$), we have again the hyperelliptic locus giving $\mu_2=(2g+2)-3=9$. The top locus $M_{r+p}=M_4$ has dimension $\mu_4=\mu_{r+p}=3g-3=15-3=12$ (Gabard is used but maybe there is an argument by hand). What about $\mu_3$? To seek an answer we refer back to the table of Fig.\[Coppens:fig\], where we traced a picture (label 223) of an uninodal quintic with gonality $\gamma=3$. Quintics depends on $\binom{5+2}{2}-1=\frac{7\cdot 6}{2}-1=20$ parameters, but modulo the collinearity group $PGL(3)={\rm Aut}({\Bbb P}^2)$ of $3^2-1=8$ dimensions, this boils down to 12 effective parameters. Of course the uninodal quintic we consider is really compelled to live on the smaller discriminant hypersurface of dimension $19$ and so our curve 223 truly depends on only 11 essential parameters. Assuming that a full neighborhood of curve $223$ consists of curves keeping the same gonality $\gamma=3$ suggests therefore the value $\mu_3=11$ (at least as a lower bound). Observe that the picture 223 is total under a pencil of lines, and it seems reasonable to expect that when the curve is slightly perturbed along the discriminant hypersurface, total reality of the pencil persists on the ground of some topological [*stability*]{}. Remember e.g., that total reality amounts to the transversality of the foliation (induced by the pencil) along the curve, and transversality is the mother of any topological stability (Thom-style philosophy). Note of course that our curve (being uninodal) represents actually a smooth point of the discriminant and so we safely dispose of the required parameters of deformation. This is perhaps worth saying if one remembers certain plane cubics (or even conics) as examples of real algebraic varieties having an isolated real point. Maybe the above stability argument adapts to situations where there are several nodes via Brusotti’s theorem describing the infinitesimal structure of the discriminant near a multi-nodal curve (with say $\delta$ nodes) as an union of smooth branches crossing transversally (normal crossing).

$\bullet$ For $(r,p)=(3,2)$ (thus $g=6$), we have no hyperelliptic locus. The top locus $M_{r+p}=M_5$ has dimension $\mu_5=\mu_{r+p}=3g-3=18-3=15$ (Gabard is used but maybe there is an argument by hand). What about $\mu_3$ and $\mu_4$? We look again back to Fig.\[Coppens:fig\], where we find curve 324. This is merely a smooth quintic with 2 nested ovals hence with gonality $\gamma=4$. Remember that smooth plane $m$-tics have in general complex gonality $(m-1)$. As quintics depends on 12 essential parameters, the above stability argument shows that the strata $M_4$ contains the locus of all such quintics, and we infer $\mu_4\ge 12$. Is this an equality? How to estimate $\mu_3$? Due to time limitation, we have to leave all this (in our opinion) exciting topic at a fragmentary stage. Perhaps a last word, if we use picture 324bis (still on Fig.\[Coppens:fig\]), which is a sextic with 4 nodes also having $\gamma\le 4$ (projection from the node), then we get a model depending on $\binom{6+2}{2}-1=\frac{8\cdot 7}{2}-1=27$ constants, minus the $8$ coming from $PGL(3)$ gives $19$, of which must be subtracted 4 units (using Brusotti’s normal crossings description). The final result is 15. Repeating the above stability argument implies that $\mu_4\ge 15$. This is a much stronger lower estimate, which in fact must be an equality since we have already attained the dimension of the full moduli space. Hence we conclude $\mu_4=15$; strikingly as big as $\mu_5$! This answer is quite intriguing in case it is correct at all? It would show that the gonality profile does not need to be strictly increasing! \[11.12.12\] Alas all of our counts are completely erroneous in view of some basic examples shown in the next section. Of course the mistake is that not all birational (conformal) equivalence giving rise to Riemann’s moduli space need to be induced by a collineation as an automorphism of the ambient plane ${\Bbb P}^2$.

\[11.11.12\] Finally, it is perhaps fruitful to keep a view on the space of all (total) circle maps. This is the [*Ahlfors space*]{} (improvised jargon) quite akin to so-called Hurwitz spaces. All what we were concerned with in this subsection is arguably just a shadow of this larger space dominating the moduli space ${\cal
M}_{r,p}$. Precisely, the [*circle maps (or Ahlfors) space*]{} $C_{r,p}$ consists for a fixed pair $(r,p)$ (number of contours and handles resp.) of all circle maps $f\colon F \to \Delta$ on a “variable” bordered Riemann surface of specified topological type $(r,p)$. Forgetting the circle map $f$ induces a natural map $C_{r,p}\to {\cal M}_{r,p}$ to the moduli space. Of course one must consider the space $C$ modulo the equivalence relation of a conformal diffeomorphism commuting with the maps to the disc. The strata $M_d$ of all surfaces of gonality $\gamma\le d$ appear then as the projections of the fibres of the degree function on $C_{r,p}$. The fibre of the map $C\to M$ (indices omitted) is the space of all total maps on a fixed bordered Riemann surface $F$.

Correcting the previous section
-------------------------------

\[11.12.12\] There are many counterexamples to our naive moduli count via plane nodal models. For instance considering curves of $g=2$, and using the projective realization as a quartic with one node, we get the dependence upon $\dim \vert 4H
\vert=\binom{4+2}{2}-1=\frac{6\cdot 5 }{2}-1=14$ parameters, of which must be subtracted one unit to be on the discriminant (due to the presence of the node) and finally one has to remove the $8$ dimensions of $PGL(3)$. The end result is $14-1-8=5$, which exceeds by $2$ units Riemann’s $3g-3=3$. Of course this excess is due to the fact that we moded out only by (linear) automorphisms of the plane whereas there might be more mysterious birational equivalence relating to configurations of our family of uninodal quartics.

This demonstrates that the estimate we got in the previous section are completely erroneous and unreliable, and one must find some completely new ideas (i.e. old stuff à la Riemann) to really penetrate the intrinsic nature of the problem. For the moment I have no idea on how to attack the problem of describing the size (=dimensions) $\mu_d$ of the varied gonality strata $$M_d=\{ F\in {\cal M}_{r,p} : \gamma (F)\le d \}.$$ Let us try anew to attack this problem of describing the gonality profile $d\mapsto \mu_d$ for each pair $(r,p)$.

\[12.12.12\] First complete information is obtained in the easy case of planar membranes ($p=0$) as a consequence of the Bieberbach-Grunsky theorem (Lemma \[Enriques-Chisini:lemma\]).

When $p=0$ the gonality profile is a skyscraper concentrated at the single place $d=r=r+p=r+2p$, i.e. $\mu_r=3g-3$, where $g=(r-1)+2p=r-1$ is the genus of the double.

This follow at once from the trivial lower bound $r\le \gamma$ on the degree of circle maps (or the allied gonality $\gamma$), and the Bieberbach-Grunsky theorem (Lemma \[Enriques-Chisini:lemma\]). (Notice that neither Ahlfors 1950 [@Ahlfors_1950] ($\gamma \le r+2p$), nor Gabard 2006 [@Gabard_2006] ($\gamma \le r+p$) is required.)

After that let us examine the cases with $p=1$. We start with:

$\bullet$ $(r,p)=(1,1)$: then we have $1=r\le \gamma\le r+p=2$ (using Gabard). However since the genus of the double is $g=(r-1)+2p=2$, the curve is hyperelliptic and we may avoid Gabard. The lower bound $r=1$ cannot be realized, since $p\neq 0$. We deduce:

For $(r,p)=(1,1)$, the gonality profile is a skyscraper concentrated at $d=2$, i.e. $\mu_d=3g-3=3$ for $d=2$ and $\mu_d=0$ elsewhere $(d\neq 2)$.

$\bullet$ $(r,p)=(2,1)$ (with $g=(r-1)+2p=3$): then we have $2=r\le \gamma\le r+p=3$ (using Gabard). However one can dispense Gabard by using the canonical embedding taking the double of the bordered Riemann surface to a Gürtelkurve $C_4 \subset {\Bbb
P}^2$, i.e. a quartic with 2 nested ovals. This proves $\gamma \le
3$ (via projection from the inner oval). So $\mu_3=3g-3=6$. Of course we have also a hyperelliptic locus, whose dimension is $(2g+2)-3=5$, so $\mu_2=5$. This proves the:

For $(r,p)=(2,1)$, the gonality profile is a “twin tower” concentrated at two places $d=2,3$, and $\mu_3=3g-3=6$ whereas $\mu_2=5$ (all other $\mu_d$ are zero).

$\bullet$ $(r,p)=(3,1)$ (with $g=(r-1)+2p=4$): then we have $3=r\le \gamma\le r+p=4$ (using Gabard). Without using Gabard, one can look at the canonical model in ${\Bbb P}^{g-1}={\Bbb P}^{3}$ of degree $2g-2=6$. This is probably a complete intersection of a cubic surface with a quadric, weighting bidegree $(a,b)=(3,3)$ on the latter, hence of genus $g=(a-1)(b-1)=2\cdot 2 = 4 $ (the expected value). One can then draw a picture by smoothing two pairs of 3 lines distributed in each ruling. When the lines are oriented in the most trivial way (each inducing the same integral homology class on the torus ${\Bbb P}^1({\Bbb R})\times {\Bbb
P}^1({\Bbb R})$) we get a total map of degree $3$ by projection on the factors of ${\Bbb P}^1\times {\Bbb P}^1$ (do a picture). Taking (somewhat cavalier) Gabard for granted we get $\mu_4=3g-3=9$. How to estimate $\mu_3$? Let us try several strategies:

\(1) [*Extrinsic plane projective realizations*]{}.—The naive idea is to look at Fig.\[Coppens:fig\] (picture 313). This is a quintic with 2 nodes and $\gamma\le 3$ (hence equal to $3$ by the trivial lower bound $r\le \gamma$). If we count the (naive) moduli of such a curve we obtain: $\dim \vert 5H
\vert=\binom{5+2}{2}-1=\frac{7\cdot 6}{2}-1=20$, of which must be subtracted $2$ for the two nodes, and $8=\dim PGL(3)$ to get $10$. This exceeds by one unit the full moduli space $3g-3$, and so we get an alienating count. Of course as already said the reason is that we only took into account linear collineations (ambient automorphisms) whereas one should mod out by all inherent isomorphisms of the family of curves. One way to remedy the situation would be to look at Cremona transformations (birational transformations of the plane), but it is not even evident that this would give the right answer on abstract moduli. Another idea is to look at higher order plane models with $\delta$ many nodes as to adjust the genus to $g=4$. For instance sextics with 6 nodes $C^6_6$, septics with 11 nodes $C_7^{11}$, octics with 17 nodes $C_8^{17}$, etc. However the same calculation shows that such family of curves depends on $\dim C^6_6=\frac{8\cdot
7}{2}-1-\delta-8=13$, $\dim C^{11}_7=\frac{9\cdot
8}{2}-1-\delta-8=16$, $\dim C^{17}_8=\frac{10\cdot
9}{2}-1-\delta-8=19$, etc. It seems that there is perpetually an increment by 3 units, and never get something realistic via naive counting.

\(2) [*Extrinsic projective realization as a branched cover of the line (or the disc), i.e. circle maps via Riemann-Hurwitz.*]{}—We fix as an Ansatz $\gamma=3$ (inside our fixed topological type $(r,p)=(3,1)$), and by the Riemann-Hurwitz relation applied to a circle map $F \to \overline{\Delta}$ of degree $3$, we find $\chi (F)= 3 \chi (\overline \Delta)-b$, so $b=3-\chi (F_{3,1})=3-(-3)=6$ many branch points. Moving those 6 points arbitrarily in the disc, and quotienting by automorphisms of the disc we arrive at $2\cdot 6 - 3=9$ (real) moduli. This looks again anomalous for we receive the same answer as for the full $3g-3$ moduli. (We already experimented this failure of Riemann-Hurwitz in the bordered setting, and we are in depressive mode.) The mystery is perhaps that we do not enjoy complete freedom in moving branch points in the bordered setting, but I lack any understanding of which sort of geometric restrictions have to be taken into account.

\(3) [*Intrinsic count à la Nielsen-Fenchel.*]{}—Another possible strategy, is to adapt the Nielsen-Fenchel count of moduli via a decomposition in pants. Remember that this works at the perfection to predict the dimension of the full moduli space (cf. e.g. our Sec.\[Nielsen-Fenchel:sec\] below). The idea would be that if we prescribe a lower gonality then an appropriate decomposition in pants (somehow calibrated on the circle map) should predict the moduli dimension of the restricted class too.

Alas for $\gamma=3$, I do not really see how to proceed, but let us first experiment the method on the simpler hyperelliptic case.

Consider e.g. a membrane with $(r,p)=(2,p)$, $p$ arbitrary, of gonality $\gamma=2$ (hyperelliptic case). On drawing the configuration, and decomposing it into pants invariant under the hyperelliptic involution (visualized as a half-twist rotation) we obtain Fig.\[Pants-hyper:fig\] (left part).

0 -5pt0

Introducing on the surface its uniformizing metric of constant curvature $-1$ (alias hyperbolic metric), we count moduli as the lengths of loops bounding pants affected by certain twist parameters. We get (reading contributions from the top to the bottom of the figure): $$2+2 \cdot 2 p-1=4p+1$$ free parameters. Indeed the first term ($2$) arises from the top loop (its length plus its twisting aptitude). Next we see $p$ shaded pants whose contours exhaust all junctures of the pants decomposition. However all bottom parts of the shaded pants are permuted via the hyperelliptic involution (half-turn rotation), hence of the same length. So each shaded pants really contributes for $2$ lengths each susceptible of a twist, whence the second term ($2 \cdot 2 p$). As to the last term ($-1$), notice that the very bottom contours of the surface have no gluing companion (to be twisted with), so one unit must be subtracted. The announced count follows.

On the other hand, such hyperelliptic curves depend (via a count à la Riemann-Hurwitz) on $(2g+2)-3$ parameters where $g=(r-1)+2p=2p+1$. Hence on $(2g+2)-3=[2(2p+1)+2]-3=4p+1$, in accordance with the result as calculated via the pants method.

A similar count works for hyperelliptic membranes with $(r,p)=(1,p)$, cf. right part of Fig.\[Pants-hyper:fig\]. In that case we obtain $$2+2\cdot 2(p-1)+1=4p-1$$ moduli, and on the other hand $(2g+2)-3=[2(2p)+2]-3=4p-1$ parameters. Both counts are again in accordance.

Of course we could even dream that the pants dissection method (cf. Fig.\[Pants:fig\], right part) affords yet another full proof of Ahlfors circle maps, but this looks a bit tricky to implement. Perhaps even more ambitious one could hope that pants dissection affords a proof of the Forstnerič-Wold 2009 [@Forstneric-Wold_2009] desideratum that each finite bordered surface embeds holomorphically in ${\Bbb C}^2$. (Notice that this is much stronger (viz. complex analytic) than the conformal embedding in $E^4$ prompted by the Garsia-Rüedy-Ko theory as implemented in Ko 1999 and 2001 [@Ko_2001]. At least the latter shows that there is no conformal obstruction to the Gromov conjecture/question (1999 [@Gromov_1999]) that any Riemannian surface should isometrically embed in $E^4$.)

Coming back to our problem of calculating $\mu_3$ for membranes of type $(r,p)=(3,1)$ we severely lack any reliable technique of calculation.

Existence of Ahlfors maps via the Green’s function (and the allied Dirichlet principle) {#Green:sec}
=======================================================================================

All what follows is extremely classical, yet the writer confesses to have assimilated (the first steps of the argument) as late as the \[04.08.12\]! First it is well-known that the solubility of the Dirichlet problem (say on a bordered Riemann surface) is tantamount to the existence of the Green’s function $G(z,t)$ with pole at $t$, for each $t$. (Actually, we primarily need that the former implies the latter.) This “Dirichlet-to-Green” mechanism will be recalled below along with the definition and some geometric (biochemical) intuition about the Green’s function. The latter has also strong electrostatic or hydrodynamic connotations. The definition of the Green’s function is somewhat easier in the case of plane domains, and its extension to bordered surface—while still laying in the range of Dirichlet—implicates some conceptual difficulties.

The [*Green’s function*]{} $G(z,t)$ with pole at $t$ (a fixed interior point) is a completely canonical function characterized by the properties: it is harmonic off $t$, vanishes along the boundary and its germ near has the singular behavior prescribed by the function $\log\vert z-t\vert$ in any local uniformizer $z$. It will be verified that $G(z,t)$ is negative on the interior of the bordered surface (consequence of Gauss’ mean value property of harmonic function and the resulting maximum principle). Then we shall try to approach the existence of the Ahlfors function by duplicating the Green-type proof of the Riemann mapping theorem (simply-connected case), which just amount to write down the magic formula $f(z)=e^{G(z,t)+i G^{\ast}(z,t)}$, where $G^{\ast}$ is the conjugate potential. Note that $G(z,t)\le 0$ ensures $\vert f(z)
\vert=e^{G(z,t)}\le 1$ with equality precisely along the boundary. The main difficulty about extending this “Green-to-Riemann” trick to the multiply-connected setting is to arrange single-valuedness of the conjugate potential $G^{\ast}$. This amounts to kill all periods of the $1$-form $dG^{\ast}$ from which $G^{\ast}$ arises through line-integration. To achieve this one is invited to introduce enough free parameters in the problem by considering a superposition of various Green’s functions $\sum_i \lambda_i G(z,
t_i)$ for several poles $t_i$ sufficiently abundant so as to enable the killing of all periods (via linear algebra). Since a planar domain with $r$ contours has $r-1$ essential cycles (up to homology) and attaching $p$ handles creates 2 new essential cycles, we need annihilating $(r-1)+2p$ periods. Taking one more pole (raising the total number to $r+2p$) supplies enough parameters for linear algebra to ensure existence of a non-trivial solution in the kernel of the period mapping. This prompts (almost) the existence of an Ahlfors circle map of degree $r+2p$ (as predicted in Ahlfors 1950 [@Ahlfors_1950]). Alas, a serious technical difficulty occurs, namely ensuring the positivity of all $\lambda_i$. Ignoring this issue, any $r+2p$ points (in the interior) could be the zeroes of a circle map. Presently, we lack a complete existence of an Ahlfors map through this procedure. Of course it would be even more challenging to arrive at Gabard’s bound (mapping degree $\le r+p$) through this classical strategy (à la Green, Riemann, Grunsky, Ahlfors, Kuramochi, etc.). In Riemann the trick of annihilating periods appears of course very explicitly in the following jargon: “[*so bestimmen da[ß]{} die Periodicitätsmoduln sämmtlich $0$ werden.*]{}” (cf. e.g. Riemann 1857 [@Riemann_1857 p.122]). The core of Heins’ argument 1950 [@Heins_1950] is also exactly in this spirit and Heins seems able to complete the program via consideration of convex geometry. Our intention is first to recall the basic procedure, and we hope to be able later to settle the positivity problem. A priori it is not evident that the latter condition is always achieved for an arbitrary selection of poles $t_i$ of Green’s functions (which will mutate into zeroes of the “Riemann-Ahlfors map” $f$ after exponentiation).

\[25.08.12\] [*Corrigendum.*]{}—The above linear superposition $\sum_i \lambda_i G(z,t_i)$ on Green’s functions is maybe somewhat too continuous in nature. This may be seen by exponentiating and looking at the local behavior of $f$. Near some $t_i$, $G(z,t)\sim \lambda_i G(z,
t_i) \sim \lambda_i \log \vert z \vert$ so that $\vert
f(z)\vert \sim \exp(\lambda_i \log\vert z\vert)=\vert z
\vert^{\lambda_i}$ so that $f$ has not the character of a holomorphic function when $\lambda_i$ is not integral.

Another way to argue in the same sense is suggested by Ahlfors 1950 [@Ahlfors_1950 p.126–7, §4.3]. Assume that $f(z)$ is a circle-map $f\colon F \to \overline \Delta$ with zeros at $t_1,\dots, t_d$ (counted with multiplicities), then upon post-composing with the function $\log\vert z \vert$ (harmonic off the origin) we get the function $\log \vert f (z) \vert$ harmonic on $F$ save at the $t_i$ where it has logarithmic poles. Therefore this function must coincide with superposition $G:=\sum_{i=1}^d G(z,t_i)$ of Green’s potentials. Indeed, the difference $\log \vert f (z) \vert-G$ is throughout $F$ harmonic (cancellation of singularities) and vanishes along the border $\partial F$, hence is identically zero. \[NB: the above remark is to be found in Ahlfors (), who (in our opinion) fails to insist on the assumption that $f$ is a circle-map (i.e. $\vert f
\vert =1$ along the border), which is crucial to ensure that $\log
\vert f (z) \vert$ vanishes along the border $\partial F$.\]

So given a circle-map $f$ with $d$ zeros $t_i$ we have the formula $$\log \vert f (z) \vert=\textstyle\sum_{i=1}^d G(z,t_i).$$ Conversely, given points $t_i$, we may consider the right-hand side of the previous equation $$\label{Green-super:eq}
G:=\textstyle\sum_{i=1}^d G(z,t_i)$$ and the following formula will define a circle-map $$f(z)=e^{G+iG^{\ast}}$$ provided $dG^{\ast}$ (the conjugate differential of $G$) has all its periods integral-multiples of $2\pi$. (It follows incidentally, that a circle-map is uniquely determined up to a rotation by the geographic location of its zeros. This can also be seen algebro-geometrically, by considering the Schottky double, where the divisor of zeros $D$ becomes linearly equivalent to its symmetric conjugate $D^{\sigma}$, spanning together a pencil $g^1_{d}$ defining a total morphism to ${\Bbb P}^1$ of degree $d$, cf. Lemme 5.2 in Gabard 2006 [@Gabard_2006].

The desired integrality of periods resembles a [*Diophantine condition*]{} (at least is qualified as a such by Ahlfors 1947 [@Ahlfors_1947 p.1]), emphasizing from the outset the relative difficulty of the problem. All of our freedom relies on dragging the points $t_i$ through the surface $F$ hoping that for a lucky constellation the $1$-form $dG^{\ast}$ acquires simultaneous integrality of all its periods along $\gamma_1, \dots, \gamma_g$ the $g:=(r-1)+2p$ many essential $1$-cycles traced on $F$ (cf. Fig.\[Green:fig\]e).

As a personal trouble, $dG^{\ast}$ seems to have singularities where $G$ does, but maybe they disappear. Bypassing this point, Ahlfors’ Diophantine problem (1947) looks well-posed and one may hope a direct attack upon arranging integrality of all periods. (Ahlfors 1950 [@Ahlfors_1950] (p.127) first reformulates the condition in term of Schottky differentials and then switches quickly to the extremal problem, so does not seem to attack directly the Diophantine question. In fact, its elementary proof on p.124–126 follows a somewhat different route by constructing a half-space map involving avatars of Green’s function with poles situated along the boundary. We shall come back to this subsequently.)

[**Trying a direct attack.**]{} Assuming the problem well-posed, we can consider a period mapping $$\wp\colon R^d\longrightarrow {\Bbb R}^g \longrightarrow ({\Bbb
R} / 2\pi {\Bbb Z})^g=:T^g,$$ where $R={\rm int} (F)$ is the interior of the bordered surface $F$, and the first map takes the periods along the fixed basis of the first homology $\gamma_i$ of the $1$-form $dG^{\ast}$ corresponding to the points $(t_1, \dots, t_d)\in R^d$ via formula . The second map is just the natural quotient map.

Now one may hope to apply the usual surjectivity criterion for a continuous map to a closed manifold (here $R^d\to T^g$) saying that if the representation induced on the top-dimensional homology of the target-manifold is non-zero then the mapping is surjective. For definiteness we recall its statement and short proof.

Let $f\colon X \to T$ be a continuous map from a (topological) space $X$ to a (target) manifold $T$ of dimension $n$, say. It is assumed that $T$ is closed (i.e. compact borderless). It is also essential to assume that $T$ is a Hausdorff manifold. If the induced homomorphism $H_n(f)$ is non-zero, then $f$ is onto.

One considers the map induced on the homology $H_n$ of dimension $n$ equal to that of the manifold $T$. If $f$ fails to be surjective, it factors through the punctured manifold $X \to T-\{t\}$ for some point $t$. Now it is a simple fact that the top-dimensional homology of a (Hausdorff) manifold vanishes, so in particular $H_n(T-\{t\})$ is trivial. By functoriality it follows that $H_n(f)=0$, violating our assumption.

In particular $0=(0,\dots, 0)\in T^g$ would be the image of some $(t_1, \dots , t_d)\in R^d$ and the corresponding potential $G$ given by would have a conjugate differential $dG^{\ast}$ meeting the Diophantine requirement.

This strategy requires a good understanding of the mapping $\wp$ perhaps in the sense that when one pole $t_i$ is dragged along the cycle $\gamma_j$ then the image winds once around the corresponding factor of the torus $T^g$. Choosing $d=g$ and in the Künneth factor of $H_g(R^d)$ the element $\gamma_1 \otimes \dots
\otimes \gamma_g$ which has the correct weight $g$ so as to be an element of $H_g(R^g)$ whose image would be the fundamental class of the torus $T^g$. This would establish the surjectivity of $\wp$ for $d=g$. Alas, this is a bit too optimistic in the planar case ($p=0$). So our argument must be foiled at some place. The reasonable result to be expected is $d=g+1$ (like Ahlfors 1950 [@Ahlfors_1950]) and boosting the method upon choosing $\gamma_1 \otimes\dots \otimes \gamma_{r-1} \otimes (\alpha_1
\star \beta_1)\otimes \dots \otimes(\alpha_p \star \beta_p)$ where the $\alpha_i, \beta_i$ are the cycles winding around the handles (cf. Fig.\[Green:fig\]e) one may expect to achieve $d=r+p$ as predicted in Gabard 2006 [@Gabard_2006].

Digression on Dirichlet (optional)
----------------------------------

The Dirichlet solution may be interpreted as the permanent equilibrium state of temperature in a heat-flow conducting medium. Arguably (physico-chemical intuition?), this phenomenology is completely insensitive to the topology. Hence Dirichlet’s problem is always soluble whatever the topological complexity of the bordered manifold is. One only requires a Riemannian metric to give a good sense to the (Beltrami) Laplacian (or the allied mean value property). Hence any metric bordered smooth manifold, say compact to stay in the reasonable realm of finiteness is suitable to pose and solve the first boundary value problem. \[Remember maybe that there is vast jungle of non-metric manifolds, those of Cantor 1883 and Prüfer 1922 being the most prominent examples, but the latter do not enter the scene of function theory at least in complex dimension 1.\] What is the most general context where the Dirichlet problem is soluble? Our guess is for any compact bordered Riemannian manifold, eventually non-orientable. Hence Dirichlet makes sense also on non-orientable manifolds, but the case of immediate interest is that of compact bordered Riemann surfaces ([*ipso facto*]{} orientable). Solid existence proofs were primarily devised by H.A. Schwarz, alternating method (ca. 1870), etc. with many subsequent extensions, e.g. Nevanlinna 1939 [@Nevanlinna_1939], several works of Ahlfors, H. Weyl 1940 [@Weyl_1940] (method of orthogonal projection), not to mention Neumann, Poincaré, Korn-Lichtenstein, etc., cf. e.g. Neumann 1900 [@Neumann_1900]). Another source is Hilbert-Courant’s book cited e.g. for this purposes in Royden’s Thesis 1950/52 [@Royden_1952]. \[For those inclined toward modern expressionism, there is surely a concept of “Dirichlet space” (Brelot, Beurling, Deny, etc.) which should englobe any bordered Riemannian manifold and much more.\] In the appropriate Hilbert space, minimizing the Dirichlet integral amounts to minimize the length of a vector lying on a certain hypersurface $M$ corresponding to the boundary data $f\colon\partial F\to {\Bbb R}$. A priori this hypersurface could spiral around the origin impeding existence of a minimum or be bumpy enough as to violate uniqueness. But one rather imagine it to be a linear manifold implying a unique minimum of the distance function (norm). Of course the hypersurface in question (corresponding to a certain boundary prescription) is readily shown to have linear character, as subtracting any member of it, its translate through the origin identifies with the set of functions vanishing along the boundary. The latter is vectorial, being the kernel of a linear mapping (restriction to the boundary). Dirichlet principle looks thus immediately imputable to an Euclid-Hilbertian conception of space, yet with difficulty concentrating on the existence question of a member (=point) in this hypersurface $M$ (i.e., of a function matching the boundary prescription having with finite Dirichlet integral). As we know Hilbert’s solution primarily involved the compactness paradigm, formalized as a such some few years later by Fréchet. The naive minimization procedure is not fairly evident, and indeed plagued by the counterexample of Hadamard 1906 [@Hadamard_1906], and the earlier one of Prym 1871 [@Prym_1871]. Prym (1871 ) describes a continuous function on the boundary of the unit disc such that the Dirichlet integral for the associated harmonic extension of the boundary function is infinite. \[The latter harmonic extension is known to exist independently of the Dirichlet principle, e.g. on the ground of Poisson’s formula which solves Dirichlet in the disc-case.\] Later Hadamard (1906 ) gave a similar example where any (continuous) function matching the boundary data has infinite Dirichlet integral. (Perhaps, any Prym data is also explosive in the sense of Hadamard?) The moral is quite subtle to grasp: roughly the Dirichlet principle fails but not the Dirichlet problem which is always uniquely soluble! Hilbert’s solution (ca. 1900 [@Hilbert_1900], [@Hilbert_1901/04]) under special hypotheses (involving only the space and not the boundary data?!) is certainly sufficient for the purpose at hand. Hilbert’s hypothesis where weakened in subsequent works by B. Levi 1906 [@Beppo-Levi_1906], Fubini 1907 [@Fubini_1907], Lebesgue 1907 [@Lebesgue_1907], compare also the historiography in Zaremba 1910 [@Zaremba_1910]). For practical purposes (e.g. for the construction of the Green’s function) one can probably restrict attention to reasonable boundary data, as those arising via geometric construction (e.g., the logarithmic charge allied to the construction of the Green’s function of a plane smoothly bounded domain). Possibly, for tame boundary data the original Dirichlet principle remains an efficient tool for a direct variational treatment of the boundary value problem.

Alternatively, of Dirichlet-Riemann-Hilbert one may use the classical but cumbersome alternating method of Schwarz (or Neumann’s variant) to solve the Dirichlet problem. To summarize we need the result:

[(Dirichlet, Riemann, Schwarz 1870, Hilbert 1900, etc.)]{} Given a compact bordered Riemann surface $F$, and a continuous boundary function $f\colon
\partial F \to {\Bbb R}$. There is a unique harmonic function $u\colon F \to {\Bbb R}$ extending $f$.

First (rigourously) obtained in Schwarz 1870 [@Schwarz_1870-alternirendes-Verfahren] via the alternating method. Variation of this technique Picard’s method of successive approximation (cf. Picard, Zaremba, Korn, Lichtenstein). Another variant of proof is Hilbert’s resurrection of the Dirichlet principle (direct variational method). Reference in book form cf. Hilbert-Courant. Another more modern trend is to use Perron’s method which affords great simplification. Compare for instance Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.138–141, esp. 11G] for an execution of Perron’s method (joint with Harnack’s principle) in the context of abstract Riemann surfaces.

From Green to Riemann {#sec:From-Green-to-Riemann}
---------------------

In term of the Green function for a simply-connected domain one may write down the Riemann map as $$f(z)=e^{G+iG^{\ast}}\,,$$ where $G^{\ast}$ is the conjugate potential (satisfying the Cauchy-Riemann equations). \[This is basically the second proof given by Riemann in 1857 [@Riemann_1857-DP], and see also e.g. Picard 1915 [@Picard_1915].\] That $f$ is a circle map follows from $G\le 0$ with vanishing precisely on the boundary, and the fact that $G^{\ast}$ is single-valued since the domain is simply-connected. Details are supplied during the next Steps, where we examine the more delicate multiply-connected domains or even general compact bordered Riemann surfaces.

The conjugate $G^{\ast}$ potential is defined by the desideratum that $G+iG^{\ast}$ is holomorphic, i.e. ${\Bbb C}$-linearizable in the small. This gives the Cauchy-Riemann equations $$\frac{\partial G}{ \partial x}=\frac{\partial G^{\ast}}{
\partial y}, \qquad
\frac{\partial G^{\ast}}{ \partial x}=-\frac{\partial G}{
\partial y}.$$ Writing formally $G^\ast$ as the integral of its differential, gives $$\begin{aligned}
G^{\ast}= \int dG^{\ast}=\int (\frac{\partial G^{\ast}}{
\partial x} dx+\frac{\partial G^{\ast}}{ \partial y}dy)
=\int (-\frac{\partial G}{
\partial y} dx+\frac{\partial G}{ \partial x}dy),\end{aligned}$$ whose integrand (a $1$-form) coincides actually with the $dG$ twisted by multiplication by $i$ on the tangent bundle. Therefore $dG^{\ast}$ is a genuine $1$-form canonically attached to the function $G$. ([*Warning*]{}.—The symbol $G^\ast$ (taken alone) as no intrinsic meaning at least as a single-valued function unless $dG^{\ast}$ is period free.)

The Green’s function
--------------------

But what is the Green’s function at all about? It is a sort of logarithmic potential attached to an electric charge placed at $t$. It is easier to define in the case of a plane domain bounded by smooth curves. The case of ultimate interest (compact bordered Riemann surfaces) will be discussed later. Given a domain $B\subset {\Bbb C}$ (smoothly bounded) marked at an (interior) point $t$ one considers the function $\log \vert z-t\vert$ which induces (by restriction) a charge (temperature) on the boundary $\partial B=C$ and one solves the Dirichlet(=first boundary-value) problem for this data. It results an (everywhere regular) harmonic function $u=u(z,t)$, which subtracted from the original logarithmic potential gives the (so-called) [*Green’s function with pole at $t$*]{} $$( \log\vert z-t \vert )- u(z)=:G(z,t). \label{Green:eq}$$ By construction, it vanishes along the contour $\partial B$ and possesses a logarithmic singularity near the point $t$. This is a canonical function attached to the sole data of $B$ and a certain interior point $t$. Note that $G(z,t)$ tends to $-\infty$ as $z$ approaches $t$, so one may think of the Green’s function as a black hole centered at $t$ with a vertiginous sink plunging into deep darkness.

One can interpret this Green’s function as some electric potential (Galvanic current) on a conducting plate. If one prefers a biological metaphor one can visualize $G(z,t)$ as the proliferation of bacteroides originating from $t$ while expanding through the medium $B$ driven by an apparent global knowledge of the shape of the universe. To be more concrete, the expansion is more rapid where more free resources are available. In particular all bacteria reach synchronously the boundary having consumed all resources of the nutritive substratum in what looks to be the most equitable way. Compare the pictures in the simply-connected case (Fig.\[Green:fig\]a) and then for a multi-connected region (Fig.\[Green:fig\]b). Trying to imagine the same proliferation occurring on a bordered surface realized say in Euclidean $3$-space we get something like Fig.\[Green:fig\]d.

0 -5pt0

Now it is clear that the above formula $f(z)=f(z,t)=e^{G(z,t)+iG^{\ast}(z,t)}$ supplies the Riemann map with $f(t)=0$ and $f(z)\in S^1$ (unit circle) whenever $z$ lies on the boundary, where $G$ vanishes. Of course the map is only defined up to rotation, coming from an arbitrary additive constant in $G^{\ast}$. \[Compare for instance Riemann 1857 [@Riemann_1857-DP], Picard 1915 [@Picard_1915], etc.\]

If one tries to adapt this proof to multi-connected domains one meets the notorious difficulty that the conjugate potential $G^{\ast}$ is not single-valued, a priori. So the efforts focus on eliminating the periods of its differential $dG^{\ast}$ by choosing appropriately some accessory parameters. \[This universally known device goes back at least to Riemann 1857 [@Riemann_1857 p.122] Schottky 1877 [@Schottky_1877], see also Picard 1913 [@Picard_1913], Koebe 1922, Julia 1932 [@Julia_1932], Grunsky 1937 [@Grunsky_1937], etc.\]

Using this idea we may concoct a circle map $B\to
\overline{\Delta}$. \[cf. Grunsky 1937 [@Grunsky_1937] or Grunsky 1978 [@Grunsky_1978] and also Ahlfors\]. The natural trick is probably to take several poles $t_i$ (say $d$ many). Those will ultimately become the zeroes of the circle map we are looking for as $e^{-\infty}=0$. One now form the combination of the corresponding Green’s functions $$G(z):=\textstyle\sum_{i=1}^d \lambda_i G(z,t_i) \quad
(\lambda_i\in {\Bbb R}).$$ This gives a (finite) constellation of black holes scattered through the domain $B$ and we shall try to choose the constants $\lambda_i$ so that $dG^\ast$ has no period. Since the combination $G$ vanishes on the contour $\partial B$ (being a superposition of Green’s functions) the allied function $f(z)=f(z;
t_i, \lambda_i):=e^{G(z)+iG^{\ast}(z)}$ will map $\partial B$ onto $S^1$. To arrange it as a circle map $f\colon B \to
\overline{\Delta} $ requires the basic remarks of the next section, plus the more delicate issue of being able to choose positive $\lambda_i>0$.

Quasi-negativity of Green
-------------------------

The following property of the Green’s function is basic, yet important.

Each Green function $G_t(z):=G(z,t)$ is quasi-negative (i.e. $\le 0$ throughout the domain and strictly $<0$ in its interior).

From its definition it is clear that $G_t(z)\to -\infty$ as $t$ approaches the pole $t$. Thus choosing a very large negative (real) constant $C<0$ the corresponding level line $L_C$ of Green $G_t^{-1}(C)$ will be a nearly circular (Jordan) curve enclosing the pole $t$ in its interior. Further it looks evident that for $C<0$ large enough (in absolute value) this Jordan curve bounds a (topological) disc in the domain. (One could uses the general Schoenflies theorem requiring just to check that $L_C$ is null-homotopic in the domain $D$.) Next it is intuitive (but need to be arithmetized) that within this sufficiently small disc-shaped domain (i.e. the inside of $L_C$ for $C<0$ sufficiently large) the Green function $G_t$ is negative (indeed $\le C$).

Cutting away from the domain $D$ the interior of $L_C$ we obtain an excised domain $D^{\ast}$ with one more contour. On this new domain, the Green’s function $G_t$ solves Dirichlet (first boundary-value) problem for the data $0$ on all contours but $C<0$ on the newly created contour $L_C$. We now conclude via the next lemma.

\[Depressive:lem\] [(Depressiveness of Dirichlet, or rather the allied harmonic functions)]{} Let $F$ be a compact bordered Riemann surface. If the (continuous) boundary data function $f\colon \partial F \to {\Bbb R}_{\le 0}$ is non-positive, then so is its Dirichlet solution $u:=u(f)$, i.e. $u\le 0$ throughout $F$.

If not then $u(z_0)>0$ (positive) at some interior point $z_0$ of the surface $F$. By compactness $u$ achieves its maximum, which is positive. Since $f\le 0$ the latter would not be achieved on the boundary violating the maximum principle (compare the next lemma).

[(Maximum principle)]{} Any harmonic function $u$ on a compact bordered surface $F$ achieves its maximum on the boundary $\partial F$. In fact, if the maximum is achieved at some interior point then the function $u$ is constant.

Assume $z_0$ to be an interior point realizing the maximum $M$ of the harmonic function $u$ defined on $F$. We trace a little (metric) circle about $z_0$ of sufficiently small radius as to lye entirely inside $F$ (together with its interior disc $D$). Harmonicity may be characterized via the [*mean-value property*]{} (Gauss, it seems): $$\int \int_D u(z) d\omega = area(D) \cdot u(z_0).
\label{mean-value-prop:eq}$$ As $u(z)\le M$, we get $M\cdot area (D)\ge \int \int_D u(z)
d\omega = area(D) \cdot u(z_0)$. Since $M= u(z_0)$, both extreme members coincide and so does the last inequality. This forces constancy on the little disc $D$ ($u$ being continuous).

It follows by ‘propagation’ that $u$ is globally constant. (Alternatively use general topology: the set of points where $u$ achieves its maximum is both nonempty (compactness), closed and open.) Indeed choosing a path from $z_0$ to any point $z\in F$ covered by a chain of little discs $D_1, \dots, D_k$, each $D_i$ centered on the border of the previous one $D_{i-1}$, one argues that two successive discs have enough overlap to ensure constancy over the next disc.

[ \[11.08.12\] [There is a Garabedian paper 1951 (A PDE..., p.486) were it is asserted that the Green’s function of a convex clamped plate need not be of one sign; but of course this is not relevant to our matter were we use the usual the Laplacian $\Delta$ and not the bi-Laplacian $\Delta^2$ corresponding to clamped plated, instead of vibrating membranes. This is the seminal work of Garabedian (but others were also involved) were the famous Hadamard conjecture on the bi-Laplacian was disproved.]{} ]{}

Killing the periods
-------------------

The previous section ensures that any superposition of Green’s functions $G:=\sum_i \lambda_i G(z,t_i)$ will be likewise quasi-negative provided all $\lambda_i$ are positive. In this circumstance the function $f=e^{G+iG^{\ast}}=e^G \cdot e^{i
G^{\ast}}$ (whose modulus is $e^G$) is a unit-circle map ($\vert f
\vert \le 1$), because the real exponential takes nonpositive values $(-\infty, 0]$ to $(0,1]$. It is consistent by continuity to send the $t_i$ on $0$.

If $r$ is the connectivity of the domain $B$ (number of its contours) then there are homologically $r-1$ non-trivial loops $\gamma_1, \dots \gamma_{r-1}$ running around the $r-1$ holes in our domain (cf. Fig.\[Green:fig\]c illustrating the case $r=3$). We consider the linear period mapping $$\begin{aligned}
\label{period-mapping:eq}
{\Bbb R}^{d} &\longrightarrow {\Bbb R}^{r-1} \cr
(\lambda_1, \dots \lambda_d) &\mapsto
(\textstyle\int_{\gamma_1} dG^{\ast}, \dots,
\int_{\gamma_{r-1}} dG^{\ast})\end{aligned}$$ By linear algebra if $d$ is large enough (precisely already for $d=r$) we have enough free constants so as to find non-trivial $\lambda_i$ extincting all periods. \[Heuristically the electric poles of the multi-battery in the electrolytic tank (nomenclature as in e.g. Courant 1950/52 (Conformal book)) are affected by suitable charges so as to generate an “ideal” potential with single-valued conjugate.\] Exponentiating gives $f=e^{G+iG^{\ast}}$ a circle map with $d=r$ zeroes, provided one is able to ensure all $\lambda_i
>0$.

Without taking care of this last proviso, one may reach too hastily the impression that we have complete freedom in prescribing the location of the $d=r$ poles (of the Green’s functions, which convert ultimately to zeroes of the related circle map). The linear-algebra argument gives only a real-line inside the kernel of the (linear) period-mapping , but a priori this line could miss the “octant” ${\Bbb
R}_{>0}^{d}$ consisting of totally positive coordinates. In fact upon letting vanish some of the $\lambda_i$ what is only required is a non-trivial penetration of this line $\ell$ into the closed octant $\overline{O}={\Bbb R}_{\ge 0}^{d}$, i.e. the intersection $\ell \cap \overline{O}$ should not reduce to the origin. A true penetration of this line in the interior of $O$, or a degenerate one where the line meet along one of its face would be enough to complete the existence-proof. The latter case amounts to extinct some Green’s “batteries” by assigning a vanishing coefficient $\lambda_i=0$. The net effect would be degree lowering of the circle map $f$. Beware, that for planar domains (which correspond to Harnack-maximal Schottky doubles) no such lowering of the degree is possible for simple topological reasons ($r\le \gamma$). However the described theoretical eventuality may well happen in the non-planar case to be soon discussed. Understanding how and why to arrange degenerate penetrations could well offer a strategy toward improving Ahlfors $r+2p$ bound.

Extra difficulties in the surface case
--------------------------------------

It is obvious that the above method via Green’s functions adapts to bordered Riemann surface $F=F_{r,p}$ of (positive) genus $p$ with $r$ contours (Rand). Remember however that at this stage we did not offered a complete treatment of the planar case ($p=0$).

First note a conceptual difficulty regarding Green’s function, which, in the plane case of a domain $B\subset {\Bbb C}$, is constructed via $\log\vert z-t \vert$ appealing to a global coordinate system. In the abstract bordered setting, there is no such ambient medium. One could try to work with a (conformal) Riemann metric and the allied logarithmic distribution $ \log \varrho (z,t), $ where $\varrho$ is the intrinsic distance (defined as usual as the infimum of lengths of rectifiable pathes joining two given points). Note however that this construction specialized to the domain case does not duplicate the former, since the intrinsic distance $\varrho(z,t)$ does not coincide with the extrinsic one $\vert z-t\vert$, unless the domain $B\subset {\Bbb C}$ is starlike about $t$.

Bypassing this difficulty \[which will be resolved later\], we first note that each handle creates two $1$-cycles yielding a total of $(r-1)+2p$ many essential loops (compare Fig.\[Green:fig\]e). Thus introducing $d:=r+2p$ poles $t_i$ we dispose of enough free parameters to arrange (via linear algebra) the vanishing of all periods of the conjugate differential $dG^{\ast}$ of the potential $G=\sum_{i=1}^{d}\lambda_i G_{t_i}$. This explains quite clearly why Ahlfors discovered (about 1948) the upper-bound $r+2p$ for the degree of a circle map. Of course there is still the subtlety of explaining why it is possible to choose all $\lambda_i>0$ at least for a clever choice of the poles $t_i$.

All this is probably when suitably interpreted the quintessence of the Ahlfors mapping (of degree $r+2p$). Again the writer does not mask his happiness after having understood this point (as late as the 04.08.12). Now it is evident to reconstruct (even if somewhat fictionally) what must have happened in Ahlfors’ brain (at least as early as 1948, and presumably much earlier, yet no record in print). With this piece of information and, on the other hand, being well-aware of the modern purely function-theoretic proofs of RMT (à la Koebe-Carathéodory, Fejér-Riesz 1922 (published by Radó 1923), Carathéodory 1928 and Ostrowski 1929) it must have seemed highly desirable (or trendy) to reinterpret the above (somewhat heuristic but fruitful potential theory) in terms of a function-theoretic extremal problem. This leads e.g. to the problem we discussed at length of maximizing either the modulus of the derivative at some inner point $t=a$, or to maximize the distance of two points $a,b$ where the first maps to $0$ and the second is repulsed at maximum distance from the origin. In both case the competing functions are analytic and bounded-by-one in modulus $\vert f \vert \le 1$. So we get the Ahlfors function $f_a$ or $f_{a,b}$. It seems obvious that all those Ahlfors functions are included in the above trick à la Green-Riemann (GR), and thus subsumed to an electrolytic interpretation. Yet the exact dependance and location of the corresponding logarithmic poles of Green’s $G$ (becoming the zero of Riemann’s $f$, after exponentiation) must be a transcendentally sublime business. Also the corresponding degree of the Ahlfors function is another mystery.

It is conceivable that less than the $r+2p$ generically required poles suffices in case the linear period mapping ${\Bbb R}^d \to
{\Bbb R}^{(r-1)+2p}$ along fundamental loops has a degenerate image permitting to economize some poles $t_i$. The task is reduced to find the lowest $d$ such that the kernel of the period map is non-trivial and contains a non-zero element all of whose coordinates are $\ge 0$. Remember, that Gabard 2006 [@Gabard_2006] showed—using another method, based on a topological argument of irrigation (Riemann-Betti-Jordan-Poincaré’s homologies, and Brouwer’s degree plus some basic Pontrjagin theory in the Jacobian torus as a very special commutative Lie group—that there is a circle map of degree $\le r+p$ (i.e. with one unit economized for each handle). Assuming that any circle map is allied to a Green-Riemann map there would be a fewer number namely $d\le r+p$ of batteries required to generate this mapping. Of course, the first part of the assertion looks evident: given a degree $d$ circle map $f$ with zeroes at $t_i$, then $\log \vert f(z) \vert$ coincides with $\sum_{i=1}^d G(z,t_i)$. This is Ahlfors formula following from the fact that both functions vanishes on the border and have the same singularities.

[*Philosophy.*]{} \[08.08.12\] Modulo elusive details, it is fair to resume the situation by saying that the Ahlfors circle maps (if not all existence theorems of function theory) derives form the Dirichlet principle (or the allied Green’s functions). \[This was of course best incarnated by Riemann, 1851 and 1857, where in bonus the whole algebraic geometry of curves was subsumed to this principle!\] Conversely one could hope that the Ahlfors function could be used to lift the Dirichlet solubility on the disc (via Poisson integral formula) to an arbitrary bordered surface. However it seems obvious that there is no way to descend the boundary function to the disc since the Ahlfors branched covering is multi-valent. We arrive at the conclusion that the true mushroom is the Dirichlet principle, while Ahlfors function being just one tentacle of the mushroom. Of course, the only paradigm susceptible of competing with Dirichlet are the function-theoretic extremal problems à la Koebe-Carathéodory-Fejér-Riesz-Bieberbach-Ostrowski, etc. For plane domains the Kreisnormierung (instead of the Ahlfors map) may be used as normal domains where the Dirichlet problem is easier to solve. This is akin to Poisson’s formula for the round disc case of Dirichlet, and quite implicit in Riemann’s Nachlass 1857 [@Riemann_1857_Nachlass] (cf. also Bieberbach 1925 [@Bieberbach_1925]). A similar reduction of Dirichlet for bordered surfaces occurs is also likely on the ground of Klein’s Rückkehrschnitttheorem (cf. Section \[sec:Ruckkehrschnittthm\]), supposed to be an extension of the Kreisnormierung.

Regarding the detailed execution of the removal of the period as to construct an Ahlfors-type mapping one should compare also the paper of Heins 1950 [@Heins_1950], Kuramochi 1952 [@Kuramochi_1952] and (albeit confined to planar domains) the paper by D. Khavinson 1984 [@Khavinson-Dimitri_1984], whose argument is considered by its author akin to the arguments of Grunsky.

The Green’s function of a compact bordered Riemann surface=CBRS
---------------------------------------------------------------

\[14.08.12\] This section examines the issue that the Green’s function $G(z,t)$ with pole at $t$ is a canonically defined function in the generality of a CBRS. This is super-classical, cf. e.g., the treatises Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] or Schiffer-Spencer 1954 [@Schiffer-Spencer_1954]. It is to be expected to find older treatments by Riemann, Schwarz, Klein, Koebe, etc. Several accounts by Nevanlinna proceed via Schwarz’s alternating method, a viewpoint which looked most convenient to adhere with.

As already noticed, the case of a plane domain $B\subset {\Bbb C}$ (bounded by smooth curves) it is easy to define Green’s function $G(z,t)$ via the (logarithmic) potential $\log \vert z-t \vert$ from which we subtract the Dirichlet solution matching the logarithmic potential restricted to the boundary $\partial B$. Alas, for a CBRS $F$ one lacks an ambient space like ${\Bbb C}$ permitting an analogous construction.

Of course, $\log \vert z-t \vert$ bears some significance only locally within a uniformizer chart about $t$. Taking another local chart, one may argue that in the small the expression will mutate into $\log \vert \alpha (z-t) \vert$ for some $\alpha\in{\Bbb
C}^{\ast}$ incarnating the derivative of the transition between the two charts. Thus the log-potential w.r.t. the new chart is $\log\vert \alpha \vert + \log \vert z-t \vert$, hence equal to the old one modulo an additive constant. Presumably some philosophical argument can corroborate the vague feeling that the asymptotic of the logarithmic pole is unaffected by such additive constant. \[Added in proof: compare Pfluger 1957 [@Pfluger_1957 p.110, 28.3] for an accurate formulation, or Farkas-Kra 1980/1992 [@Farkas-Kra_1980/1992 p.182, Remark].\] It seems then meaningful to set:

The Green’s function of a CBRS $F$ with pole at $t$ (an interior point of $F$) is the unique harmonic function on $F$ save $t$ with singularity $\log\vert z-t \vert$ near $t$ which vanishes continuously on the boundary $\partial F$.

Compare (modulo a different sign convention) Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.158, 4B]. Uniqueness is considered as evident there. Indeed, a chart change affect the logarithmic potential by an additive constant and harmonic functions are quite rigid (being determined by their values on any open disc). Hence knowledge of the function on any punctured chart about $t$ via $\log \vert z-t \vert$ determines it uniquely. The delicate point is existence. Choose around $t$ a nice analytic Jordan curve $J$ and via RMT construct a holomorphic chart taking $D$ (the “sealed” interior of $J$, i.e. $J$ included) to the unit disc $\overline{\Delta}$. Consider $\log\vert z\vert$ in the unit circle and transplant to $D\subset
F$ and then after adding an additive constant we try to solve a Dirichlet-Neumann problem on $F-D$ piecing together smoothly the logarithmic piece with the Dirichlet-Neumann solution. In this procedure the Green’s function looks highly non-unique depending on the “ovaloidness” of the Jordan curve $J$ chosen. In fact $J$ cannot be chosen at will but must somehow be a level-line of Green (still undefined). Infinitesimally $J$ should be a perfect circle, and this is perhaps the key to put the naive pasting argument on a sound basis via a convergence procedure. (Infinitesimal circles are well-defined on Riemann surfaces via the conformal structure.) Existence and uniqueness look then plausible, but involve a considerable sophistication over the plane-case where the Green’s function reduced straightforwardly to the Dirichlet problem.

Let us paraphrase the above more formally. Take any chart $\varphi\colon U \to \Delta$ about the “pole” point $t$ (sending $t$ to the origin $0\in {\Bbb C}$), write down $\log\vert z\vert$ in that chart and shrink gradually attention to the (round) disc $\Delta_{\varepsilon}$ of radius $\varepsilon$. Let $D_{\varepsilon}$ be $\varphi^{-1}(\Delta_{\varepsilon})$. For each (positive) value of $\varepsilon$ one can solve the Dirichlet problem in $F-{\rm int} D_{\varepsilon}$ with boundary value $0$ on $\partial F$ and $\log \varepsilon$ on $\partial
D_{\varepsilon}$. Denote by $u_\varepsilon $ the corresponding solution. By construction $u_\varepsilon$ pasts continuously with the $\varphi$-pullback of the log-potential (i.e. $(\log \vert
z\vert) \circ \varphi$). Of course this glued function is a Frankenstein creature lacking a smooth juncture. For instance, if $\varepsilon=1$ then $u_{\varepsilon}$ is identically zero, whereas in $D_1$ we have the logarithmic “trumpet” with derivative $1$ along the normal direction. However as $\varepsilon$ decreases from $1$ to $0$, $u_\varepsilon$ becomes $\le 0$ (having prescribed the negative value $\log \varepsilon$ on $\partial D_{\varepsilon}$) and the dependence of $u_{\varepsilon}$ is perhaps monotonic. So it seems arguable (Harnack?) that while $\varepsilon \to 0$ (say via dyadic numbers $\varepsilon_n=1/2^n$) the $u_n$ converges to a harmonic function on $F-t$ which is the desired Green’s function $G(z,t)$. It seems evident (since $\partial D_n$ becomes more and more circular in $F$ as $n$ grows to infinity) that the limit is harmonic and independent of the gadgets used along the way (chart $\varphi$, dyadic sequence $\varepsilon$).

This vaguely explains existence and uniqueness can maybe be derived by a similar trick (combined with a “leapfrog” argument). Try to locate a reference along this naive line: maybe Schwarz?, Klein? Koebe? Weyl? Pfluger? and otherwise try Ahlfors-Sario [@Ahlfors-Sario_1960], Sario-Oikawa [@Sario-Oikawa_1969]. (Sometimes Sario’s formalism of the normal/principal operator is a bit awkward to digest.) For treatments of the Green’s function on a CBRS cf. Schiffer-Spencer 1954 [@Schiffer-Spencer_1954 p.33, and 93–94]. See also Sario-Oikawa 1969 [@Sario-Oikawa_1969 p.49–50]. We summarize the discussion by the

Given a CBRS $F$ and an interior point $t$, there is a uniquely defined Green’s function $G(z,t)$ with pole $t$ which is characterized by the following conditions: it is harmonic on $F-t$, vanishes (continuously) on the boundary $\partial F$ and it has the prescribed singularity $\log \vert z-t \vert$ near $t$.

For complete details, compare several sources:

$\bullet$ first Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.158] and Sario-Oikawa 1969 [@Sario-Oikawa_1969 p.50] (both via Perron’s method, and Sario’s formalism of the normal operator).

$\bullet$ Then also Pfluger 1957 [@Pfluger_1957 p.110, end of §28.2, as well as p.110, §28.3 and last 3 lines of p.111]

$\bullet$ Schiffer-Spencer 1954 [@Schiffer-Spencer_1954 §4.2, p.93–94]

$\bullet$ Nevanlinna 1953 [@Nevanlinna_1953-Uniformisierung] via Schwarz’s alternating method (SAM). We detail this argument in the next section.

Schwarz’s alternating method to construct the Green’s function of a compact bordered surface (Nevanlinna’s account)
-------------------------------------------------------------------------------------------------------------------

\[15.08.12\] As promised, in this section we attempt to understand Nevanlinna’s exposition of the existence of Green’s functions on compact bordered surfaces. All pagination given refers to Nevanlinna 1953 [@Nevanlinna_1953-Uniformisierung], the book “Uniformisierung”. Nevanlinna follows Schwarz’s alternating method (SAM) quite closely. The argument is a bit tedious but quite elementary. It uses merely the following facts on a bordered surface $F$:

\(1) if $f\le g$ on $\partial F$ then the associated Dirichlet solution $u(f)\le u(g)$ (compare Lemma \[Depressive:lem\]).

\(2) maybe Harnack’s theorem is required?

In Nevanlinna’s book the relevant information is scattered at two places (at least) so we attempted to compactify the presentation for our own understanding.

First Nevanlinna introduces a concept of “Kreisbereich”. Alas the jargon is not very fortunate being already consecrated by Koebe in a different context, so let us speak rather of a “celluloid” (or a “Kreisgebilde”). This is \[cf.p.142\] a connected finite union of (closed) discs in a Riemann surface (whose images by a (parametric) chart are round discs in ${\Bbb C}$). On each such disc the first boundary value problem (abridged DP=Dirichlet problem) is soluble via Poisson’s formula. Assuming the contours of each pair of discs to have finite intersection, SAM enables one to solve DP on the union, hence on any celluloid.

So for instance it is clear that any CBRS is a celluloid. (A formal proof certainly requires Radó’s triangulation theorem 1925 [@Rado_1925].) To absorb the boundary in one stroke one could add annular regions where the DP is also soluble by an explicit recipe, sometimes ascribed to Villat 1912 [@Villat_1912].

The Green’s function will be obtained by specializing the following technical lemma \[cf.p.148\] \[due to Schwarz and probably related to what Koebe’s calls the “gürtelförmige Verschmelzung”($\approx$belt-shaped fusion)\]. Intuitively, the lemma amounts to construct a harmonic function $u$ with prescribed boundary values and with prescribed singularity $u_0$ near a point $t$, or rather on a ring enclosing the pole $t$. (At first, it is not perfectly transparent how to deduce the Green’s function from the lemma, but we shall try elucidate this issue later.)

Let $F$ be a compact bordered Riemann surface and $t\in {\rm int} F$ an interior point. Let $U$ be a neighborhood of $t$ mapped to the unit-disc $\Delta=\{ \vert z
\vert <1\}$ via a chart. In $U$, let $K$ be the ring corresponding to $r_1\le \vert z \vert \le r<1$. [\[It seems that $r_1=0$ is permissible and needed for the application to the Green’s function.\]]{} Let further $X$ be a celluloid containing the external contour of the ring $K$ $(\vert z \vert=r)$ in its interior, as well as the boundary $\partial F$ but missing the small disc in $U$ corresponding to $\vert z\vert \le r_1$. Set finally $A:=X
\cup K$ [(compare Fig.\[Nevanl:fig\]]{}).

Then given $u_0 \in H (K)$ (harmonic on the ring $K$) and $f\colon \partial F \to {\Bbb R}$ continuous, there is a unique $u \in H(A)$ such that $u_{\vert \partial F}= f$ and with $u-u_0$ extending harmonically to $B$, the disc corresponding to $\vert z\vert\le r$.

-5pt0

-5pt0

Before detailing the proof, let us see how this helps defining $G(z,t)$ the Green’s function. \[The difficulty is just so trivial to be not completely explicit in Nevanlinna \[p.198–199, §2, Art.6.4\].\] First we impose $f\equiv 0$. Then we choose the singularity function $u_0=\log\vert z\vert$ which has to be defined on $K$, hence we shrink $r_1$ to $0$ via a sequence of dyadic radii $r_n=1/2^n$. On applying the lemma we get a sequence of solution $u=u_n$ defined on $A_n$ a sequence of expanding subsurfaces (the outsides of the shrinking discs $\vert z\vert<
1/2^n$). Now observe that $u_n$ for a large $n$ solves the problem of the lemma for all smaller values of $n$: just take the restriction (and use uniqueness). Consequently all the $u_n$ form a telescopic system of functions (each restricting to all its predecessors) defined on larger and larger compact subregions $A_n$ ultimately expanding to the punctured surface $F-t$. The very constant (indeed completely monotone) limit of those $u_n$ gives the desired Green’s function $G(z,t)$.

It is harmonic on $F$ save $t$, vanishes on the boundary and $G(z,t)-\log\vert z \vert$ extends harmonically through $t$ (on a little neighborhood). It remains to check that those 3 properties defines $G(z,t)$ unambiguously. This is again the same sort of argument. Assume there were two Green’s solutions $G_1,G_2$, then $G_i-u_0=:h_i$ harmonic on some neighborhoods $V_i$ of $t$. So $G_1-G_2=(h_1+u_0)-(h_2+u_0)=h_1-h_2$ which is harmonic on the intersection $V_1 \cap V_2$. Hence the difference $G_1-G_2$ is harmonic throughout $F$, but with vanishing boundary value on $\partial F$. Consequently it must be identically zero (by the uniqueness part of Dirichlet) which follows from the maximum principle.

This is a matter of implementing Schwarz’s alternating method \[see p.148–150\] and we follow exactly Nevanlinna’s text (annotating our copy by the symbol $\bigstar$ to indicate the sole cosmetic difference).

$\bullet$ [*Uniqueness*]{} Assuming the existence of two functions $u_1, u_2$ solving the problem, their difference $u_1-u_2$ will be harmonic on $A$, and 0 on $\gamma:=\partial F$. But each difference $u_i-u_0=:h_i\in H(B)$ extends harmonically across $B$ ($i=1,2$). Hence on $B$, $u_1-u_2=(h_1+u_0)-(h_2+u_0)=h_1-h_2 \in
H(B)$, and therefore $u_1-u_2 \in H(A\cup B)$, and $A\cup B$ is all of $F$. It follows (Dirichlet’s uniqueness) that $u_1-u_2$ vanishes identically.

$\bullet$ [*Optional remark.*]{} It is clear that the case $f\equiv 0$ is typical, since the general case just requires adding the Dirichlet solution for the data $f$. \[This explains why I had the impression to find many misprints!\]

$\bullet$ [*Existence (after Schwarz)*]{} First it is observed that DP is solvable on both $A$ and $B$ ($B$ is just a ball and $A$ is a celluloid, yet of the general type involving a ring). Of course $A$ is also a compact bordered surface and therefore one is ensured of Dirichlet solvability, thereby bypassing the concept of a celluloid, and accordingly one can shorten slightly the statement of Nevanlinna’s lemma, with the direct bonus that one can make abstraction of all the little discs drawn on the picture. \[CAUTION: here it is perhaps NOT permissible to take $r_1=0$\]

We denote by $\alpha$ and $\beta$ the internal resp. external contour of $K$, and let $\gamma:=\partial F$. Set first $v_0\equiv 0$. We define inductively sequences $u_n\in H(A)$ and $v_n\in H(B)$ by their boundary values ($n\ge 1$) $$\label{SAM:eq}
u_n=\begin{cases} v_{n-1}+u_0 \quad &\textrm{on } \alpha, \cr
0 [\bigstar or f] \quad &\textrm{on } \gamma,
\end{cases}$$ and $$v_n=u_n-u_0 \quad \textrm{on } \beta.$$ \[This Ansatz comes a bit out of the blue, but notice that passing to the limit both definitions leads to the identity $u-u_0=v$ holding on $\alpha \cup \beta$ which is the full contour of the ring $K$, so that anticipating harmonicity this will hold throughout $K$, and $v$ will afford the required extension of $u-u_0$ (only defined on $A\cap K=K$) to the disc $B$ containing the ring $K$. Of course, it is also crucial to notice that both sequences $u_n,v_n$ are “interlocked” or “leapfrogged” requiring an alternating progression of one term to go one step further with the other.\]

The successive differences are given by $$\label{success-diff:eq}
u_{n+1}-u_n=\begin{cases} v_{n}-v_{n-1} \quad &\textrm{on }
\alpha \cr 0  \quad &\textrm{on } \gamma
\end{cases}$$ and $$v_{n+1}-v_n=u_{n+1}-u_{n} \quad \textrm{on } \beta.$$ Let us write $$M_n:=\max_{\beta} \vert u_{n}-u_{n-1}\vert=\max_{\beta} \vert
v_{n}-v_{n-1}\vert,$$ then by the maximum- and minimum-principle $ \vert
v_{n}-v_{n-1}\vert\le M_n$ in $B$, and so in particular on $\alpha$. Hence by , $ \vert
u_{n+1}-u_{n}\vert\le M_n$ on $\alpha$. Further, the difference $u_{n+1}-u_{n}$ vanishes on $\gamma$ (cf. ), and so it is bounded on the boundary of $A$ (and therefore throughout $A$) by the potential $M_n
\cdot \omega$, where $\omega$ is the harmonic function vanishing along $\gamma$ and equal to $1$ on $\alpha$. Hence $$\label{diff:eq}
\vert u_{n+1}-u_n \vert\le M_n \cdot \omega\quad  \textrm{in }
A.$$ In the interior of $A$, one has $0<\omega < 1$. If $q$ is the maximum of $\omega $ on $\beta$, then $0<q<1$. Further on $\beta$ we have $$\vert u_{n+1}-u_n \vert\le q \cdot M_n,$$ and also (by definition of $M_n$) $$M_{n+1}\le q \cdot M_n.$$ By induction, it follows that $$M_{n+1}\le q^{n} \cdot M_{1},$$ and recalling again the definition of $M_n$ we get (first on $\beta$ and thus on $B$) $$\vert v_{n+1}-v_n \vert \le M_{n+1}\le q^{n} \cdot M_{1}.$$ When particularized to $\alpha$, this implies in view of $$\vert u_{n+1}-u_n \vert \le q^{n-1} \cdot M_{1} \quad \textrm{in }
\alpha,$$ and by the maximum principle this extends to $A$ (recall that $\partial A= \alpha \cup \gamma$ and the function $u_{n+1}-u_n$ vanishes on $\gamma$). Consequently, both series $\sum_{n}(u_{n+1}-u_{n})$ and $\sum_{n}(v_{n+1}-v_{n})$ converges uniformly on $A$ resp. $B$.

The limiting functions $u$ and $v$ of $u_n$ resp. $v_n$ are therefore harmonic on $A$ resp. $B$, and taking the limit in the definition of $u_n$ (see ) we see that $u$ vanishes on $\gamma$ \[$\bigstar$ equals $f$ on $\gamma$\].

We show finally that $u-u_0=v$ on $B$ \[$\bigstar$ $K$ probably?\]. Indeed, taking the limit in the first line of gives $u=v+u_0$ on $\alpha$, and the definition of $v_n$ pushed to its limit gives $v=u-u_0$ on $\beta$. Therefore the same identity $u-u_0=v$ holds on both contours of the ring $K$, and consequently its validity propagates throughout $K$.

Finally, as $v$ is harmonic on $B$ we are happy to conclude that $u$ fulfills all of our requirements: namely $u\in
H(A)$, $u=f$ on $\gamma=\partial F$ and $u-u_0$ defined on $A \cap
K=K$ coincide there with $v$ defined on the larger set $B\supset
K$, yielding the asserted harmonic extension.

\[NAIVE AND WRONG—see rather the argument given above\] Finally, \[compare p.198–199\] one obtains the Green’s function $G(z,t)$ by taking $u_0=\log \vert z \vert$, $f\equiv
0$ and $r_1=0$ \[Caution: this point is not made explicit in Nevanlinna\]. For this choice of $r_1$, note that $A=F-t$. The lemma supplies a unique $u\in H (F-t)$ such that $u_{\vert
\partial F}=0$ and so that $u-\log\vert z\vert=:h$ is harmonic on $F$. The function $u$ is the desired Green’s function $G(z,t)$.

Little green’s men dreams (extraterrestrial applications of Green’s)
====================================================================

The following three subsections are optional reading containing more questions than answers. The reader interested primarily in the Ahlfors map should preferably skip them.

From Green to Gromov? (directly bypassing Riemann and Löwner) {#sec:Green-to-Gromov}
-------------------------------------------------------------

To mention once more a deep frustration (the Gromov filling conjecture) it looks not completely crazy to hope that a careful examination of the Green’s function and the allied isothermic coordinates could prompt a solution of this problem. We tried quickly the \[14.08.12\] but failed dramatically as usual (along with circa 10 attempts of essentially the same vein). Roughly the idea would be to look at the streamlines of Green and its equipotentials, and remove every trajectory ending to the (finitely many) critical points of Green while attempting to estimate area via this (isothermic) parametrization. Of course, Schwarz’s inequality enters into the game but I only arrived at weak estimates like $\pi$ or $\pi/2$ (in place of $2\pi$!) upon doing highly fallacious calculus.

Schoenflies via Green? {#Schoenflies:sec}
----------------------

A notorious topological paradigm is the so-called Schoenflies theorem to the effect that a reasonably embedded sphere $S^{n-1}$ in ${\Bbb R}^n$ bounds a topological ball $B^n$. (There is a large debate (cf.e.g.Siebenmann 2005 [@Siebenmann_2005]) about who (and more broadly speaking which community) proved first the case $n=2$. In the topological-combinatorial realm there is a contribution of Schoenflies reaching full maturity ca. 1906, and somewhat earlier there is the contribution of Osgood which may have reached full stability with Carathéodory 1912 [@Caratheodory_1912]. Of course the statement (for $n=2$ and maybe even $n=3$) was largely anticipated heuristically by other workers, e.g. Moebius 1863 [@Moebius_1863]. Schoenflies’s theorem was extended to higher dimensions by J.W. Alexander ($n=3$ ca. 1922), B. Mazur and M. Brown (all $n$ ca. 1960) for any locally flat (e.g. smooth) hypersphere in ${\Bbb R}^n$. From Thom or Smale’s $h$-cobordism theorem (early 1960’s) it is inferred that the closed ball $B^n$ carries a unique smooth structure when $n\neq 4$ (the case $n=4$ being still largely unsettled). It follows that the interior of the smoothly embedded sphere is a ball differentiably. Another unsolved problem of longstanding is the truth of the same conclusion for $n=4$ (the so-called [*smooth Schoenflies*]{} in dimension $4$, SS4, see e.g. papers by Scharlemann). Naive physical (or bacteriological, cf. Fig.\[Green:fig\]) intuition about the Green’s function makes hard to visualize why there should be any anomaly for $n=4$, yet nobody ever succeeded to prove or disprove SS4. This belongs to the charming mysteries of low-dimensional differential topology at the critical dimension $n=4$. One may speculate about a naive approach to SS4 through the ca. 200 years older potential theory (of Laplace, Poisson, Green, Gauss, Dirichlet and Riemann’s era). Alas, there is few records in print of analysts feeling confident enough about the explorative aptitudes of the Green’s function (compare Fig.\[Green:fig\]) to claim the required diffeomorphism with $B^4$. Of course in the very small vicinity of the pole $t$ the levels of $G(z,t)$ (now $z\in {\Bbb R}^n$) look alike round spheres, and by the synchronization principle stating that each bacteria reaches the boundary at the same moment it may look immediate how to write down the diffeomorphism. Can somebody explain why this Green’s strategy fails to establish SS4. Less ambitiously can somebody reprove SSn (for $n \neq 4$) via the Green’s function. If yes with some little chance his/her proof will possibly include the case $n=4$.

Green, Schoenflies, Bergman and Lu Qi-Keng
------------------------------------------

\[06.08.12\] As discussed in the previous section, a dream would be to show SS4 (smooth Schoenflies conjecture) via the Green’s function in 4D-space ${\Bbb R}^4$. On reading an article by Boas 1996 (PAMS), where Suita-Yamada 1976 [@Suita-Yamada_1976] is cited we see a potential connection between both problems.

The problem of Lu Qi-Keng asks for domains where the Bergman kernel is zero-free (so-called Lu Qi-Keng=LQK-domains). Since Schiffer 1946 [@Schiffer_1946], there is an identity connecting the Bergman kernel to the Green’s function. It seems that the zeros of Bergman corresponds to the critical points of Green. Of course the latter is forced to have critical points as soon as the topology is complicated (not a disc). Suita-Yamada’s result that the Bergman kernel necessarily exhibits zeroes for membranes which are not discs looks nearly obvious. Hence LQK-bordered surfaces are precisely those topologically equivalent to the disc.

Now Boas in 1986 found a counterexample showing that no topological characterization of LQK-domains holds in higher dimensions: there exists in ${\Bbb C}^2$ a bounded, strongly pseudoconvex, contractible domain with $C^{\infty}$ regular boundary whose Bergman kernel does have zeroes. \[Addendum \[18.09.12\]: in fact upon reading Boas original paper (1986), Boas’ domain is diffeomorphic to the ball $B^4$.\]

[**Optimistic scenario (Green implies Schoenflies)**]{} It would be interesting to know what the topology of Boas’ hypersurface $S=\partial \Omega$ is. In view of Poincaré-Alexander-Lefschetz duality $S$ must be a homology sphere, if I don’t mistake. Now upon speculating that SS4 is true (by naive geometric intuition), and even more that it is provable via the streamlines of Green’s function, and granting a persistence of Schiffer’s Green-Bergman identity (in the realm of two complex variables), it may seem that Boas’s counterexample must have an “exotic” boundary (not diffeomorphic to $S^3$). \[Of course, not so in view of the just given Addendum.\]

[**Pessimistic scenario (Green does not implies Schoenflies).**]{} The other way around, assuming that Boas’ boundary is the 3-sphere, there would be critical points of the Green’s function $G(z,t)$ and Boas’s example may foil any naive attempt to reduces SS4 to the streamlines of the Green’s function. But even so maybe the Green-Bergman identity of Schiffer is specific to one complex variable, leaving some light hope that there is a potential-theoretic proof of the differential-topology puzzle of SS4.

So a bold conjecture (somewhat against Boas’ philosophy that there is no topological characterisation of LQK-domains) would be that any domain in ${\Bbb C}^2$ bounded by a smoothly embedded 3-sphere is a LQK-domain (i.e. its Bergman function is zero-free). \[This is wrong in view of Boas 1984 (addendum just mentioned)\]. However it could be true that the Green’s functions $G(z,t)$ for any center $t$ located in the inside of $\Sigma$ is critical point free, whereupon an elementary integration of its gradient flow should establish a diffeomorphism of the inside the spheroid with the ball $B^4$ with its usual differential structure. (Recall that it is yet another puzzle of low-dimensional topology, whether the $4$-ball has a unique smooth structure! All others balls (maybe except the five-dimensional one) do enjoy uniqueness by virtue of Smale’s $h$-cobordism theorem.) Note that the Bergman kernel is defined without reference to a basepoint whereas Green’s function requires a basepoint (its pole).

Arithmetics vs. Geometry (Belyi-Grothendieck vs. Ahlfors) {#sec:Belyi-Grothendieck}
---------------------------------------------------------

\[10.08.12\] Closed Riemann surfaces are subsumed to the (alienating) theorem of Belyi-Grothendieck, that [*a surface is defined over $\Qbar$ iff it admits a morphism to the line ${\Bbb
P}^1$ ramified at only $3$ points*]{} (so-called [*Belyi map*]{}). Another characterization (due to Shabat-Voevodsky 1989 [@Shabat-Voevodsky_1989/89]) is the possibility to triangulate the surface by equilateral triangles (with or without respect to the hyperbolic uniformizing metric). Basically this follows as one may sent homographically the 3 points to the vertices of the regular tetrahedron inscribed in the sphere. (Compare Belyi 1979/80 [@Beyli_1979/80], Grothendieck 1984 [@Grothendieck_1984/1997-esquisse-d'un-programme] “Esquisse d’un progamme”, Shabat-Voevodsky 1989 [@Shabat-Voevodsky_1989/89], Bost 1989/92/95 [@Bost_1989/92/95] (p.99–102), Colin de Verdière-Marin, etc.)

Is there an analog of this result for bordered surfaces in the context of Ahlfors (circle) mapping to the disc, and if so what is its precise shape? In the Riemann sphere any 3 points are transmutable through a Moebius rigid motion. The analog statement in the disc involves either one boundary point plus one interior point or 3 boundary points. Those are of course just the (heminegligent) hemispherical trace of real triads on the equatorial sphere corresponding to ${\Bbb P}^1$ with its standard real structure. (Remember that there is an exotic twisted real structure projectively realized by the invisible conic $x_0^2+x_1^2+x_2^2=0$.) This lack of canonical choice of a real triad on ${\Bbb P}^1$ could plague slightly an appropriate bordered version of Belyi-Grothendieck. \[12.11.12\] More seriously the ubiquity of real points in both those triads of the disc looks incompatible with Ahlfors maps lacking real ramification (when Schottky doubled to the realm of Klein’s orthosymmetric curves). Of course since bordered surfaces are in bijective correspondence with real orthosymmetric curves, one may expect first an answer along the line: [*a real orthosymmetric curve is defined over $\Qbar\cap {\Bbb R}\supset {\Bbb Q}$ iff it admits a totally real map ramified solely at $3$ real points or at one real point and $2$ imaginary conjugate points*]{}. Remember yet that total reality means that the inverse image of the real line is the real locus of the (orthosymmetric) curve, and since such maps lack real ramification our naive real version of Belyi-Grothendieck looks foiled. There seems to be a structural incompatibility between Belyi-Grothendieck and Klein-Ahlfors. Of course our desideratum of a simultaneous realization of Belyi-Grothendieck and arithmetization of Ahlfors may well just be a nihilist folly. By an “arithmetization of the Ahlfors map” we just mean something in much the same way as Belyi-Grothendieck arithmetizes Riemann’s existence theorem (any closed Riemann surface admits a morphism to the sphere ${\Bbb P}^1({\Bbb C})$). Possibly, one should be content with a reality version of Belyi-Grothendieck without bringing Ahlfors’ total reality into the picture. Then we have something like [*a real curve is defined over $\Qbar$ iff it admits a real morphism to the line ramified above only one of the two real triads, i.e. $0,1, \infty$ or $0,\pm i$.*]{} A priori this statement tolerates both types of real curves (ortho- and diasymmetric) and thus be more liberal than Ahlfors theorem (which tolerate only orthosymmetric curves). Adhering instead to the geometric interpretation of Belyi-Grothendieck (due to Shabat-Voevodsky 1989/89 [@Shabat-Voevodsky_1989/89]) in terms of equilateral triangulations might be more appealing. For instance one can imagine an orthosymmetric real curve with an equilateral triangulation invariant under (complex) conjugation. A such would according to BG be defined over $\Qbar$. It is clear that such a triangulation would contain the real circuits as subcomplex of the triangulation. In particular what is the significance of the corresponding vertices, e.g. as rational points of the curve. Also the tetrahedron plays some rôle in Belyi-Grothendieck-Shabat-Voevodsky and what are the rôle of the other Platonic solids? In particular the octahedron looks particularly well suited for getting pull-backed by the Ahlfors map? etc. \[14.11.12\] Of course invariant equilateral triangulability is not reserved to orthosymmetric patterns, as shown e.g. by the sphere acted upon by the antipodal map endowed with a Platonic triangulation invariant under the involution (octahedron and icosahedron). One can also consider in genus $1$ a rhombic lattice in ${\Bbb C}$ leading to a diasymmetric (non dividing) curve with $r=1$ real circuit. When the lattice is equilateral say spanned by 1 and $\omega$ a cubic root of $-1$, we have an obvious invariant equilateral triangulation by 8 triangles (with vertices at $0,
1/2,1,\omega/2,\omega/2+1/2,\omega$ and their conjugates).

\[10.08.12\] Back to the closed case, we know (Mordell-Faltings ca. 1981) that when the genus is $g\ge 2$ then the curve has finitely many rational points in any number field (finite extension of $\Bbb Q$). Of course this fails if we raise up to the full $\Qbar$ (as slicing a plane model by rational lines gives infinitely many $\Qbar$-points on the curve). One can dream on a connection between the “canonical” equilateral triangulation (ET) and the finitely many rational points evaluated in the various number fields.

Of course given an ET of an arithmetic (Riemann) surface we can imagine a subdivision into another ET. Given a Euclidean equilateral triangle it is obvious how to subdivide it in 4 smaller equilateral triangles (bisecting the edges). Is there an equivalent subdivision for hyperbolic equilateral triangles? (I cannot see one...) Thus maybe there is some rigidity. At any rate among all ET of an arithmetic $\Qbar$-surface there is one involving the least number of triangles. This gives an integer invariant for any Riemann surface defined over $\Qbar$. Can this value be related to the finitely many rational points when $g\ge
2$?

By Gauss(-Bonnet) \[$\alpha+\beta+\gamma=\pi+\int_T K dA$\] which reduces to $3\alpha=\pi-area$ for an equi-triangle in constant negative curvature equal to $-1$ we see a direct relation between the area and its angle of an equi-triangle.

\[11.12.12\] For a more lucid Real Belyi theory than our vague ideas, compare the account in Köck-Singerman 2006 [@Koeck-Singerman_2006], where however the Ahlfors maps does not seem not enter the arena.

Ahlfors’ proof {#Ahlfors-proof:sec}
==============

\[27.08.12\] This section is our modest attempt to examine and understand Ahlfors’ existence proof of a circle map (of degree $\le r+2p$). Alas we failed this basic goal, but it is perhaps of some interest to discuss the original text while trying to capture some mental pictures (made real) which may have circulated in Ahlfors’ vision. More objectively we also try to identify if Ahlfors argument can be boosted to reassess the prediction of maps with smaller controlled degree $\le r+p$ (Gabard 2006 [@Gabard_2006]). We emphasize once more that Gabard’s result is potentially false, but even if so, it is evident that for low values of the invariants $(r,p)$ Ahlfors bound $r+2p$ fails sharpness. Near its completion, Ahlfors proof takes a geometric “tournure” (convex geometry) where there seems to be some free room suitable for improvements. We tried to imagine some (topological) strategy which could possibly sharpen Ahlfors result along his method (at least for low invariants). This is, apart from didactic interest, the only original idea of the present section.

In the original paper Ahlfors 1950 [@Ahlfors_1950 p.124–126], the existence proof, we are interested in, occupies only a short 2 pages argument which looks essentially self-contained albeit not quite easy to digest. I would (personally) be extremely grateful if someone understanding Ahlfors proof could publish a more pedestrian account than Ahlfors’, explaining it in full details. Some of the background required is dispatched earlier in the text (esp. p.103–105 in ), hence trying some rearrangement could improve readability. We were personally not able to follow all the (boring) computations or formulas required by Ahlfors.

Alas, big masters tend to give only cryptical output of boring computations. Ahlfors is further typical for his annoying (arrogant?) style “it is clear that”, etc. and one often suffers a lot just to fill some details. Of course, nothing is clear in mathematics especially when it comes to follow mechanical computations. Maybe the presence of those just reveals a lack of conceptual grasp over the underlying geometry. Trying to be more optimistic and less severe due to frustration, it would be nice—I repeat myself intentionally—if somebody could take the defense of Ahlfors by presenting an argument as close as possible to the original (meaning perhaps just eradication of misprints, if any?) which further would be completely mechanical, i.e. where each identity is decorated by the appropriate tag referring to the formula under application.

Of course, Ahlfors’ proof seems to involve nothing more than the formalism of differential forms (à la Cartan, de Rham, etc., which he learned from A. Weil’s visit in Scandinavia during World War II), plus Stokes’ formula (already a nightmare to prove, at least for Bourbaki) and the allied integration-by-part formula (consequence of Leibniz’s rule). We were personally unable to produce a perfectly pedestrian (accessible to anybody, in particular myself!) exposition of Ahlfors’ account, lacking both intelligence and patience to make his text perfectly intelligible. The writer probably read this Ahlfors’ argument several times in diagonal (since ca. 2001/02), but never completely understood the details. My motivation for looking at it more closely became more acute, after realizing (August 2012) that it is not completely trivial to complete the Green’s function strategy to the problem (cf. previous Section \[Green:sec\]). It should be noted that Ahlfors’ argument does not employ exactly the Green’s function, but a close relative cousin with pole located on the boundary instead of the interior. As a matter of joking we refer to it as the [*Red’s function*]{}, and as far as we know there is no (standardized) terminology to refer to this object! Accordingly, Ahlfors rather constructs an half-plane map instead of a circle map. Of course both moneys are ultimately convertible, yet both geometrically and analytically this implies a little alteration of the viewpoint. One may then may get a bit confused about wondering on the optimal strategy.

Finally, remember that several workers in Japan or the US seem to have found necessary to rework Ahlfors’ proof in a more do-it-yourself fashion. Several other authors, having to cite Ahlfors work, often cross-cited those alternative proofs, like those produced by Heins 1950 [@Heins_1950] or Royden 1962 [@Royden_1962] (cf. e.g. Stout 1972 [@Stout_1972] or Gamelin 1973 [@Gamelin_1973-Extremal-I p.3], who both cite Royden for the piece of work originally due to Ahlfors). For a more complete list of “dissident” authors drifting from Ahlfors’ account as the optimal source compare Sec.\[dissident:sec\]. The latter tabulation is supposed to illustrate that I may not be isolated in having missed the full joy of complete satisfaction with Ahlfors’ output. Yet, personally we still would like to believe that Ahlfors account is superior in geometric quintessence to all of what followed, but only regret to have missed some crucial details. As far as we know, nobody ever raised a fatal objection against Ahlfors’ proof. (Personally, I only criticize a lack of details in the execution, plus a matter of organization[^13] and finally a lack of geometric visualization.) It may also be speculated that the argument published by Ahlfors 1950 [@Ahlfors_1950] (and reproduced below) is not the way Ahlfors originally discovered the statement (as early as 1948, cf. Nehari 1950 [@Nehari_1950]), which looks more intuitive when approached from the Green’s function viewpoint, or just bare Riemann-Roch theorem (yet with dangerous probability of collision, cf. the remark in Gabard 2006 [@Gabard_2006 p.949]). In the sequel we shall attempt to conciliate Ahlfors’ analytic treatment with the geometric intuition behind it.

The goal is (as usual) to prove:

[(Ahlfors 1950 [@Ahlfors_1950 p.124–126])]{} Let $\overline{W}$ be a compact bordered Riemann surface of genus $p$ with $r\ge 1$ contours. Then there exists a circle map $f\colon \overline{W}\to \overline{\Delta}$ of degree $\le
r+2p=g+1$, where $g:=(r-1)+2p$ can be either interpreted as the genus of the (Schottky) double or as the number of essential $1$-cycles on $F$ considered up to homologies (the so-called Betti number).

The core of Ahlfors’ argument
-----------------------------

For the proof Ahlfors uses the concept of a Schottky differentials. Those are differentials on the bordered surface which extends to the Schottky double. The following subclass plays a special rôle: $$S_r=\textrm{ the space of analytic Schottky differentials
which are real along $C=\partial \overline{W}$}.$$

\[bipole:lemma\] Given $g+1$ distinct points $z_j$ on the contour $C=\partial \overline{W}$ and corresponding reals $A_j \in
{\Bbb R}$, it is possible to construct an analytic differential $\theta_0$ which is real on[^14] $C$ and whose only singularities are double poles at the $z_j$ with singular parts: $$A_j \frac{dz}{(z-z_j)^2},$$ where the local variable $z$ at $z_j$ is chosen so as to map $C$ onto the real-axis ${\Bbb R}$ and inner points of $W$ into the upper half-plane.

Further such a differential $\theta_0$ is uniquely determined up to a differential $\theta\in S_r$, and for a proper choice of the latter we can make vanish the periods and half-periods of the imaginary-part $\Im \theta_0$.

Ahlfors prefers to construct instead of a circle map a upper half-plane mapping $F\colon \overline W \to \overline H=\{ \Im z
\ge 0\}$ which will ultimately arise through the equation $\theta_0=dF$, after arranging exactness of $\theta_0$ for a suitable location of the $z_j$ and some $A_j\ge 0$.

Once this is achieved we may write $\theta_0=dF$ for some analytic function $F$ on $\overline W$. The latter is uniquely defined modulo an additive constant and can be chosen real on $C=\partial \overline W$, except at the $z_j$ where $\Im F$ becomes positively infinite. The maximum principle ensures $\Im F >0$ on the whole interior $W$, and therefore $F$ is the desired half-plane mapping of degree $\le r+2p$.

This is the bare strategy of the argument, but it is time to adventure into the details.

A first ingredient is the fact (compare the second Corollary on p.109):

\[g-dimensional:lemma\] The real vector space $S_r$ (of Schottky differentials real along the border) has real dimension $g$.

This looks rather plausible upon thinking with the Schottky double and explains the second (uniqueness) clause of the above lemma. Notice indeed that there is $(r-1)$ half-periods corresponding to pathes on the bordered surface $\overline W$ joining a fixed contour $C_1$ to the remaining ones $C_2,
\dots, C_r$ and $2p$ full periods arising by winding around the $p$ handles.

To arrange exactness of $\theta_0$, Ahlfors employs the inner product $(\theta_0, \theta)$ and a corresponding criterion for exactness in terms of orthogonality to the space $S_r$ (cf. Lemma \[orthogonality:lemma\] below). (The reader can skip the proof of the next two lemmas to move directly to the core of the argument which in our opinion is Lemma \[clever-choice:lemma\].)

Before attacking the proof we first recall the pertinent definitions. The [*inner product*]{} of two differentials on a Riemann surface is defined by: $$(\omega_1, \omega_2)=\int_W \omega_1
\overline{\omega_2}^{\ast},$$ where the star denotes the [*conjugate differential*]{} and the bar is the [*complex conjugate*]{} (compare Ahlfors, p.103). (Locally if $\omega=a\, dx+b\,dy$ then $\omega^{\ast}=-b\, dx+a\,dy$ and $\overline\omega=\bar a\,
dx+\bar b\, dy$)

Further we need probably Stokes $$\int_W d\omega=\int_C \omega,$$ which combined with Leibniz $$d(f \omega)=df \cdot \omega + f d \omega.$$ gives the so-called integration by parts formula $$\int_W (df \cdot \omega + f d \omega)\buildrel{\rm
\tiny{Leibniz}}\over=\int_W d(f\omega) \buildrel{\rm
\tiny{Stokes}}\over{=}\int_C f\omega,$$ which can be rewritten as $$\int_W df \cdot \omega =\int_C f\omega- \int_W f d \omega,$$ which is hopefully the exact form used (subconsciously) in the sequel.

Further he requires an expression of this inner product in term of local variables. Namely the following:

\[loc-formula:lemma\] If $\theta=\alpha dz$ near $z_j$, then we have the following formula for the inner product $$\label{loc-formula:eq} (\theta_0,
\theta)=-\pi\textstyle\sum_{j=1}^{g+1}A_j \alpha(z_j),$$ where $\theta_0$ is the differential of Lemma \[bipole:lemma\].

As in the first lemma, once we have arranged vanishing of the period and the half-period of the imaginary part $\Im
\theta_0$ we may write something like $$\theta_0-\overline{\theta_0}=i \, dG,$$ where $G$ vanishes on $C$ except at the $z_j$. Then brute-force computation gives $$\label{inner-product:eq}
(\theta_0,
\theta)\buildrel{?}\over{=}(\theta_0-\overline{\theta_0},
\theta)=(i \, dG, \theta)=\dots=-\int_C G \bar \theta,$$ where the “dots” indicates steps left un-detailed by Ahlfors. Of course one should first apply the definition of the inner product and then use integration-by-part, as we just recalled, while noticing that the second term vanish involving the differential of an analytic function. \[Alas, the writer had not the energy to complete the detailed computation.\]

Now writing $\theta=\alpha dz$ near $z_j$, Ahlfors claims the following local expression for $G$ $$G \sim i\, A_j (\frac{1}{z-z_j}-\frac{1}{\bar z-z_j}),$$ whereupon he claims that the singularity at $z_j$ contributes the amount $-\pi A_j \alpha(z_j)$ to the last integral of . The announced formula should follow easily.

\[orthogonality:lemma\] $\theta_0$ is exact iff $(\theta_0, \theta)=0$ for all $\theta
\in S_r$.

A priori we could expect to save forces by proving only sufficiency (i.e. the implication $[\Leftarrow]$), but alas Ahlfors’ proof requires the direct sense as well, plus the previous lemma involving the rather (unappealing) computation in local coordinate. Enough philosophy and lamentation, and let us follow along Ahlfors’ exposition.

$[\Rightarrow]$ Write $\theta_0=dF$. Then Ahlfors write cryptically $$(\theta_0,\theta)\buildrel{?}\over{=}(\theta_0,\theta+\bar
\theta)=\int_W dF \overline{\cdots}=i\int_C F(\bar \theta-
\theta)=\pi \textstyle\sum_{j=1}^{g+1 } A_j \alpha(z_j),$$ and comparison with Equation  shows that $(\theta_0, \theta)=0$, as required.

$[\Leftarrow]$ Conversely, suppose $(\theta_0, \theta)=0$ for all $\theta\in S_r$, and let $\varphi$ be the analytic Schottky differential making $\theta_0-\varphi$ exact. Then by the former implication[^15] $( \theta_0-\varphi, \theta)=0$ and so $(\varphi,
\theta)=0$ for all $\theta \in S_r$. This implies $\varphi=0$, and we conclude that $\theta_0$ is exact.

Combining both those lemmas, the exactness of $\theta_0$ is reduced to the following (tricky) lemma, involving a mixture of convex geometry and Stokes formula (which Ahlfors calls the [*fundamental formula*]{} probably due its anticipation by Green or Gauss and others).

\[clever-choice:lemma\] It is possible to choose the $z_j$ and the $A_j\ge 0$ so that $$\label{Ahlfors_sum-Aj:eq}
\textstyle\sum_{j=1}^{g+1}A_j \alpha(z_j)=0$$ for all $\theta \in S_r$ locally expressed as $\theta=\alpha dz$.

Let $\theta_i\in S_r$ ($i=1, \dots ,g$) be a basis of the $g$-dimensional space $S_r$ (cf. Lemma \[g-dimensional:lemma\]). Locally we can write $\theta_i= \alpha_i dz$ near $z_j$. Equation  can be satisfied with $A_j\ge 0$ iff the simplex with vertices $$(\alpha_1(z_j), \dots, \alpha_g(z_j))\in {\Bbb R}^g
\qquad\textrm{ for } j=1, \dots, {g+1}$$ contains the origin $0\in {\Bbb R}^g$.

If this condition is not full-filled for any choice of the $z_j$, the convex-hull of the set of points $$K:=\{(\alpha_1(t), \dots, \alpha_g(t)) : \textrm{ for } t\in C
\}$$ would fail to contain $0$. (One can think of this set as a sort of link (in the sense of knot theory) traced in ${\Bbb
R}^g$ with $r$ components. However the latter is not perfectly canonical since the $\alpha_i(t)$ depends on the local chart.

[*Expressing some naive doubts.*]{} So here Ahlfors argument looks a bit fragile (or at least sketchy) as one probably requires to fix a finite system of holomorphic charts covering the full contour of the bordered surface). \[We do not have a specific objection, yet it should be noted that the whole Ahlfors theory even that of the refined extremal problem depends on the non-emptiness of the class of bounded functions, hence upon the present argument! In principle even if there should be a global crash of Ahlfors’ proof here, then the theorem should conserves its validity in view of several subsequent treatments hopefully logically more reliable, we cite:

$\bullet$ Kuramochi 1952 [@Kuramochi_1952] (alas quite unreadable?),

$\bullet$ Mizumoto 1960 [@Mizumoto_1960],

$\bullet$ Royden 1962 [@Royden_1962] (alas a bit functional-analytic, whereas the statement sentimentally belongs to pure geometric function theory), and maybe

$\bullet$ Gabard 2006 [@Gabard_2006] (hopefully reliable, at least it first part not improving Ahlfors’ $r+2p$).

However it is likely that the set $K$ can be defined according to the totality of possible $\alpha_i(t)$ arising through a fixed system of permissible charts covering the contour $C$.

Now a (Euclidean) set of ${\Bbb R^g}$ whose convex-hull misses the origin is contained in a closed half-space \[maybe even an open half-space?\]. Thus there exists scalars $a_1, \dots, a_g
\in {\Bbb R}$ (not all zero) so that $$\textstyle\sum_{i=1}^g a_i \alpha_i(t)\ge 0 \quad \textrm {for
all } t\in C.$$ (Geometrically, this is to be interpreted as the scalar product with the vector $(a_1, \dots, a_g) \in {\Bbb R}^g$ orthogonal to the hyperplane whose half contains the set $K$.) Hence the corresponding differential $\theta= \sum_{i=1}^g a_i
\theta_i$ is $\ge 0$ along $C$. \[Maybe strict???\] However this violates the fact that $\int_C \theta=0$, as prompted by Stokes’ formula $$\int_{C=\partial \overline W} \theta=\int_{\overline W} d
\theta,$$ and the fact that $\theta$ belongs to $S_r$, hence analytic, and thus closed, i.e. $d\theta =0$.

Geometric interpretation as dipoles
-----------------------------------

\[28.08.12\] Let $F$ be a membrane (=compact bordered Riemann surface), then Ahlfors constructed (cf. previous subsection) a half-plane map $F\to \overline H:=\{\Im z\ge 0 \}$ to the closed upper-half plane. We get a circle map after post-composing with the natural conformal map to the unit-disc $ \overline H \to
\overline{\Delta}$. Under such a map, the horizontal lines transforms to a pencil of circles tangent to the boundary and vertical lines mutate to arc of circles orthogonal to the boundary. (cf. Fig.\[Dipole:fig\]a). One recognizes essentially the so-called Hawaiian earrings (cf. Fig.\[Dipole:fig\]b).

Given a circle map, one can pull-back the isothermic (=right-angled) Hawaiian bi-foliation to obtain a graphical representation of the circle map.

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Starting with with a (doubly-connected) ring, one obtains Fig.\[Dipole:fig\]c or Fig.\[Dipole:fig\]d. Going to higher connectivity on gets for instance Fig.\[Dipole:fig\]e. The Bieberbach-Grunsky theorem (or just Riemann-Roch, cf. e.g. Lemma \[Enriques-Chisini:lemma\]) tell us that we can prescribe a point on each contour and there is a circle map taking all those points to the same image in the unit-circle $S^1=\{ \vert z \vert=1\}$. Hence, we enjoy complete freedom in picturing the isothermic bi-foliation of circle maps, at least in the planar case. This situation is to be contrasted with the situation for the zeros, where some hidden symmetry requires to be fulfilled (compare e.g. Gabard 2006, where we have the condition $D\sim D^{\sigma}$ of linear equivalence of the divisor with its conjugate, an also Fedorov 1991 [@Fedorov_1991] who speaks of an opaque condition that must be satisfied to prescribe the zeros).

Of course, the contemplation (and manufacture) of such pictures raises more questions than clarifying the perception of Ahlfors’ theorem. One can hope some guidance via physical intuition (if one feels comfortable with the mineral world) or appeal again to the metaphor about proliferation of bacteria in some nutritive medium. We do not repeat the long discourse we made already for Green (cf. Sec.\[Green:sec\], esp. Fig.\[Green:fig\]) where one had radial expansions emanating from an inner point. Presently, the bacteria are rather located on the boundary, whereupon their local expansion is more of the Hawaiian type, or if you prefer look alike the Doppler effect at the critical speed of sound. The dipole of our title would occur upon considering the symmetric Schottky double of the membrane. This new Hawaiian/Doppler mode of expansion can again be explained via lacking nutritive resources caused by the boundary where the world stops.

On Fig.\[Dipole:fig\]f, we have attempted to picture the pull-back of the Hawaiian foliation under a circle-map of degree $r+p=1+1=2$ (for the value $r+p$ predicted by Gabard). This picture looks anomalous for the following reason. Letting grow the population, there is a first junction of the 2 populations right “under” the handle, then there is 2 self-junction at 2 points aside the handle. From now on the bacteria starts invading the handle from both “sides” and will actually merge on the core circle of it. This is problematic since ultimately the expansion should finish along the boundary contours (by definition of a circle-map). It easy to manufacture a picture where no such anomaly occurs (cf. e.g. Fig.\[Dipole:fig\]g which admittedly requires some little effort of concentration to contemplate its morphogenesis). Of course, similar pictures can be made by prescribing less boundary points than the degree of circle-maps predicted by Ahlfors $r+2p$ or $r+p$, e.g. by a choosing a single dipole, cf. Fig.\[Dipole:fig\]h and Fig.\[Dipole:fig\]i. However those patterns cannot correspond to circle-map due to obvious topological obstructions: first the degree of a circle-map must be $\ge r$ impeding Fig.\[Dipole:fig\]h to be allied to a circle-map. As to Fig.\[Dipole:fig\]i the degree would be one, implying the circle-map to be unramified and covering theory (of the simply-connected disc) implies the membrane $F$ to be the disc, violating its genus $1$ nature.

We stop this graphical discussion at this primitive stage, yet it is to be hoped that a deeper study of such figures could lead to some theoretical results complementing the understanding of the Ahlfors maps. Perhaps such (dipole) isothermic drawings are of some relevance to Gromov’s filling conjecture, as we already suggested in the case of Green’s function (Sec.\[sec:Green-to-Gromov\]).

\[29.08.12\] In fact there is a another more convincing obstruction impeding Fig.\[Dipole:fig\]f to represent a circle-map. This consists in identifying the counter-images of the growing Hawaiian circles past the critical levels while checking if they contribute to the correct numerical multiplicity permissible with the degree of the branched covering. To be concrete we enumerate a series of typical smooth levels on Fig.\[Dipole:fig\]f. The first one denoted $1$ consists of $2$ little circles. Past the first critical level, we see the curve $2$ with $1$ component. After the next critical level, we pick a curve $3$, which has $3$ components. This is too much for our mapping to be of degree 2. This proves that Fig.\[Dipole:fig\]f do not correspond to a circle-map.

In contrast repeating the same counting exercise for Fig.\[Dipole:fig\]g, no such excess occurs. The level $1$ has 2 components, level 2 (chosen after the first critical level) has one component, level 3 has 2 components and finally level 4 has 1 component. Thus the picture looks topologically coherent, but it is evident that it is far from metrically realist. Naively speaking we were forced to distort the propagation so has to have a virtually planar mode of depiction for the levels.

Trying to recover Ahlfors from the Red’s function {#Red's-function:sec}
-------------------------------------------------

\[29.08.12\] Let $F$ denote a finite (=compact) bordered Riemann surface of genus $p$ and with $r$ contours. From the previous section, it seems evident that there is some canonical function akin to the Green’s function yet with pole pushed to the boundary (dipole singularity when doubled). Call them perhaps the [*Red’s function*]{} as an [*ad hoc*]{} acronym honoring writers like Riemann, Schwarz, Klein, Koebe, Ahlfors, etc. Such a Red’s function denoted $R(z,t)=R_t(z)$ with (di)pole at $t\in \partial F$ (a boundary-point) is defined by the property of being harmonic, null along $\partial F$ save at $t$ where it becomes positively infinite according to a specific local singularity (maybe like ${\rm Re}(1/z^2)$). \[18.10.12\] As a more intrinsic definition one can define $R_t$ as the unique positive harmonic function vanishing continuously along $\partial F-\{
t\}$. The function then looks unique up to scalar multiple. Note however that Heins (in e.g. Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF p.241, right after Thm3.1]) defines the function $u_{\zeta}$ our $R_t$ by adding the requirement of minimality (in the sense of Martin 1941 [@Martin_1941]). A positive function $u$ is minimal if whenever there is a smaller function $0<v<u$, $v$ is a constant multiple of $u$.

The sudden explosion of $R_t$ at just one boundary point looks at first almost paradoxical, but see again our previous Fig.\[Dipole:fig\]b-h-i) for a depiction of their levels and one can of course imagine such a function just as a “borderline” degeneration of the usual Green’s function. Now one can attempt to construct a half-plane-map (HP-map, for short), by considering a superposition $R(z):=\sum_{i=1}^{d} R(z, t_i)$ of such Red’s functions $R(z,t_i)$ for several points $t_i$ on the border. The formula $$\varphi:=R+i R^{\ast},$$ where $R^{\ast}$ is the conjugate function would then define the HP-map provided the conjugate potential is single-valued in other word that the conjugate differential of $R$, $(d{R})^{\ast}$ is period free. Since $F$ has $(r-1)+2p=:g$ essential cycles (homologically independent), a parameter count suggests that if $d=g+1$ there is enough freedom to annihilate all the $g$ periods of $d{R}^{\ast}$.

Maybe this approach (which presumably differs not very much from Ahlfors’) has some technical advantage over the Green’s technique (presented in Sec.\[Green:sec\]). First it seems that the dipole singularity has some linear character contrasting with the arithmetical rigidity of the logarithmic singularity. Thus it is permissible to form a more general linear combination $$R(z):=\textstyle\sum_{i=1}^{d} \lambda_i R(z, t_i),$$ with some reals $\lambda_i$ which must however be $\ge 0$. Hence killing the periods essentially reduces to linear algebra. Another advantage over the Green’s approach stems from the fact that in the interior we meet no singularity thus the period mapping looks less dubious.

As usual we write down the period mapping by integrating the $1$-form $dR^{\ast}$ along the $g$ many 1-cycles $\gamma_1, \dots,
\gamma_g$ and obtain for each fixed $t_1, \dots t_{g+1}\in
\partial F$ a linear map ${\Bbb R}^{g+1} \to {\Bbb R}^g$. Thus there is some non-zero vector in the kernel, and the corresponding $(\lambda_i)$ would solve the problem, provided one is able to check that they can be chosen $\ge 0$. This is non-trivial and a priori it is not evident (and nobody ever asserted) that this can be done for any choice of the $(g+1)$-tuple $t_i$.

So it is just here that the difficulty starts, and that some idea is required to complete the proof.

\[04.09.12\] Due to a lack of creativity/energy, I was blocked here for a couple of days. So let me make a list of writers who seem to have grasped the geometric quintessence of Ahlfors’ argument:

$\bullet$ Gamelin-Voichick 1968 [@Gamelin-Voichick_1968 p.926]: “According to \[1, §4.2\](=Ahlfors 1950 [@Ahlfors_1950]), there exist $r+1$ ($r=g$ in our notation) points $w_1, \dots, w_{r+1}$ on $bR$ such that if $B_j$ is the period vector of the singular function $T_j$ corresponding to a unit point mass at $w_j$, then $B_1, \dots, B_{r+1}$ are the vertices of a simplex in ${\Bbb R}^r$ which contains $0$ as an interior point.” \[10.09.12\]

$\bullet$ Fisher 1973 [@Fisher_1973 p.1187/88]: “By a theorem of Ahlfors \[A1; §4.2\] there is a set of $r+1$ points $p_j$ in $\Gamma$ such that if $v_j$ is the period vector of a unit mass at $p_j$, then $v_0,\dots, v_r$ form the vertices of a simplex in ${\Bbb R}^{r}$ which contains the origin as an interior point.” \[this looks alike verbatim copy of the previous source, yet reinforce confidence in the viewpoint\]

\[07.09.12\] In fact some little hope to complete the argument is raised by borrowing ideas of convex geometry used by Ahlfors, yet in our context which is perhaps not so reliable (albeit it seems to match with the Gamelin-Voichick twist of Ahlfors). Alas, we failed to recover Ahlfors statement, but we see obvious room for improving upon Ahlfors by using essentially his method of proof augmented by some further geometric tricks. Ideally one would like to recover the bound predicted in Gabard 2006 [@Gabard_2006] by using an argument very close to Ahlfors’. Let us now be more concrete.

Again we fix some $d$ points $t_1, \dots t_d$ on the boundary $\partial F$, with at least one point one each contour $C_i$ (forming the boundary $\partial F$). For any point $t\in
\partial F$ the function $R_t(z):=R(z,t)$ is uniquely defined once a chart around $t$ is specified (otherwise it is unique only up to a positive scaling factor). Let us assume $R_t$ fixed once for all with a continuous dependence over the parameter $t$. (Alas the writer has no clear-cut justification of this possibility. \[09.09.12\] Maybe use a boundary uniformizer for an annular tubular neighborhood of each contour, cf. e.g. Hasumi 1966 [@Hasumi_1966 p.241], also Gamelin-Voichick 1968 [@Gamelin-Voichick_1968 p.926]. \[18.10.12\] Of course since $R_t$ is unique up to scalar multiple, we are somehow choosing a section of a ray-bundle and even if after winding once around an oval of $\partial F$ the $R_t$ should not return to its initial position $R_{t_0}$, it seems easy to apply a sort of “closing lemma” so that $R_t$ comes back to the original choice.)

We now introduce $\Pi(t)$ the period of $(dR_t)^{\ast}$ along the fixed representatives $\gamma_1, \dots, \gamma_g$ of the first homology, that is, $$\Pi(t)=(\textstyle\int_{\gamma_1} (dR_t)^{\ast}, \dots,
\int_{\gamma_g}(dR_t)^{\ast}) \in {\Bbb R}^g.$$ We seek $R$ of the form $R=\sum_{i=1}^{d} \lambda_i R_{t_i}$ with $\lambda_i>0$ such that the conjugate differential $(dR)^{\ast}$ is period-free. Period-freeness amounts to say that [*the simplex of ${\Bbb R}^g$ spanned by the $\Pi(t_1), \dots, \Pi(t_d)$ contains the origin in its interior*]{}[^16]. Then positive masses $\lambda_i$ can be assigned to the $\Pi(t_i)$ so that the origin occurs as barycenter of this masses distribution.

The italicized condition is equivalent to saying that the convex-hull of the set $X:=\Pi(\partial F)$ contains the origin (say then that the set $X$ is balanced). Balancedness paraphrases also into the condition that the set is not contained in a half-space delimited by a hyperplane through the origin.

Ahlfors derives his result from the following simple lemma applied to $X=\Pi(\partial F)$.

Let $X$ be a subset of some number space ${\Bbb
R}^g$. Any point in the convex-hull of $X$ is the barycenter (=convex combination involving positive coefficients) of at most $g+1$ points of $X$.

Of course the lemma is sharp in general: consider $X\subset {\Bbb
R}^2$ a set of 3 points in general position (not collinear) then any point chosen in the interior of the convex-hull of $X$ (a simplex) requires all 3 points in a barycentric combination. However if $X$ is a more continuous shape like a topological circle in ${\Bbb R}^2$ it is clear that 2 points situated on $X$ will suffice (cf. Fig.\[Convex:fig\]a). Indeed, imagine first that $X$ is a Jordan curve and that the point lies in its interior. Any line through the point intercepts the Jordan curve in at least 2 points which can be used for a convex combination of the given point. If the point is not in the interior, one can meet an “U-shaped” Jordan curve where the point is situated near the top of the “U” (Fig.\[Convex:fig\]b), yet still expressible as the barycenter of 2 points on the top of the “U”. This already raises some hope upon improving Ahlfors, and optimistically a careful inspection could recover the $r+p$ bound of Gabard 2006 [@Gabard_2006].

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Let us summarize the situation. The lemma shows is that if the convex-hull of $\Pi(\partial F)$ contains the origin $0$, then one can certainly find $g+1$ points $t_i$ (eventually fewer) and corresponding $\lambda_i>0$ such that $R=\sum \lambda_i R_{t_i}$ has a period-free conjugate differential. This implies the existence of a half-plane map (via $f=R+iR^{\ast}$) of degree $\le
g+1=r+2p$, recovering therefore Ahlfors’ result of 1950.

Thus the problem splits in two parts:

$\bullet$ Step (1): explain why the convex-hull of $\Pi(\partial
F)$ contains the origin $0$ (implying Ahlfors’ $r+2p$ bound); (Ahlfors is able to do this, yet hopefully the ambient context of his argument can be slightly simplified to our present setting which is closer say to Heins’ accounts in 1950 [@Heins_1950] or 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF])

$\bullet$ Step (2): try to lower Ahlfors degree $r+2p$ by taking advantage of the fact that $X=\Pi(\partial F)$ is not an arbitrary set but the continuous image of $r$ circles; (ideally try to recover the $r+p$ upper bound predicted in Gabard 2006 [@Gabard_2006], or at least partial improvements of Ahlfors bound $r+2p$ for low values of the invariants $(r,p)$).

As to the first point (1), we notice that if it is violated then the set $\Pi(\partial F)$ is contained in a half-space of ${\Bbb
R}^g$. Thus there is a non-zero vector $a=(a_1,\dots, a_g)\in
{\Bbb R}^{g}$ such that the scalar product $(a, \Pi(t))>0$ for all $t\in \partial F$. This means $$\sum_{i=1}^g a_i \int_{\gamma_i}(dR_t)^{\ast}>0 \textrm{ for
all } t\in \partial F.$$ Alas, the writer failed to find a reason why this should be a contradiction. (In Ahlfors’ presentation Stokes’ theorem plays a crucial rôle.)

Even if the present geometric strategy (cooked by the writer via slow assimilation of the very classical strategy of annihilating periods) should be impossible to complete, nothing forbids to switch again to the original treatment of Ahlfors, and apply our Step (2), whose tangibleness relies on Fig.\[Convex:fig\]. The essential point is that ultimately the geometric setting is invariably the one and same problem of convex geometry, whether we start from Ahlfors “analytic” approach or from our more geometric reformulation via the Red’s functions.

Let us be more explicit. We have a map $\Pi\colon
\partial F=:C \to {\Bbb R}^g$. (“$C$” for contours, like in Ahlfors notation.) In Ahlfors’ paper this occurs as the map $C\ni t \mapsto (\alpha_1(t), \dots ,
\alpha_g(t))$ cf. p.125 of his article. (From the algebro-geometric viewpoint this must probably be the vectorial lift of the so-called [*canonical map*]{} $\varphi\colon C \to
{\Bbb P}^{g-1}$ (usually ascribed to Noether or Klein) allied to the canonical series $\vert K \vert$ living over the curve $C$, obtained by doubling the bordered Riemann surface.)

We try to address the second issue (2). The setting is a map $\Pi\colon C \to {\Bbb R}^g$ whose image is balanced (i.e. the convex-hull of the image contains the origin, or equivalently the set $\Pi(\partial F)$ is not contained in any open half-space of ${\Bbb R}^g$ delimited by a hyperplane through the origin). The whole problem is then reduced to the following geometric question.

\[problem:Ahlfors-circuit\] Given two integers $r\ge 1$ and $p\ge 0$. Let $g:=(r-1)+2p$, and suppose given in the corresponding Euclidean space ${\Bbb R}^g$ a collection of $r$ (possibly singular) circles $C_1, \dots, C_r$. It is assumed that the union of all these circles is balanced. Find the minimum cardinality of a group of $d$ points with at least one point on each $C_i$ spanning a simplex containing the origin.

The previous lemma solves the problem for degree $d=g+1=r+2p$ (recovering Ahlfors’ result). To do better we start from such a group and try to move the vertices, while taking care that the simplex still contains $0$. From the $r+2p$ points, we imagine $r$ many as essentially fixed and the other coupled in $p$ many pairs. The initial simplex is top-dimensional matching the dimension $g$ of the ambient number-space ${\Bbb R}^g$. Moving vertices, it looks reasonable that we may coalesce two points of the $g$-simplex to get a $(g-1)$-simplex still containing $0$. This presupposes both coalescing points being located on the same circuit $C_i$ (try to argue with the pigeon hole principle). After $p$ such collisions (one for each pair) we reach the degree $r+p$ predicted by Gabard 2006 [@Gabard_2006].

Alas this “piano mover” argument is not easy to believe, nor to prove. Perhaps a less naive variant involving an adequate trick (most probably of a topological nature akin say to the Borsuk-Ulam proof of the ham-sandwich theorem) could recover the $r+p$ bound. Less optimistically, it may happen that the above problem is not always soluble with $d\le r+p$, but only for circuits $C_i$ arising from bordered Riemann surfaces via the period map recipe.

At any rate, we see the prominent rôle of convex geometry in the question of the least possible degree of the Ahlfors function. In principle there is a canonically defined set $\Pi(C)\subset {\Bbb
R}^g$ (we shall call the [*Ahlfors figure*]{}) whose spanning simplices going through the origin affords a complete understanding (in theory at least) of the minimal degree of a circle map concretizing the given bordered surface $F$.

\[11.09.12\] Perhaps one can solve the above problem (\[problem:Ahlfors-circuit\]) for $d=r+p$ by an inductive procedure. Let us sketch an attempt that fails (reasonably close to the goal). Recall that given two integers $(r,p)$ and a balanced configuration of $r$ circles $C_i$ in ${\Bbb R}^g$, where $g:=(r-1)+2p$. We would like to show that the origin is the barycenter of at most $r+p$ points with at least one on each $C_i$. Of course the assertion holds true when $p=0$, because we know (by the lemma) that $d\le g+1=r+2p=r$ and on the other hand we have the trivial lower-bound $r\le d$ imposed by the fact that each circle supports at least one point. It follows that $d=r=r+p$, and the claim is vindicated.

Thus one can try an induction reducing to the “planar case” $p=0$. This can be done in several ways via the moves $(r,p)\mapsto (r,p-1)$, or $(r,p)\mapsto (r+1,p-1)$ or finally $(r,p)\mapsto (r+2,p-1)$. The latter of which has the advantage that the new value of $g$, denoted $g'$ stays invariant. Now given a geometric configuration of type $(r,p)$ in the number-space ${\Bbb R}^g$ we construct one of type $(r+2, p-1)$ in the same ${\Bbb R}^g$, maybe naively just by duplicating two of the circles (i.e., assigning them a multiplicity). This new configuration is still balanced, so by induction hypothesis the origin is expressible as the barycenter of $r'+p'=(r+2)+(p-1)=r+p+1$ points located on the $C_i$. Alas, this exceeds by one unit the desired $r+p$.

\[18.10.12\] Low-dimensional examples may help to give some weak evidence toward solving Problem \[problem:Ahlfors-circuit\] with Gabard’s bound $d=r+p$. Let us discuss this aspect. If we take $(r,p)=(1,1)$, then $g=2$. So geometrically we have one circuit in the plane ${\Bbb R}^2$. In this situation our Fig. \[Convex:fig\] prompts solubility of the problem with $d=2$. Note the agreement with Gabard’s bound $r+p$. This proves the (modest) theorem that [*a bordered surface with one contour and of genus one always admits a circle map of degree $2$*]{}, whereas Ahlfors only predicts degree $r+2p=1+2=3$. Another evidence comes from the well-known hyperellipticity of genus 2 curves. Indeed the double of such a membrane having genus 2, it is hyperelliptic and can therefore be visualized in 3-space as something like Fig.\[Convex2:fig\]a. Doing a rotation of angle $\pi$ we find the required circle map of degree 2 (look at the Figs.\[Convex2:fig\]b and \[Convex2:fig\]c).

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Let us next examine the case $(r,p)=(2,1)$, then $g=(r-1)+2p=1+2=3$. So we have 2 circuits in space ${\Bbb R}^3$ (like in knot or link theory). Since the set of circuits is balanced, we have something like Fig.\[Convex2:fig\]d (assuming no knotting for simplicity). Balancing amounts picturesquely to say that if you dispose of a 180 degrees angular vision (like any respectable homo sapiens) you will never be able from the origin to contemplate the full link. Paraphrased differently, whatever the direction you choose to focus your vision the link will always move in your back. It seems plausible that, instead of the 4 points prompted by Ahlfors’ top-dimensional $3$-simplex, 3 points actually suffices to span a $2$-simplex passing through the origin (see again Fig.\[Convex2:fig\]d). Justifying this intuition could again corroborate the $r+p$ bound (at least for low invariants). Of course the genus (of the double) being now $g=3$ there is no hyperelliptic reduction, yet appealing to the canonical map $C_g\to {\Bbb P}^{g-1}$ (an embedding precisely when the curve is not hyperelliptic) our curve is concretized as a plane quartic (the canonical divisor $K$ having degree $2g-2$). Some basic knowledge of Klein’s theory then prompts that our orthosymmetric real quartic with $r=2$ must consist of two nested ovals. Projecting from a real point on the inner oval gives a totally real morphism of degree $4-1=3$, in accordance again with the $r+p$ bound.

All these little experiments raise the hope that Ahlfors original approach suitably sharpened by a geometric lemma about balanced collections of circuits in ${\Bbb R}^g$ should enable some improvements, and eventually confirm the prediction of the $r+p$ bound. However we confess that the required positive solution to Problem \[problem:Ahlfors-circuit\] with $d=r+p$ looks difficult to obtain and perhaps only true for special circuits arising through period maps. It is quite hard to connect Ahlfors method with the one in Gabard 2006 [@Gabard_2006] in which Abel’s map was exploited more systematically. Since both maps, $\Pi$ an Abel, involve periods, a natural guess is that [*Ahlfors’ figure*]{}, that is the set $\Pi(\partial F)\subset {\Bbb R}^g$, is closely related to the Abel map or at least the so-called (Noether-Klein) canonical map $C\to {\Bbb P}^{g-1}$ which is just the Gauss map of the Abel map: each tangent to the curve seen in its Jacobian is reported to the origin via translation in the Jacobi torus. If so interpretable, it is perhaps no surprise that Ahlfors approach is cumbersome because one is working in the Plato cavern where the essence (embedded-ness) of things is lost.

Still, the Ahlfors figure is perhaps useful for other questions. For instance if we take a top-dimensional spanning simplex with $g+1=r+2p$ vertices containing $0$ in its interior, it is clear that we may perturb slightly the vertices keeping the origin inside the simplex. This shows a sort of topological stability of Ahlfors maps having degrees $r+2p$. (This phenomenon is not new, compare Černe-Forstnerič 2002 [@Cerne-Forstneric_2002].) The same stability cannot be expected with the more economical $r+p$ bound, for a slight perturbation of our hypothetical simplex will generally miss the origin. Ahlfors’ figure also shows existence of circle maps for each degree $\ge r+2p$. For those of degrees $>r+2p$ there is a menagerie of convex combinations expressing $0$ and accordingly plenty of circle maps having the same fibre above a boundary point. Such results look not easily accessed via Gabard’s method (in Gabard 2006 [@Gabard_2006]).

Trying to make the last “menagerie” point more accurate could lead to interesting result. For simplicity imagine ${\Bbb R}^g$ as the plane ${\Bbb R}^2$ and in it a $2$-simplex spanning the origin. If we have more than $(g+1)$ points, say $g+2=4$ then we may interpret the convex-hull of those 4 points as the shadow (projection) of a $3$-simplex living in ${\Bbb R}^3$. Hence above the origin there is a segment in this higher $3$-simplex each elements of which is a convex sum of the 4 vertices. Hence we get $\infty^1$ circle maps having the same 4 points as prescribed value. This requires of course to be better presented but should be straightforward application of Ahlfors method.

Strip mappings (Nehari, Kuramochi)
----------------------------------

\[31.08.12\] As we saw instead of a circle map, Ahlfors 1950 [@Ahlfors_1950] prefers to construct a half-plane map. Ultimately this amounts to the same except that the disc instead of being decorated by the polar coordinates it is by the Hawaiian dipole (Fig.\[Dipole:fig\]a). A third option is to envisage (as Nehari and Kuramochi 1952 [@Kuramochi_1952]) a strip mapping to the strip $S:=\{z: -1\le\Re(z)\le 1\}$. When rectangular coordinate on the strip are transplanted to the disc we obtain a dipole looking like a mitosis. This yields yet another isothermic system on the disc.

To synthesize, the disc can be decorated by 3 types of isothermic coordinates (systems):

\(1) the monopole attached to an inner point of the disc, which when the pole is the center is just the foliation by concentric circles plus the orthogonal rays. We may from here drag the pole away from the center to get other isothermic systems best interpreted as the geodesic expansion w.r.t. to the hyperbolic metric on the disc. Upon letting degenerate the pole to the boundary circle we get:

\(2) the dipole depicted on Fig.\[Dipole:fig\]a and finally upon disintegrating this source of multiplicity 2 into two separate elements of multiplicity one we get:

\(3) a genuine dipole which ultimately can be the mitosis about antipodal points of the circle.

In principle to each of these geometric decoration of the disc corresponds an existence-proof of the Ahlfors function differing so-to-speak just in the “cosmetic details”.

Finally, each isothermic system suggests an angle of attack to Gromov’s filling conjecture. Eventually, it seems plausible that the totality of those isothermic systems could be exploited collectively upon using an averaging process (somehow reminiscent to Löwner-Pu’s trick).

Hurwitz type proof of Ahlfors maps? {#Hurwitz-type}
===================================

\[21.10.12\] This section wonders about an elementary existence-proof of circle maps via a continuity method reinforcing some naive moduli count. As we noted (in Sec.\[Minimal-sheet:sec\]) the disaster with bordered surfaces is that their gonality is not prompted by a naive moduli count, and thus the project looks from the scratch a bit hazardous. However it is not impossible that we missed something crucial.

The general philosophy would be not to fix a surface and try hard to find a map, but rather to look at all possible maps and lift the complex structure of the disc while hoping that if the degree is large enough there are sufficiently many free parameters to paint the full moduli space. Hence any Riemann surface would be expressible as a branched cover of the disc of some controlled degree. (Natanzon suggested to me this strategy during an oral conversation at the Rennes conference 2001, and I came again to this idea by reading Natanzon et al. 2001 [@Natanzon-Shapiro-Vainshtein_2001/XX].)

The basic idea may be formalized as follows. We fix a topological type $(r,p)$ encoding the number of contours and the genus. We introduce the (Hurwitz) space $$H_{r,p}^d:=\textrm{ set of all circle maps from surfaces of type }
(r,p) \textrm{  having degree } \le d.$$ An element of this natural set (hence a space!) is a branched cover of which we may keep in mind only the “total space”. This gives a map $$\tau\colon H_{r,p}^d\to {\cal M}_{r,p},$$ to the moduli space of bordered surfaces of type $(r,p)$. We want to show that this mapping is surjective for $d$ sufficiently large (but controlled à la Ahlfors). First, we know (since Klein essentially) that ${\cal M}_{r,p}$ is connected. Thus it would be enough to find a suitable $d$ so that the $\tau$-image is closed, open and nonempty.

As $(r,p)$ is fixed we may omit it from the notation. Of course $H^d:=H_{r,p}^d$ is empty when $d<r$. The example of rotational surfaces (cf. Fig.\[Chambery:fig\]) shows that $H^d$ is non-void for $d=r$ or $d=r+1$ when $r$ is even resp. odd.

It seems also trivial (since we have defined $H^d$ by the condition $\deg(f)\le d$) that the image $\tau(H^d)$ is closed for any $d$. Intuitively a map can degenerate to a map of lower degree, but will never degenerate to one of higher topological complexity. Observationally, this is well seen on the example of the Gürtelkurve (plane quartic with two nested ovals): when projected from a point in the interior of the oval we get a total map of degree 4, which can degenerate to one of degree 3 if the center of projection is specialized toward the inner oval. However, if we take a sequence of maps of degree 3 given by such projections the limit will be a similar projection (the oval being closed) and we never reach a map of degree 4. Of course an abstract explanation requires be given (perhaps just by compactness of $H^d$).

The hard part is to show that $\tau$ is open for some large $d$.

Naively one could hope to do this via Brouwer’s invariance of the domain requiring something like $\tau=\tau_d$ being étale for a suitable $d$.

Another idea is perhaps to factorize $\tau$ by taking the fibre of the circle map $f\colon F \to \Delta$ ($\Delta$=closed disc, here!) over the origin $0$ of the disc to get a surface marked by a group of $d$ points. The nice feature is that $(F,f^{-1}(0))$ permits one to recover uniquely (up to rotation) the map $f$ (cf. Lemma 5.2 about unilateral divisors in Gabard 2006 [@Gabard_2006]). Taking instead the fibre over the real unit $1\in \Delta$ gives a surface marked by a group of $d$ distinct along the boundary. Taking simultaneously the fibre over $0$ and $1$ gives a surface marked by $d$ points on both the interior and the border.

So we have 3 natural spaces of marked surfaces living above the moduli space ${\cal M}={\cal M}_{r,p}$, namely $I^d$ (interior marking); $B^d$ (bordered marking); and $M^d$ (mixed marking). Forgetting the markings gives varied arrows descending to $M$. The map $\tau$ factorizes through all these marked moduli space.

An idea could be to show that the lift of $\tau$ (which is an embedding especially when we factor through the mixed marking) is sufficiently horizontal w.r.t. to the fiber bundle projection afforded by the forgetful map. Alas, this is not very evident and should of course hold for some special value of $d$.

Another route to explore is to make a Lüroth-Clebsch/Hurwitz type analysis of trying to understand from ramification and monodromy how one reconstruct the Riemann surface.

Miscellaneous
=============

Moduli counts via dissection in pants (Klein, Fricke, Nielsen, Fenchel, etc.) {#Nielsen-Fenchel:sec}
-----------------------------------------------------------------------------

\[25.11.12\] This section presents a well-known argument to count moduli of Riemann surfaces, which applies both to the closed and bordered cases. The argument uses a decomposition in pants and the hyperbolic metric, so differs somewhat from the original arguments of Riemann 1857 [@Riemann_1857] and Klein 1882 [@Klein_1882], respectively.

\[Nielsen-Fenchel:thm\] [(Riemann 1857 [@Riemann_1857], Klein 1882 [@Klein_1882], Teichmüller 1939 [@Teichmueller_1939])]{} The closed genus $g$ surface $F_g$ depends on $3g-3$ complex moduli or $6g-6$ real moduli, while compact bordered Riemann surfaces $F_{r,p}$ with $r$ contours and $p$ handles depend upon $3g-3$ real moduli, where $g=(r-1)+2p$ is the genus of the double.

First consider the closed case. Introduce on $F_g$ a uniformizing metric of curvature $K\equiv -1$ and choose a decomposition in [*pants*]{} (alias [*trinion*]{} by Möbius 1860/63 [@Moebius_1863]). Each pant is a bordered surface with $3$ contours and of genus $p=0$. The conformal structure is unambiguously defined by the lengths of the contours, plus some twisting parameters (rotation like) permissible at the junctures of pants. Looking at the left-hand side of Fig.\[Pants:fig\], we count $(g-2)$ shaded pants each contributing for 3 lengths, and one must add one loop on the top and two on the bottom part of the figure. We arrive at a total of $$1+3(g-2)+2=3g-3$$ many loops. Since each such loop is a juncture we add as many twisting parameters to get finally the dependence upon $$2(3g-3)=6g-6$$ real moduli.

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In the bordered setting, we proceed similarly by looking at a pants decomposition of the bordered surface $F_{r,p}$ as depicted on the right-hand side of Fig.\[Pants:fig\]. Counting from the top to the bottom we get $$1+3(p-1)+(r-1)+r=3p+2r-3$$ many loops. Each of these loops is twistable by a parameter, except the $r$ boundary loops which have no companion loops. So we get $3p+r-3$ additional parameters, hence a total of $$6p+3r-6$$ real moduli. On the other hand the genus of the double of $F_{r,p}$ is $g=(r-1)+2p$, so that the above quantity is nothing but $3g-3$. This completes the proof.

Of course a more algebro-geometric count do as well the job while using the reality paradigm of the Galois-Riemann Verschmelzung. More concretely inside the complex moduli space one defines an antiholomorphic involution, and the moduli of “real surfaces” appears as the real (=fixed-point) locus of that involution so has half dimensionality. Such an argument has the advantage of encompassing directly the diasymmetric case, which leads to non-orientable Klein surfaces. For more details, cf. Klein 1882 [@Klein_1882], Teichmüller 1939 [@Teichmueller_1939], Earle 1971 [@Earle_1971-On-the-moduli], Seppäla 1978 [@Seppala_1978-Teich-spaces-of-Klein-surfaces], Huisman, etc.

Part II: Hilbert’s 16th {#Hilbert's16th-PartII:sec}
=======================

General overview
----------------

\[26.03.13\] As announced in the introduction, we enter now in the second part to our text dealing with Hilbert’s 16th problem. The switch from Ahlfors to Hilbert’s 16th flashed us when reading in more details Rohlin’s work of 1978 (cf. the next Sec.\[Klein-Rohlin-conj:sec\]). Since our assimilation of the material evolved in slow organical mode (with several mistakes of ours), it seems worth summarizing which waters were investigated and what seems to be urgent open problems in the field. This section should thus replace the reading of all the sequel which needs severe reorganization at several places. Further what we understood is still miles away of the fine jewellery reached by Russian scholars in this field, but to defend our messy text we also feel that the philosophy à la Ahlfors or Rohlin has not yet been fully exploited, nor elucidated.

First, Hilbert’s 16th includes the topological classification of real algebraic (smooth=non-singular) curves in $\RR P^2$. In its original formulation the critical degree was $m=6$ (sextics), where Hilbert’s intuition produced both the best (the Ansatz that an $M$-curve[^17] cannot have all its $11$ ovals unnested) as well as a misconception that persisted through several decades, until being refuted through Gudkov’s seminal 1969 construction of the curve $\frac{5}{1}5$ (5 ovals enveloped in a larger oval, plus 5 ovals outside). This was a big surprise as Hilbert expected that $M$-curves appear only along the scheme $\frac{1}{1}9$ discovered by Harnack 1876, and the one constructed by himself $\frac{9}{1}1$ in Hilbert 1891 (compare the top-row of Fig.\[Gudkov-Table3:fig\]). The Gudkov symbol $\frac{x}{1}y$ encodes a distribution of ovals where $x$ ovals are directly nested in one oval, while $y$ unnested ovals are lying outside (compare again Fig.\[Gudkov-Table3:fig\] if necessary). Petrovskii’s own scepticism about the unexpected twist of Gudkov’s solution, launched the work of Arnold 1971, and Rohlin 1972, where Gudkov hypothesis $\chi \equiv_8 k^2$ went verified through revolutionary insights on the “complexification”. Here $\chi$ always denotes the Euler characteristic of Ragsdale’s orientable membrane bounding the curve from “inside”, while $k=m/2$ is the semi-degree of a curve of even degree $m=2k$. The modern era of real algebraic geometry was launched, characterized by deep interconnections with 4D-differential topology (Rohlin’s early work on spin 4-manifolds, etc.)

What has this topic to do at all with Ahlfors maps? To say the least very few factual links have been tied up presently, but we can dream of a big connection. The sequel is our attempt to enhance the rôle which Ahlfors theory could play in Hilbert’s 16th. We should warn the reader that our viewpoint is much partisan (biased by what produced such masters as Ahlfors and Rohlin) and it may well be the case that the real mathematical terrain is not as plastic and smooth as the expressed in the next lines. First, we should stress that there is no anachronism in expecting such a connection with Hilbert since (modulo technical details) the quintessence of Ahlfors theory truly goes back to the Riemann-Schottky-Klein era (resp. 1851/57–1875/77–1876/82), which is much prior to Hilbert (1862–1943), and a fortiori to Hilbert’s geometrical period ca. 1891—when he left Algebra, Invariant theory, Number theory—to move in the softer realms of geometry, calculus of variations, or “functional” analysis, especially Dirichlet, Fredholm, etc.

To be honest, our connection was already envisioned by Rohlin 1978 [@Rohlin_1978] (apparently completely unaware of Ahlfors work, as we were ourselves ca. 2001 when rediscovering the result independently), but who also used implicitly what we call [*total reality*]{} as a tool detecting the dividing character of curves. More strikingly, in a genius stroke without any antecedents, Rohlin asserts a phenomenon of total reality for certain $(M-2)$-sextics explaining a posteriori (nearly) all prohibitions of Gudkov’s table of periodic elements (=Fig.\[Gudkov-Table3:fig\]). The latter table affords nothing less than the complete solution to Hilbert’s 16th by way of a curious pyramidal structure encoding all possible distributions of ovals realized by algebraic curves of degree 6 with real coefficients. Rohlin’s (unproved) synthetical assertion stayed dormant for more than 3 decades until Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics] recently managed to establish a slightly weaker form thereof. We can thus now feel confident in expecting that Ahlfors theory will have to play some major rôle in the future destiny of Hilbert’s 16th, i.e. for curves of degree $m\ge 8$. (The case $m=8$ looks nearly settled if one is expert enough in the field and willing to sacrifice a long period of his time to assemble many bits of knowledge scattered through the literature.) Hilbert’s problem (like any existential puzzle) splits naturally into constructions versus prohibitions. Now the rôle of Ahlfors could be as follows. If one has a distribution of ovals (à la Hilbert) such that all curves representing it are dividing(=type I=orthosymmetric) in the sense of Klein (what Rohlin calls [*a scheme of type I*]{}), then it seems a reasonable folly to expect the phenomenon of [*total reality*]{}, namely existence of a pencil of “adjoint” curves cutting only real points on the given curve. At least Ahlfors theorem implies no conformal obstruction to the scenario.

, while general topology (i.e. the image of a connected set is connected) implies an absence of topological obstruction to existence of such a total pencil.

Incidentally, it should be no surprise that both Ahlfors 1950 and Rohlin’s maximality claim (1978) refers back to a common denominator, namely works of Felix Klein. In Ahlfors’ case this is indirect since reference is more readily confessed to Schottky’s results somewhat prior to Klein’s (but also more schlichtartig than Klein’s). Via Teichmüller 1941 [@Teichmueller_1941] some return to Klein is implicit though poorly cross-referenced. In Rohlin’s case the analogy with Klein is inherent though disputed in Viro’s survey 1986 [@Viro_1986/86-Progress] via Marin’s assessment of Klein’s assertion that curves of type I cannot gain an oval by crossing the discriminant. Apart from those details it is evident that Klein (and before him Riemann) gave the impulse for all what followed, and the fusion awaited upon is probably merely a matter of reunifying the original conception of Riemann-Klein before it diverged into pure conformal geometry (Schwarz, Schottky, Klein, Koebe, Bieberbach, Grötzsch, Ahlfors, Grunsky, Teichmüller, Ahlfors again) versus plane curves in Hilbert’s 16th (Harnack, Hilbert, Ragsdale, Rohn, Brusotti, Petrovskii, Gudkov, Arnold, Rohlin). One can wonder how much knowledge went lost just through older generations passing away and how much time consuming it will be for us to revive old wisdoms that are probably the key to most of our naive questions.

Our 1st fundamental problem is to decide if Ahlfors theorem particularized to the setting of plane curves implies existence of such a total pencil. I personally always thought this being a triviality (see optionally Gabard 2004 [@Gabard_2004 p.7]), but recently Marin warned me that life might not be so easy (cf. letter in Sec.\[e-mail-Viro:sec\]). Alas, meanwhile I forgot nearly all the little I ever knew about the foundations of algebraic geometry, so that what I thought to be trivial is now floating in some suspense (“ombre propice” as would say Thom).

Even if not true (or rather implementable), synthetical procedures à la Rohlin-Le Touzé (=RLT) could redeliver the phenomenon of total reality [*ab ovo*]{} (independently of Ahlfors conformal geometry). This seems to require a vertiginous and lengthy verification process climbing ad infinitum. In more gently slope, one can expect a gradual propagation of RLT from degree 6 to 8, and so on, that could be relevant to detect new prohibitions in Hilbert’s 16th.

Why so optimistic? As exemplified by the case of sextics $m=6$ (much influential upon Gudkov, Arnold, Rohlin, etc. and as demonstrated by those smart guys fairly typical of the general case $m=2k$) it is likely that a scheme of type I is totally real under a suitable pencil (or viceversa) and this should in turn imply the scheme being maximal in the hierarchy of all schemes. This produces prohibitions in Hilbert’s 16th, which a priori could be new, and governed by an uniform paradigm.

So a 2nd fundamental problem is to decide if Rohlin’s maximality conjecture (RMC) positing that “type I implies maximal” is true. At first sight, it seems that a positive solution to the 1st problem implies this as a byproduct, but there seems to be severe obstacles in completing the programme. For explicitness, it is worth sketching the (naive, uncomplete) argument. Given a scheme of type I, there is by Ahlfors a total pencil, which cuts only real points on the curves. Hence the curve is already saturated, and cannot be enlarged by adding an additional oval without violating Bézout. The difficulty however is that the enlargement is not a priori involving the same (or even a nearby curve) augmented by some other ovals, but can be a priori very distant of the original curve. ([*Added in proof*]{} \[13.03.13\], for a loose strategy using isotopies, cf. Sec.\[RMC-via-Mangler:sec\]\].)

However a 3rd route is that whenever we encounter a synthetic phenomenon of total reality à la Rohlin-Le Touzé looks (akin to a concretization of Ahlfors abstract theorem within the Plato cavern of Hilbert’s 16th involving only plane curves), then it seems evident (via Bézout-saturation) that Rohlin’s maximality conjecture will hold true for this specific scheme. Again the proof is not easy to formalize, but it is perhaps realist to expect a positive solution in the case of curves of degree 6, and hopefully somewhat higher as to produce new truths.

This brings us to the 4th problem. How to extend Rohlin’s total reality claim to high-degree curves $m> 6$. Is there any algorithm telling one where to assign basepoints in order to assure total reality of the corresponding pencil? One could dream that this can be done from the sole knowledge of Rohlin’s complex orientations, cf. optionally Sec.\[Galton-brett:sec\].

For plane $M$-curves of degree $m$, we prove below a basic Theorem \[total-reality-of-plane-M-curves:thm\] stipulating a total pencil swept by curves of order $(m-2)$. This merely traduces the so-called Bieberbach-Grunsky theorem, which (apart from phraseological details like Dirichlet’s illness) truly belongs to Riemann 1857, Schottky 1875–77, Enriques-Chisini 1915, and only then Bieberbach 1925, Grunsky 1937, Wirtinger 1942 [@Wirtinger_1942], etc. A structural asymmetry appears: while $M$-curves are crudely-put reputed hardest-to-construct within Hilbert’s 16th, their conformal geometry is most trivial, due to the planar=schlichtartig character of the half of the curve. Total reality is simplest to ensure in the $M$-setting, just because it is like having one train on each track, hence no risk of collision. Precisely, the trick is just to choose one point on each oval getting so a group of $g+1$ points which moves (by Riemann-Roch or Abel), and total reality is automatically granted (cf. Lemma \[Enriques-Chisini:lemma\] for more details). Making this abstract argument concrete proves the theorem.

Can we extend this to non-maximal curves? The risk is then an overpopulation of $g+1$ points scattered on $r<g+1$ ovals, hence 2 of them are forced to cohabit on the same oval (pigeonhole principle due to Dirichlet apparently) exposing us to a possible collision jeopardizing total reality! So what is demanded is controlling a dextrogyration of points when moving along linear equivalence. Ahlfors 1950 [@Ahlfors_1950] or Gabard 2006 [@Gabard_2006] affords basic tricks to achieve dextrogyration in the abstract setting. Can we transplant them directly inside the Plato cavern of plane curves, as we just managed to do for $M$-curves? As yet we never succeeded, but this should not discourage more serious attempts.

If we think more concretely à la Rohlin-Le Touzé (or perhaps to go back earlier in history à la Brill-Noether), numerological reasons make evident that $(M-2)$-curves of type I (or even schemes of type I) and degree $m$ will have their total reality exhibited by a pencil of curves of degree $(m-3)$. Evidence is given later in this text, but readily follows by analogy with Rohlin-Le Touzé and a simple constants count (cf. Remark \[M-2-curve-degree-like-Gabard:rem\], which is just the end product of numerological coincidences observed for $m=6,8,10,
\dots$). Remind perhaps at this stage old Italian works (recognized as possible competitors to Ahlfors 1950), like Matildi 1945/48 [@Matildi_1945/48], Andreotti 1950 [@Andreotti_1950]. Those could already anticipate our present desideratum. Since already Rohlin’s proof (which is lost) and that of Le Touzé (2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]) is quite delicate (or the sequel of this text which contains ca. 30 pages of unsuccessful attempts to prove Rohlin’s original claim), we are not claiming that total reality will be easy to prove in full generality but perhaps for degree $8, 10$ this remains manageable (at least within the next 4 decades). By experience we are accustomed in the field to slow progresses (remind Hilbert, Gudkov, etc.), and it is quite unlikely (but not impossible) that a new Abel or Riemann will crack the full puzzle in a single stroke.

Some part of our text tries to take the census of all such schemes of type I in degree 8. Alas our optical faculties tend to be much more limited than those of aliens, like insects with 8 eyes looking at their preys via pencil of cubics, chameleons with mobile ocular systems, or any sort of creature with 19 eyes (when it comes to look at the world through a pencil of quintics, …), and generally, $M-3$ basepoints (i.e. 19 when $m=8$, $34$ for $m=10$). Accordingly, we are presently (and probably for the rest of our life) confined to deduce total reality not from optical skills but via boring arithmetics, namely the subliminal (Rohlin?)-Kharlamov-Marin congruence $\chi\equiv k^2+4 \pmod 8$, which forces type I under this “shifted” Gudkov-style congruence mod 8.

This is a crucial weapon (whose proof we have not yet studied in full details). This harpoon detects for us a menagerie of schemes of type I, all possibly subsumed to total reality, and conjecturally (via RMC=Rohlin’s maximality conjecture) acting prohibitively upon all schemes(=distribution of ovals) pretending to enlarge the given one. Such schemes crystallize therefore Bézout-extremal (or saturated) shapes of Hilbert’s 16th, which as would say Klein cannot develop further without exploding the latent degree.

Call an RKM-scheme any scheme satisfying the RKM-congruence $\chi\equiv_8 k^2+4$. It is not clear to the writer, and the experts were a bit silent on this aspect as yet, if conversely any $(M-2)$-scheme of type I is forced to respect the RKM-congruence. This deserves perhaps to be clarified at the occasion. \[[*Added in proof*]{} \[13.04.13\] An answer is probably implicit in Rohlin 1978, Art.3.5, on p.93 (extremal property of Zvonilov-Wilson).\]

With some sloppiness, we arrive at some big picture along the following philosophy (in our opinion fairly implicit in Rohlin 1978). Any scheme of type I is detected:

$\bullet$ either trivially because it is an $M$-scheme whose total reality is exhibited à la Bieberbach-Grunsky (yet no direct impact upon Hilbert’s 16th by virtue of Harnack’s bound (1876), or more simply its intrinsic variant $r\le g+1$ due to Klein 1876 proved via retrosections à la Riemann), \[but some indirect impact by using satellites!! (13.04.13)\]

$\bullet$ or it is an $(M-2)$-scheme verifying the RKM-congruence, in which case total reality is flashed by a pencil of $(m-3)$-tics.

$\bullet$ or finally it arises as “satellite” of a scheme of lower degree dividing the given degree.

The idea of satellites arises simply by noting (or expecting) that total reality propagates when the curve is doubled, tripled and so on, by replicating several copies of the curve within a tube-neighborhood of it (\[satellite-total-reality:sec\]). For the conic with a single oval (unifolium) this just leads to the series of deep nests total under a pencil of lines, while for a quartic with 4 ovals (quadrifolium in the jargon of Zeuthen 1874 [@Zeuthen_1874] who inspired much Klein 1876) this leads to the series of curves in degrees multiple of $4$, totally real under a pencil of conics. The case of degree 8 is explicitly mentioned in Rohlin 1978, being just the double of the quadrifolium. We expect that satellites do extend to curves of odd degrees (\[Satellite-odd-degree:sec\]), yielding some interesting prohibitions on schemes of degree 10 when applied to an $M$-quintic doubled. Likewise doubling the Rohlin-Le Touzé sextic $\frac{6}{1}2$ (or its mirror $\frac{2}{1}6$) gives a scheme of degree 12 which should be maximal (hence killing all extensions of it).

The general philosophy is now clear. Total reality (basically due to Ahlfors 1950, though Teichmüller 1941 ascribes it to Klein directly) acts as an upper-bound on the complexity of Hilbert’s 16th problem, by killing all distributions of ovals adventuring above one totally flashed by a pencil. In substance everything boils down to a phenomenon of Bézout-saturation, with in the background of the scene an extension of the Riemann mapping theorem to surfaces of higher topological structures (so-called Ahlfors maps).

This looks a fundamental truth (or philosophy?) since it seems robust, and implementable when the flashing is as explicit as Rohlin-Le Touzé’s as opposed to the abstract nonsense of Ahlfors. If skeptical, the just predicted maximality of satellites in degree 10 and 12 should be tested against highbrow methods of constructions of the modern era (Viro-Itenberg). If the Ahlfors-Rohlin philosophy resists the shock against this structural test, then some experimental evidence is gained that the Ahlfors-Rohlin Verschmelzung is a deep reality governing a substantial part of Hilbert’s 16th at the universal scale (all degrees). If not, then the whole story of the 16th problem could be even more chaotical and unruly than it presently is, i.e. just a combinatorial mess only worth deserving the attention of computing machines. Of course the latter are quite likely to show us hidden patterns of symmetries, maximality, etc. that were not yet appreciated due to a lack of experimental data. More pragmatically, it must feasible to inspect if in degree 8, the Ahlfors-Rohlin scenario of total reality and the allied extremal principle of saturated schemes is compatible with factual data, and optimistically even able to preclude schemes that were not yet prohibited. Alas, we are not expert enough in the field to tell an answer, but peoples like Viro, Fiedler, Korchagin, Orevkov, Le Touzé, must already have a clear-cut vision along this idea. In degree 6 it is clear that the saturation principle of Rohlin is entirely covered (or re-explain) by the congruences mod 8 due to Gudkov and Gudkov-Krakhnov-Kharlamov, but it is not clear to me if the same subordination holds true in degree 8 (maybe in general). On the other hand if the RKM-congruence fails to detect a type I scheme, then there could be some sporadic phenomenon of total reality explaining it, and this would be a new source of saturation (perhaps not covered by the congruences mod 8).

This is the main-body of our quest, but during the trip we went sidetracked to other connected topics. Here are some aspects perhaps worth putting in evidence centering around the theme of rigid-isotopy, and the allied contraction principles where the end-point of the path is permitted to touch the discriminant (parameterizing all singular curves).

Total reality takes its simplest incarnation for the deep nests swept out by a total pencil of lines. A theorem by Nuij 1968 [@Nuij_1968] (later revisited by Dubrovin 1983 [@Dubrovin_1983/85]) states that such deep nests are rigid in the sense than one can pass from any 2 curves representing it by a continuous deformation of the coefficients without encountering any singular curve during the deformation. Such large deformation pertains to what is called [*rigid-isotopies*]{}, which actually refines Hilbert’s 16th problem. This topic always attracted geometers even prior to Hilbert’s era, e.g. Schläfli (apparently known for having the most massive human brain ever weighted with ca. 1.936 kg for only 157 cm of body height), or Zeuthen and Klein adding several contributions regarding curves and surfaces of low orders (quartic and cubics resp.). For quartic curves Klein established (1876 [@Klein_1876_Verlauf]) that the rigid-isotopy class is fully determined by the real scheme already.

It seems natural to ask if Nuij’s rigidity result (for deep nests) has equivalents whenever total reality holds true. Alas this fails by the Marin-Fiedler locking technique which refutes this Ansatz for $M$-curves of degree 7 (cf. Fig.\[Marin:fig\], for a hopefully lucid exposition of Marin’s trick). Despite this disruption of the naive scenario, it seems to us likely that rigidity holds true for satellites of the quadrifolium. We confess however to have not yet studied Nuij’s proof, nor do we know (a fortiori) if his proof extends mutatis mutandis.

Though rigid-isotopy merely involves the $\pi_0$ (=nullest homotopy group measuring the arcwise-connected components) of the space of curves excised along the discriminant, very little is known on such problems. A naive conjecture of us—based essentially on the failure of the Marin-Fiedler locking device, plus the fact (subsequent to Rohlin’s formula) that curves of type I have $r\ge m/2$ ovals (also valid if $m$ is odd)—postulates that curves with few ovals are necessarily [*rigid*]{}, i.e., are unambiguously determined up to a large deformation by their sole real schemes. Precisely this could hold true as soon as the curve has strictly less ovals than $DEEP+2$, where $DEEP:=\Delta:=[(m+1)/2]$ is the number of circuits of the deep nest of degree $m$. The intuition is simply that by Rohlin’s formula (\[Rohlin-formula:thm\]) this is the first dividing scheme encountered (as $r$ the number of ovals increases), and two units above this ($r=\Delta+2$) it is a simple matter to exhibit a scheme of indefinite type (Rohlin’s jargon to say that there is curves of both types I.vsII realizing a prescribed configuration of ovals).

This conjecture (called LARS, for low-altitude-rigidity-conjecture) is merely a cavalier extension of:

\(1) Nuij’s theorem of 1968, which is not specific to curves (but valid for algebraic hypersurfaces, where there is an evident notion of deep nest via concentric spheres, plus an eventual pseudo-plane).

\(2) Nikulin’s rigid classification of sextics in 1979/80 [@Nikulin_1979/80] implying the case $m=6$ of our LARS, and telling much more namely the fact that the real scheme (as tabulated on Gudkov’s table) enhanced by the data of Klein-Rohlin’s types affords a complete system of invariants under rigid-isotopy. Hence for $m=6$, Nikulin is stronger than LARS as it prompts rigidities at all altitudes. However as soon as $m\ge
7$ this is foiled (cf. again Marin’s example=Fig.\[Marin:fig\]).

\(3) A unofficial conjecture of Rohlin (reported in a Viro letter in Sec.\[e-mail-Viro:sec\]) that curves of odd degrees with a single (pseudoline) component are rigid-isotopic, cf. also Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical]. By analogy, curves of even degrees with a single oval could be rigid-isotopic. Those questions are settled in degrees $m\le 6$ ($m=4$ Klein 1876 [@Klein_1876_Verlauf], $m=5$ Kharlamov 1981 [@Kharlamov_1981/81], $m=6$ Nikulin 1979/80 [@Nikulin_1979/80]), but still resist in degrees $\ge 7$. Hence our conjecture LARS appears very presumptuous, and it may be a more reasonable challenge trying to disprove it. Alas the locking method of Fiedler-Marin looks (as far as we experimented in the sequel) quite impuissant to destroy LARS.

What techniques could be used to prove LARS or more modest rigidity conjectures? Our naive idea is that geometric flows (amounting to look at orthogonal trajectories of suitable functionals like calculating the length or area of ovals) could prove this and related results of rigid-isotopies. This would give some intrusion of differential-geometric methods in problems of rigid-isotopies, a priori of a purely algebraic nature. Presently we were never able to complete any serious proof along this way, but our text contains ca. 20 pages of (dubious) trials along such lines. Viro’s survey 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical] also contains a brief desideratum to know more about geometric properties of curves, and this could evidently pertain to rigid-isotopies, in a way perhaps reminiscent of the Ölfleck of H.A. Schwarz (where the Riemann mapping theorem is visualized by an oil-flake restoring to the circular shape), or the eclectic Ricci flow of Yau-Hamilton-Perelman, where a similar phenomenology appears in the abstract Riemannian setting (convergence to the round metric, with the well-known bonus about Poincaré’s conjecture).

Affiliated to those rigid-isotopy questions there is a conjecture of Itenberg-Viro (cf. Viro’s preface of the volume containing Itenberg 1994 [@Itenberg_1994]) to the effect that some empty oval of any curve can always be shrunk toward a solitary node. \[[*Added in proof*]{} \[13.04.13\].—Similar (but more vague) ideas are actually ubiquitous in Klein, e.g. 1892 [@Klein_1892_Realitaet].\] This is still wide open, but Itenberg’s article just cited establishes the case $m=6$. Again one could hope that the flow minimizing the length of an oval could achieve such a contraction. Inspired by this conjecture we advanced a strengthened version CCC(=collective contraction conjecture) saying that all empty ovals can simultaneously contract toward solitary nodes. (This is like a perfect landing in Flight-Simulator v.18.5 with aircraft Antonov 72 having its 94 wheels touching the ground simultaneously!) If this (unlikely) miracle is true, one gets e.g. a 2-seconds proof of Hilbert’s Ansatz of the non-existence of an $M$-sextic without nesting by reduction to Bézout. Of course this is also more hygienically derived from Rohlin’s formula (or Arnold’s congruence mod 4), which involves softer homological intersection theory à la Poincaré-Lefschetz, etc. We were not as yet able to disprove our strong CCC-version of Itenberg-Viro. Its real impact being still obscure we did not pursued this issue in any serious fashion. A philosophical consequence, of large deformations is that they should (like total reality) act prohibitory, whereas small perturbations are classically exerting their swings at the constructive level (Harnack 1876, Hilbert 1891, Brusotti 1914/21, Gudkov 1969/72, Viro 1980, Itenberg 1993, etc.) Of course a clear-cut realm of where to corrupt CCC could be a dividing curve without nesting, for those could after [*strangulation*]{} be split into two complex-conjugate halves intersecting in as many points as there were ovals initially. Alas either Thom (\[Thom-Ragsdale:thm\]) or better Rohlin’s formula (\[Rohlin-formula:thm\]) forces such a curve to have $\chi=r\le
k^2$ resp. exactly $r=k^2$ ovals, hence we fail to corrupt Monsieur Étienne Bézout. It looks so quite challenging to kill CCC, albeit its truth looks very fragile, as it incarnates an extreme flexibility of algebraic objects reputed “rigid” in the large. Yet it should be remembered (though at some more local viz. regional scale) that Brusotti’s theorem gives via Riemann-Roch-Brill-Noether-Severi a remarkable flexibility of algebraic curves (independence of the smoothing of nodes). So one should not be surprised at last, if sometimes algebraic curves appear more plastic than expected a priori. However, as we shall soon discuss, Shustin disproved (in degree 8) a flexibility conjecture of Klein (1876) that nondividing curves can always acquire a solitary node through continuous variation of the coefficients ([*champagne bubble phenomenon*]{}). In slight contrast, building over the previously cited works of Nikulin 1979, Itenberg 1994, and the whole diagrammatic of the Gudkov-Rohlin table (1969–78), we think that Klein’s intuition of champagne bubbling is correct in degree 6 (cf. Prop.\[Klein-vache-deg-6:prop\]). The philosophical impact of Shustin’s disproof of Klein (though his aim was refuting a related assertion of Rohlin) is that we cannot expect to have solely topological obstructions regulating large algebraic deformations.

A last theme involves the impact of Thom’s conjecture (meanwhile Kronheimer-Mrowka’s theorem 1994) upon Hilbert’s 16th (Sec.\[Thom:sec\]). A classical trick (called the Arnold surface) is to fill Klein’s half of a dividing curve $C_m$ (of degree $m=2k$) by the real Ragsdale orientable membrane bounding the curve from inside. This gives a homology class of half-degree $k=m/2$, smoothly represented (after rounding corners, if necessary). This object looks ideally suited to an application of Thom’s genus estimate. Taking for granted orientability of the Arnold surface, we found an erroneous estimate $\chi\le k^2$ (for all dividing plane curves of degree $2k$ (cf. (\[Thom-Ragsdale:thm\])). Albeit wrong in general (as Fiedler kindly pointed out to us) it holds in special cases when all (primitive) pairs of ovals are positive in the sense of Rohlin, i.e. when complex and real orientations match together. Real vs. complex orientations may even disagree yet along pieces not connected by the Ragsdale membrane (cf. Lemma \[Arnold-surface-orientable-iff-oddly-charged:lem\]). If optimistic Thom or even Rohlin’s formula gives a way to attack the (still open) Ragsdale’s conjecture for $M$-curves, which amounts to $\vert \chi \vert \le k^2$ (\[Thom-implies-one-half-of-Ragsdale:lem\]). Is the Arnold surface (=Klein’s half glued with the Ragsdale membrane) of an $M$-curve always orientable? If yes, then the proof of our (erroneous) Theorem \[Thom-Ragsdale:thm\] implies Ragsdale’s conjecture via Thom’s estimate on the genus. Unfortunately, Arnold’s surface is nonorientable already for Hilbert’s $M$-sextic, cf. Lemma \[disproof-orientability-Arnold-M-curve:lem\].

Maybe the theorem à la Bieberbach-Grunsky specialized to plane $M$-curves (i.e. our Theorem \[total-reality-of-plane-M-curves:thm\]) could give (via dextrogyration[^18]) enough control on complex orientations of $M$-curves as to imply Ragsdale, either via Thom or directly via Rohlin’s formula $2(\pi-\eta)=r-k^2$ (where $\pi:=\Pi^+$, $\eta:=\Pi^-$ to abridge notation). (This amounts then to check that $\pi-\eta\le
n$, the number of negative ovals.) This admittedly looks naive, but we cannot exclude such a coarse strategy for the moment.

More modestly, it may be noted that filling Klein’s half with Ragsdale’s membrane of an $M$-sextic without nesting reduces Hilbert’s nesting Ansatz to the “baby” case of Thom for homology classes of degree 3, acquitted by Kervaire-Milnor 1961 [@Kervaire-Milnor_1961] building upon Rohlin’s early work 1951 on spin $4$-manifolds.

As said, our erroneous estimate $\chi\le k^2$ was corrected by Fiedler in a series of letters where he learned us the Petrovskii estimates on $\chi=p-n$, and Arnold’s strong avatars thereof involving hyperbolic ovals. This is again closely connected to the Ragsdale conjecture, which is still a [*pièce de resistance*]{} in the case of $M$-curves. Moreover though our estimate $\chi\le k^2$ was erroneously founded it turned out to be quite difficult to find an explicit counterexample. At least we failed via classical methods (cf. Figs.\[HilbGab1:fig\]–\[HilbGab4:fig\]), which rather inclined to think that $\chi\le k^2$ was sharp if true at all. Namely using Hilbert’s construction we find an infinite series of $M$-curves or $(M-2)$-curves such that $\chi=k^2$ exactly, but failed to beat $k^2$. We presume this is exactly the sort of experiments that led Ragsdale to her conjecture. However the story does not finish here, and the big surprise arrives now.

It is notorious that a marvellous construction of Viro-Itenberg (patchwork and $T$-construction, cf. Fig.\[Itenberg:fig\]) killed the Ragsdale conjecture in degree 10 (even in its relaxed shape of Petrovskii), yet leaving intact the $M$-curves case. The Itenberg-Viro construction supplied us with the apparently simplest counterexample to our erroneous estimate $\chi \le k^2$ (for type I curves of degree $2k$). It produces namely an $(M-2)$-curves of degree 10 with $\chi=29
\nleqslant k^2=25$, hence necessarily dividing by the RKM-congruence $\chi\equiv_8 k^2+4$. This was the fatal stroke (coup de grâce) against our estimate $\chi \le k^2$, which is quite robust as it seems incorruptible via Harnack-Hilbert and challenging to refute in the $M$-case.

Last but not least, there is a disproof due to Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin] of one-side of Rohlin’s maximality conjecture namely “type I$\Leftarrow$ maximal”. This disproof is not so dramatic for Rohlin’s prohibitive programme which uses rather the converse (still hypothetical) implication “type I$\buildrel{?}\over\Rightarrow$ maximal”. Shustin’s note looks historically pivotal as it kills the second part of Klein’s intuition, pertaining to large deformations of nondividing curves as always admitting the apparition of a champagne bubble created by crossing the discriminant through a solitary node. Due to its extreme concision we had first not understood Shustin’s argument (and [*unduly*]{} mistrusted his result for a while). Finally, we understood its logic, but confess to have not yet assimilated all the results required to complete its proof. It suffices to say that Shustin’s work exploits Viro’s construction on the one side, and also advanced Bézout-style obstructions due to Fiedler and extended by Viro. Some details perhaps assisting beginners to grasp the structure of Shustin’s proof are to be found in Sec.\[Shustin-understood:sec\].

This a brief summary of the territories we managed to explore in ca. 3 months of investigation. Besides our text may have some didactic value on the following aspects.

(1).—We give a self-contained account of Gudkov solution to Hilbert’s 16th problem in degree $m=6$, by exposing the original constructions of Harnack, Hilbert and Gudkov. Those issues are well-known and described in Gudkov’s seminal survey (and at several other places like A’Campo’s Bourbaki survey 1979 [@A'Campo_1979], etc.). Yet not all species are always accompanied by decent pictures requiring sometimes clever twists of Harnack’s construction (oft messy to implement if one wants to realize a type given in advance). So we had long hours of trials with computer-assisted depictions. This can hopefully be of some use to some nonspecialist readers. Our intention was to reproduce all (including the infructuous) trials, but that generated “microfilm” pictures often too heavy for the purpose of arXivation. By the way our microfilm though still readable in pdf-format at 600 dpi resolution will still be hard to contemplate on the screen. \[[*Added in proof*]{} \[13.04.13\] This technical problem was settled by shrinking the size of pictures in the Adobe software, permitting so to economize much memory space, yet without altering the optical size of pictures.\]

(2).—We give also full details (and a graphical view=Fig.\[Gudkov-Table3:fig\]) of Rohlin’s enhancement of Gudkov’s census of sextics by adding the complex topological characteristics of Klein (that were much neglected during the era of Hilbert, Ragsdale, Rohn, Petrovskii, Gudkov) up to the Arnold-Rohlin revival of the complexification (which turned to be the conceptual key to explain Gudkov’s experimental phenomenology). This is merely a simple exercise yet that can be quite time-consuming if one starts from zero-knowledge. Of course an excellent account of this, differing form ours only in the minor details, is already given in the masterpiece Marin 1979 [@Marin_1979]. (Our account differs just in using more primitive configurations of 3 ellipses.)

(3).—We give in Sec.\[Prohibitions:sec\] a reasonably exhaustive list of classical obstructions, especially a (nearly complete) proof of Rohlin’s formula (\[Rohlin-formula:thm\]). In the original source (Rohlin 1974 [@Rohlin_1974/75]) this is not presented in its full generality (only $M$-curves), though the adaptation to general dividing curves is very minor. This Rohlin’s formula looks extremely fundamental as it appears as the most universal obstruction that can be derived by nearly abstract nonsense (i.e. using very little from the assumption of algebraicity), yet still affording strikingly precise information while staying completely elementary. For instance it covers Hilbert’s Ansatz of nesting, and extends it to all degrees $m\ge
6$. It also formally implies the Arnold congruence mod 4, which is a weak form of Gudkov hypothesis for $M$-curves, yet an extension thereof to arbitrary dividing curves. Then there is a series of avatars of the Gudkov congruence mod 8, that truly requires more advanced topological tools, essentially in the spirit of Rohlin 1952 [@Rohlin_1952-4-manifolds]. Those more advanced results are not proved in our text, and we hope to be able to offer a lucid view on them in the future. Hence, to assimilate the marvellous congruences due to Rohlin, Gudkov-Krakhnov/Kharlamov, Kharlamov-Marin, etc., our reader is invited to consult the original sources (Rohlin 1972 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov] (with a gap but essentially correct and repaired by Marin-Guillou, e.g. 1986 [@Guillou-Marin_1986] or Marin 1979 [@Marin_1979]), and also Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91], giving a synthesized view).

Challenging vs. less challenging open problems {#Challenging-open-prob:sec}
----------------------------------------------

\[28.03.13\] This section summarizes what looks to us major open questions in the field investigated. The reader is warned that our list is a mixture of hard Soviet conjectures of longstanding with newcomers (due to myself), therefore probably much easier to settle down when not ill-posed. To distinguish among them the symbol $\bigstar$ marks venerable Russian conjectures, while our more modest variants are marked by “$\bullet$”.

$\bigstar$ (R6) Can somebody reconstruct Rohlin’s lost proof that the $(M-2)$-sextics with schemes $\frac{6}{1}2$ or $\frac{2}{1}6$ are totally real under a pencil of cubics assigned to pass through 8 points distributed on (or inside) the empty ovals. This is nearly solved in Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics], but she uses the RKM-congruence (\[Kharlamov-Marin-cong:thm\]) to infer a priori the curve being dividing. It could be more natural to draw dividingness from total reality via a purely synthetical procedure [*a priori*]{}. At any rate the conjunction of the RKM-congruence ($\chi\equiv_8 k^2+4$) with Le Touzé’s result implies that Rohlin’s assertion is true. Hence, it should be already “safe terrain” to explore. If much more pessimistic Rohlin’s claim is wrong and then either RKM, or Le Touzé is false, which is very unlikely.

$\bigstar$ (RLT6$\to$RMC6) Can someone complete the proof that the Rohlin-Le Touzé phenomenon of total reality (RLT6) prevents all sextic schemes extending those described in the previous problem (R6), so as to infer nearly all obstructions of Hilbert’s 16th via the paradigm of total reality (TR) and the allied phenomenon of Bézout-saturation. Cf. the diagrammatic of the Gudkov table (Fig.\[Gudkov-Table3:fig\]) to appreciate this issue in degree $m=6$. Of course this problem can be considered as very implicit in Rohlin 1978 [@Rohlin_1978], but in our opinion not solved there.

$\bullet$ (RLT$m>6$) How does the Rohlin-Le Touzé phenomenon described in (R6) above extend to higher degrees $m>6$? Cf. Sec.\[total-(M-2)-schemes:sec\] for a germ of answer.

$\bullet$ (A50$\to$R78) How valuable is the abstract theory of Ahlfors to assess Rohlin’s vision of total reality? Cf. e.g. Sec.\[Esquisse-dun-prog-deja-esquiss:sec\] for some scenarios. In particular is it true that any dividing plane curve admits a total pencil (i.e. whose real members cut only real points)? If yes, is it always of degree $\le (m-2)$ when the given curve has degree $m$? For the case of $M$-curves, cf. (\[total-reality-of-plane-M-curves:thm\]) which gives a total pencil of order $(m-2)$.

$\bullet$ (R78$\to$G13) Is it true as conjectured in our text (\[M-2-curve-degree-like-Gabard:rem\]) that any curve belonging to an $(M-2)$-scheme of type I and degree $m$ has its total reality exhibited by a pencil of curves of degree $(m-3)$. Further what is the exact rôle of Riemann, and Brill-Noether adjoint curves, in this game? Notice still in (\[M-2-curve-degree-like-Gabard:rem\]) a strange concomitance between Rohlin-Le Touzé’s role of cubics and Gabard’s $r+p$ bound on the gonality. The latter improves Ahlfors by replacing $g+1$ by the mean-value of Harnack’s bound $g+1$ and the number $r$ of real circuits. All this numerology looks to match too nicely for this being merely a fortuitous coincidence. In particular for quartics, quintics, sextics, etc. the total reality of $(M-2)$-curves of type I seems always exhibited by such a pencil of degree $(m-3)$.

$\bullet$ (RKM$\leftarrow$type I) The RKM-congruence $\chi\equiv_8
k^2+4 $ detects many $(M-2)$-schemes of type I, but does it detect all of them? The answer is yes for $m=6$ (cf. the Gudkov-Rohlin table Fig.\[Gudkov-Table3:fig\]). Hence $m=8$ is the first place to look for a counterexample. Assuming there is one, then it could be that total reality detects type I schemes at places where RKM fails. \[[*Added in proof*]{} \[13.04.13\].—The answer to this question must be implicit in Rohlin 1978 [@Rohlin_1978 p.93, Art.3.5] (and due to Zvonilov-Wilson).\]

$\bigstar$ (RMC) Is Rohlin’s maximality conjecture true, i.e. all schemes of type I are maximal in the hierarchy of all schemes of fixed degree? Can this be disproved by the Viro-Itenberg patchwork, as it was possible to refute the converse sense of Rohlin’s conjecture (cf. Shustin’s note 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]).

$\bullet$ (SAT) Are satellites of schemes of type I still of type I? For instance what about the 2nd satellite of the Rohlin-Le Touzé sextics of point (R6). Can this be disproved via patchwork? Assuming a positive answer to the first question (even in a special case) points to potential place where to corrupt RMC. Personally, we would be much more happy if RMC holds true, as then, and only then, there is some chance to make a big Riemann-Hilbert or Ahlfors-Rohlin synthesis.

$\bigstar$ (H8) Complete the solution of Hilbert’s 16th in degree 8, and analyze objectively if Rohlin’s maximality principle (RMC) has some things to say in this realm, as it did in degree $m=6$. In particular does Rohlin’s maximality conjecture (RMC) still persists in degree $8$. (Some hints are given in Orevkov’s letter in Sec.\[Orevkov:sec\].)

$\bullet$ (LARS) Can someone disprove our low-altitude rigidity speculation (LARS) positing that a curve with less ovals than 2 units above the deep nest is entirely determined up to large deformations by its real scheme. Cf. (\[LARS:conj\]).

$\bullet$ (URS) \[02.04.13\] The unnested rigidity speculation (URS) is akin to LARS, and posits that any unnested curve is rigid provided the number of ovals is not the square of the semi-degree ($r\neq k^2$, and assume $m$ even). Motivation comes from Rohlin’s formula (which forces such curves being of type II), and the case $m=6$ which follows from Nikulin. Another (weak) evidence comes from the fact that the locking technique of Fiedler-Marin seems to have little grip on such schemes as there is no way to choose a canonical triangle (moving frame).

$\bigstar$ (OOPS)=(One oval postulation).—In particular what about the much more modest (than LARS or URS, yet still wide open) rigidity conjecture for curves having only one component. Are such unifolium curves rigid as conjectured by Rohlin, Viro, etc. How useful are geometric flows to do this? Cf. (\[OOPS:one-oval-rigid-isotopic:conj\]). Actually Viro ascribes to Rohlin (cf. e-mail in Sec.\[e-mail-Viro:sec\]) the rigidity of curves of odd degree with a unique component, but the case of even degrees looks likewise open. Further it seems evident (at least for Viro, cf. the same letter) that OOPS is implied by CC, i.e. the Itenberg-Viro contraction conjecture for empty ovals. By analogy it seems evident that our CCC (cf. right below) implies URS. Sketch of proof: contract all ovals simultaneously (which are all supposed empty) as to reach the connected empty chamber, and do this twice. Of course when $r=k^2$ the real scheme can be of both types, and this case has to be ruled out (or optimistically the type is the sole obstruction to rigid isotopy).

$\bullet$ (CCC)=(Collective contraction conjecture).—Can someone disprove our strong version CCC (of the Itenberg-Viro contraction principle for empty ovals) positing a simultaneous and collective contraction of all empty ovals toward solitary nodes. Cf. (\[CCC:conj\]).

$\bigstar$ (CC) \[30.03.13\] There is a (still open) conjecture of Itenberg-Viro (cf. Itenberg 1994 [@Itenberg_1994] and Viro’s preface of the same volume) positing that [*any*]{} empty oval of a real plane curve can be contracted to a point (solitary node). This is true in degree 6 as proved by Itenberg (), and weaker than our (CCC) above (cf. (\[CCviaCCC-Brusotti:lem\])).

$\bigstar$ (CC vs. TR) A noteworthy consequence of CC is that all obstructions in degree 6 derived (clumsily) via total reality (TR) are likewise derived by this contraction principle of Itenberg (CC6) (modulo knowledge of the RKM-congruence and Klein’s Thesis which is fairly easy to prove since Marin 1988 [@Marin_1988]). The problem is first to decide which method “total reality” versus “contraction” is more easily implemented in degrees $\ge
8$, while trying to make a comparative study of the prohibitions resulting from both procedures. In particular one may wonder if the Itenberg-Viro conjecture implies (formally or not) Rohlin’s maximality conjecture. Sketch of proof: Take any scheme of type I, and a curve enlarging it. Contract an empty oval so as to recover the initial scheme (note here an obvious difficulty, namely the additional oval of the extended scheme is not necessarily an empty one!), and conclude via Klein’s Thesis (a curve of type I cannot champagne-bubble).

$\bullet$ (Refuting CC via Shustin?) \[31.03.13\] By the proof of Prop. \[Klein-vache-deg-6:prop\], we see that the Itenberg contraction principle combined with the diagrammatic of the Gudkov-Rohlin table (of all typed-schemes) implies [*Klein-vache*]{} (KV), i.e. the possibility for diasymmetric curve to acquire a solitary node and then a new oval ([*comme surgit du néant*]{}). Now as Klein-vache is disproved in degree 8, it seems that it is just a matter of waiting completion of Hilbert’s 16th problem in degree 8, until the Itenberg-Viro contraction conjecture get refuted. This is merely a crude scenario but of course one needs to keep track of a massive diagrammatic to get an extension in degree 8 of Rohlin’s theorem (\[Rohlin-type:thm\]) classifying all sextics according to their types.

$\bullet$ (GR8)=(Gudkov-Rohlin census in degree 8).—Assume someone has completed Hilbert’s 16th in degree 8 (i.e. isotopy classification of real schemes), how difficult will it be to complete the corresponding Rohlin table enhancing schemes by their types I or II. Assume this information available, does it follow (by analogy with our proof of Prop.\[Klein-vache-deg-6:prop\]) that under the contraction principle (CC), Klein-vache holds true in degree 8? If yes, then Shustin 1985 would refute the Itenberg-Viro contraction conjecture in degree 8 (CC8).

$\bigstar$ (RAG)=(Ragsdale).—While our erroneous Thom-style estimate $\chi\le k^2$ (cf. \[Thom-Ragsdale:thm\]) is disproved by the Itenberg-Viro $(M-2)$-curve (Fig.\[Itenberg:fig\]), is this estimate still true for $M$-curves? This amounts to one-half of Ragsdale conjecture $\vert \chi\vert \le k^2$ (still open in the $M$-context). A priori a “random” computer-assisted search along the Itenberg-Viro method could detect an $M$-curve refuting Ragsdale. How difficult is it to program a machine adventuring blindly and by brute force in such a random quest? In contradistinction, how difficult is it to write down a proof of Ragsdale’s conjecture in case it should be true. Could it be that a clever use of Thom, or Rohlin’s formula and even some knowledge of complex orientations derived maybe from our synthetic version (\[total-reality-of-plane-M-curves:thm\]) of the Bieberbach-Grunsky theorem (planar case of Ahlfors) assesses the full puzzle. If feasible this would be a spectacular application of conformal geometry to the Hilbert-Ragsdale-Petrovskii 16th problem, boiling down in quintessence to Riemann’s Nachlass 1857 [@Riemann_1857_Nachlass]. Of course we do not claim this to be an easy project.

$\bullet$ In degree 6, it may be observed that among the trinity of congruences mod 8 (due to GR, GKK, RKM, where G=Gudkov, R=Rohlin, K=Krakhnov or Kharlamov (twice), M=Marin, cf. (\[Gudkov-hypothesis:thm\]), (\[Gudkov-Krakhnov-Kharlamov-cong:thm\]), (\[Kharlamov-Marin-cong:thm\])), the latter, i.e. RKM, implies the 2 formers, when combined with Rohlin’s maximality principle (RMC). Is this subsuming a general feature due to trivial geographical/arithmetical reasons? If yes can we condense, i.e. proceed to an unification of forces by reducing nearly all prohibitions of Hilbert’s 16th to the phenomenon of total reality.

$\bullet$ Can we write down an explicit ternary form with integral coefficients $F\in \ZZ[x_0,x_1,x_2]$ whose real locus is Gudkov’s curve $\frac{5}{1}5$, and estimate the smallest size of the coefficients involved? As discussed in Sec.\[Diophantine-and-proba:sec\], can we compute the natural masses (w.r.t. Lebesgue measure on the space of coefficients) of each of the 64 chambers (past the discriminant) of smooth sextic curves given by the census of Gudkov-Rohlin-Nikulin (i.e. Fig.\[Gudkov-Table3:fig\]).

$\bullet$ \[31.03.13\] A more modest but fundamental problem is to publish (in the West side of Ural) an avatar in degree 8 of the Gudkov-Rohlin table (Fig.\[Gudkov-Table3:fig\]). For a partial depiction of just the simplest planar face of this 4D-pyramid, cf. Fig.\[Degree8:fig\]. I presume that one can by mean of an Atlas consisting of ca. 20 pages dress a list of all combinatorially possible schemes after taking into account the obvious Bézout-style obstructions (Bézout, Zeuthen, Hilbert’s bounds on the depth of nest, Gudkov, plus the total reality obstructions allied to the deep nest and doubled quadrifolium, etc.). Once this atlas of all octics is made available it should be a trivial matter to appreciate:

—how far/close we are to solve Hilbert’s 16th in degree 8 (soft-isotopy);

—how the paradigm of total reality (resp. the contraction principle) explain the prohibitions, and finally,

—whether the contraction principle (CC) implies Klein-vache, in which case CC would be disproved by Shustin’s refutation of Klein-vache in degree 8.

$\bullet$ (CG6)=(Contiguity graph for $m=6$) \[01.04.13\] Can we describe all the contiguity relation realizable via algebraic Morse surgeries on the Gudkov-Rohlin table of periodic elements (in degree 6). To be more specific, is some result along our Conjecture \[eversion-and-other surgeries:conj\] true. The proof of this could be merely a matter of adapting the work by Nikulin, and Itenberg, yet it seems quite challenging to decide precisely which eversions are realized algebro-geometrically.

$\bullet$ (KV7) Klein-vache (KV) was disproved in degree 8 by Shustin 1985 via a conjunction of Viro’s method and advanced Bézout obstructions due to Viro (and Fiedler). On the other hand, we prove below (\[Klein-vache-deg-6:prop\]) that KV is true in degree 6. So one may wonder about the case $m=7$, where to my knowledge KV is undecided.

$\bullet$ (II/II) Is the “toutou” conjecture true? This posits that any scheme of type II and even degree $2k$ augmented by a pseudoline to a scheme of degree $2k+1$ is of type II too. Cf. (\[toutou:conj\]) for some surgical motivation (à la Fiedler) and inspiration coming from reading Gross-Harris 1981, who were unable to settle the case of quintics with 2 unnested ovals (and a pseudoline of course). Perhaps a general solution of this problem merely follows from a conjunction of Rohlin’s and Mishachev’s formulae. If not, then one could use a large deformation principle.

$\bullet$ (Klein’s bipolarity conjecture). Is it possible for two real plane curves to have distinct distributions of ovals, yet conformally equivalent underlying symmetric Riemann surfaces (under an equivariant diffeomorphism). This can be paraphrased in the algebro-geometric language as the quest of two real planes curves with distinct distribution of ovals, but bi-rationally equivalent over $\RR$ as abstract curves. For more see (\[Klein-bipolarity:conj\]), where it is explained that the first place where to look for this (hypothetical but likely) phenomenon is degree $m=6$. It would be interesting to see if this question due to Klein 1922 (safe misunderstanding on my side) can be settled via Cremona transformations not inducing diffeomorphisms of $\RR P^2$.

$\bullet$ (RIG/SAT) Is rigidity stable under satellites? This is a wild speculation based on Nuij’s rigidity of the deep nest caricatured as reducible via satellite to the rigidity of the conic (known since time immemorial). Likewise the more highbrow rigidity result of Klein 1876 for quartics could induce rigidity of all satellites of the 6 possible quartic schemes, in particular of the quadrifolium (whose satellites are totally real under a pencil of conics). Further Nikulin’s rigidity result for sextics could imply also a vast array of rigidity results in degrees $6k$, by satellitosis of all schemes of degree 6 which are not of indefinite type (and which are explicitly known $64-2\cdot 8=48$ types by the Gudkov-Rohlin table=Fig.\[Gudkov-Table3:fig\]). Perhaps all this stability of rigidity under satellites has to be combined with total reality, in which case the analogy with Nuij’s rigidity is still deeper. In that case we would only take satellites of Rohlin’s $(M-2)$-schemes of degree $6$, cf. (\[satellite-Rohlin-(6)-schemes-rigid:conj\]), and for quartics only the quadrifolium (and the deep nest) would be permissible.

$\bullet$ (ANTI-GAB) It seems that the case of $(M-4)$-sextics of type I offers a possible corruption of Gabard’s bound $r+p$. Compare Scholium \[(M-4)-sextics-corrupt-Gabard:scholium\]. If not, this is at least a [*pièce de résistance*]{} against the principle that any abstract pencil is concrete, and therefore Ahlfors abstract theorem is unlikely to apply without friction in Hilbert’s 16th problem. In other words Riemann’s canary feels claustrophobic in the Plato cavern of Brill-Noether-Hilbert.

$\bullet$ (LETOUZE-SCH) Inspired by a Scholium of Le Touzé 2013 (\[LeTouze-quintic:scholie\]), we extended her result to all $M$-curves of odd degrees, cf. Theorem \[Le-Touzé-extended-in-odd-degree:scholium\]. It seems of interest to extend her method to even degrees as well. We had just the time to treat the case of degree 6, cf. Lemma \[Le-Touzé-scholium-deg-6:lem\] which uses imaginary basepoints yet without affecting total reality. It could be challenging to see if this method of total pencil (becoming more and more explicit) could be used to reprove the deep prohibitions for $M$-curves due to Hilbert-Rohn-Gudkov-Rohlin.

$\bullet$ (RMC via Mangler 1939 and Ahlfors 1950, maybe implicit in Rohlin 1978).—-Rohlin’s maximality conjecture looks nearly implied by Ahlfors, safe for the difficulty that the enlargement of the type I scheme is a priori very distant from the enlarged curve realizing the orthosymmetric scheme. Using triviality of the mapping class group of $\RR P^2$ (Mangler 1939, probably a student of H. Kneser?), one can try to isotope the distant enlargement to make it identic with the original curve. The latter being swept out by a total pencil (Ahlfors 1950, plus epsilon!), one could get a corruption of the homological version of Bézout (i.e. intersection theory à la Poincaré, Lefschetz, etc.) Of course one requires a procedure to extend the (Mangler) isotopy to $\CC
P^2$, and one may object that our sketch of proof equally well applies to curves of type I whose scheme is however of indefinite type (but non-maximal). So there is perhaps some obstruction to extend Mangler’s isotopy as to preserve positivity of intersection-indices. Understanding this obstruction, and supposing one able to show its vanishing in case of a scheme of type I, could procure a proof of the elusive RMC. It seems very likely that Rohlin thought about this strategy, but never wrote something down. Perhaps experts like Marin can complete this game? Cf. Sec.\[RMC-via-Mangler:sec\] for slightly more details.

To keep some slight control on all these conjectures, see Fig.\[CCvsCCC:fig\] showing how they interact and their validity range.

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\[30.03.13\] Let us conclude with a historical remark. It should always be remembered, and amazing to rediscover everyday, that “everything” in this topic goes back to Klein. Himself expected that the type of the symmetric Riemann surface (underlying a real curve acted upon by complex conjugation) has some interplay with Hilbert’s problem on the distribution of ovals. Compare a footnote added ca. 1922 in his Ges.Math.Abhdl., reproduced as Quote \[Klein-1922-immer-vorsgeschwebt:quote\], but of which we now reproduce the most prophetical side:

This is worth translating (in the poor English of the writer):

It always puzzled me, to infer more about shapes of real plane curves of arbitrary degrees by pursuing considerations of the text, not only regarding the number of circuits, but also their mutual dispositions. I do not abort this hope, but must alas confess, that the reality theorems on curves of arbitrary genus (which I deduce from the general theory of Riemann surfaces, specially that of symmetric Riemann surfaces), are not sufficient for this purpose, affording instead merely a framework for the menagerie of possibilities to be investigated.

It is striking to notice how this Kleinian prose remains very much actual, reflecting best our own frustration to make the Ahlfors-Rohlin Verschmelzung, we are dreaming about, a true reality. It shows also how much Klein would have appreciated the developments made possible in the 1970’s by Gudkov, Arnold, and especially Rohlin, etc., and perhaps even more, something like anticipating the vision of total reality by Rohlin.

\[05.04.13\] The last sentence of this same footnote, reads:

Da man über die Natur dieser Bedingungen zunächst wenig weiss, kann man noch nicht von vornherein sagen, dass alle die Arten reeller Kurven, die man gemäss meinen späteren Untersuchungen für $p={ n-1 \cdot n-2 \over 2}$ findet, bereits im Gebiete besagter ebener Kurven $n$-ter Ordnung vertreten sein mü[ß]{}ten, auch nicht, da[ß]{} ihnen immer nur [*eine*]{} Art ebener Kurven entspräche. K.

Here, one realizes that Klein anticipated the simple phenomenon of what Rohlin calls schemes of indefinite type, i.e. that the real scheme alone (i.e., distribution of ovals) does not need to determine the type (i.e. dividingness or not). Klein also emphasizes the issue that not all topologically permissible symmetric Riemann surfaces have to appear in the plane. In both cases the first examples appear in degree 5, and then massively in degree 6. For instance a quintic with only one pseudoline cannot be of type I , albeit since its genus is even (namely 6) the corresponding Riemann surface exists. (Compare (\[Klein-Marin-quintic:lem\]) which is based on Klein-Marin, or Gross-Harris argument via theta-characteristics discussed at the same place that was probably known to Klein in 1892.

Finally, it is also puzzling to see that Klein 1892 anticipated somewhat the contraction conjecture of Itenberg-Viro, cf. historical note right after (\[Itenberg-Viro-contraction:conj\]).

Albeit Klein missed some basic modern tricks (like Rohlin’s formula, or Fiedler surgical smoothing law), he also mastered perfectly the Riemannian theory (conformal maps, the allied circle maps and total reality as credited by Teichmüller 1941, theta-characteristics, allied deep enumerative problems of bitangents to quartics à la Plücker-Zeuthen). Further he appealed to contraction principles, as well as his own singular geometric method to represent complex loci as multiple cover of the projective plane as to infer the “complexified” topology of real curves (in the 1874–76 articles “Über eine neue Art der Riemannschen Flächen”). Hence Klein’s legacy on the topic is massive (ca. 300 pages, if one counts the Göttingen lectures [@Klein_1892_Vorlesung-Goettingen]), and much remains to be learned from it.

The Klein-Rohlin conjecture on real schemes of type I {#Klein-Rohlin-conj:sec}
=====================================================

\[01.01.13\] A fascinating question is raised by (the master) V.A. Rohlin in his 1978 [@Rohlin_1978 p.95] survey looping back directly to a (prophetic) allusion of Klein. Remember that Rohlin was fluent with German language, being involved during World War II as translator on the front-line, cf. Guillou-Marin’s book 1986 [@Guillou-Marin_1986 p.ix]: “[*En 1941, quand l’Allemagne attaqua l’U.R.S.S., Rohlin rejoignit le corps des volontaires du Peuple (unités militaires non entraînées). Son unité fut encerclé et Rohlin fait prisonnier par les allemands. Ensuite il réussit à s’échapper, à rejoindre l’armée soviétique et finit la guerre comme traducteur militaire (Rohlin parlait couramment l’allemand). Immédiatement après la guerre Rohlin fut emprisonné par la sécurité de l’armée (comme ce fut le cas pour de nombreux anciens prisonniers de guerre) mais fut libéré à la fin de l’année 1945.*]{}”

. [A study of the available factual material suggests that possibly a real scheme belongs to type I iff it is [*maximal*]{}, that is, it is not part of a larger real scheme of the same degree. This conjecture is true for $m\le 6$, and there is much to be said in its favour[^19] for $m>6$. There is an allusion to it in Klein: see \[4\], p.155 (=Klein 1922=Ges. Math. Abh. II [@Klein-Werke-II_1922]).]{}

The passage Rohlin had in mind is unambiguously identified as the following (going back actually to Klein 1876 [@Klein_1876]), which is worth reproducing albeit it is first quite hard to interpret (cf. also Viro 1986/86 [@Viro_1986/86-Progress p.67–68] or Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.785], and especially Marin 1988 [@Marin_1988], clarifying earlier work partially incorrect of Cheponkus 1976 [@Cheponkus_1976]):

\[Klein\_1876-niemals-isolierte:quote\]

\[17.01.13\] It is essential to note that Klein’s quote contains two very distinct parts. The first part on which Klein is affirmative may be translated as the assertion that a dividing (=type I or orthosymmetric) plane curve cannot acquire a new oval by transgressing the discriminant at a solitary node (with imaginary conjugate tangents like the germ $x^2+y^2=0$). With the strong word “kann …niemals” (=never never!!), Klein emphasizes his complete self-confidence about the truth of his assertion. Alas no proof (as far as I know) were ever given by him, even in his Göttingen lectures 1892 [@Klein_1892_Vorlesung-Goettingen]. The first proof had to wait 112 years until Marin 1988 [@Marin_1988] write down a two-lines argument (of a somewhat stronger assertion). A recent e-mail exchange with Viro suggested that Klein’s Ansatz may easily be deduced from the Ahlfors map (cf. Lemma \[Klein-via-Ahlfors(Viro-Gabard):lem\]).

The second part of Klein’s text, starting with “während die Kurven …”, is pretty subtle to interpret and definitively less categoric. It is suggested that curves of type II are in contrast susceptible of acquiring new ovals springing [*ex nihilo*]{} from a solitary double point like a champagne bubble. The vague wording “sozusagen noch entwicklungsfähig” (=“so-to-speak still developable”) emphasizes that Klein did not saw any nondividing curve champagne-bubbling, but merely that he found no (topological) obstruction to such an eventuality. As we shall see, this second clause which we shall call “Klein-vache” is refutable (in degree $8$) via the disproof of one half of Rohlin’s conjecture by Shustin 1985 [@Shustin_1985]. Alas, we have not yet completely digested Shustin’s work, which relies on deep Bézout-style obstructions due to Fiedler-Viro. In the positive sense, we proved via a cocktail of Russian results that “Klein-vache” holds true in degree 6 (cf. Prop.\[Klein-vache-deg-6:prop\]).

\[01.01.13\] Rohlin’s conjectural criterium looks a pearl of observational skills. What does it mean, or rather how practical is it if true at all? One should make a list of all real schemes (i.e. isotopy classes) of curves of a given order. Then assuming one competent and patient enough to have tabulated the exhaustive list one could detect the dividing types by inspecting maximum elements in the lattice ordered by inclusion. ([*Insertion*]{} \[28.03.13\].—This is not really what happens in practice, especially since Shustin’s disproof, and it seems more likely that the residual half of Rohlin’s conjecture acts by means of prohibitions, that are anyway required to dress a table of all schemes.)

Let us work out low-order examples to gain some experimental evidence Rohlin is referring to.

First in degree $1$ there is just the line, which is of type I (=dividing). Then in degree 2, there is two isotopy classes represented either by the circle $x^2+y^2=+1$ and the invisible conic $x^2+y^2=-1$ (empty real locus). (This follows e.g. from Sylvester’s law of inertia, alias diagonalization of quadratic forms, also to be found earlier by Jacobi, and presumably many others? and on the case of 2 variables this can safely goes back to ancient Greeks, Euclid, etc.) In order 3 we have cubics (extensively studied by Newton and Plücker), but up to isotopy the story becomes much simpler and we have two isotopy classes differentiated merely by the number of real circuits $r=1,2$; the latter being dividing while the other is not. This follows readily from the abstract Klein-Weichold classification of symmetric surfaces (the latter being merely a mirror image of the Möbius-Jordan classification of abstract topological surfaces).

The little zoo of all quartics (Plücker 1839, Zeuthen 1874, Klein 1876, Rohlin 1978)
------------------------------------------------------------------------------------

\[01.01.13\] Next it comes to quartics (order 4). Here the number of real circuits $r$ fails to classify isotopy classes for there exist quartics with 2 ovals being either nested or not. The first basic thing-to-do (going back apparently to Plücker 1839 [@Plücker_1839]) is exploring varied examples by smoothing a pair of conics with 4 intersections (cf. Fig.\[KleinRohlin-quartic:fig\]).

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Among all those curves only those marked by the attribute “dividing” are dividing as they result from sense-preserving smoothings. One can also remember Klein’s congruence $r\equiv g+1
\pmod 2$ in the dividing case to detect nondividing curves, e.g. those with $r=3,1$. Also under Harnack-maximality, i.e. $r=g+1$ (here $4$) then dividingness is automatic (by Riemann-Klein 1876 [@Klein_1876] or the more tedious synthetic argument of Harnack 1876 [@Harnack_1876]). So $r=2$ is the only ambiguous value. Here, as shown on the figure, the 2 ovals can either be nested or not. In the first case the curve is dividing (due to total reality under a pencil of lines through the innermost oval), while the unnested case is always nondividing (as Klein knew as early as 1876).

\[Klein-unnested-quartic-nondividing:lem\] [(Klein 1876)]{} All quartics with $r=2$ unnested ovals (and not just the two traced above) are nondividing.

This is already a nontrivial result. We sketch several proofs:

\(1) How Klein derived the result? Maybe as follows. Klein knew as early as 1876 [@Klein_1876_Verlauf] (basing himself on deep works of Schläfli and Zeuthen on cubic surfaces and their apparent contours which are quartics) that quartics are rigidly classified by the real scheme. This is to mean that any two quartic curves having the same distribution of ovals can be continuously deformed through a large deformation of the coefficients without ever meeting a singular curve. Hence Klein had only to check the nondividing character of a specific quartic to get that of all curves with 2 unnested ovals. Klein used a special device of representation of the curve as a branched cover of the projective plane by assigning to each point (of the complexification) the unique real point of the tangent and so could see the curve. Nowadays the surgical recipe of Fiedler looks also best suited to do this. For an elementary graphical proof compare our Fig.\[Guertel-genetic:fig\] earlier in this text.

\(2) Perhaps one way to argue could involve Ahlfors theorem, yet some nontrivial details deserve being worked out. \[03.01.13\] Assuming Gabard there is a total map of degree $r+p=3$ if dividing (and not less via the complex gonality), yet since the ovals are unnested it cannot be induced by a pencil of lines. So the auxiliary curves are of order at least two. Assume first the order to be two, so we have a pencil of conics. Since the degree of the morphism is 3, we have $2\cdot 4-3=5$ basepoints on the $C_4$, but a pencil of conics has only 4 basepoints by Bézout. (Note that Ahlfors bound $r+2p=g+1=4$ would not be strong enough for this purpose!) If the auxiliary curve are of order $3$, then we must have $3\cdot 4-3=9$ basepoints on the $C_4$. No basic corruption is detected?

\(3) Another argument via theta-characteristics is implicit in Klein 1892 [@Klein_1892_Realitaet] (see also his Göttingen lectures 1891/92 [@Klein_1892_Vorlesung-Goettingen]) and appears in modernized form in Gross-Harris 1981 [@Gross-Harris_1981].

\(4) Another more elementary (and purely topological) proof follows from Rohlin’s formula 1974–78 (valid for dividing curves), cf. Sec.\[Rohlin-formula:sec\]. This formula reads $2(\pi-\eta)=r-k^2$, where $\pi$, $\eta$ are the number of positive, resp. negative pairs of ovals. This distinction appears by comparing the orientation induced as boundary of the half of the Riemann surface underlying the curve, with that of the annuli bounding a pair of nested ovals. In our case there is no nesting hence $\pi=\eta=0$, and so $r=k^2=4$ violating the assumption $r=2$.

\(5) A related proof involves Arnold congruence 1971 [@Arnold_1971/72] for $M$-curves of degree $2k$ (with an obvious extension to dividing curves in Wilson 1978 [@Wilson_1978]). This reads $\chi:=p-n \equiv k^2 \pmod 4$ and suffices. Here $p, n$ are notation coined in Petrovskii 1938 [@Petrowsky_1938 p.190] for positive and negative ovals, also interpretable as the number of even and odd ovals. An oval is said to be [*even*]{} if it is lying within an even number of consecutive ovals. For the case at hand (2 unnested ovals), both are even (being subsumed to zero ovals), hence $p-n=2-0\equiv
k^2=4 \pmod 4$ is violated, and the nondividing character of the curve follows. The difference $p-n$ is readily interpreted as the Euler characteristic $\chi$ of the Ragsdale membrane bounding (orientably) the curve from inside. In the case at hand, the Ragsdale membrane is the disjoint union of 2 discs, whence obviously $\chi=2$.

Now using the theorem of Bézout, it is clear (cf. Zeuthen 1874) that our picture above (Fig.\[KleinRohlin-quartic:fig\]) exhaust all possible shapes traced by quartics. For instance a such cannot have 2 ovals nested in a third one, etc. So it is a simple matter to convince that we have listed all real schemes of quartics (with all of them safe the empty curve $x^4+y^4=-1$ arising through small perturbation of 2 ellipses). Of course a priori a quartic could have 5 ovals but this was precluded by Harnack 1876 [@Harnack_1876], and of course already by Zeuthen 1874 [@Zeuthen_1874 p.411] using a prototype of Harnack’s device. Indeed if a quartic had 5 ovals (or more) the conic through them would cut it in $5\cdot 2=10>8=2\cdot 4$ overwhelming Bézout. At any rate Klein’s argument of 1876 via the underlying Riemann surface gives the general Harnack bound $r\le g+1$ in some more intrinsic fashion.

Even stronger is the following result (due to Klein 1876 [@Klein_1876_Verlauf], though his proof makes a détour through surfaces and it could be interesting to find a more direct argument). Perhaps the transition through cubic surfaces is necessary as it rationalize the irrationality of quartics curves, though Klein in 1876 seems to have add a direct argument staying in the realm of curves, but he did not exposed details.

The real scheme is a complete invariant for rigid-isotopy classes of quartics.

(Rigid isotopy refers to the morcellation of the space of all curves of some fixed order effected by the discriminant hypersurface parametrizing singular curves.) Modulo such knowledge one can draw the lattice of all real schemes (right part of Fig.\[KleinRohlin-quartic:fig\]) on which the (Klein-)Rohlin intuition is verified: a real scheme is dividing (or of type I) iff it is maximal.

\[28.03.13\] In fact it is tempting to make a baby Gudkov table in degree 4 (inspired by the case of degree 6, cf. Fig.\[Gudkov-Table3:fig\]) as to visualize the situation. Here the Gudkov symbol $\frac{x}{1}y$ is merely a symbolical way to mean that $x$ ovals are nested in a big oval (the denominator $1$), while $y$ ovals are lying outside. It is noteworthy that Rohlin’s maximality principle is fully validated here and prohibits all the schemes lying above the configuration $\frac{1}{1}$ of the nest of depth 2, which is already Bézout-saturated. It is also pleasant to notice the presence already of the highbrow Gudkov-Rohlin sawtooth (dashed on the picture) so typical of the solution of Hilbert’s 16th in degree 6 (cf. again Fig.\[Gudkov-Table3:fig\]). This extends to all degrees by the congruences of Gudkov-Rohlin $\chi\equiv_8 k^2=4$ for $M$-curves (\[Gudkov-hypothesis:thm\]), Gudkov-Krakhnov-Kharlamov for $(M-1)$-curve and Kharlamov-Marin for $(M-2)$-curves of type I). This sawtooth, which looks like a piecewise linear sine-curve, forces the scheme below its depressions, to be of type I. Of course it tends to pass unnoticed here ($m=4$) as it is such a trivial consequence of Bézout with lines.

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As already announced we conjecture in general that the whole sawtooth can be explained by invoking the phenomenon of total reality for $(M-2)$-curves via adjoint curves of order $(m-3)$. At this stage a comparison with degree 6 (Fig.\[Gudkov-Table3:fig\]) makes it puzzling to wonder if all schemes lying below the sawtooth are always realized, yielding a sort of denseness below the sawtooth (alias Gudkov line). The answer is no in degree 8 (cf. Fig.\[Degree8:fig\]), where Petrovskii’s estimate of 1933/38 (\[Petrovskii’s-inequalities:thm\]) starts to act prohibitively. Yet taking this and other conjectural hypothesis of Ragsdale into account, one can still wonder about the question of denseness of schemes, namely the issue as to whether prohibition are essentially confined to the high level of the pyramid (i.e. above $(M-2)$-curves), or if in contrast there is some sort of porism (or lacunae) killing schemes at low altitudes. As we shall see latter if the conjectural maximality principle of Rohlin as well as our stability of type I under satellites holds true, it is likely that for high degree $m$ (especially when the integer $m=2k$ as a rich factorization into primes) then there will be a myriad of cone-like [*no man’s land*]{} zone where schemes are killed because they extend a Bézout-Ahlfors-Rohlin saturated scheme subsumed to the paradigm of total reality.

Of course the situation of low degrees $m=4,6$ may give the wrong impression that the whole paradigm of obstruction by the saturation allied to total reality are already explained by the trinity of Russian congruences mod 8 (of all the workers already cited starting with Gudkov-Arnold-Rohlin). Yet in reality this is not even true for low degrees because under the disguise of Bézout it is already total reality which assures the planar character of the lowest Gudkov tables $m\le 6$. Otherwise we had to consider a menagerie of other schemes with more nested structures. So this gives some intuition a priori that Rohlin’s maximality principle (in our opinion much allied to Ahlfors) will not be subsumed to the trinity of congruence mod 8, albeit the lowest of it pertaining to $(M-2)$-curves may act as vivid generator of total reality phenomena.

Quintics (Klein 1892?, Rohlin-Mishachev 1976, Fiedler 78, Marin 79, Gross-Harris 1981) {#quintic-table-Klein-Gudkov:sec}
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\[04.04.13\] This survey has some repugnancy against curves of odd order for reasons hard-to-explain, perhaps allied to the cumbersomeness of the avatar of Rohlin’s formula (due to Mishachev). However the theory especially our main focus of total pencils works as well in this case. Let us take a small look at the “Gudkov-Rohlin” table in degree $m=5$. We recommend however to skip this section on first reading as our understanding is lacunary (in part because we do not discuss Mishachev’s formula, or because we do not entered into the Klein-Gross-Harris theory of real theta characteristics). Yet, the case of quintics and more generally curves of odd degrees (especially those of the shape $m=5+4n$, else Klein’s congruence suffices) offer a pleasant application of the Klein-Marin principle, when it comes to check that the scheme with only one circuit is of type II (see Lemmas \[Klein-Marin-quintic:lem\] and \[Klein-Marin-odd-degree:lem\]).

First, in degree $m=5$, Harnack’s bound is $M=g+1=7$, since $g=\frac{(m-1)(m-2)}{2}=(4\cdot 3)/2=6$. In odd degrees there is always a unique pseudoline (Möbius 18XX [@Moebius_18XX], von Staudt 18XX [@von-Staudt_18XX], Zeuthen 1874 [@Zeuthen_1874], Harnack 1876 [@Harnack_1876], Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege], etc.), and we may omit it from the Gudkov symbols. Hence on the table below (Fig.\[Gudkov-Table-quintic:fig\]) we suppress the pseudoline from the real scheme depiction. As usual one of the most noteworthy configuration is the deep nest, here $\frac{1}{1}=(1,1)$ which is total under a pencil of lines, hence of type I.

[*Bibliographical puzzle*]{}.—This argument looks to us much more elementary than the one of Gross-Harris 1981 [@Gross-Harris_1981 p.175] via theta-characteristics which goes back probably to Klein 1892. Despite its extreme elementariness, it seems also to have escaped Klein’s attention. In fact Klein uses it in 1892 (p.177 of Ges. Math. Abh, II) but only after contracting the empty oval of the Gürtelkurve. We cite the relevant passage:

Sollen wir diese geometrischen Verhältnisse durch Beispiele belegen, so nehmen wir vielleicht zunächst den Fall der Gürtelkurve $p=3$. Hier hat es ersichtlich keine Schwierigkeit, das innere Oval auf einen Punkt zusammenzuziehen. Von diesem aus projizieren wir jetzt die Kurve auf eine gerade Linie. Die Gerade wird dann nach ihrer ganzen Erstreckung von den Bildpunkten doppelt überdeckt, so zwar, da[ß]{} dabei kein reeller “Scheitel” auftritt[^20]. Das entspricht in der Tat dem orthosymmetrischen Falle $\lambda=1$ des Geschlechtes $p=2$.

Later, the argument of total reality seems to have escaped the attention of another great master, namely Alexis Marin 1979 [@Marin_1979], compare especially on p.56 his complicated argument for “N’existe pas” in the bottom-right angle of the tabulation, as well as the question p.59: “Est-ce qu’une courbe ayant cette disposition sépare sa complexifiée.”

This total reality (or Bézout-saturation) kills all schemes enlarging it (cf. unframed white-colore schemes on Fig.\[Gudkov-Table-quintic:fig\]). Apart from this obstruction there is essentially no other. First we can construct an $M$-curve necessarily unnested (by Bézout-saturation) with symbol $6$ (again the pseudoline $J$ is omitted). This is constructed via a Hilbert-method on Fig.\[Gudkov-Table-quintic:fig\]a (for a less schematic picture cf. Fig.\[Harna0:fig\]). We presume that nobody knew existence of such a curve prior to Harnack 1876 (but this is just a historical challenge, perhaps Plücker, Zeuthen, Klein before but not sure).

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Next Fig.b. shows a quintic with 5 ovals (and one pseudoline), which arises by slight perturbation of 2 ellipses plus a line. This was probably known to Plücker 1839 [@Plücker_1839], or earlier. The curve constructed is of type II by Fiedler’s signs law, or just by Klein’s congruence $r\equiv_2 g+1$. The latter forces actually all the schemes $5,3,1$ being of type II.

Below we have the scheme $4$. This admits realizations of both types (as knew Rohlin 1978, and perhaps Klein?), as shown by Fig.c and Fig.d (using Fiedler’s smoothing law). The curve of Fig.c has actually a total pencil of conics assigned to pass through the 4 ovals (which was depicted earlier in this text). Again it is interesting to note that Gross-Harris (p.176–177) used a somewhat more synthetic and complicated argument than just Fiedler’s law, to show existence of curves in both types I/II.

[*Historical note*]{}.—The above phenomenon is the first instance of where the type of a curve is not determined by the distribution of ovals. It admits as a simple consequence the fact that there exists obstructions to rigid-isotopy lying beyond the real scheme (remark due to Rohlin 1978). It is clear however that Klein knew (or at least suspected) this basic phenomenon, compare his footnote of 1922:

\[Inserted 05.04.13\].—In fact this can be interpreted either à la Rohlin, by saying that a real scheme can have realizations in both types (I/II=ortho- or diasymmetric). Somewhat more crazy would be the following interpretation.

\[Klein-bipolarity:conj\] [(Kleinian bipolarity—Klein 1922, Gabard 2013)]{}.—An abstract symmetric Riemann surface (SRS) can admit plane realizations with distinct distributions of ovals.

Actually I do not know if this phenomenon of “bipolarity” can occur. Of course it does trivially occur, with a line and a conic both representing the Riemann sphere with its standard real structure (equatorial involution). In degree 3, it cannot occur since the real scheme determines the type, and likewise in degree 4 (cf. previous section). In degree 5, predestination of the type by the real scheme is not true any more, yet the combinatorics of Fig.\[Gudkov-Table-quintic:fig\] is sufficiently simple as to preclude bipolarity. Indeed if the SRS is fixed, hence in particular the number $r$ of real circuits, the only height at which there are several distributions of ovals is $r=3$, where we have the nested ($\frac{1}{1}$) and the unnested ($2$) schemes. Yet both of them are differentiated by the type. Hence the first place to look for bipolarity is degree 6. Here we have (see the Gudkov-Rohlin table Fig.\[Gudkov-Table3:fig\]) a myriad of sextics having the same underlying (topological) symmetric surface. It is unclear if they can be conformally equivalent while exhibiting different distributions of ovals. It could be imagined that a Hilbert sextic is sometimes conformally diffeomorphic to one of Harnack, or even Gudkov. This question looks a bit artificial or puzzling, yet has perhaps of some importance if one likes to link with the abstract theory of Ahlfors taking into account only the abstract Riemann surface. To settle the bipolarity question we can look at the natural map from the hyperspace of smooth plane curves to the real moduli space of SRS, i.e. $\mH-\disc \to M_g$. Perhaps then two chambers may have overlapping images. As noted by the old Felix Klein (aged 73 at the moment of his 1922 footnote) plane curves have specialized moduli. Hence the images in question are fairly small subloci of the moduli space, but this does not prevent overlap. Another approach is to use Cremona transformations of the plane defined over $\RR$ which do not induce diffeomorphisms of the plane $\RR
P^2$ (this remembers works by Ronga-Vust ca. 2002, or their student J. Blanc). By this procedure we can perhaps alter the distribution of ovals, yet without distorting the conformal structure, as the curve and its image are in birational equivalence. Can this vague idea be implemented? Otherwise the approach can be the Teichmüller-theory of the map described above from concrete plane curves to the moduli space of abstract real(=symmetric) Riemann surfaces, while trying to study exactly the coincidences of this mapping. One could try to determine exactly which among the 64 chambers of sextics (Nikulin’s theorem (\[Nikulin:thm\])) are in bipolarity, i.e. contains conformal replicas of the same symmetric Riemann surface. This defines an additional graph structure on the Gudkov-Rohlin table where edges are traced whenever two vertices(=chambers) contains curves abstractly isomorphic over $\RR$. Of course the edges have to preserve the height $r$ on the Gudkov pyramid, as well as the types. Those (topological) obstructions to bipolarity could be the sole ones, in which case the graph in question would show plenty of edges. In particular it restricts to the complete graph on each levels at height $r$ not congruent to $g+1 \pmod 2$ where diasymmetry reigns ubiquitously (Klein’s congruence). Few other levels have pure types too, e.g. $M$-curves (type I only), and the level $r=3$ (types II only) (via Rohlin’s formula), see again Fig.\[Gudkov-Table3:fig\]. At those levels the bipolar graph could be complete too.

After this digression, we return to the basic classification of quintics. The scheme $3$ is of type II by Klein’s congruence (as we already noted), and exists as shown by Fig.e.

Next the scheme 2 exists in type II as shown by Fig.f. It is however more tricky to prove that the scheme $2$ is of type II. This follows either from the avatar in odd degree of Rohlin’s formula (i.e. Mishachev’s formula). Perhaps there is a more elementary argument, say by using a pencil of lines while trying to permute 2 imaginary points during a sweeping. Also Gross-Harris have probably an argument via theta-characteristics, but alas those authors confess being not able to prove this compare p.175 of Gross-Harris 1981 [@Gross-Harris_1981] where we read: “In the non-nested case, we suspect that $a(X)$ is always 1 \[i.e. type II, or nondividing\] but have no proof”. Was Felix Klein (1892 paper [@Klein_1892_Realitaet] or lectures [@Klein_1891--92_Vorlesung-Goettingen]) able to tackle this case?

A crude principle that do this work is the postulation that whenever we add a pseudoline to a scheme of type II, it remains of type II. Recall that we know since Klein that the quartic scheme $2$ is of type II, cf. Lemma \[Klein-unnested-quartic-nondividing:lem\]. Some evidence comes from surgeries on the Riemann surface while noticing that diasymmetry is a dominating character in the genetical sense. This is implicit in Fiedler’s law of smoothing and really a simple matter of visualizing the corresponding Riemann surfaces. So let us posit the:

\[toutou:conj\] [(Gabard 2013, but probably standard by Rohlin-Fiedler, if not erroneous)]{}.—When a scheme of even degree $2k$ is of type II, then the same scheme of degree $2k+1$ augmented by a pseudoline is of type II too. (Il y a trop de toutous dans la langue anglaise, mon ostie!)

Proving this could again involve a large deformation principle (as discussed in the sequel) like minimizing the length of the pseudoline as to make it a line. There will then be a strangulation of the Riemann surface and we are reduced to Fiedler’s genetic law. Perhaps there is an elementary proof of the conjecture based on a conjunction of Rohlin’s and Mishachev’s formulae. Note also that the conjecture holds true for the empty scheme by Lemma \[Klein-Marin-odd-degree:lem\] below.

Another idea to show that the quintic (unnested) scheme $2$ is of type II could be to use Klein’s 1876 remark that a curve of type I cannot acquire a solitary node, cf. below for a proof (\[Klein-Marin:lem\]) essentially along the lines of Marin 1988. Then we are reduced to showing that any quintic with scheme 2 (again we omit the pseudoline $J$) can indeed acquire a new oval. This looks a priori hard, but in view of the diagrammatic of the table Fig.\[Gudkov-Table-quintic:fig\] (mostly prompted by Bézout) we could just make a deformation along a pencil spanned by the curve plus a curve with more ovals (e.g. Harnack’s or just $5$ of Fig.b). The difficulty however is that the deformation is not forced to raise immediately the number of ovals, as it may first lower down the number of ovals. Incidentally if this argument via Klein-Marin would have worked it would also have prohibited the type I realization of the scheme 4.

Next we have the scheme $1$ forced to be of type I, by Klein’s congruence, and easily constructed (e.g. by a slight alteration of the picture Fig.\[Gudkov-Table-quintic:fig\]f above).

Finally, the scheme $0$ poses again a little problem, but can also be shown to be of type II. This follows either from Rohlin-Mishachev, or via theta-characteristics. In this case Gross-Harris 1981 [@Gross-Harris_1981 p.175] were able to conclude type II via theta-characteristics (see their proof of Prop.7.1, p.173, which contains some minor misprints, namely “Prop.4.1” should be “Prop.5.1”, and “$h^0({\bf
a})=(d^2-1)/2$” should be “$h^0({\bf a})=(d^2-1)/8$”).

A somewhat more conceptual argument can be based on Klein’s Thesis (as Viro calls it) of 1876 to the effect that [*a curve of type I cannot gain an oval (at least when crossing a solitary node)*]{}. This was perhaps historically the first known proof, albeit Klein did not mentioned this consequence explicitly in print (1876 paper, nor later). On writing down the proof below, we realized that one needs the stronger version due to Marin 1988 of Klein’s Thesis relaxing the parenthetical proviso above. Hence our claim of historical priority is somewhat sloppy, but in substance Klein could have anticipated it.

\[Klein-Marin-quintic:lem\] [($\approx$Klein 1876, 1892, Rohlin-Mishashev ca. 1974–76, Gross-Harris 1981, Marin 1988, Gabard 2013 trying to assembly all this today)]{}.—Any quintic with only one pseudo-line is necessarily of type II (i.e. nondividing or diasymmetric).

Take such a curve $C_5$ (with only a pseudoline) and any auxiliary (smooth) curve with at least one oval (and so $r\ge 2$). Pass a line through both curves (in the hyperspace of curves) and perturb it slightly to ensure transversality w.r.t. the discriminant. Since the initial curve $C_5$ has the least possible number of real circuit (namely one), the first contact (along one of the 2 possible pathes inside the pencil) with the discriminant will be a “Morse” surgery (jargon Thom-Milnor) [*forced to increase the number of ovals*]{} ($\bigstar$).

This last (italicized) assertion ($\bigstar$) requires perhaps more substantiation. Let us admit it to conclude quickly. If the new oval raises from a solitary-node then Klein’s Thesis of 1876 (alas left unproven by the great geometer) suffices to conclude. If not, e.g. if the pseudoline self-collides with itself as to split off a new oval (Fig.\[Eversionpseudo:fig\]a), then Marin’s version of Klein completes the proof, cf. (\[Klein-Marin:lem\]) or Marin 1988 [@Marin_1988].

To justify better ($\bigstar$) we should check that all Morse surgeries of a pseudoline forces an augmentation of the number of circuits. In the case of an oval this is not true due to “eversions” (cf. Sec.\[Eversion:sec\] especially Fig.\[Eversion:fig\]), whence our extreme prudence. However doing naive experimental pictures deforming a pseudoline, it seems impossible to evert a pseudoline (Fig.\[Eversionpseudo:fig\]b). It remains of course to find a theoretical explanation.

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It is clear that the above lemma extends to all other odd degrees:

\[Klein-Marin-odd-degree:lem\] Any curve with a unique real circuit is of type II, safe if it is a line or a conic (degree $m=1,2$).

[*Remark*]{}.—The argument of (Klein-)Gross-Harris only works under the (extraneous and stringent) assumption $m \equiv 5 \pmod
8$ (cf. their Prop.7.1, p.173).

The case of odd degrees follows by the same method using the Klein-Marin theorem. The case $m=3$ is of course more elementary and can be reduced to the uniformization of elliptic curves e.g. à la Weierstrass via the doubly-periodic $\wp$-function defined on a rhombic lattice. Sorry, it suffices actually to use Klein’s congruence, or to remember—if you do not want to sell your soul to the devil of arithmetics—that a symmetric torus with one fixed circuit is forced to be $S^1\times S^1$ acted upon by exchange of both factors (while fixing the diagonal circle).

Actually Klein’s congruence $r\equiv_2 g+1$ settles the lemma whenever $m=3+4n$. Indeed then $g=\frac{(m-1)(m-2)}{2}=\frac{(2+4n)(1+4n)}{2}=(1+2n)(1+4n)=1+6n+8n^2$, which is odd, and so Klein’s congruence (forced by type I) is corrupted, whence type II.

For the other cases $m=1+4n$, Klein’s congruence tells nothing and one make appeal to the Klein-Marin argument instead.

For even degrees, one can again treat half of the cases via Klein’s congruence, namely when $m=4, 8, 12, \dots$, i.e. $m=4n$ as then $g=\frac{(4n-1)(4n-2)}{2}=(4n-1)(2n-1)$ which is odd, and so Klein congruence is violated for $r=1$. For the other cases $m=2+4n$, the congruence tells nothing. However we can still conclude type II (of course provided $m\ge 4$), either via Rohlin’s formula (\[Rohlin-formula:thm\]) or maybe a variant of the Klein-Marin argument. However now the configuration with one circuit has not the minimal number of circuits and so we may first descend to the empty chamber and the Klein-Marin method looks impuissant. Of course perhaps some extra trick can ensure that we can increase the number of component immediately yet I do not see any obvious argument.

The impressive landscape of all sextics (Harnack 1876, Hilbert 1891/00/09, Rohn 1911/13, Petrowskii 1933/38, Gudkov 1948/54/69, Arnold 1971, Rohlin 1972/74/78)
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\[31.12.12\] Perhaps the Klein-Rohlin conjecture follows from Ahlfors theorem interpreted in terms of total reality. Intuitively having a total pencil, no real circuit can be added without corrupting Bézout (more on this in Sec.\[Rohlin-via-Ahlfors\]). Yet perhaps this is too naive as shown by an example of order 6 to be found in Gabard’s Thesis 2004 [@Gabard_2004 p.8] (I should acknowledge Kalla-Klein 2012 [@Kalla-Klein_2012-Computation-cite-Gabard] for reminding me that my Thesis contained this example).

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This shows that the real scheme alone fails to determine the dividing character (alias type I=erster Art in Klein 1876 [@Klein_1876]). At first I thought this corrupts Rohlin’s assertion that his conjecture is true in degree 6. Of course Rohlin 1978 [@Rohlin_1978] knew very much this phenomenon, which he calls “real schemes of indefinite type” (on p.94 of ), i.e. real schemes admitting representatives of both types (dividing or not). Hence it was first puzzling to wonder why he made such a basic mistake, or more likely why we first failed to interpret correctly his simple message?

Understanding the Klein-Rohlin conjecture requires some more mature thinking. One should list all schemes dominating the “nine unnested ovals” scheme. On p.95 Rohlin 1978 [@Rohlin_1978] refers to the census (tabulation) set up by Gudkov 1974 [@Gudkov_1974/74 p.40] listing all the logically possible (real schemes of) sextic curves (taking into account Bézout for lines). This is worth reproducing as Fig.\[Gudkov-Table3:fig\]. Recall that in Gudkov’s symbolism, $\frac{x}{1} y$ denotes the scheme consisting of $x$ ovals enclosed by one “big” oval, while there is $y$ ovals living outside. This gives a total of $1+2+3+\dots+11=\frac{12 \cdot
11}{2}=6\cdot 11=66$ logically possible curves (counting inside the “triangle”), to which must be added the empty real scheme (denoted $0$) and the deep nest of depth $3$ (denoted $(1,1,1)$ or $\frac{1}{{\frac{1}{1}}}$). We get so the 68 schemes ([*année érotique*]{}) mentioned by Gudkov (p.40).

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Not all of those 68 schemes are actually realized. If they would this would roughly mean that all obstructions are Bézout-like prompted by tracing a single line. However the plane is swept out by a myriad of other curves. Quite eclectically, the architecture of Hilbert-Gudkov’s table of elements is a bit like a pharaohs pyramid (turn Fig.\[Gudkov-Table3:fig\] upsidedown) and those are known to have sanctuary galleries forming tunnels. The first to have spotted this porousness of the pyramid is no less an authority than Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege], albeit it took several decades until his work got consolidated (especially by Rohn 1911, Petrovskii 1933/38) and pushed forward to its ultimate perfection (thanks to the efforts of Gudkov). Soon afterwards, Arnold and Rohlin offered quite dramatic simplifications based on pure topology, and extensions of the prohibitions to all degrees. We suspect this Hilbert-Gudkov pyramid to have some overlap—not yet much elucidated except perhaps for allusions in Rohlin 1978 [@Rohlin_1978 p.94]—with some more ancient force, namely Abel-Riemann-Klein-Teichmüller-Ahlfors and their circle maps. The latter are of course specific to dividing curves concomitant with the paradigm of total reality. In the case at hand (real plane smooth sextics), total reality is exhibited according to a Rohlin’s claim 1978 (not yet fully understood by the writer) via total pencil of cubics for certain schemes, e.g. $\frac{6}{1}2$ and its mirror $\frac{2}{1}6$. More trivial is the nest of depth $3$, totally real under a pencil of lines. This can be interpreted as a prohibition of diasymmetry for those schemes. Likewise the porous portion of Gudkov’s pyramid (=white cases above the broken line on Fig.\[Gudkov-Table3:fig\]) can be prohibited (and this is how I understand vaguely the Hilbert-Rohn method) by pure synthetic geometry. Paraphrasing, not merely linear Bézout obstructions do exist, but also those via the menagerie of all other curves grooving nonlinearly the plane. More than that, not just static curves but dynamical collections of such (e.g., pencils) have to be considered. It is charming to note a strong parallel between Hilbert’s and Rohlin’s claims that pure geometry is able to prohibit schemes, especially as both look insufficiently justified, but intuitively plausible. How much Rohlin’s synthetic proof of the type I of the schemes $\frac{6}{1}2$ and its mirror has in common with Hilbert-Rohn’s method?

A last word of caution for pyramids builders: one of the first ever constructed in Ancient Egypt had a somewhat pathetic destiny. Once arriving near the $2/3$ of the planned final size, fissures started to appear menacing the whole foundations to crack under pressure. The only reasonable option left to the engineers was to diminish the slope for the last third as to lower pressure. It is not known if this sufficed to ensure immortality of the Pharaoh. Thus, it should be no surprise that the most telluric part of the pyramid (near the funerary chamber where the pressure is highest) is the most secrete part of the edifice. This needed to wait the contribution of Gudkov 1969 [@Gudkov-Utkin_1969/78] who exhibited the most elusive schemes $\frac{5}{1}5$ of Fig.\[Gudkov-Table3:fig\] (cf. also Fig.\[GudkovCampo-5-15:fig\] for the explicit construction). At this stage Hilbert’s 16th problem was completely solved (at least for sextics which is arguably the official context of Hilbert’s question).

Now let us be more formal. As explained by Gudkov (1974 ), Kahn 1909 [@Kahn_1909] and Löbenstein 1910 [@Löbenstein_1910] published dissertations under Hilbert’s direction—(if I understood well the story, vgl. Hilbert 1909 [@Hilbert_1909-Ueber-die-Gestalt-sextic], both were feminine candidates)—attempting to prohibit sextics with 11 unnested ovals. (Challenge: try to prove this via Ahlfors 1950 or rather via Bieberbach-Grunsky (1925/1937). Philosophically, this would just, as it should, put Little Hilbert in the baskets of Big Riemann!) This follows also from Rohlin’s formula of 1974–78, cf. Sec.\[Rohlin-formula:sec\], or from Arnold’s congruence of 1971. Soon later Rohn 1911–1913 [@Rohn_1913] devoted two articles attempting by the same method to exclude sextics of type $\frac{10}{1}$ or $11$, making a big contribution to the development of Hilbert’s idea. The resulting prohibition method was christened by Gudkov (1974 ) the [*Hilbert-Rohn method*]{}. In Gudkov’s view, even Rohn’s proof is not perfectly sound due to some messy combinatorics impeding Rohn to take care of all logically possible cases. Gudkov then mentions several more Western attempts, by Wright 1907 (same idea as Hilbert, but not rigorous prohibition of type $11$). In Donald 1927, the same non-rigorous attempt is repeated (apparently without knowledge of Kahn, Löbenstein or Rohn’s work). Hilton 1936 devoted a paper criticizing Donald’s work.

The next step is essentially Gudkov’s work (yet do not miss what did Petrovskii 1933/38 though its impact upon the case of sextics is nearly covered by Hilbert-Rohn). Ultimately Gudkov was able to prohibit in 1969 and probably much earlier (Gudkov-Utkin 1969 [@Gudkov-Utkin_1969/78]) all schemes above the broken line of Fig.\[Gudkov-Table3:fig\]. This breakthrough originated in 1948 when Andronov suggested (to Gudkov) applying the concept of [*roughness*]{} (also known later as [*structural stability*]{} in the West since Lefschetz, and adhered to by Thom, etc.) to the topology of real algebraic surfaces. Petrovskii’s advice (1950) suggested focusing rather on the case of sextic curves. Combining those novel Russian ideas with the Hilbert-Rohn method, enabled Gudkov in 1954 [@Gudkov_1954] to get solid prohibitive proofs above the critical line, and even beyond (sic! cf. lilac schemes $\frac{5}{1}y$, $3\le y \le 5$ on Fig.\[Gudkov-Table3:fig\]) but that turned out to be too massive amputation for the pyramid to support its own structural mass. Nowadays there are simpler proofs from the Arnold-Rohlin era (early 1970’s) or via Rohlin’s complex orientation formula 1974–78 which prohibit only a portion of those (namely those [*not*]{} lying on the continuation of the lattice by blue rhombs and red circles on Fig.\[Gudkov-Table3:fig\]). However Rohlin’s proof (1972/72 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov]) of the Gudkov hypothesis inhibits all $M$-schemes above the broken line, but the price to pay is highbrow differential topology à la Rohlin from the early 1950’s (cf. Sec.\[Gudkov-hypothesis:sec\]). Related work by Gudkov-Krakhnov/Kharlamov prohibits all the four $(M-1)$-schemes above the broken line.

What happens under the critical line? Short-cutting a century of efforts, the answer is rapid: all of them are realized. In fact all specimens (except the 3 lilac-colored ones) are easily construct by (slight variants) of Harnack’s and Hilbert’s method. At least this is what we read in Gudkov’s survey 1974 [@Gudkov_1974/74], yet the cases of $\frac{4}{1}5$, $\frac{3}{1}5$ are a bit tricky (but see our Fig.\[HarnaGudkov4-15:fig\] and Fig.\[HarnaGudkov3-15XXL:fig\] resp.). The three remaining ones $\frac{5}{1}5$, $\frac{5}{1}4$, $\frac{5}{1}3$ needed to wait until Gudkov’s trick (1969–1973) of using some Cremona transformations. Beware yet that historically, the very first argument of Gudkov’s Thesis 1969 was a pure existence proof along the line of Hilbert-Rohn’s method (ca. 20 pages long and extremely hard-to-follow according to Russian experts, cf. Polotovskii 1996 [@Polotovskii_1996-D-A-Gudkov], Viro, etc.), and was not constructive at all. Remind also that Gudkov himself at some early stage, in 1954, asserted incorrectly inexistence of those 3 difficult birds, quite in line with Hilbert’s intuition at the Paris Congress of 1900. As like to emphasize Arnold, it seems that Petrovskii himself was at first very skeptical about the twist taken by Gudkov’s solution.

Harnack’s and Hilbert’s constructions
-------------------------------------

First Fig.\[Harnack-original:fig\](left) recalls Harnack’s method of construction (trying to keep reasonably close to the original 1876 [@Harnack_1876 p.195], but making more explicit pictures (assisted by Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical p.188]). The top-right of Fig.\[Harnack-original:fig\] reproduces Hilbert’s more expeditious way to realize this scheme. The bottom-right is the new scheme discovered by Hilbert in 1891 [@Hilbert_1891_U-die-rellen-Zuege] (yet no pictures until Hilbert 1909 [@Hilbert_1909-Ueber-die-Gestalt-sextic], who traces only the top-right picture, whose scheme is Harnack’s). If The bottom nice picture is borrowed from A’Campo 1979 [@A'Campo_1979 p.08–09].

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Once all the knowledge synthesized in Gudkov’s table (Fig.\[Gudkov-Table3:fig\]) is understood (or admitted for short) one starts appreciating Rohlin’s maximality claim. Indeed having listed all real schemes (there remains $68-12=56$ many below the broken line) it is an easy matter to spot maximal elements in the lattice. We find 6 types red-circled on Fig.\[Gudkov-Table3:fig\] corresponding to

$\bullet$ the three $M$-schemes $\frac{9}{1}1$, $\frac{5}{1}5$, $\frac{1}{1}9$ of Hilbert, Gudkov, Harnack respectively,

$\bullet$ plus two $(M-2)$-schemes namely $\frac{6}{1}2$ and $\frac{2}{1}6$,

$\bullet$ and finally one $3$-schemes $(1,1,1)$ corresponding to the deep nest.

Rohlin’s assertion is that those (distinguished) 6 schemes are precisely those which are definite of type I (i.e. universally orthosymmetric). Of course the assertion is trivial for the 3 possible $M$-schemes (since Harnack 1876 or via Klein’s 1876 intrinsic proof of Harnack’s inequality via the topology of Riemann surfaces). The deep nest of weight 3 is likewise trivially of type I, for it is enough to sweep it out by a total pencil of lines. It remains thus to analyze the two $(M-2)$-schemes with $r=9$, i.e. $\frac{6}{1}2$ and $\frac{2}{1}6$ by showing that they are definite of type I. At first one can imagine to prove this via pencil of conics (or maybe cubics pencils?). A complete argument must be given in Rohlin. [*Insertion.*]{}—\[28.03.13\] In fact Rohlin claimed a proof which is now lost via pencils of cubics, so that there is strictly speaking presently still only one known proof which involves a congruence modulo 8 due to Rohlin-Kharlamov-Marin (\[RKM-congruence-reformulated:thm\]). Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics] was able to validate the total reality assertion of Rohlin, yet only after supposing the curve dividing. It should however not be impossible that methods of Le Touzé suitably modified establish the full Rohlin claim. This seems to be an urgent problem to deal with. We personally tried a lot but failed dramatically. This sort of problem seems to require extreme cleverness.

Rohlin 1978 [@Rohlin_1978] states the following theorem summarizing all those efforts (up to Gudkov plus his own input reconciliating with Klein’s viewpoints):

\[Rohlin-type:thm\] [(Rohlin 1978)]{} The $56$ possible real schemes for sextics (Harnack, Hilbert, Rohn, Gudkov) split as follows according to Klein’s types:

$\bullet$ There are $6$ schemes of type I (red-circles on Gudkov’s table=Fig.\[Gudkov-Table3:fig\]);

$\bullet$ There are $42$ schemes of type II (green-squares on Gudkov’s table=Fig.\[Gudkov-Table3:fig\]);

$\bullet$ There are $8$ schemes of indefinite(=mixed) type (blue-rhombs on Gudkov’s table=Fig.\[Gudkov-Table3:fig\]).

Note that the resulting distribution of types to be nearly symmetric (on Fig.\[Gudkov-Table3:fig\]), modulo some anomaly at the place $5$ (five unnested ovals). A similar remark is made in Fiedler 1981 [@Fiedler_1981 p.13]: “[*Bemerkung. Eventuell ist die Tabelle nicht vollständig. Aber es ist schon ersichtlich, da[ß]{} die Tabelle der zerteilenden Kurven im Unterschied zur Tabelle aller existierender Typen von singularitätfreien Kurven sechster Ordnung (vgl. [\[1\](=Gudkov 1974 [@Gudkov_1974/74])]{}) nicht symmetrisch ist.*]{}”. Indeed the asymmetry we (and Fiedler) notice (but of course implicit in Rohlin’s survey) is the (unique) symmetry breaking occurring between the schemes $\frac{4}{1}$ and the scheme ${5}$. The latter turns out to be of type II, as it cannot satisfy Rohlin’s formula (\[Rohlin-formula:thm\]), whereas the former scheme is easily seen to be indefinite (cf. Fig.\[R4-1:fig\]). Note that Arnold’s congruence $5-0=p-n=k^2
\pmod 4=3^2=9=5 \pmod 4$ is not fine enough to detect this break of symmetry.

Let us try to understand this spectacular statement of Rohlin (\[Rohlin-type:thm\]).

\(1) From the easy Klein congruence $r\equiv g+1 \pmod 2$ if type I, we draw that all schemes with an even number $r$ of ovals belong to type II (this explains all the green-squares at heights $r=0,2,4,6,8,10$, cf. Fig.\[Gudkov-Table3:fig\]).

\(2) As already explained all $M$-schemes (here $r=11$) are trivially of type I (Harnack’s inequality or Klein’s argument of 1876 [@Klein_1876]). For another reason the scheme $(1,1,1)$ (deep nest of profundity 3) is easily shown to be of type I (total pencil of lines).

\(3) For similar reasons (but deeper) is the assertion that the 2 circled schemes with $r=9$ belongs to type I. This is truly the work of Rohlin, albeit philosophically akin to Klein-Teichmüller-Ahlfors’ total reality. [*Insertion*]{}—\[28.03.13\] As just said this is still unproven synthetically, and the only proof available involves deep differential topology (Rohlin-Kharlamov-Marin).

\(4) Appurtenance to the indefinite type is usually easy requiring merely exhibiting two curves, one in each type. So for instance the scheme $9$ (consisting of $9$ outer ovals without nesting) is indefinite (cf. our Fig.\[KleinRo-sextic:fig\]). Of course here the basic theoretic tool is Fiedler’s observation that the type is governed by the smoothing effected in the Plücker-Klein-Brusotti method of small perturbation. Full details are worked out in Sec.\[indefinite-types:sec\].

\(5) Another piece of information (now purely Rohlinian) is Rohlin’s inequality $r\ge m/2$ for a smooth plane curve of degree $m$. (Remind this to follow for Rohlin’s formula, in turn derived by a intersection theory argument of halves of the dividing curve capped off by real ovals and brought into general position by perturbing via a vector field normal to the real locus). Conceptually this involves Poincaré homology theory, and the allied intersection theory (e.g. by Lefschetz, etc.). From Rohlin’s inequality, one deduces that the scheme with $r=1$ is of type II.

\(6) Using the stronger Rohlin formula, one must be able to treat all schemes with $r=3$ to belong to type II (except the deep nest) and likewise assess type II for all other schemes. [*Insertion*]{} \[28.03.13\].—Yes this is essentially true. More precisely Rohlin’s formula admits the Arnold congruence as corollary (cf. \[Rohlin-implies-Arnold:lem\]), and the latter $\chi\equiv_4 k^2=9\equiv 1$, forces a curve of type I to live on the grid formed by blue rhombs (and red-circles) of Fig.\[Gudkov-Table3:fig\]. So Rohlin’s assertion is evident safe for the scheme $5$, $\frac{1}{1}1$ and $1$. But all those cases are prohibited by Rohlin’s formula $2(\pi-\eta)=r-k^2$. Indeed in case of no-nesting Rohlin’s formula reduces to $r=k^2=9$, hence rules out the schemes $1$ and $5$. For $\frac{1}{1}1$, we have only one pair so $\pi+\eta=1$, while Rohlin’s formula says $\pi-\eta=-3$, whence $2\pi=-2$, which is impossible as $\pi$ is a cardinal (namely the number of positive pairs).

At this stage the proof of Rohlin’s theorem (\[Rohlin-type:thm\]) is complete.

What can be proved via Ahlfors?
-------------------------------

\[17.01.13\] From the viewpoint of our survey, it is of some interest to decide which results of the theory aroused from Hilbert’s 16th problem (Hilbert-Rohn, etc. and the Russian school Gudkov-Arnold-Rohlin, just to quote the 3 supermassive black holes) can be (re)proved via the Ahlfors map.

As we said already all this section was actually motivated by the guess that Ahlfors could be used to prove the still unsettled Rohlin maximality conjecture (at least what remains thereof post Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]). However as yet we failed to complete this grandiose project.

Another more didactic aspect (yet perhaps not to be neglected as a first step toward subsequent progresses) would be to see if Ahlfors implies the (Gudkov-)Arnold congruence mod 4: $\chi=p-n=k^2 \pmod 4$ for dividing curves of degree $2k$.

[*Insertion*]{} \[02.04.13\].—This game looks quite artificial since Arnold’s congruence is, e.g., a fairly trivial consequence of Rohlin’s formula, cf. (\[Rohlin-implies-Arnold:lem\]).

The method would be to examine the Ahlfors foliation, i.e. that induced by the total pencil of curves while trying to apply Poincaré(-Bendixson-Kneser-Hamburger) index formula for foliations. Recall that $\chi$ is the Euler characteristic of the “Ragsdale-Petrovskii” (orientable) membrane of ${\Bbb R}P^2$ bounding the ovals. Of course it looks hard for Ahlfors to beat the elegance of Arnold’s argument based on intersection theory and the divisibility by 8 of the signature of spin manifold (as prompted by the algebra of integral quadratic symmetric form). However, as just mentioned, there is an even simpler proof of Arnold based on Rohlin’s formula.

Of course there is a myriad of sub-Arnoldian truths that could be treated via the Ahlfors foliation, e.g. Hilbert-Rohn prohibition of an $M$-sextic without nesting, or the type II of a quartic with two unnested ovals (all these assertions being implied by Arnold’s congruence).

Another game out of reach to Arnold, but proved via Rohlin’s formula is the prohibition of the sextic scheme $5_I$ of five unnested ovals in the type I case. This could perhaps also be proved via the Ahlfors map.

The only point which we managed (presently) to prove via Ahlfors is the (easy sense) of Klein’s Ansatz that a dividing curve cannot gain an oval while crossing the discriminant through a solitary node (with a complex conjugate pair of tangents). Compare for this Lemma \[Klein-via-Ahlfors(Viro-Gabard):lem\] suggested by a letter of Viro. However, we always use as a premiss the issue that for a plane curve the abstract total map of Ahlfors extends to the ambient projective plane. We should acknowledge a letter of Marin (cf. Sec.\[e-mail-Viro:sec\]) for having made us aware of this subconscious short cut. We still hope this to be true via basic algebraic geometry, of which we forgot all the foundations.

Klein’s Ansatz (1876) can also be proved without Ahlfors by using some Picard-Lefschetz and Dehn stuff, or rather just some “Anschauung” that might have been folklore as early as 1876.

Here is an argument (cf. also the next Sec.\[Klein-Marin:sec\]). At the level of the complexification, one can only explain the apparition of a solitary node as the strangulation of some vanishing cycle $\beta$ on the Riemann surface. Then we analyze all possibilities. By the reality of our deformation, the cycle $\beta$ must be invariant under conjugation $\sigma$, hence either be a real circuit (or “oval”[^21]) (pointwise invariant under $\sigma$), or an ortho-cycle (2 fixed points under $\sigma$) or a dia-cycle (no fixed point under $\sigma$). This is exhaustive via the classification of involutions on the circle, which via the quotient map and covering theory, reduces to the classification of one-dimensional manifolds (Hausdorff and metric).

As we already noted a dia-cycle cannot exist in the orthosymmetric case (Lemma \[antioval:lem\]). For an “oval” it can indeed shrink to a point (hence a solitary node) but then disappear of course (cf. Fig.\[Klein-Marin:fig\], right). If we have an ortho-cycle then two cases are to be distinguished. It can either cross two distinct “ovals”, in which case both ovals merges together after the Dehn twist (cf. Fig.\[Klein-Marin:fig\], left). The last possibility is an ortho-cycle cutting only one “oval”. In this case Fig.\[Orthoovals2:fig\] shows that $r$ stays constant, and of course we do not cross a solitary node in that case, but rather a non-isolated one with 2 real tangents.

This “proves” Klein’s Ansatz (modulo some Picard-Lefschetz theory), and even Marin’s stronger assertion that a dividing curve cannot increase its number component when crossing the discriminant. Indeed in all 3 cases analyzed, either $r$ drops by one unity (first two cases), or stays constant. Remark however that Marin 1988 [@Marin_1988] has a more conceptual proof.

What is Picard-Lefschetz theory in our context? Since any crossing of the discriminant can be interpreted as a smooth arc traversing the discriminant transversally, we may (in the small) always replace this little arc by a linear pencil, and are reduced to classical Picard-Lefschetz theory, where in our case we have holomorphic fibration of the plane by a pencil of curves. The theory in question tell us the geometric monodromy when winding around a singular member of the pencil, but also gives the Dehn twist description of what happens when we (more cavalier) cross frontally the singularity.

Recall that Picard’s thesis (the first work of Picard on another subject) is dated 1879 [@Picard_1879], while Klein’s Ansatz (no proof but probably Klein had one) is dated 1876 (3 years younger). So clearly our approach is somewhat historically contorted. Still, it is not impossible that Klein (and many others) were aware of the geometry behind our argument (via Dehn twists, ca. 1910). It is also possible that Klein’s argument was closer to Marin’s, albeit the latter result is perhaps slightly different (and of course stronger).

Rohlin’s conjecture almost implied by Klein-Marin (Klein 1876, Marin 1988, Viro 1986) {#Klein-Marin:sec}
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\[04.01.13\] This section presents another tactic (pseudo-proof) of Rohlin’s conjecture that probably everybody had in mind (especially Rohlin and Marin), yet nobody write it down as it fails blatantly. Although being a “pot-pourri” it is worth presenting as it helps clarifying the relation (or absence thereof) between Klein’s original assertion 1876 [@Klein_1876] as interpreted by modern workers, notably Marin 1988 [@Marin_1988] and Viro 1986/86 [@Viro_1986/86-Progress] (the latter being based on a “private communication” of the former).

First there is a remarkable observation of Klein 1876 that Marin 1988 [@Marin_1988] was probably the first to supply with a proof. To be perfectly accurate we believe that Marin’s result is slightly stronger than Klein’s original asserting only that a dividing curve cannot acquire a new oval like a champagne bubble emanating from a solitary node (compare Klein’s Quote \[Klein\_1876-niemals-isolierte:quote\] especially the phraseology “isolierte reelle Doppeltangente”). In Marin 1988 article, full credit is ascribed to Klein, either by over-modesty or because Marin overlooked to notice the little nuance between his and Klein’s weaker assertion. (Compare the recent e-mail exchanges in Sec.\[e-mail-Viro:sec\].)

\[Klein-Marin:lem\] [($\approx$ Klein 1876, but in the formulation of Marin 1988)]{}.—A (plane) dividing curve cannot increase its number of ovals when crossing a node (non-degenerate double point).

For Viro (1986 ) the curve does not actually need to be plane.

Perhaps Klein gained evidence from the case of quartics. Imagine a Gürtelkurve (quartic with 2 nested ovals), then there cannot be created a new oval without violating Bézout. Hence either both ovals amalgamate or the inner oval evanishes. In both cases the number of ovals decreases (by one unit). \[02.04.13\] This is not an exhaustive discussion, for there can be also an eversion (Sec.\[Eversion:sec\]), keeping $r=2$ constant.

Bringing into the picture the Riemann surface (of orthosymmetric type) underlying the dividing curve, then, as the latter traverses the discriminant (at some smooth point of it) our curve becomes uninodal via some vanishing cycle pinching the Riemann surface.

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This vanishing cycle can actually be an oval of the Riemann surface: once shrunk to a point it disappears and one oval gets lost (cf. Fig.\[Klein-Marin:fig\], right). This is not the only possibility as shown by the Gürtelkurve whose 2 ovals may coalesce. It is just a little harder to visualize the corresponding surgery on the Riemann surface. The key is to imagine an anti-invariant vanishing cycle $\beta$ whose contraction is depicted on Fig.\[Klein-Marin:fig\], left. The two ovals traversed by the cycle $\beta$ have merged together.

[*Insertion*]{} \[02.04.13\].—Further there is a 3rd possibility, of when the ortho-cycle $\beta$ intersects only one oval (cf. Fig.\[Orthoovals2:fig\]). Then the corresponding Morse surgery is an eversion keeping $r$ constant but destroying the dividing character of the curve.

(Marin’s proof in the dirty fingers of Gabaredian[^22]) Marin’s proof is somewhat different and in substance as follows (please refer to the French original for faithfulness). The initial curve is orthosymmetric. Such curves satisfy the Klein congruence $r\equiv g+1 \pmod 2$. On the other hand when traversing the discriminant the curve is uninodal and the real part undergoes a “Morse surgery” which alter the number of ovals by one unit.

[*Insertion*]{} \[02.04.13\].—Warning. Possibly $r$ can stay constant in case of an eversion, cf. Fig.\[Orthoovals2:fig\], but then the post-critical Riemann surface becomes diasymmetric.

At any rate, the new curve (past the discriminant) is necessarily diasymmetric, either by Klein’s congruence when $r$ moves by one, or by the Dehn-twist argument of Fig.\[Orthoovals2:fig\] when $r$ is kept constant. Marin concludes by arguing that a path between two conjugate points avoiding the real locus subsists in all nearby curve. (Alas I confess to have not properly understood this argument which is presumably much superior to the above via vanishing cycles.) Compare also Marin’s e-mail, where he explained us more details. (NB: Marin’s argument is also repeated in Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.785, 4.6.8], and was apparently always easily digested by Russian scholars, Viro included.)

Armed with the lemma, let us try to attack Rohlin’s conjecture.

(Pseudo-proof of Rohlin via Klein-Marin) Suppose $S_1$ to be a scheme of type I which is not maximal, say embeddable in $S_2$. Take algebraic models $C_i$ of each $S_i$ ($i=1,2$). By general position we may assume that the line $L$ through $C_1$ and $C_2$, in the hyperspace of all curves of degree $m$ (à la Cayley, etc.), crosses transversally the discriminant hypersurfaces (in smooth points of it) at uninodal curves. This unique node is necessarily real (when we look at real members of the pencil $L$). Hence when we join $C_1$ to $C_2$ we get real curves (finitely many of them being singular). Whenever we cross the discriminant the real locus undergoes a “Morse surgery” which is (up to reversing time) is either

$\bullet$ the death of an oval (shrinking to a point)

$\bullet$ the fusion of two unnested ovals

$\bullet$ the fusion of two nested ovals.

Each operation effects a fluctuation of $\pm 1$ on the number $r$ of ovals. (Warning \[02.04.13\].—This is not even true due to eversions!) So we can imagine a staircase starting from $C_1$ to $C_2$ recording the history of the varied fluctuations of $r$ during the transition from $C_1$ to $C_2$ along the pencil $\lambda C_1 + \mu C_2=0$ (cf. Fig.\[Klein-Marin:fig\], center-bottom). By Klein-Marin the first staircase is moving downwards, and as we ultimately reach $C_2$ having more ovals, we are naively inclined to claim that we shall revisit the same scheme $S_1$ at some step after which $r$ only increases. This would be true if a scheme would be completely encoded by its number $r$ of circuits. In this naive world, we get an intermediate curve $C_1'$ also representing the scheme $S_1$ (hence dividing since $S_1$ is of type I) and after which $r$ only increases. This would violate the Klein-Marin theorem.

The moral is that Klein-Marin seems to imply, but fails implying, the Rohlin maximality conjecture (for an explicit objection see the little diagrammatic of ovals on Fig.\[Klein-Marin:fig\], bottom). Nonetheless, the Klein-Marin lemma certainly implies the:

The chambers past the discriminant corresponding to orthosymmetric curves are local maxima of the function $r$ counting the number of real circuits. Further all chambers adjacent to an orthosymmetric chamber are diasymmetric.

The last assertion follows directly from Klein’s congruence $r\equiv g+1 \pmod 2$. [*Insertion*]{} \[02.04.13\] This is true when $r$ varies (by one unit), but if it stays constant one has to invoke the Dehn-twist argument of Fig.\[Orthoovals2:fig\].

Hence orthosymmetric chambers are never contiguous (along a wall of codimension 1), but a priori they could still have closures with non-void intersections.

Back to degree 6: Rigid isotopy (Nikulin 1979 via K3’s, Torelli of Pyatetsky-Shapiro-Shafarevich 1971)
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\[05.01.13\] Even if Rohlin’s conjecture (type I $\Rightarrow$ maximal) looks out of reach, it might be easier in degree 6 (we mean by a theoretical argument independent of Rohlin’s census). In that case granting orthosymmetry of the schemes $\frac{6}{1}2$ and $\frac{2}{1}6$ one recovers all of Gudkov’s obstructions (prohibition of the semi-hexagons above those schemes, cf. Fig.\[Gudkov-Table3:fig\], safe those that were established by Hilbert and Rohn, namely the schemes $11$ and $\frac{10}{1}$). Of course this is nothing new, yet methodologically distinct from the topological arguments à la Arnold-Rohlin explaining the Gudkov hypothesis. So we are asking for a fighting interplay between pure geometry and topology.

Also in view of the Morse surgery inherent in the Klein-Marin theorem, one can ask several questions about the contiguity of chambers in the space of all sextics and correlate this with the diagrammatic of Gudkov’s table (Fig.\[Gudkov-Table3:fig\]). The first basic point is that when we cross a wall $r$ fluctuates by $\pm 1$. Hence we do not have complete freedom to random-walk on the triangular lattice underlying Gudkov’s table (all horizontal edge cannot be used).

[*Insertion*]{} \[02.04.13\].—This is a naive misconception, since in fact there is also eversion (cf. Sec.\[Eversion:sec\]) keeping $r$ constant!

Define the [*distance*]{} between two chambers as the minimum number of walls needed to be crossed to join them (by a path transverse to the discriminant). Another “distance” is defined by restricting to pathes along (linear) pencils of curves.

A great miracle (specific to order 6) is the following result due to joint efforts of Kharlamov and Nikulin 1979/80 [@Nikulin_1979/80]:

[(Nikulin 1979)]{}\[Nikulin:thm\] The real scheme enhanced by the (Klein-Rohlin) type affords a complete invariant of the rigid-isotopy class of sextics. Thus via Rohlin’s classification (Theorem \[Rohlin-type:thm\]) there is $6+42+2\cdot 8=56+8=64=2^{8}$ “typed” schemes and so many rigid-isotopy classes. (This number being a power of 2 is perhaps just good fortune? probably because for quartics the number of chambers is $6$.)

This is yet another “tour de force”. It uses (but strangely does note cite!) the topological classification of Rohlin 1978 [@Rohlin_1978] (making already a fusion between Klein 1876 and Hilbert 1891/1900’s 16th problem as solved by Gudkov 1969 [@Gudkov-Utkin_1969/78]). But that is not all! It also combines this with the complex geometry of K3 surfaces (Kummer-Kähler-Kodaira as coined by Weil), especially the contribution of Pyatetsky-Shapiro–Shafarevich 1971/71 [@Pyatetsky-Shapiro-Shafarevich_1971/71] on the global Torelli theorem, as well as the surjectivity of the period mapping (Kulikov 1977 [@Kulikov_1977]). Rohlin, and especially Kharlamov’s rôle in this proof seems to have been quite pivotal (and acknowledged as a such).

The Gudkov-Rohlin-Nikulin pyramid and the contiguity graph
----------------------------------------------------------

[*Warning*]{} \[02.04.13\].—This section is a miscellany of mistakes about the combinatorial structure of the hyperspace of all sextics. We kept our text intact in its original shape (modulo Insertions and corrections) since we think that it is more important to avoid the basic mistake than to reach the ultimate verity of what is quite likely to become a combinatorial mess if pushed to its ultimate perfection. We still encourage the indulgent reader to follow our output as it may contain interesting problems. In particular is it possible for a pencil of sextics to visit only a single chamber? This could be the case if there is a chamber contiguous to itself. You move in your chamber and tries to get out of it by traversing a wall, but alas fall again trapped in the same room as you started where.

\[05.01.13\] Looking (once again) at Gudkov’s table Fig.\[Gudkov-Table3:fig\] (while duplicating all the blue-rhombs) we get a complete picture of all chambers past the discriminant (i.e. rigid-isotopy classes of smooth curves). One would like to understand their contiguity relation to enhance this set into the [*contiguity graph*]{}.

Basically we have 6 moves prompted by the equilateral lattice underlying Gudkov’s table. But as all Morse surgeries amounts to the creation or destruction of an oval we can rule out the two horizontal moves (keeping $r$ unchanged and corresponding resp. to the evasion or encapsulation of an oval). Next certain Morse surgeries are of course incompatible with Bézout (e.g. that depicted on the top-right of Fig.\[Gudkov-contig:fig\]). A little moment thought shows that all admissible Morse surgeries correspond to one of the $4$ legal moves.

A further obstruction comes from the Klein-Marin theorem (\[Klein-Marin:lem\]) impeding the $2$ creationist moves (going up $r\mapsto r+1$) as soon as the chamber is of type I.

-1.2cm0 -5pt0

-5pt0

Naively one is tempted to say that this is a complete list of legal moves. If so is the case then we would have a complete description of the contiguity graph (whose edges are depicted by red strokes on Fig.\[Gudkov-contig:fig\]). As we shall see later (eversions) this answers is quite unlikely to be the definitive answer.

(Exercise: count the number of edges of this graph: counting edges going to the North-West: we have $2\cdot 9+6+3\cdot 5+2+3\dot
1=18+6+15+2+3=44$. This must be doubled by symmetry to $88$. Then adding those edges going to the $8$ indefinite types adds $2\cdot
8-3=13$ edges. Next the empty scheme $0$ gives one edge and the deep nest for 2. In definitive, $88+13+3=104=2\cdot 52=2^2\cdot
26=2^3 \cdot 13$ edges. This is the number of contiguity zones of the discriminant hypersurface.)

It seems also that the most remote pair of vertices are the schemes $\frac{9}{1}$ and $10$ lying at distance 18 apart. On the other hand the discriminant has degree $\delta=3(m-1)^2$. (This can be proved via a Euler characteristic count in the fibration induced by a pencil of $m$-tics after blowing up the basepoints, and is also implied by the so-called Zeuthen-Segre formula in the algebro-geometric community, which is merely the avatar of Riemann-Hurwitz in one more dimension.) For sextics $m=6$ this gives $\delta=3 \cdot 5^2=75$. It follows that the linear distance between two chambers (as measured inside a linear pencil) is at most $[75/2]=[37.5]=37$ (the temperature of the human body). Probably this bound is far from sharp (except of course if there is a line hitting the discriminant 75 times on the reals), and one could try to find a least upper bound.

Given two chambers (=rigid-isotopy classes) define their distance $\delta$ as the minimum number of wall-crossings separating them. This is also the combinatorial distance in the contiguity graph.

$\bullet$ Define also their écart $\varepsilon$ as the minimum number of wall-crossings in a generic pencil (transverse to the discriminant) through two curves belonging to the given chambers.

We have always $\delta \le \varepsilon$; and when $m=6$, $\delta
\le 18$ and $\varepsilon \le 37$.

Since the degree of the discriminant is 75 ($\delta=3(m-1)^2$ is odd whenever $m$ is even) any pencil of sextics intersects the discriminant (in a real point) and so the curves undergo at least one Morse surgery, and assuming genericity there is an odd number of such surgeries. However the structure of the graph only permits loops of even length (the girth=systole of the graph is 4). This is almost a contradiction in mathematics. How to resolve it?

[*Insertion*]{} \[21.01.13\].—Just look at eversions, cf. Sec.\[Eversion:sec\]. Further using eversions it is likely that $\delta$ is much smaller than above, and we predict rather something like $\delta\le 11$ (cf. \[eversion-and-other surgeries:conj\] and the semi-conjectural Prop.\[Erdos-number-of-sextics=11:prop\]).

Then there is a host of combinatorial-geometric questions arising. E.g. is there a pencil cutting 75 times the discriminant (a sort of total reality of Bézout). If not what is the maximal number $\mu$ of real intersections a line can have with the discriminant? (Since the distance between the extreme $(M-1)$-schemes is 18, taking the line joining them we get a pencil with $18+18=36$ (aller-retour, no one way ticket!) real intersections, to which one can safely add one unit due to oddness of the degree, so $\mu
\ge  37$ the temperature of the human body.)

[*Insertion*]{} \[01.04.13\].—Alas this argument is foiled as it does not take into account eversions. With eversions the maximal distance seems to be 11 (between $M$-curves and the empty one), and so arguing as above gives only $\mu\ge 22+1=23$.

What is the least (resp. maximum) number of chambers visited by a pencil of sextics? Denote them $\alpha$ resp. $\omega$. Naively a pencil could stay entirely inside a chamber, but this is precluded by Bézout as $\deg \disc=75$ is odd (so $\alpha\ge 2$). A priori among the 64 chambers all could be visited since the discriminant has degree 75. Same question for the length of the loop induced in the contiguity graph by a (generic) pencil. (A priori this length can be as long as $75$, but not longer.) Is this loop always non-contractible (in the contiguity graph)? Can it be embedded (i.e. visits only once each chambers it visits)? Is there a pencil visiting only diasymmetric chambers? (It is evident from Klein’s congruence and surgeries affecting $r$ by $\pm 1$ that a pencil cannot visit only orthosymmetric chambers.) What is the maximum number of orthosymmetric chambers visitable by a pencil? (Of course at least two, take the line spanned by 2 points in two ortho-chambers, but probably some lucky Stonehenge alignment exists.)

What is the minimum height of a pencil? The height being just the invariant $r$, number of real circuits. An interesting result is the theorem of Cheponkus-Marin (cf. Marin 1988 [@Marin_1988 p.192]):

[(Cheponkus-Marin 1988)]{} In any generic (linear) pencil of curves of even degree $m> 2$ there is a curve having at most $M-3$ components ($r\le M-3$).

Looking at Gudkov’s table (Fig.\[Gudkov-Table3:fig\]) this looks almost evident, but is not. Since $M-3$ is the highest line full of squares, this Marin result implies that a pencil cannot confine its visits in one of the 3 regions lying above this line. One can define the depth $d$ of a pencil as the lowest value of $r$. So Marin’s result implies $d\le M-3$. Is this sharp at least for $m=6$?

Define three chambers as aligned if there is a line hitting them simultaneously. In view of Gudkov’s table enhanced by the Klein-Marin theorem one sees that there are $3+1=4$ special chambers which are contiguous to a single chamber (vertices of valency 1 in the contiguity graph), namely those of type I with schemes $\frac{8}{1}$, $9$, $\frac{4}{1}$ as well as the empty scheme $0$ ($\bigstar$). It follows that the triad consisting of any of those four, plus its unique neighbor and any chamber are aligned. It would be interesting to find a triad of chamber which are [*not*]{} aligned.

[*Insertion*]{} \[02.04.14\] Again the assertion right before the ($\bigstar$) above, looks foiled due to eversions. It looks more realist to expect that the empty scheme is the unique chamber contiguous to a single chamber. It could be interesting to describe the chambers adjacent to only 2 chambers. By the theory of eversion (developed latter), those includes the three $M$-schemes, and 2 Rohlin maximal $(M-2)$-schemes, plus apparently the 3 orthosymmetric chambers corresponding to symbols on the “boundary” of the pyramid, namely $\frac{8}{1}$, $9$, $\frac{4}{1}$. However it should not include a scheme like $10$, which by eversion is potentially related to $\frac{9}{1}$ (cf. Fig.\[Gudkov-eversion:fig\]). The above list could be exhaustive, but beware that the median schemes in type I, like $\frac{4}{1}4$, $\frac{3}{1}3$, $\frac{2}{1}2$ have also only two connections except for being potentially related to themselves under eversion. Yet later we shall see that eversion necessarily destroy the orthosymmetry, so that those schemes are eversively related to their type II twins lying below the sheet of paper. So those schemes have really valency 3. This raises however the question if a chamber can be contiguous to itself. By the diagrammatic of all Morse surgeries (eversion included) a necessary condition is that the chamber lies on the median line of the Gudkov table. By what as been said (orthosymmetry destroyed by eversions), the sole candidate for self-contiguity are $\frac{1}{1}1$ and $1$. Under this phenomenon of self-contiguity it could be the case that a pencil of sextics stays entirely within such a chamber safe for a quick perforation of the discriminant (forced by by Bézout), yet bringing us directly back to the same chamber. In that case the invariant $\alpha$ discussed above could be as low as $1$.

\[13.01.13\] Another aspect of Nikulin’s isotopic classification of sextics via the Rohlin pyramid is that it affords a broad generalization of the Nuij 1968 [@Nuij_1968] and Dubrovin 1983/85 [@Dubrovin_1983/85] theorem stating that the deep nest schemes represents a unique rigid-isotopy class of curves. (Actually Nuij’s theorem holds in arbitrary dimension.) By Nikulin’s theorem this uniqueness determination by the real scheme holds true more generally for all sextic schemes which are not hermaphrodite (i.e. of indefinite type).

All this just amount to the connectivity of the chambers, yet one may wish to know more on their individual topology (in the large). One obvious tool is the monodromy representation $$\pi_1(\textrm{ some chamber} ) \to {\frak S}(\textrm{ovals})$$ acting upon the ovals by permutation while following a loop inside some fixed chamber. Now for a scheme having both inner and outer ovals there is an obvious constraint preventing the permutation to shuffle inner ovals with outer ovals. In other words for a scheme of type $\frac{k}{1}\ell$ the range of monodromy would lye inside ${\frak
S}_k\times {\frak S}_{\ell}$. A (naive?) conjecture would be that this are the sole restrictions on the monodromy (i.e. the restricted morphism is epimorphic). If so is the case then all chambers are not simply-connected, except perhaps the 5 ones corresponding to the non-permutable schemes, i.e. $0$, $1$, $\frac{1}{1}$, $\frac{1}{1}1$ and $(1,1,1)$. For those schemes the monodromy of ovals is a trivial representation, and so there is no obstruction for those chambers to be simply-connected. Can one of those chambers even be contractible? A natural tactic is to ask if it can be starlike, in the sense of having a special viewpoint (curve) inhabiting the chamber so that each curve of the same chamber is accessible by the half-circle of the line joining the base curve to the “variable” one. Some obvious candidate are the Fermat equations $x^6+y^6=-1$ for chamber $0$ and $x^6+y^6=+1$ for chamber $1$, yet it is not clear at all if those are “visibility curve”.

[*Insertion*]{} \[02.04.13\].—Much sharper and complete results of the monodromy of sextics are due to Itenberg 1994 [@Itenberg_199X-monodromy-deg-6] extending results of Kharlamov. We shall come back this this latter.

Weak reformulation à la Marin-Viro of the Klein-Rohlin maximality conjecture
----------------------------------------------------------------------------

\[13.01.13\] After some discussions with Marin (12–13 Jan. 2013, cf. Sec.\[e-mail-Viro:sec\]), the following issue came quite clear. First let us contemplate once more the Gudkov-Rohlin pyramid as depicted as the contiguity figure \[Gudkov-contig:fig\]. On it we imagine the blue rhombs schemes doubled with a “$\Lambda$” shaped pair of edges raising to the orthosymmetric chambers (provided not on the periphery of the pyramid), whereas the diasymmetric chamber have generically a $X$-shaped quadruplets of edges in the contiguity graph.

We can consider the POSET of all real schemes enhanced by the type I/II of Klein. This is essentially what did Rohlin 1978, safe that instead of declaring indefinite those “hermaphrodite” schemes tolerating both type of representatives (type I and II) we duplicate those schemes to see them as independent elements. This amounts considering all Gudkov’s symbols decorated by signs $\pm$ telling the ortho/dia-symmetry, and of course only those realized algebro-geometrically. This is a well-defined finite set of $64=2^8$ elements. How to define an ordered structure to make it into a POSET? Answer just as the picture Fig.\[Gudkov-contig:fig\] suggests, namely a type I scheme (alias ortho-scheme) has two legs going down (some leg may be amputated if the scheme is peripheral), whereas dia-schemes have two legs (going down) and two arm (going up), except if it lies in the periphery. For instance the scheme $0$ is maximally amputated having one arm but no legs.

Of course this order structure looks somewhat ad hoc yet quite in line with the remarks of Klein and the theorem of Marin 1988, which is a stronger variation thereof (apparently Marin did not noticed that his statement looks stronger than Klein’s original statement, compare our discussion in Sec.\[e-mail-Viro:sec\]). Let me call the purified pyramid this poset. Paraphrasing Rohlin’s census (diagrammatically encoded in Fig.\[Gudkov-contig:fig\]) we plainly have:

The maximal elements of the purified pyramid of sextic ortho- and dia-schemes are exactly the orthosymmetric ones.

It seems evident that this statement extends trivially to all degrees as a mere paraphrase of Marin’s theorem (1988 [@Marin_1988]). Yet is non trivial to make a picture even for degree $8$. Yet this is worth trying to depict at the occasion (cf. Fig.\[Degree8:fig\]).

An eclectic proof of Rohlin’s conjecture via Ahlfors {#Rohlin-via-Ahlfors}
----------------------------------------------------

\[04.01.13\] Let us summarize the situation. Rohlin in 1978 [@Rohlin_1978] advanced (taking some indirect inspiration by Klein 1876) the bold conjecture that $$\textrm{a scheme is of type~I iff it is maximal,}$$ in the hierarchy of all real schemes of some fixed degree.

One sense of the conjecture turned wrong, in degree 8 by a conjunction of Polotovskii 1981 [@Polotovskii_1981] and Shustin 1985/85 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin] works (see also the remarks in Viro’s survey 1986/86 [@Viro_1986/86-Progress p.67–68]). Namely Shustin showed existence of a maximal scheme in degree 8 of type II. So the “$\Leftarrow$” implication of Rohlin’s conjecture is disrupted. It remains the hope that the “$\Rightarrow$” implication is correct (still open in 2013):

[(Rohlin’s maximality conjecture—post Shustin)]{} Fix any integer $m\ge 1$, and consider only schemes of that degree $m$. If a real scheme is of type I, then it is maximal in the lattice (POSET) of all real schemes.

This is perhaps a trivial consequence of Ahlfors theorem:

[(Gabard 31.12.12 and 04.01.13)]{} If a real scheme is of type I, then it is maximal (among all schemes of the same degree).

Suppose the given scheme, say $S_1$, to be of type I. By contradiction assume it non-maximal so that it embeds in some larger scheme $S_2$ as a strict subset. But our schemes are algebraically realized by real algebraic curves say $C_1$ and $C_2$ (defined over ${\Bbb R}$) so that the inclusion $C_i({\Bbb R})\subset {\Bbb P}^2({\Bbb R})$ belongs to the respective isotopy classes of $S_i$ ($i=1,2$). Since $S_1$ is of type I, $C_1$ is dividing, and thus there is by Ahlfors 1950 [@Ahlfors_1950] a total pencil $\pi$ of auxiliary curves all of whose real members cut only real points on $C_1$ (at least as soon as they are mobile). Now $C_2$ has at least one extra real circuit over $C_1$ (which in fact must be an oval, as curves of odd order necessarily have a pseudoline). Naively, one would like to choose any point $p_0$ on $C_2({\Bbb R})$ and let pass through it a curve of the pencil $\pi$, say $\Gamma_0\ni p_0$ while arguing that this curve has supernumerary intersection with $C_2$, violating thereby Bézout. This works (effortlessly) if we could assume $C_1({\Bbb R})\subset C_2({\Bbb R})$, but this corrupts rigidity of algebraic curves.

We see that Ahlfors nearly implies Rohlin, but some gigantic gap requires to be filled. Obviously the problem has to be embedded in some more flexible medium so as to bridge the gap between algebraic rigidity and softness of isotopy classes à la Hilbert-Rohlin.

\[10.01.13\] The little flash on how to complete the argument came to me ca. \[05h20\] in the morning after some too early waking up. It is as follows. Suppose our curve $C_1$ to be of type I. By Ahlfors there is a total pencil of curves. If the scheme of $C_1$ is not maximal it can be enlarged, so there is a curve $C_2$ with larger scheme. But $C_1$ is transverse to the foliation induced by the total pencil, and transversality is a robust feature (structural stability à la Thom, etc.) Accordingly a small perturbation of $C_1$ towards $C_2$ is still maximally cut by the curves of the pencil. Propagating this so forth we see (assuming genericity of the pencil) that the first Morse surgery decreases the number of ovals. (This was anticipated by Viro yesterday \[09.01.13\] (cf. Sec.\[e-mail-Viro:sec\]), and goes back to Klein 1876.) However one would more, namely that $C_2$ cannot have more ovals than $C_1$. To be more precise one should compare the pencil $L$ spanned by $C_1, C_2$ to the total pencil for $C_1$, while understanding perhaps the filmography of the deformation in reference to this foliation. Thinking of the latter as locally vertical, the first Morse surgery is like the hyperbola $(x-y)(x+y)=x^2-y^2=\varepsilon<0$ transverse to the vertical foliation while degenerating to the pair of lines of slope $\pm 1$ and then becoming another hyperbola $x^2-y^2=\varepsilon>0$ no longer transverse to the vertical foliation. (In fact this is only the scenario of when the node is not a solitary one.) The effect of crossing the first critical level is that some member of the pencil loose their total reality. Yet not all of the total reality is lost. In fact merely an interval of “imaginariness” is inserted.

Pushing the analysis in the large (several Morse surgeries), while also treating the other case one may hope to ensure that when $C_2$ is reached still some totally real curve persists in the pencil.

In fact Ahlfors theorem (only) implies Klein’s thesis (cf. Klein’s Quote \[Klein\_1876-niemals-isolierte:quote\]):

\[Klein-via-Ahlfors(Viro-Gabard):lem\][(Klein 1876, in a presentation of Gabard inspired by Viro, while using Ahlfors)]{} When a dividing curve crosses the discriminant it cannot acquires a solitary double point. In particular all (discriminantal) walls bounding an orthosymmetric chamber correspond to nodes with real tangents.

(This was anticipated the \[10.01.13\] by Oleg Viro (e-mail communication in Sec.\[e-mail-Viro:sec\]), but I only understand it now after ca. 10 hours of delay and some sleep in between.)

Imagine the curve moving, with suddenly, a solitary double point appearing in the real locus, like an ufo(=unidentified flying object) raising into the blue sky. If not believing in extraterrestrial flying saucers, think of this as the ex-nihilo creation of a champagne bubble from a point inflating slightly to a little oval. The initial curve $C_{-1}$ is dividing, hence admits by Ahlfors 1950 [@Ahlfors_1950] a total pencil of curves[^23]. The later induces a (mildly singular) foliation of the real projective plane, which we call the [*Ahlfors (or total) foliation*]{}. It can be assumed transverse to the given curve $C_{-1}$. A priori the (Ahlfors) foliation may be singular along the curve, yet upon dragging away the center(s) of perspectives it should always be possible to avoid this (compare upper row of Fig.\[Tube:fig\] for an implementation on the Gürtelkurve). W.l.o.g. suppose the curve (already) close to the discriminant and the deformation $C_t$ ($t\in[-1,1]$) to be a small one traversing that hypersurface. By continuity, transversality to the Ahlfors foliation persists after the critical level, and so a nascent champagne bubble would violate Bézout. Indeed, passing a curve of the total pencil through a point inner to the newly created oval (or on that oval) gives an excessive intersection with the post-critical curve $C_{+1}$.

Formalizing requires to fix a tubular neighborhood of the initial curve while noticing that its product structure is the trace of the total foliation. If the degree $m$ is odd then there is one pseudoline whose tubular neighborhood is a twisted bundle (Möbius band).

-5pt0

-5pt0

As long as the discriminant is avoided, a slight continuous variation of the coefficients engenders a small perturbation of the real locus (continuity lemma for rigid-isotopies). When the discriminant is crossed at a solitary node (ordinary double point with 2 imaginary conjugate tangents), the real locus acquires a (single) champagne bubble, while the rest of the curve is isotoped within the prescribed tube (compare Fig.\[Tube:fig\] bottom). It can be assumed (for simplicity, but not vital) that the deformed curves $(C_t)_{t\in[-1,+1]}$ stay transverse to the foliation. Inside of the tube, the intersection of the curve $\Gamma$ of the total pencil through the solitary node, with the post-critical curve $C_{+1}$ is in natural bijection with that of the pre-critical curve $C_{-1}$ . Yet the former intersection $C_{-1} \cap \Gamma$ is totally real, whereas the second contains two additional points when $\Gamma$ cuts the new created oval of $C_{+1}$ (better ask $\Gamma$ to pass through a point of this new oval). Bézout is violated, and our (Viro inspired) proof of Klein’s assertion is complete.

Klein’s thesis (as discussed in Marin 1988 [@Marin_1988] or Viro 1986) is the stronger assertion that a dividing curve cannot see its number of circuits increase when crossing the discriminant. This probably also follows from Ahlfors after some suitable thinking, yet perhaps is less close to Klein’s original statement. It is only now that Klein’s allusion (Quote \[Klein\_1876-niemals-isolierte:quote\]) appears to me quite transparent (yet via the powerful Ahlfors theorem). [*Insertion*]{} \[30.03.13\].—It is unlikely that this was the original proof of Klein (despite the fact that Teichmüller 1941 ascribes to Klein the theorem usually ascribed to Ahlfors). Klein’s original reasoning (alas unpublished in details, but only claimed in Klein 1876 [@Klein_1876]) might have been rather purely topological, essentially like Marin’s (though the latter’s statement is somewhat stronger). More on this will be discussed below, especially in (\[Klein-Marin:lem\]).

\[11.01.13\] One may wonder if, conversely, a nondividing curve can always acquire a solitary node and so a new oval. This is also implicit in Klein 1876 intuition, and probably true up to degree $6$ (cf. Gudkov’s table=Fig.\[Gudkov-Table3:fig\] for some evidence, while a rigorous proof probably rests on Nikulin’s rigid-isotopy classification via Rohlin’s enhanced Gudkov table by complex characteristics). (For a verification of Klein’s intuition in degree $6$, see Prop.\[Klein-vache-deg-6:prop\], and for a disproof in degree 8, cf. Shustin 1985 and our accompanying comments in Sec.\[Shustin-understood:sec\].)

Gudkov’s table shows however that the location for the apparition of a bubble cannot be chosen in advance. Indeed starting say from the scheme $10$ we could by bubbling create the scheme $11$ violating the Hilbert-Rohn-Petrovskii-Gudkov theorem, that such a scheme is not realized algebraically. Another more obvious argument is just to take any scheme (not on the “visible faces” of Gudkov’s pyramid, equivalently, such that $\frac{1}{1}1$ is a subscheme) and create a bubble inside the outer oval so that the new real scheme contains the subscheme $\frac{1}{1}\frac{1}{1}$ consisting of 2 nests of depth 2, which violates Bézout.

Prohibitions {#Prohibitions:sec}
============

\[28.03.13\] From now on, we do not follow historical order, but rather logical necessity. Admittedly there is no universal measure of simplicity as it depends much on the background of the investigator. From a radical viewpoint, the unique measure of simpleness could be the natural historical time-arrow. Yet sometimes big surprises arise. Arguments extremely powerful and strikingly simple (nearly stemming from nowhere) tend to trivialize much of the past efforts. Such an example is Rohlin’s formula discussed below, which bears some antecedents only by Arnold, plus the topological heritage of Riemann, Betti, Poincaré, Lefschetz, Weyl, Pontryagin, etc (homological intersection theory).

The source of prohibitions in Hilbert’s 16th problem, are multiple. Albeit we are not expert in the field let us brush a brief historical sketch.

First there are evident restrictions coming from Bézout. Those were used by Zeuthen 1874 [@Zeuthen_1874], and exploited in full in Hilbert 1891 (boring bounds on the depth of ovals). A major prohibition (directly affiliated to Zeuthen) is the Harnack inequality $r\le g+1=\frac{(m-1)(m-2)}{2}$ of 1876, in no way specific to plane curves. Klein 1876, then aged 27, (but already Harnack’s teacher) gave a more intrinsic justification boiling down to a basic fact on the topology of surfaces directly imputable to Riemann’s definition of the genus (or rather its allied connectivity). Recall that the jargon of the genus is due to Clebsch. Harnack’s inequality is something very robust, as it extends to all dimensions via Smith theory, as was noticed by Thom and Milnor, yielding something like $b_\ast(\RR X)\le b_\ast (\CC
X)$, for $b_\ast$ the total Betti number. We shall not need this as we confine attention to curves (where enough work remains to be done).

As discussed above (\[Klein-unnested-quartic-nondividing:lem\]), Klein 1876 also used large deformations (rigid-isotopies), to prove e.g. that a quartic with 2 unnested ovals is nondividing. Later he also exploited theta-characteristics. The first method is unlikely to extend to curves of higher orders (despite Nikulin’s rigid classification), while the second has been poorly explored further since Klein 1892, and Gross-Harris 1981 [@Gross-Harris_1981], and does not seem able to compete seriously with information distilled by Rohlin’s formula. Perhaps those old Jacobi-Riemann-Klein methods deserve to be revived. As to Nikulin, it seems at first that it will tell nothing being rather built upon the Gudkov-Rohlin classification by types (cf. Fig.\[Gudkov-Table3:fig\]). However in the fingers of Itenberg 1994 [@Itenberg_1994] (contraction theorem of empty ovals), we can expect (at least if this strengthens to our CCC=(\[CCC:conj\])) to rederive via strangulation the diasymmetry(=type II) of the schemes $1$ and $5$ of degree 6 (gaining so some analogy with Klein’s rigid-isotopy argument for the bifolium $2$ in degree 4). Yet, the difficulties are so great that this looks quite artificial as compared to the topological straightforwardness of Rohlin’s formula. By the way this would miss the scheme $\frac{1}{1}1$. Further this seems much limited to degree 6 as we lack precise information on rigid-isotopy in high-degrees. Then the connection with K3-surfaces is lost, and so the tool making Nikulin’s theorem possible (deep transcendental algebraic geometry, global Torelli theorem, etc.).

After Zeuthen-Harnack-Klein, came Hilbert’s 1891 intuition[^24] that an $M$-sextic is forced to nest. This has no antecedents (as far as I know), yet it could be challenging to reprove it via conformal geometry (i.e. the Riemann-Schottky-Bieberbach-Grunsky theorem). This has never been implemented and is probably a hard game, if feasible at all.

After Hilbert came several things like Ragsdale 1906, and Rohn 1911 who consolidated Hilbert’s method. This involves a deep analysis of the stratification of the space of curves and the usage of pencils. In Gudkov’s fingers, this produced an exhaustive list of prohibitions in degree 6. Perhaps an extension of this method also implies (or rather converges) with Rohlin-Le Touzé’s phenomenon of total reality. At least the diagrammatic of the Gudkov table (Fig.\[Gudkov-Table3:fig\]) strongly suggests this. In degree 6, all the information gained via Hilbert-Rohn is recovered for topological reasons à la Gudkov-Arnold-Rohlin (and extended to all degrees). It is not clear to me if this subsuming of HR to GAR is specific to degree 6 or a general feature. Probably not if I remember well a seminal talk by Orevkov (Geneva, ca. 2011) where Hilbert-Rohn was still much on the appetizer. After all it is unlikely that deep geometrical methods get completely phagocytozed by topological ones.

Enriques-Chisini 1915 [@Enriques-Chisini_1915-1918] gave a proof of Harnack’s inequality based on Riemann-Roch and a continuity argument (compare our Lemma \[Enriques-Chisini:lemma\]). This is much akin to the phenomenon of total reality, and need to be extended to less trivial cases. Recall that from the viewpoint of total reality, $M$-curves constitute the trivial case. This desideratum is the main motivation of the present text yet we still have very few factual things to present.

The next great step is Petrovskii 1933/38, who seems to be the first to find universal obstructions (valid in all degrees). This is based on Euler-Jacobi-Kronecker’s interpolation formula plus some Morse theory.

Then there is Gudkov breakthrough (apparition of congruences mod 8 as opposed to mere estimates), and the theorists Arnold, Rohlin, etc. validating them via 4D-topology or Atiyah-Singer. In this move we have the trinity of congruences modulo 8 for $M$, $(M-1)$ and $(M-2)$-curves due to GR, GKK, RKM, respectively. Here G=Gudkov, R=Rohlin, 1st K=Krakhnov, 2nd and 3rd K=Kharlamov, while M=Marin. The importance of those can hardly be underestimated. First, the conjunction of GR and GKK explains all prohibitions in degree 6 on real schemes (i.e. Hilbert’s 16th), while the 3rd GKK (forcing orthosymmetry(=type I) of schemes with $\chi\equiv_8
k^2+4$) seems even to imply (via the hypothetical Rohlin maximality conjecture=RMC) the conjunction of GR+GKK. Even without the elusive RMC, it can be that explicit instances of total reality (e.g., Rohlin-Le Touzé’s) imply in low-degrees (say $m=6,8$) the truth of RMC in special situations. This looks after all plausible, since totality involves a geometrization of the type I topological condition by a stronger geometric property (total pencil). Here and in the sequel, we shall often abridge “total reality” by “totality”.

Then appears Rohlin’s formula 1974–78. This is very strong and completely elementary. In degree 6, it rules out all schemes above the broken-line of Gudkov’s table (Fig.\[Gudkov-Table3:fig\]) safe 6, namely the 2 triangles involving the symbols $\frac{7}{1}3$, $\frac{7}{1}2$, $\frac{6}{1}3$ and its mirror $\frac{3}{1}7$, $\frac{2}{1}7$, $\frac{3}{1}6$. Rohlin’s formula is very powerful, yet somewhat too elementary to grasp the full mystery. It need therefore to be complemented by more advanced weapons like the Gudkov congruence (GR), and GKK, or by Rohlin’s maximality principle allied to total reality.

In 1978, we have Rohlin’s maximality principle (RMC), still conjectural and not yet fully exploited in our opinion. This could loop-back to conformal geometry à la Riemann, Schwarz, Schottky, Klein, Koebe, Bieberbach, Grunsky, Teichmüller, Ahlfors. As said above, if RMC looks impossible to implement in universal generality it could be verifiable in special cases by using totality as a geometric strengthening of the (topological) type I-condition. For instance Rohlin-Le Touzé’s totality should suffice (either with or without RKM) to kill all expansions of the 2 orthosymmetric $(M-2)$-schemes of degree 6. This would unify all prohibitions in degree 6 safe the schemes $11$ and $\frac{10}{1}$ (easily ruled out via Rohlin’s formula).

Ca. 1978–80, we have advanced Bézout-style obstructions à la Fiedler-Viro (\[Viro-Fiedler-prohibition:thm\]) that really pertains to curves of degree $8$. Those plays a pivotal rôle in Shustin’s disproof of Klein’s champagne bubbling principle for nondividing curves, as well as the disproof of the reverse implication of Rohlin’s maximality principle. More generally those look indispensable in the higher cases $m=7,8$ of Hilbert’s 16th.

We have also the locking trick of Marin-Fiedler (also founded on Bézout for lines) that provides obstruction to rigid-isotopy on $M$-curves of degree $\ge 7$. Here the idea is that if we have a triangle (3 lines) which is Bézout-saturated and canonically attached to a scheme (typically a disc with 3 holes), then during a rigid-isotopy ovals cannot traverse this moving frame. Hence the distribution of ovals past such a fundamental triangle is an invariant of the rigid-isotopy class. Of course this method is not a method of prohibition of schemes, but prohibits the existence of pathes in the hyperspace of smooth curves.

Finally, we have probably a rôle of Thom’s conjecture on genus-bound (verified since Kronheimer-Mrowka 1994 [@Kronheimer-Mrowka_1994]), yet whose role is not so clear-cut as initially expected. The simple case of Thom, due to Kervaire-Milnor 1961 [@Kervaire-Milnor_1961], may be used to settle Hilbert’s nesting “theorem” for $M$-sextics. In general the role of Thom, is perhaps marginalized by Rohlin’s formula and other strong results, yet seems to give new information in the work of Mikhalkin 1994 [@Mikhalkin_1994-adjunction-Thom] when it comes to split curves (communication of Fiedler, not yet digested by the writer=Gabard).

This is a brief overview of nearly all what exists. In contrast one can ask for more conciseness when it comes to explain all the prohibitions of Hilbert’s problem (in degree 6) to a classroom. As often repeated, nearly everything could reduce to the (Klein-Ahlfors-)Rohlin-Le Touzé’s phenomenon of total reality. Remind that 2 technical points are still obscure, but philosophically trivial. The first is a complete proof of Rohlin’s claim (preferably without employing the RKM congruence mod 8). The second is to verify that Rohlin-Le Touzé’s total reality is strong enough to imply maximality of the two Rohlin’s $(M-2)$-schemes. Assuming this settled, we still miss the prohibition of Hilbert’s unnested scheme $11$, and Rohn’s maximally nested scheme $\frac{10}{1}$. This is paradoxical inasmuch as those 2 guys were historically the first ruled out by the Hilbert-Rohn method. The 1st scheme $10$ can be killed by the Kervaire-Milnor 1961 [@Kervaire-Milnor_1961] elementary case of Thom’s conjecture in degree $k=3$, but the second $\frac{10}{1}$ fails to succumb under Thom. However both of them are killed by Rohlin’s formula. Hence a good cocktail (for the classroom or the economical reader) is to mix total reality with Rohlin’s formula. This reduces all prohibitions in degree 6 to only 2 paradigms. As far as we know, apart form the Hilbert-Rohn method (as developed by D.A. Gudkov) there is no universal force unifying all prohibitions in a single one (even in degree 6). A substitute to Thom-Kervaire-Milnor is to use Petrovskii 1933/38 (\[Petrovskii’s-inequalities:thm\]). This rules out $11$ but not Rohn’s scheme $\frac{10}{1}$. The latter is not even killed by the strong Petrovskii inequality of Arnold (\[Strong-Petrovskii-Arnold-ineq:thm\]), i.e., $n-p^-\le
\frac{3}{2}k(k-1)=\frac{3}{2}3\cdot 2=9$, where $p^{-}=1$ is the number of hyperbolic positive ovals, so $n\le 10$ while Rohn’s scheme has $n=10$.

Obstructions via Rohlin’s formula (Rohlin 1974, 1978) {#Rohlin-formula:sec}
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\[03.01.13\] We repeat the proof of the following pivotal result (whose proof puzzled me a lot as I was young, and still imbues some suitable respect[^25] when getting older). Crudely put, Rohlin’s formula is nothing less than the most universal obstruction that one may derive by abstract non-sense (i.e. using virtually nothing from the algebraicity assumption).

[(Rohlin 1974–78)]{} \[Rohlin-formula:thm\] For any (real, smooth, algebraic, plane) dividing curve of even order $m=2k$ (odd orders were treated by Rohlin’s student Mishachev), the following equation holds: $$2(\Pi^+ -\Pi^-)=r-k^2, \label{Rohlin-formula:eq}$$ where $r$ is the number of ovals, while $\Pi^{\pm}$ are the number of positive (resp. negative) pairs of nested ovals. Each pair of nested ovals bounds a ring=annulus in ${\Bbb R}P^2$, and upon comparing with the complex orientation (as the border of the semi-Riemann surface) one defines a positive pair when both orientations (complex vs. real) agree, and a negative pair when they disagree (cf. Fig.\[Rohlinsformula:fig\]).

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The idea involves computing the self-intersection of the half of the dividing curve after capping off by discs bounding the ovals in the real projective plane, or rather the intersection with the conjugate capped off membrane. This argument seems inspired by a similar device used by Arnold in 1971 [@Arnold_1971/72], but now slightly more punch is acquired. The proof is very elementary using merely intersection of homology classes (available since the days of Poincaré, Lefschetz, Hopf, Pontryagin, etc.) and Poincaré’s index formula (available since Gauss?, Kronecker 1868, Poincaré 1885), plus some basic trick about the “Lagrangian property” of real parts of algebraic varieties (jargon used in Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000]). One can then nearly wonder why such a formula escaped Felix Klein’s attention, but this is of course just historical slowness of the revelation of brutal combinatorial truths.

\[06.03.13\] The detailed proof is given in Rohlin 1974/75 [@Rohlin_1974/75 p.332], but stated there only for $M$-curves. The adaptation to the general case requires only minor notational changes, even simplifying a bit Rohlin’s original. For convenience let us thus copy Rohlin’s prose (while adapting it to the broader context, brackets are our additions):

Denote by $B_C\subset \RR P^2$ the bounding disc for the oval $C$. Complete the half $C^+$ of the dividing curve to a closed surface $\Sigma$ by adding nonintersecting copies of the disc $B_C$. Let $T$ be the closed surface obtained from the other half $C^{-}$ by the same procedure, and let $\varphi \colon \Sigma \to \CC P^2$ and $\psi \colon T \to \CC P^2$ be mappings fixed[^26] on $C^+$ and $C^-$ and superimposing copies of $B_C$ onto these discs. Further, let $\xi$, $\eta$ be elements of the (integral) homology group $H_2(\CC P^2)$ determined by the mappings $\varphi$ and $\psi$ and the natural orientations on the pretzels $\Sigma$ and $T$ (i.e. the orientation obtained from $C^+$ and $C^-$). We shall establish Eq. by computing the intersection index $\xi \eta$ by two procedures \[geometrically and algebraically\].

1.—The first procedure is based on the fact that $\xi \eta$ can be interpreted as the algebraic number of points in the intersection of the oriented singular pretzels $\varphi\colon
\Sigma \to \CC P^2 $ and $\psi \colon T \to \CC P^2 $. This number cannot be determined directly, since the intersection consists of wholes disks, and we begin by applying a deformation to $\varphi$, making the intersection more regular\[=nearly transverse\]. Let $u$ be some tangent vector field on $\RR P^2$ with a finite number of zeros, not having zeros on $A:=C_m(\RR)$ and normal to $A$ on $A$. Since the field $iu$ is normal to $\RR P^2$ in $\CC P^2$ and normal to $\CC A:=C_m(\CC)$ on $A$, it can be normally extended to some field $v$ on $\RR P^2 \cup C^+$ (the latter, of course, will have zeros inside of $C^+$); let $\gamma\colon \RR P^2 \cup C^+
\to \CC P^2$ be a geodesic translation defined by the field $\delta v$, where $\delta$ is a sufficiently small positive number, and $\varphi'\colon \Sigma\to \CC P^2$ be the mapping defined by the formula $\varphi'(x)=\gamma(\varphi(x))$. For $\varphi'$ the algebraic number of points of intersection with $\psi$ is determined directly and can be found in the following way. Since the sum index of the singularities of $u$ in each of the disks $B_C$ is equal to 1 and multiplication by $i$ anti-isomorphically maps the tangent bundle of $\RR P^2$ onto its normal bundle in $\CC P^2$, the sum index of $v$ on each of the disks $B_C$ is equal to $-1$. Consequently, the contribution added by the pair of disks $B_C$ and $B_C'$ to the algebraic number \[$\xi \eta$\] of intersection points that we are interested in is equal to:

$\bullet$ $+1$ if $C=C'$;

$\bullet$ $+2$ if the pair $C,C'$ is negative, and equal to

$\bullet$ $-2$ if the pair $C,C'$ is positive.

This number itself is thus equal to $r-2(\Pi^+-\Pi^-)$. Since $\varphi'$ is homotopic to $\varphi $, the index $\xi \eta$ is also like that and thus $$\xi \eta=r-2(\Pi^+-\Pi^-).$$

2.—The second procedure reduces to two remarks. First the class of $\xi + \eta$ is realized by the surface $\CC A$ and therefore coincides with $2k \alpha$, where $\alpha$ is the natural generator of the group $H_2(\CC P^2)$. Second, since the homomorphism $conj_\ast\colon H_2(\CC P^2)\to H_2(\CC P^2)$ represents multiplication by $-1$ \[as it flips the orientation of the generator interpreted as the fundamental class of a line defined over $\RR$\] and takes $\xi$ to $-\eta$, we have $\xi=
\eta$. From these remarks it follows that $\xi=k\alpha$, $\eta=k\alpha$ and $\xi \eta=k^2$.

Comparing the last equations obtained along each procedure, we obtain the announced formula .

We list some consequences. First a (promised) remark about quartics:

[(Klein 1876, Rohlin 1978)]{} Any quartic with $2$ unnested ovals is nondividing.

Since there is no nesting there in no pairs of ovals and the left-side of Rohlin’s formula vanishes, while the right-side is equal to $r-k^2=2-2^2=2-4=-2$.

The sextic scheme $5$ (five unnested ovals) is of type II. More generally the sextic scheme $r$ ($0\le r\le 11$ excepted $r=9$) is of type II (actually $11$ is not realized by Hilbert, Kahn 1909, Löbenstein 1910, Rohn 1911–13, Petrovskii 1938, Gudkov, but a more limpid proof follows from Rohlin’s formula).

(due to Rohlin 1978 [@Rohlin_1978], also in Fiedler 1981 [@Fiedler_1981 p.13]). Since there is no nesting $\Pi^{\pm}$ are both zero, while the left-side $r-k^2=r-3^2$ of Rohlin’s formula vanishes only for $r=9$.

\[Rohlin’s-inequality:cor\] [(Rohlin’s inequality)]{} A dividing plane curve of (even) order $m$ has at least $r\ge m/2$ ovals. Further if equality $r=m/2$ holds (and the curve is dividing) then its real scheme must be a deep nest (i.e. $m/2$ ovals each pair of them being nested).

(explicit in Marin 1979 [@Marin_1979], or Gabard 2000 [@Gabard_2000 p.148], but due to Rohlin). Let $\Pi=\Pi^+ +\Pi^-$ be the total number of nested pairs of ovals. We have $$\Pi\le \textstyle\binom{r}{2},$$ (binomial coefficient counting the number of pair of a finite set of size $r$). Equality occurs only for a deep nest! Rohlin’s formula gives: $$r=k^2+2(\Pi^+-\Pi^-)\ge k^2-2\Pi^-\ge k^2-2\Pi\ge k^2-2
\textstyle\binom{r}{2}=k^2-r(r-1),$$ whence (looking at the extremities) $r^2\ge k^2$, i.e. $r\ge k$. If an equality each intermediate estimates crunch to equality, in particular the estimate $\Pi\le \binom{r}{2}$, which is fulfilled only for a deep nest.

The sequel studies Rohlin’s consequence in degree 6. This is a bit pedestrian, and can be omitted as we gave a somewhat more conceptual explanation before, by noticing that Rohlin implies Arnold, etc.

Assume again no-nesting ($\Pi=0$). Then Rohlin’s formula gives $0=2(\Pi^+-\Pi^-)=r-k^2=r-9$, it follows $r=9+ 0= 9$ (in accordance with Fig.\[Gudkov-Table3:fig\]). It is quite remarkable to notice that this gives an instant proof of Hilbert’s conjecture (and semi-theorem of his students Kahn-Löbenstein and Rohn, etc.) to the effect that there is no $M$-curve with 11 unnested ovals.

Next we assume that there is one pair of nested ovals ($\Pi=1$). Then Rohlin’s formula gives $\pm 2=2(\Pi^+-\Pi^-)=r-k^2=r-9$, it follows $r=9\pm 2= 11, 7$ (in accordance with Fig.\[Gudkov-Table3:fig\]).

Next suppose $2$ nested pairs ($\Pi=2$). Hence $\{ 4, 0,-4 \} \ni
2(\Pi^+ -\Pi^-)=r-k^2=r-9$, it follows $r=9+ \{ 4,0,-4 \}= 13,
9,5$ (in accordance with Fig.\[Gudkov-Table3:fig\]).

For 3 nested pairs, $\{ 6, 2,-2, -6 \} \ni 2(\Pi^+
-\Pi^-)=r-k^2=r-9$, it follows $r=9+ \{ 6, 2,-2, -6 \}= 15, 11,
7,3$ (in accordance with Fig.\[Gudkov-Table3:fig\]).

For 4 nested pairs, $\{ 8, 4, 0, -4,-8 \} \ni 2(\Pi^+
-\Pi^-)=r-k^2=r-9$, it follows $r=9+ \{ 8, 4, 0, -4,-8 \}= 17, 13,
9,5,1$ (in accordance with Fig.\[Gudkov-Table3:fig\]).

For 5 nested pairs, $\{ 10, 6, 2, -2,-6, -10 \} \ni 2(\Pi^+
-\Pi^-)=r-k^2=r-9$, it follows $r=9+ \{ 10, 6, 2, -2,-6, -10 \}=
19, 15, 11, 7, 3, -1$ (in accordance with Fig.\[Gudkov-Table3:fig\]).

Etc., at this stage it is clear how to link the arithmetics of Rohlin’s formula to the geometry of Gudkov’s table enhanced by Rohlin’s data, and we have proven Rohlin’s claim:

All green-squared schemes on Gudkov’s table=Fig.\[Gudkov-Table3:fig\] are of type II.

One noteworthy feature of the diagrammatic is that Rohlin’s formula gives a tiling by squares rooted on our blue-rhombs plus the red circles of Fig.\[Gudkov-Table3:fig\]. Hence all the schemes not situated on this grid are necessarily of type II. In particular since $M$-schemes are forced to type I it follows from Rohlin that all $M$-schemes not on the square grid are prohibited as real schemes (yielding a significant contribution to Hilbert’s 16th problem). Explicitly we have:

All the $M$-schemes outside the grid are not realized (algebraically), that is $\frac{10}{1}$, $\frac{8}{1}2$, $\frac{6}{1}4$, $\frac{4}{1}6$, $\frac{2}{1}8$, $11$.

However Rohlin’s formula alone fails to prohibit the schemes $\frac{7}{1}3$ and $\frac{3}{1}7$ (which are situated on the grid). Those are however prohibited either by the Hilbert-Rohn-Gudkov method, or by the Gudkov hypothesis proved by Rohlin 1972/72 (as detailed in the next Sec.\[Gudkov-hypothesis:sec\]).

This last corollary helps the beginner to catch the substance of the following remark by Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.736]: [*“Another fundamental result difficult to overestimate is Rokhlin’s formula for complex orientations. The notion of complex orientation of a dividing real curve (see below), as well as Rokhlin’s formula and its proof, seem incredibly transparent at first sight. The formula settles, for example, two of Hilbert’s conjectures on 11 ovals of plane sextics, which Hilbert himself tried to prove in a very sophisticated way and then included in his famous problem list (as the sixteenth problem).”*]{}

To remind it seems that Hilbert conjectured (wrongly) that only the schemes $\frac{9}{1}1$ and $\frac{1}{1}9$ do exist among $M$-schemes. This turned out to be wrong when Gudkov exhibited the scheme $\frac{5}{1}5$. So Rohlin’s formula settles actually six (!) of Hilbert’s conjectures (if taken as individual prohibition). Presumably what Degtyarev-Kharlamov had in mind were the extreme schemes $\frac{10}{1}$ and $11$ (eleven unnested ovals). The philosophical outcome of this spectacular Rohlin formula is how much information can be derived by basic topological methods, basically emanating from the Riemann-Betti-Poincaré tradition, yet to which workers like Klein or Hilbert were not enough familiar with. Of course a first class topologist like Rohlin was needed to reveal this truth.

Gudkov hypothesis (Gudkov 1969, Arnold 1971, Rohlin 1972, etc.) {#Gudkov-hypothesis:sec}
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\[07.01.13\] For $M$-curves, the congruence $\chi\equiv_8 k^2$ was conjectured by Gudkov on the basis of experimental data gathered along his Hilbert-Rohn approach for sextics, and of course by looking as well to higher degrees via the Harnack-Hilbert construction. Figs.\[HilbGab1:fig\], \[HilbGab2:fig\], \[HilbGab4:fig\] below illustrate with which metronomic precision the Hilbert construction always produce $M$-curves respecting the congruence $\chi\equiv_8 k^2$. As pointed somewhere in Viro’s writings, nothing could thus have impeded Miss Ragsdale to detect this congruence in 1906 already. Yet it is the full-credit of Gudkov to have spotted this regularity. Once Arnold knew this, it was just a matter of hard-work toward elaborating the right strategy of proof, and some extra-skills of Rohlin turned to be indispensable.

So the full proof belongs to Rohlin 1972/72 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov] (alas contain a little bug), boosting ideas initiated by Arnold 1971 [@Arnold_1971/72] (who got the weaker congruence mod 4). Rohlin’s proof extract his punch not just from algebra (divisibility by 8 of an even integral unimodular quadratic form) but from the deeper divisibility by 16 coming from his own old “grand cru” of 1952 (Rohlin 1952 [@Rohlin_1952-4-manifolds]) on the signature of spin smooth $4$-manifolds. It is notorious that Rohlin’s proof (1972/72 ) contains a mistake that was repaired by Guillou-Marin 1977 [@Guillou-Marin_1977] (compare e.g. Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.736] and also Wilson 1978 [@Wilson_1978], who seems to have noticed (the same?) gap).

\[Ragsdale-Petrovskii:notatio\] [(Ragsdale 1906 [@Ragsdale_1906], Petrovskii 1938 [@Petrowsky_1938]) Given a plane curve of even order (or more generally a real scheme of ovals), it is customary to denote by $p$ the number of [*even ovals*]{} (those included in an even number of ovals) and by $n$ the number of [*odd ovals*]{} (defined analogously). The difference $p-n$ can always be interpreted as the Euler characteristic $\chi$ of the orientable membrane of $\RR P^2$ bounding the curve. The notation $p,n$ are Petrovskii’s, intended to stand for positive and negative ovals.]{}

\[Gudkov-hypothesis:thm\] [(Gudkov hypothesis/conjecture =Rohlin’s theorem of 1972, modulo a correction by Guillou-Marin 1977), and another proof in Rohlin 1974]{} A plane $M$-curve of degree $m=2k$ satisfies the Gudkov-Rohlin congruence: $$\chi=p-n\equiv k^2 \pmod 8.$$

The technique is akin to the subsequent Rohlin’s complex orientation formula of 1974–78, namely fill the halves of the orthosymmetric curve to a closed membrane and calculate the resulting intersection. However here the proof use (an extension of) the seminal Rohlin theorem 1952 [@Rohlin_1952-4-manifolds] on the divisibility by 16 of the signature of spin 4-manifolds. (At this stage there is a huge constellation of coincidence around Hilbert’s heritage: the 16th problem and as well his student, H. Weyl, whose “Analisis situs combinatorio” of 1922 is the first place where the signature of $4k$-manifolds is defined). So a breathtaking connection between differential topology and the more rigid algebraic geometry is accomplished in the Arnold-Rohlin era.

Logically this proof is a bit tricky to implement for one is warned of some mistakes in Rohlin’s initial paper. Hence pivotal is the Guillou-Marin extension of Rohlin’s signature formula, for a full exposition cf. Guillou-Marin 1986 [@Guillou-Marin_1986]. Once this is understood its application to Gudkov’s hypothesis is exposed in A’Campo 1979 [@A'Campo_1979] (following a presentation due to Marin).

Among all sextic $M$-schemes only those of Harnack, Hilbert and Gudkov exist.

Originally the proof was achieved by Gudkov 1954 [@Gudkov_1954] via the Hilbert-Rohn(-Gudkov) method, i.e. supplemented by the concept of roughness coming from the Andronov-Pontrjagin theory of dynamical (structural stability). However here we derive it rather from the above theorem (Rohlin 1972). For schemes of degree 6, written in Gudkov’s notation $\frac{k}1 \ell$, we obviously have $p=\ell+1$ and $n=k$. Some boring computation is required to check that this prohibit all $M$-schemes above the broken line in Gudkov’s table. Indeed:

$\bullet$ for $\frac{10}{1}$, $p-n=1-10=-9=-1\neq +1=k^2 \pmod 8$.

$\bullet$ for $\frac{9}{1} 1$, $p-n=2-9=-7=
+1=k^2 \pmod 8$ (no obstruction),

$\bullet$ for $\frac{8}{1} 2$, $p-n=3-8=-5\neq
+1=k^2 \pmod 8$,

$\bullet$ for $\frac{7}{1} 3$, $p-n=4-7=-3\neq
+1=k^2 \pmod 8$, etc. (progression by 2 units),

so the rest is better done mentally on looking at Gudkov’s table (Fig.\[Gudkov-Table3:fig\]) of which we reproduce the top portion below (Fig.\[Gudkov-TableTop:fig\]).

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Interestingly, prohibiting sextic $M$-schemes is much easier (no deep differential topology à la Rohlin) for the schemes not situated at the two centers of the semi-hexagon of Gudkov’s table, i.e. $\frac{k}{1} \ell$ with $k$ even, whereas the “hexagonal” schemes $\frac{7}{1}3$ and its mirror $\frac{3}{1}7$ are much harder to disprove (at least in the modern Arnold-Rohlin theory). For sextics, one may wonder what is more elementary: Hilbert-Rohn-Gudkov or Rohlin 1952–1972.

Remember that Arnold 1971 [@Arnold_1971/72] proved the weaker congruence modulo 4 of Gudkov’s hypothesis:

[(Arnold 1971, Wilson 1978)]{} For $M$-curves of degree $2k$ (or more generally dividing curves, cf. [Wilson 1978 [@Wilson_1978 p.67–69]]{}), we have $$\chi=p-n\equiv k^2 \pmod 4.$$

This theorem of Arnold is more elementary than Gudkov-Rohlin, while prohibiting exactly the same $M$-schemes as those excluded by Rohlin’s formula (i.e. fails to exclude the $\frac{7}{1}3$ and its mirror $\frac{3}{1}7$). (Of course this is not so surprising as Rohlin owed some inspiration from Arnold.) Note also that Arnold’s congruence forces all schemes on the square-grid (extending the red-hexagons where $p-n\equiv -1 \pmod
4$) to be of type II as do Rohlin’s formula. The latter is however a bit stronger for ascribing type II to the schemes $5$, $\frac{1}{1}1$, and $1$. \[06.03.13\] In fact, as suggested in Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.737]:

\[Rohlin-implies-Arnold:lem\] Rohlin’s formula [(\[Rohlin-formula:thm\])]{} implies straightforwardly the (extended) Arnold congruence $\chi\equiv k^2
\pmod 4$ (for dividing curves).

This involves some abstract nonsense (yet pleasant) combinatorics. Using the usual notation of Petrovskii (cf. \[Ragsdale-Petrovskii:notatio\]), we have $\chi=p-n$ (Euler characteristic of the Ragsdale membrane), $r=p+n$ (total number of ovals split into $p$ even ones and $n$ odd ones), and Rohlin’s formula $2(\Pi^+-\Pi^-)=r-k^2$. Assembling this gives $$\begin{aligned}
\label{Rohlin-to-Arnold:eq}
\chi=p-n=(p+n)-2n&=r-2n\cr
&=[2(\Pi^+-\Pi^-)+k^2]-2n\cr
&=k^2+2(\Pi^+-\Pi^- -n).\end{aligned}$$ It remains to check that the “corrector term” $(\Pi^+-\Pi^- -n)$ is even. Modulo 2 we have, $ \Pi^+-\Pi^- \equiv_2 \Pi^+ +\Pi^- =
\Pi$. Hence we can ultimately ignore Rohlin’s complex orientations. The following lemma concludes the proof, via the usual construction (like on Fig.\[Stalin3:fig\]) assigning to a plane curve its nested hierarchy of ovals ordered by inclusion of their insides (i.e. the unique bounding disc of the oval afforded by “Schoenflies theorem” applied in $\RR P^2$). Recall that a Jordan curve on any surface is null-homotopic iff it bounds a disc. (Cf. e.g. Reinhold Baer’s proof ca. 1927, Thesis under H. Kneser, reproduced in Gabard-Gauld 2010 [@Gabard-Gauld_2010-Jordan-and-Schoenflies].)

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\[Stalin:lemma\] Given a finite tree with a directed structure upward so that the tree really looks like the roots of a tree (or better a mushroom in Arnold’s metaphor). Formally we have a finite POSET where each element admits at most one superior, i.e. an element larger than it and minimal with this property (like in capitalistic or feodal hierarchies). Then the number $\Pi$ of pairs $x<y$ and the number $n$ of vertices lying at odd depths are congruent modulo $2$: $$\Pi\equiv n \pmod 2.$$

By additivity we may assume the tree connected. Then there is a unique maximal element in the hierarchy (Stalin), and we can draw from him all his subordinated elements as a “tree” growing downwards (cf. Fig.\[Stalin:fig\]) with several elements lying at different depths(=combinatorial distance to Stalin).

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Let $p_0, p_2, p_4, \dots$ be the number of elements at even depths $0, 2, 4, \dots$, and $n_1, n_3, n_5, \dots$ be those at odd depths $1,3,5, \dots$ respectively. To count $\Pi$ the number of subordinations of the hierarchy, we range them by order of importance (proximity to Stalin). Since an element has as many superiors as its depth, this gives $$\Pi=n_1+2p_2+3n_3+4p_4+5n_5+\dots\equiv_2 n_1+n_3+n_5+\dots=n.$$ This enumeration clearly exhausts all possible hierarchical pairs, and the proof is complete.

Gudkov-Rohlin congruence via Rohlin’s formula? {#Gudkov-hyp-via-Rohlin's-formula:sec}
----------------------------------------------

\[11.03.13\] In fact the programme of this section looks extremely dubious, just by virtue of the diagrammatic of the Gudkov table in degree 6 (=Fig.\[Gudkov-Table3:fig\]). Indeed for a scheme like $\frac{7}{1}3$ the Rohlin equation is trivially soluble (cf. e.g. Theorem \[no-chance-to-reduce-Gudkov-to-Rohlin:thm\]). Therefore there is little chance to reduce Gudkov hypothesis to Rohlin’s formula and the sole signs-law on the Rohlin tree, unless one is able to infer sharper information on complex orientations from geometrical considerations, maybe via total pencils that are fairly easy to construct (cf. Theorem \[total-reality-of-plane-M-curves:thm\]) but it is probably another matter to visualize their dynamics. Hence we recommend to skip reading this section.

\[08.03.13\] In view of the previous reduction of Arnold’s mod 4 congruence to Rohlin’s formula, an evident idea is to get better control on the residue modulo 4 of the term $(\Pi^+- \Pi^- -n)$ occurring in Equation  to draw sharper congruences (than Arnold’s). In the above proof we ignored completely (and could do so) the sign of Rohlin’s pairs, yet there is an evident composition law when the Hilbert tree of the scheme is decorated by signs (dictated by Rohlin’s pairs with complex orientations, see Fig.\[Rohlinsformula:fig\]). It seems however unlikely that one can boost the method up to include a proof of Gudkov hypothesis based on Rohlin’s formula. (If feasible, this would have certainly been mentioned in the Degtyarev-Kharlamov survey [@Degtyarev-Kharlamov_2000]).

If optimistic one could use a total pencil (like in Theorem \[total-reality-of-plane-M-curves:thm\]) as to control complex orientations. This could give an elementary proof of Gudkov hypothesis via basic algebraic geometry instead of highbrow topology (like Rohlin 1952, or Marin 1977–79). Let us look if Rohlin’s formula implies the Gudkov-Rohlin congruence. As above, we start from the Rohlin-to-Arnold equation $$\chi=k^2+2(\Pi^+-\Pi^- -n),$$ and try now to control the residue modulo $4$ of the corrector term $(\Pi^+-\Pi^- -n)$ under the assumption that the curve $C_{m=2k}$ is an $M$-curve. (En passant, it seems that this corrector term is always $\le 0$ by Thom’s conjecture, cf. Theorem \[Thom-Ragsdale:thm\]. \[29.03.13\] Warning: this is false!) If one is able to show that this corrector term is $0$ modulo 4 then Gudkov hypothesis follows.

As in the previous reduction of Arnold-to-Rohlin, we consider the hierarchy of the scheme (alias tree or mushroom), but now one takes into account complex orientations. The latter induce a distribution of signs on all injective pairs of the tree. (An injective pair is any hierarchical pair $x>y$ like in Hegel’s dialectic “du maître et de l’esclave” where $x$ is not necessarily the direct superior of $y$.)

\[Signs-law:lem\] [(Signs-law)]{}.—In the Rohlin tree with (injective) pairs decorated with signs $\pm 1=\sigma_{x,y}$ we have given two (composable) pairs $x<y$, and $y<z$ the following “twisted” signs-law: $$\sigma_{x,z}=(-1)\sigma_{x,y} \cdot \sigma_{y,z}.$$ This looks a priori exotic, as it amounts to say that $+\times
+=-$, $-\times-=-$, i.e. consanguinity is bad, while mixing the genes is good, i.e. $+\times -=+$, $-\times+=+$.

This exotic signs law is justified by looking at Fig.\[Signs-law-dyad:fig\]:

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Of course the boring sign $(-1)$ in the lemma could be avoided if we flipped the convention in Rohlin’s definition, but we are too conservative to risk such a modification. Actually, Rohlin’s convention is perfectly sound, cf. again Fig.\[Rohlinsformula:fig\].

Next, it may be observed that the difference $\Delta
\Pi=\Pi^+-\Pi^-$ computed locally in reference to a pair of consecutive edges is $\sigma_{x,y}+\sigma_{y,z}+\sigma_{x,z}=
\sigma_{x,y}+\sigma_{y,z}+(-1)\sigma_{x,y} \cdot \sigma_{y,z}$ which is always either $+1$ or $-3$ (compare again Fig.\[Signs-law-dyad:fig\]), hence always $+1$ modulo 4.

Globally on the whole Rohlin tree, we have the formula $$\Delta \Pi:=\Pi^+-\Pi^-= \sum_{all\; edges (x<y)} \sigma_{x,y}.$$ One could hope via the signs-law to evaluate this modulo 4, and all should boil down to $n$ modulo $4$ under the assumption of an $M$-curve. Along each triad $x<y<z$ the contribution is $+1$ modulo 4. However it becomes soon a combinatorial mess, and one cannot hope this in full generality, as otherwise the Gudkov-Rohlin congruence mod 8 would hold for all dividing curves and not merely for $M$-curves. This violates experimental knowledge, e.g. the Gudkov-Rohlin table in degree 6 (Fig.\[Gudkov-Table3:fig\]).

A naive idea is to write down a cumbersome formula evaluating $\Delta \Pi$, but alas this still does not use the $M$-curve assumption. Maybe this is a first necessary step unless one has some better idea.

Denote as before $p_0,n_1,p_2,n_3,p_4,\dots$ the number of ovals at depths $0,1,2,3,4,\dots$ respectively. Using Rohlin’s signs we define an oval at depth $\ge 1$ as positive if the edge immediately above it is a positive pair and as negative otherwise. Accordingly we get splittings: $$\begin{aligned}
\label{splitting-rel:eq}
n_1&=n_1^+ + n_1^-\cr
p_2&=p_2^+ + p_2^- =p_2^{++}+p_2^{+-}+p_2^{-+}+p_2^{--} \cr
n_3&=n_3^+ + n_3^-
=n_3^{++}+n_3^{+-}+n_3^{-+}+n_3^{--}=n_3^{+++}+n_3^{++-}+etc.,
$$ where $n_1^+$ is the number of oval at depth 1 which are positive, $p_2^{++}$ is the number of ovals at depth 2 such that the 2 edges right above it are positive, while $p_2^{+-}$ is the number of ovals at depth 2 surmounted by 2 edges of signs $+$ and $-$ (in this order when moving up), and so on. Once all this notation is introduced we can write down a cumbersome formula for $\Delta \Pi$ enumerating all edges (=injective pairs) weighted by their signs according to the depth of their starting-point: $$\begin{aligned}
\Delta \Pi =& n_1^+ - n_1^-\cr
&+p_2^+ -p_2^- + (-p_2^{++}+p_2^{+-}+p_2^{-+}-p_2^{--})\cr
&+n_3^+ -n_3^- + (-n_3^{++}+n_3^{+-}+n_3^{-+}-n_3^{--})\cr
&+(+n_3^{+++}-n_3^{++-}-n_3^{+-+}+n_3^{+--}
-n_3^{-++}+n_3^{-+-}+n_3^{--+}-n_3^{---})\cr &+etc.\end{aligned}$$ Using the splitting relations above this can be somewhat condensed as $$\begin{aligned}
\Delta \Pi =& n_1^+ - n_1^-\cr
&+ 2p_2^{+-}-2p_2^{--}\cr
&+(+n_3^{+++}-n_3^{++-}+n_3^{+-+}-n_3^{+--}
-n_3^{-++}+n_3^{-+-}-n_3^{--+}-3n_3^{---})\cr
&+etc.\end{aligned}$$ alas some intelligence is required to decipher the hidden structure. Even if properly done we still require to put into action the $M$-curve assumption. Since each non maximal vertices of the Rohlin tree defines a unique edge above it we have the relation $r=p_0+ number\; of\; edges$. This is only a weak grip.

All this mess is just given as to motivate someone to arrange a combinatorial proof of the Gudkov hypothesis $\chi\equiv k^2 \pmod
8$ on the basis of Rohlin’s formula alone. This seems a serious combinatorial challenge. Since all classical proofs—(i.e., the first erroneous one of Rohlin 1972, the latter one by Rohlin 1974 via Atiyah-Singer, plus the Marin-Guillou rescue of Rohlin’s original misproof, yet still via an extension of Rohlin’s deep result on signatures of spin 4-manifolds)—use some deep results it is quite unlikely that our naive programme can be completed. Still someone gifted in combinatorics with a clever idea on how to exploit the $M$-curve assumption (Harnack maximality) can perhaps crack the problem in a very elementary fashion. A vague suggestion is to exploit the total reality result for $M$-curves given in Theorem \[total-reality-of-plane-M-curves:thm\] prompting perhaps some information on complex orientations via the usual dextrogyration argument.

Another project along this reductionism to Rohlin’s formula would be to attack our naive converse conjecture to the RKM-congruence, cf. Conjecture \[RKM-converse:conj\]. But this is merely a naive conjecture quite unlikely to hold true.

A trinity of congruences: Gudkov-Krakhnov-Kharlamov and (Rohlin)-Kharlamov-Marin
--------------------------------------------------------------------------------

\[07.01.13\] To prohibit $(M-1)$-schemes (above Gudkov’s broken line) one can use (beside the Hilbert-Rohn-Gudkov method) the following analogue congruence (paralleling that of Gudkov-Rohlin) due to Kharlamov 1973/73 [@Kharlamov_1973/73] and independently Gudkov-Krakhnov 1973/73 [@Gudkov-Krakhnov_1973/73]:

[(Kharlamov 1973, Gudkov-Krakhnov 1973)]{} \[Gudkov-Krakhnov-Kharlamov-cong:thm\] A plane $(M-1)$-curve of degree $m=2k$ satisfies the congruence $$\chi=p-n\equiv k^2\pm 1 \pmod 8.$$

Several proof are available:

$\bullet$ The original sources just referred to.

$\bullet$ Since $(M-1)$-curves are not dividing the technique is different from the capping-off trick à la Arnold-Rohlin. However Marin is able to get an unified proof (à la Rohlin) by using the Guillou-Marin extension of Rohlin’s signature formula. For an exposition cf. A’Campo 1979 [@A'Campo_1979].

\[11.01.13\] We have also the following remarkable congruence (due independently to Kharlamov and Marin (first reported in print in Rohlin 1978 [@Rohlin_1978 3.4] and the first detailed proof is given in Marin 1979/80 [@Marin_1979]):

[(Kharlamov 197?, Marin 1979/80, first reported in print in Rohlin 1978)]{} \[Kharlamov-Marin-cong:thm\] A plane $(M-2)$-curve of degree $m=2k$ and type II satisfies the congruence $$\chi=p-n\equiv k^2 \textrm{ or } k^2\pm 2 \pmod 8.$$ This can be paraphrased by saying that an $(M-2)$-curve with $\chi\equiv k^2+4 \pmod 8$ is necessarily of type I.

Compare Rohlin 1978 [@Rohlin_1978 3.4] or Marin 1979/80 [@Marin_1979], or also Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91] for an unified account (and various approaches). For the paraphrase either look at the Gudkov table in degree 6, or more seriously do some boring arithmetics, cf. (\[RKM-congruence-reformulated:thm\]), which we reproduce quickly. The paraphrase follows from the fact that [*an $(M-2)$-curve of order $m=2k$ verifies universally $\chi \equiv
k^2 \pmod 2$.*]{} This is easy to prove using the relation $\chi=p-n$, $r=p+n=M-2$ $$\chi=p-n=(p+n)-2n\equiv_2 p+n = r=M-2,$$ while by Harnack’s bound and the genus formula $g=\frac{(m-1)(m-2)}{2}$ we have $$M=g+1=\textstyle\frac{(2k-1)(2k-2)}{2}+1=(2k-1)(k-1)+1=2k^2-3k+2,$$ whence $$\chi\equiv_2 M-2=2k^2-3k \equiv_2 k \equiv_2 k^2.$$

Specializing to sextics ($m=6$, so $k=3$) implies the following (compare Fig.\[Gudkov-TableTop:fig\]):

[(Rohlin 1978)]{} The two real sextic schemes $\frac{6}{1}2$ and $\frac{2}{1}6$ are of type I.

It seems that this result had no classical counterpart à la Hilbert-Rohn-Gudkov prior to the Arnold-Rohlin revolution. Nonetheless Rohlin 1978 [@Rohlin_1978] mentions the possibility of a synthetic argument involving pencil of cubics. More on this in Sec.\[total-(M-2)-schemes:sec\].

\[10.03.13\] In fact this synthetic argument of Rohlin is now lost but was partially reconstructed by Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]. This issue of Rohlin-Le Touzé should have a strong interaction with Ahlfors theorem, while affording the first nontrivial phenomenon of total reality. We will have the occasion to dwell more on this later in this text.

\[10.03.13\] Another little remark (valid in degree 6 but perhaps more universally) is the:

Once we know Arnold’s congruence and the Gudkov-Krakhnov-Kharlamov(=GKK) congruence then Gudkov hypothesis follows formally.

Indeed contemplating Gudkov’s table (Fig.\[Gudkov-Table3:fig\]), the Arnold congruence prohibits all while-colored $M$-schemes safe those at the center of the semi-hexagons (i.e. $\frac{7}{1}3$ and it mirror $\frac{3}{1}7$). Further the GKK-congruence prohibits all white-colored $(M-1)$-schemes on that same Gudkov table. Hence if one of the two schemes $\frac{7}{1}3$ (or its mirror) existed, it would appear as an isolated island in the ocean. Yet, by transversality (à la Bertini-Morse-Whitney-Sard-de Rham-Thom[^27]) a generic pencil of curves through an (hypothetical) algebraic representant and any other curve with less oval, e.g. the anti-Fermat (invisible) curve with zero oval (equation $x_0^6+x_1^6+x_2^6=0$) would produce a combinatorial path on the Gudkov table. This violates isolation. (NB: even an eversion (Sec.\[Eversion:sec\]) can only take the scheme to its mirror, and so elementary Morse surgeries necessarily create an elementary path on the Gudkov table).

\[10.03.13\] A more radical intuition of Rohlin 1978 (now partially justified by Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]) is that owing to their total reality the $(M-2)$-schemes $\frac{6}{1}2$ (and its mirror) are maximal. This explains all the prohibitions materialized by the (white) semi-hexagons on the Gudkov Table (Fig.\[Gudkov-Table3:fig\]), safe the 2 schemes $11$ and $\frac{10}{1}$ that were prohibited since the Hilbert-Rohn era (at least modulo some German sloppiness, made perfectly rigorous by Academician D.A. Gudkov). Nowadays prohibiting them is a trivial consequence of either Arnold’s congruence or Rohlin’s formula (\[Rohlin-formula:thm\]). So at least in degree 6, we see that the phenomenon of total reality acts as a strong unifying principle for classical prohibitions. Rohlin probably had the intuition that this phenomenon perpetuates in higher degrees. More along this vertiginous idea (potentially allied to Ahlfors theorem) will be discussed in Sec.\[Esquisse-dun-prog-deja-esquiss:sec\].

Total reality of the two maximal sextic $(M-2)$-schemes (Rohlin 1978, Le Touzé 2013) {#total-(M-2)-schemes:sec}
------------------------------------------------------------------------------------

\[03.01.13\] This is akin to Ahlfors’ theorem, yet somewhat different and actually the hard part of the game. Rohlin 1978 [@Rohlin_1978 p.94] writes the following cryptical note:

\[Rohlin1978-total-reality:quote\] [(Rohlin 1978)]{}

After a suitable modification, these arguments can be applied to some other schemes. For example, when we replace real lines by real curves of degree 2 we can establish that a real scheme of degree 8 consisting of 4 nests of depth 2 lying outside one another belongs to type I[^28], and when we apply it to curves of degree 3, we can establish (in a rather complicated way) that the schemes $\frac{6}{1}2$ and $\frac{2}{1} 6$ of degree 6 \[considered below in §3.8\][^29] belong to type I. However, all the schemes that we have so far succeeded in coping with by means of these devices are covered by Theorem 3.4 and 3.5.

What is crucial here is the parenthetical comment “(in a rather complicated way)”. This is highly reminiscent of some Hilbertian prose “[*freilich auf einem au[ß]{}erordentlich umständlichen Wege*]{}”, cf. Hilbert 1891 (p.418, in Ges. Abh., Bd.II)): “[*Diesen Fall $n=6$ habe ich einer weiteren eingehenden Untersuchung unterworfen, wobei ich—freilich auf einem au[ß]{}erordentlich umständlichen Wege— fand, da[ß]{} die elf Züge einer Kurve 6-ter Ordnung keinesfalls sämtlich au[ß]{}erhalb un voneinander getrennt verlaufen können. Dieses Resultat erscheint mir deshalb von Interesse, weil er zeigt, da[ß]{} für Kurven mit der Maximalzahl von Zügen der topologisch einfachste Fall nicht immer möglich ist.*]{}” Of course both problems are slightly different but perhaps there is some common difficulty in both games, while it is not impossible that Rohlin’s made a direct winking at Hilbert’s prose.

So Rohlin claims being able to prove the following (on the basis of pure geometry):

[(Rohlin 1978, no published proof)]{} The two real sextic $(M-2)$-schemes $\frac{6}{1}2$ ($6$ ovals encapsulated in one oval and $2$ outsides) and $\frac{2}{1} 6$ ($2$ ovals encapsulated in one oval and $6$ outsides) are of type I, i.e. any smooth real curve realizing one of those schemes is necessarily orthosymmetric (=dividing) in the sense of Klein.

On the basis of Rohlin’s Quote (right above) one guesses that the proof involves looking at a pencil of cubics through 8 points inside the deep ovals while checking total reality of the resulting morphism to the line. (As usual it results a circle map à la Ahlfors, which is of degree $3\cdot 6-8=10$ after degenerating the basepoints on the ovals. This quantity coincides with Gabard’s bound $r+p=\frac{r+(g+1)}{2}=\frac{(g-1)+(g+1)}{2}=g=10$.) Of course, it is quite sad that Rohlin did not found the place to write down the details.

Naively the proof could be as follows. Take a cubic in the pencil based at some 8 points inside the $2+6=8$ deep ovals (equivalently those containing no ovals). If the cubic is connected then it visits all 8 points. Counting intersection we have $2\cdot 8=16$ intersections coming from the deep ovals, plus two intersections coming from traversing the enclosing oval. This gives 18 the maximum permitted by Bézout, whence the desired total reality.

This looks simple, but this by no mean a complete argument. What to do if the cubic is not connected? One could of course try to arrange a pencil of connected cubics. Recall that the discriminant parametrizing singular cubics has degree $3(k-1)^2=3 \cdot 2^2=12$ of even degree. Thus there is no “Galois” obstruction to finding a line in the space of cubics $\vert 3 H \vert \approx {\Bbb P}^9$ missing the discriminant.

Another objection is that our simpleminded proof equally well applies to all other $(M-2)$-schemes excepted $9$. Indeed this is clear for all of them since the ovals are split in two packets by the enclosing oval. In the case of $\frac{8}{1}$ the enclosing oval is also necessarily cut twice, since $C_3({\Bbb R})$ is not null-homotopic. This would imply that all $(M-2)$-schemes safe the unnested one ($9$) are of type I. This is however too radical and incompatible with experience (or with theory, e.g. Arnold’s congruence). For instance it is easy to alter Hilbert’s method to get the scheme $\frac{7}{1}1$ in a nondividing way as switched some signs of smoothing (cf. Fig.\[TypeII:fig\]).

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\[08.01.13\] Maybe a fruitful idea is to look at special pencils of cubics spanned by 2 reducible (split) cubics well-understood. For instance one could try with reducible cubics aggregating a conic and a line. (This looks too rigid as a conic can pass only through 5 points while the 3 remaining one are not necessarily aligned.) Another idea is to look for rational cubics (uninodal).

Inside any cubics pencil (e.g. one based on the 8 deep ovals), we have by the theorem of Cheponkus-Marin 1988 (cf. Marin 1988 [@Marin_1988 p.192]) a curve with at most $M-3=(g+1)-3=(1+1)-3=-1$ components. This looks exotic and wrong for cubics for it would lack real circuits violating thereby Galois-Bézout! Shame on me, I forgot that Cheponkus-Marin assume the pencil to be even degree $>2$!

However it is a well-known fact that a generic pencil of cubics has a curve with $r=1$, which sounds quite likely (though not forced by the even degree 12 of the discriminant). Even accepting this we get only one curve of the pencil which is connected, then of course a family of such, but we are by no mean ensured that all will be so.

The case of the Gürtelkurve, or that of the sextic having a nest of depth 3, inclines to believe that the total reality of Rohlin’s sextic schemes enjoy some structural stability in the sense that it is enough to assign the 8 basepoints inside the deep nests to ensured total reality. So we are looking for something quite rare yet reasonably robust.

Suppose we have some connected cubics in our pencils. When moving in the pencil it may splits in two components. This can occur (assuming genericity, i.e. transversality to the discriminant) either through the birth of an oval after crossing a solitary node or by a self-coalescence of the pseudoline of the cubic crossing in this case a non-solitary node (with two real branches).

$\bullet$ In the first case the newly created oval cannot contribute to additional intersections.

Here is our argument requiring some hypothesis. First our generic pencil will exhibit at most 12 uninodal curves (either solitary or non-solitary nodes). By general position it may be assumed that those 12 curves (as well as the allied 12 nodes) are not located on the given sextic $C_6$. When crossing the discriminant, the solitary node will inflate into a little surrounding oval (or at least nearby ovals). Thus by continuity this little oval do not interact with the $C_6$. In fact what is important is that our 12 nodes are not on the 8 basepoints. Hence our just born oval do not contribute visiting the base locus, which is therefore entirely visited by the residual pseudoline. This is connected and so total reality is preserved (even after crossing the wall), at least at some instantaneous future right after the crossing. Then the little bubble (oval) can inflate, yet as the number of real intersections is already maximum via the pseudoline, the oval cannot cross the $C_6$. Its motion is in some sense confined to its complement (of the $C_6$). Then two scenarios are possible, either the oval deflates again to some solitary points or it merges with the pseudoline. In both cases we come back to a curve with one circuit and total reality is ensured for topological reasons. This story has to be repeated perhaps 12 times but we seem finished, modulo analyzing the other case.

$\bullet$ In the second case (real normal crossing) our pseudoline of the moving cubic self-collides with itself and then splits in two branches. Of course it may then result a loss of total reality. Imagine the crossing (non solitary node) to be located on the $C_6$, then at the critical time there are locally two intersections and soon afterwards these may disappear loosing two intersections. Yet assuming by general position that our nodes are never located on the $C_6$ (after eventually perturbing the 8 basepoints) we still have right after the critical time two real intersections, and total reality is conserved in the short run. Now our curve is decomposed in two branches, and accordingly so are the basepoints. If the pseudoline (at the post-critical level) contains a mixture of points both inside and outside the nonempty oval of $C_6$ we are happy for two extra intersections are created while entering in and evading out this separating oval. If not, then the oval of $C_3$ visits the 6 inner ovals of $C_6$ and the pseudoline the 2 outer oval of $C_6$. (The other situation is also possible but then total reality is obvious for the pseudoline must evade the surrounding oval.) Now our oval in the long run may loose two intersections. However as the intersection $C_3\cap C_6$ was totally real, this can only occur by a retraction of a tongue slipping inside the separating oval, and then the 6 inner ovals of $C_6$ are trapped inside an oval of a cubic. The latter is reasonably rigid and convex. This oval still has $2\cdot 6=12$ intersection with the 6 inner ovals. In particular it cannot shrink to a point. It cannot also evolve to an ellipse, as otherwise the 6 inner basepoints would be co-elliptic, which can be avoided by general position. OF COURSE the proof is still not finished and some idea need to be discovered. Perhaps let pass a conic through 5 of the inner basepoints, while noting that a 6th intersection is created by Galois-Bézout, etc... Of course all this needs much more substantiation!

\[08.01.13\] The end of Rohlin’s Quote (above) shows that there must be alternative non-synthetic but crudely speaking topological proofs of the theorem. This is indeed what was discussed in the previous section.

E-mail correspondence {#e-mail-Viro:sec}
=====================

\[09.01.13\] This section gathers some responses given by experts (Viro, Marin, Orevkov, Kharlamov, Shustin, Le Touzé, etc.) to some naive questions of mine about the work of Rohlin. Here are the original messages in chronological order (inserted with the tacit approval of their authors). I acknowledge most sincerely their authors for the stimulating atmosphere it created and their generous sharing of knowledge.

\[09.01.13\] Two naive questions on Rohlin 1978

Dear Viatcheslav, Alexis, Oleg, Stefan and Grisha,

Sorry for disturbing so many experts among yours with some little aspect of the work of academician Vladimir Abramovich. (I should have written this message in French, yet cannot remember exactly about Oleg’s progresses over the last 6 years in that language.) I was those last days quite fascinated by reading Rohlin’s 1978 survey on complex topological characteristics of real curves in some more detail. As you all know, he gave a quite spectacular enhancement of Gudkov’s pyramid for all schemes of sextics by enriching it with the data of Klein’s type I/II (1876). (Compare optionally the attached pdf file giving a graphical snapshot view of Rohlin’s classification.)

My two questions are as follows.

\(1) First Rohlin (1978) claims to have a certain synthetic argument (via pencils of cubics) able to show the type I of the schemes 6/1 2 and 2/1 6. He confesses however his argument to be a complicated one. Let me cite Rohlin exactly:

“...when we apply it to curves of degree 3, we can establish (in a rather complicated way) that the schemes $\frac{6}{1}2$ and $\frac{2}{1} 6$ of degree 6 belong to type I. However, all the schemes that we have so far succeeded in coping with by means of these devices are covered by Theorem 3.4 and 3.5.”

My first question is whether Rohlin’s synthetic argument has ever been published (assuming its truth of course)? I suspect the proof to be quite beautiful, but I am myself not able to prove it for the moment. Did one of you ever worked out the argument in detail, or remember about some exposition during Rohlin’s lectures? Is it of the same order of difficulty as the Hilbert-Rohn method, requiring “roughness” á la Andronov-Pontrjagin to turn round? Would it be didactically useful to publish (on the arXiv) an account of Rohlin’s argument if one is able to reconstruct it?

\(2) The second question is of course the general Rohlin’s maximality conjecture (a scheme is of type I iff it is maximal in the hierarchy of all real schemes of some fixed degree). As reported in Viro’s survey (1986 Progresses over the last 6 years) it seems that one implication was disproved by Polotovskii and Shustin (combined efforts ca. 1982, 1985). Yet one implication looks still possible, namely type I implies maximal (if I am not wrong). It seems to me that this (last vestige of the) Rohlin conjecture could be proved somewhat eclectically in two lines via Ahlfors theorem (1950) on the total reality of orthosymmetric curves (alias type I). Namely if the curve is of type I, then there is a pencil of curves cutting only real points on the curve, so its real scheme cannot be enlarged without violating Bezout. q.e.d. Some more thinking shows of course this argument to be insufficient but maybe there is a (clever) way to complete it. Qu’en pensez-vous?

Many thanks for your attention, and also for all your fantastic papers (I am presently trying to digest, so do not take the pain to answer me properly if my questions look too naive.) I apologize again for this collective message, but as the material is quite old, most of you probably forgot some details. So I hoped to maximize some chance of getting an answer from a collective chat room.

Best regards, Alex

PS: In attachment I send you a copy of an informal text of mine on the Ahlfors map. Section 24 (pp. 205–229) is more specifically devoted to Rohlin’s conjecture, yet contains nothing original (while being quite poorly organized).

$\bullet$ On Wed, 9 Jan 2013 13:33:23 +0100 (alexandregabard@hotmail.com) wrote to Kharlamov, Marin, Viro, Fiedler, Orevkov, and Mikhalkin a collective e-mail titled “Two naive questions on Rohlin 1978”:

Dear Viatcheslav, Alexis, Oleg, Thomas, Stepan and Grisha,

Sorry for disturbing so many experts among yours with some little aspect of the work of academician Vladimir Abramovich. (I should have written this message in French, yet cannot remember exactly about Oleg’s progresses over the last 6 years in that language.)

I was those last days quite fascinated by reading Rohlin’s 1978 survey on complex topological characteristics of real curves in some more detail. As you all know, he gave a quite spectacular enhancement of Gudkov’s pyramid for all schemes of sextics by enriching it with the data of Klein’s type I/II (1876). (Compare optionally Fig.71 on page 208 of the attached pdf file giving a graphical snapshot view of Rohlin’s achievement.)

My two questions are as follows.

\(1) First Rohlin (1978) claims to have a certain synthetic argument (via pencils of cubics) able to show the type I of the schemes 6/1 2 and 2/1 6. He confesses however his argument to be a complicated one. Let me cite Rohlin exactly:

“...when we apply it to curves of degree 3, we can establish (in a rather complicated way) that the schemes $\frac{6}{1}2$ and $\frac{2}{1} 6$ of degree 6 belong to type I. However, all the schemes that we have so far succeeded in coping with by means of these devices are covered by Theorem 3.4 and 3.5.”

My first question is whether Rohlin’s synthetic argument has ever been published (assuming its truth of course)? I suspect the proof to be quite beautiful, but I am myself not quite able to write it down for the moment. Did one of you ever worked out the argument in detail, or remember about some exposition during Rohlin’s seminar? Is it of the same order of difficulty as the Hilbert-Rohn method, requiring “roughness” à la Andronov-Pontrjagin to turn round? Would it be didactically useful to publish (on the arXiv) an account of Rohlin’s argument if one is able to reconstruct it? Many thanks if you have some ideas (or recent references) on those or related questions...

\(2) The second question is of course the general Rohlin’s maximality conjecture (a scheme is of type I iff it is maximal in the hierarchy of all real schemes of some fixed degree). As reported in Viro’s survey (1986 Progresses over the last 6 years) it seems that one implication was disproved by Polotovskii and Shustin (combined efforts ca. 1982, 1985). Yet one implication looks still possible, namely “type I implies maximal” (if I am not wrong). It seems to me that this (last vestige of the) Rohlin conjecture could be proved (somewhat eclectically) in two lines via Ahlfors theorem (1950) on the total reality of orthosymmetric curves (alias type I). Namely if the curve is of type I, then there is a pencil of curves cutting only real points on the curve, so its real scheme cannot be enlarged without violating Bézout. q.e.d. Alas, some more thinking shows of course this argument to be insufficient but maybe there is a (clever) way to complete it. Qu’en pensez-vous?

Many thanks for your attention, and also for all your fantastic papers (I am presently trying to digest). So do not take the pain to answer me properly if my questions sound too naive. I apologize again for this collective message, but as the material is quite old, most of you probably forgot some details. So I hoped to maximize some chance of getting an answer from a collective chat room.

Best regards, Alex (Gabard)

PS: The attachment[^30] is a copy of an informal text of mine on the Ahlfors map. Section 24 (pp. 205–229) is more specifically devoted to Rohlin’s conjecture, yet contains nothing original (except being poorly organized).

$\bullet$ \[Viro’s answer the same day (09.01.13) ca. 20h00, additional footnotes are mine (Gabard)\]

Dear Alexandre,

Thank you for your message and manuscript. I was not aware about the Ahlfors theorem[^31]. It seems to be very interesting.

I doubt though if it can be used for proving the half of Rokhlin conjecture. It gives a proof for impossibility of raising the number of components of a type I curve by a single algebraic Morse modification (what I called Klein’s thesis).

I do not remember if I even ever heard about Rokhlin’s proof that you ask about, but the fact follows from the congruence. Slava[^32] did not mention it when he proved the corresponding congruence (at the moment the type was not yet considered). I learned this theorem from Slava in September 1977 and wrote down Slava’s proof to my notebook then. I guess the first proofs was[^33] published by Slava Nikulin (among many other statements) and Alexis Marin. Marin’s proof looks simpler, but requires Pin- structures.

Best regards, Oleg

$\bullet$ Gabard’s reply \[Same day (09.01.13) ca. 21h00\]

Dear Oleg,

Many thanks for your rapid and illuminating responses, plus all the historical details. If you see no objection, I would be very happy to cut-and-paste them in my survey. I still need to assimilate some congruences of the early phase (Rohlin, Gudkov-Krakhnov-Kharlamov, etc.) Hence you cannot imagine how your hints are illuminating my modest understanding of that golden period. Regarding Ahlfors, as you say, there is little hope to crack the big fish, yet of course I shall keep you informed if I get not too depressed by the immense difficulty.

All the best, and so many thanks again, Alex

$\bullet$ 10 Jan 2013 (Marin’s answer)

Cher Gabard

En plein déménagement, je met un peu plus de temps à vous répondre que Viro.

Comme Viro, je ne connais pas la preuve de Rohlin pour votre première question (c’est pourquoi j’avais imaginé la preuve dont parle Viro qui est dans “Quelques remarques sur les courbes algébriques planes réelle”, votre référence 742) Cependant ce séminaire de Paris VII est dans un carton et y restera tant que je n’aurai pû trouver un nouvel appartement assez grand pour contenir ma bibliothèque et, n’ayant le temps d’aller à la bibliothèque, ma mémoire ne me permet pas de vous en dire plus que Viro.

Pour la seconde question par contre je peux vous répondre, c’est à dire lever votre aveux d’incompréhension en fin (p. 226) de preuve du Lemme 24.20[^34].

Soit une courbe séparante gagnant un ovale de plus après franchiment d’un point quadratique ordinaire. Un argument de congruence (utilisant $d > 2$ dans le cas plan ou une hypothèse dans le cas général donnant que la désingularisée de cette courbe de franchiment est irréductible : l’ensemble de ses points complexe est connexe) donne que cette désingularisée de la courbe de franchiment est non séparante.

Ainsi deux points non réels conjugués de la courbe de franchiment sont lié par un arc évitant la partie réelle, en particulier le point singulier, et par extension des isotipie[^35] un tel arc subsite dans toute déformation vers l’un des des deux côtés du discriminant, en particulier avant le franchiment la courbe est non séparante ce qui contredit l’hypothèse.

Par contre si le franchiment du discriminant se fait en un point singulier plus compliqué il me semble que l’on peut augmenter le nombre de composantes connexes d’une courbe séparante. Je crois me souvenir que selon les constructions de Viro (ou peut être seulement après avec la présentation Itenbergienne de cette méthode de Viro) il y a une courbe singulière de degré 6 dont tout voisinage contient tous les types.

N’étant plus familier du sujet depuis plus de 20 ans je ne peux vous en dire plus, par contre pour les surfaces de degré 3 vous trouverez dans le second tome des oeuvres de Klein un magnifique article illustré de non moins magnifiques figures où il établi que tous les types de surface cubique s’obtiennent par déformation de la (unique à changement projectif de coordonnées) surface cubique qui a 4 points quadratiques ordinaire.

Merci de votre long article que j’essayerai de lire quand déménagement, vente,.... seront terminés.

Bien cordialement et bonne année.

Alexis Marin

PS 1 Je trouve Viro un peu “oublieux” d’écrire “ (at the moment the type was not yet considered)”: en parcourant le second tome des oeuvres de Klein vous vous appercevrez qu’un sciècle avant Viro “tout” était chez Klein!

2 Vous trouverez un article historique, beaucoup plus court\* et sur un autre sujet en mettant dans la boite de recherche d’Arxiv le mot clef “troupeau”

\*il fait 6 pages table des matières comprise et tout est dit (de façon “autocontenue” ) dans le résumé en français de la première page, mais si vous remontez à toutes les références\*\* des commentaires bibliographiques celà peut vous prendre un peu de temps.

\*\*accesibles à travers la "bibliothèque des sophomores http://alexis.marin.free.fr/BIB/

$\bullet$ Gabard’s answer \[12.01.13 ca. 23h00\]

Cher Alexis, C’est avec une immense joie que j’ai reçu votre message. N’ayant pas d’internet à la maison, je l’ai seulement découvert ce soir en visitant mon père, qui lui est connecté. Je vais donc tenter d’assimiler toutes vos remarques savantes, et si vous le permettez, de les intégrer dans mon survey, en spécifiant bien sûr qu’il s’agit de vos contributions. De mon côté, je me demande si une courbe non-séparante peut toujours acquérir un point double ordinaire solitaire. (C’est semi-implicite dans Klein 1876 qui écrivait “noch entwicklungsfähig”, mais il me semble que ça contredit le résultat de Shustin 1985 (contre-exemple à la conjecture de Rohlin), dont la logique m’échappe quelque peu, mais j’ai sûrement raté une subtilité).

Grâce à vos commentaires je devrais pouvoir produire prochainement une version plus solide et limpide de la section correspondante du survey, que je vous enverrai dès que possible. L’interaction avec Ahlfors me semble aussi prometteuse...

Amitiés, et merci infiniment pour votre message, Alex

PS J’espère que le déménagement se passe bien. Restez-vous à Grenoble, ou bien s’agit-il d’une opération plus conséquente?

PPS: J’ai bien à la maison votre article de Paris VII, qui a toujours été mon meilleur compagnon (en 1999-2000), et je suis content de le retrouver pour ce point encore plus profond.

PPPS: je me suis procuré une copie de l’article sur “il capo”, qui me semble fabuleux. Merci beaucoup. C’est exactement l’analyse que l’on rencontre à proximité de Dirichlet, etc jusqu’à Ahlfors, et Rogosinski, et que je dois essayer à l’occasion d’apprivoiser...

PS 1 Je trouve Viro un peu “oublieux” d’écrire “ (at the moment the type was not yet considered)”: en parcourant le second tome des oeuvres de Klein vous vous appercevrez qu’un sciècle avant Viro “tout” était chez Klein!

Vous avez parfaitement raison, et je suis moi même très “spécialisé ” dans l’oeuvre de Klein. Cependant le gros quiz, c’est l’assertion de Teichmüller 1941, qui prétend que Klein 18XX? anticipe Ahlfors 1950, de 70 ans environ. Toute courbe séparante (ou surface de Riemann orthosymmétrique, pour reprendre le jargon kleinéen) admet un morphisme réel vers la droite dont les fibres au dessus des points réels sont toutes exclusivement formées de points réels. C’est cet énoncé fondamental qui me semble être sous-exploité! Evidemment comme la noté Viro, il implique la partie facile de l’assertion de Klein (1876): une courbe séparante ne peut gagner un ovale spontanément comme une bulle de champagne surgit du néant.

$\bullet$ Réponse de Marin (le lendemain 13 Jan 2013 ca. 09h00)

de les intégrer dans mon survey, en spécifiant bien sûr qu’il s’agit de vos contributions.

A part l’explication de votre doute (où relativement à l’article que vous citez il n’y a que les mots “extension des isotopies” en plus) ce ne sont que de très vagues souvenirs que je vous conseille de vérifier (éventuellement auprès de plus compétent : Viro, Itenberg,... avant de les intégrer)

De mon côté, je me demande si une courbe non-séparante peut toujours acquérir un point double ordinaire solitaire.

voulez-vous dire dont les deux directions tangentes sont complexes conjuguée? celà me parait très très optimiste.

(C’est semi-implicite dans Klein 1876 qui écrivait “noch entwicklungsfähig”,

Êtes vous sûr que c’est ce que pensait Klein, ou incluait-il dans ce terme les modification par franchiment d’une courbe ayant un unique point double qui est quadratique ordinaire à tangentes réelles “apparu en rapprochant deux points d’un même ovale”?

mais il me semble que ça contredit le résultat de Shustin 1985 (contre-exemple à la conjecture de Rohlin), dont la logique m’échappe quelque peu, mais j’ai sûrement raté une subtilité).

PS J’espère que le déménagement se passe bien.

oui mais c’est long, à ce propos, vous trouverez sur

http://alexis.marin.free.fr/BIB/papier/

la liste des livres que j’ai en plusieurs exemplaires et (sauf ceux dont la colonne “héritier” est remplie (par Vinel et/ou Guillou)) qui sont à la disposition de qui (en particulier vous) les demande. Restez-vous à Grenoble, ou bien s’agit-il d’une opération plus conséquente?

Je reste près de Grenoble (mon adresse est dans la signature électronique ci-dessous

PPPS: je me suis procuré une copie de l’article sur “il capo”,

Voulez vous dire “Le capo”?

Cependant le gros quiz, c’est l’assertion de Teichmüller 1941, qui prétend que Klein 18XX? anticipe Ahlfors 1950, de 70 ans environ. Toute courbe séparante (ou surface de Riemann orthosymmétrique, pour reprendre le jargon kleinéen) admet un morphisme réel vers la droite dont les fibres au dessus des points réels sont toutes exclusivement formées de points réels.

Voulez-vous dire revêtement d’espace total l’ensemble des ovales? Il y a-t-il quelque chose de plus précis sur le degré et sa répartition parmis les ovales? Les références sont-elles dans votre article?

C’est cet énoncé fondamental qui me semble être sous-exploité! Evidemment comme la noté Viro, il implique la partie facile de l’assertion de Klein (1876): une courbe séparante ne peut gagner un ovale spontanément comme une bulle de champagne surgit du néant.

Soyez plus précis pourquoi un tel morphisme admettrait-il une déformation le long de la modification d’adjonction d’un ovale?

Amitiés.

Alexis

– http://le-tonneau-de-thales.tumblr.com/

Alexis Marin, chez Danielle Bozonat 6 Allée de la roseraie, 38240 Meylan fixe : 04 76 00 96 54 port. : 06 38 29 33 99, 00351925 271 040

$\bullet$ Gabard 13 Jan 2013 ca. 13h30

Cher Alexis, Merci pour votre message. Je vais en effet essayer d’intégrer vos commentaires de manière ciblée et prudente. De toute manière avant d’arXiver une nouvelle version d’ici six mois environ, j’aurai l’occasion de vous montrer précisement la prose que je vous aurez emprunté. J’essaye maintenant de répondre à vos questions:

De mon côté, je me demande si une courbe non-séparante peut toujours acquérir un point double ordinaire solitaire.

voulez-vous dire dont les deux directions tangentes sont complexes conjuguée? celà me parait très très optimiste.

REPONSE: Oui, exactement à tangentes imaginaires conjuguées. Cela me parait aussi très optimiste. Klein semble le prétendre semi-implicitement (du moins qu’il n’ y a a priori pas d’obstruction topologique à la formation de telles bulles de champagne). Cependant si ce truc fou (“Klein-vache”) est vrai alors un des sens de la conjecture de Rohlin 1978 (type I iff maximal real scheme) est vérifié. Malheureusement, ce que donne “Klein-vache” est le sens de Rohlin détruit par Shustin 1985 (dont je n’ai cependant pas compris l’argument). Mais vous avez surement raison “Klein-vache” est probablement beaucoup trop optimiste...

Êtes vous sûr que c’est ce que pensait Klein, ou incluait-il dans ce terme les modification par franchiment d’une courbe ayant un unique point double qui est quadratique ordinaire à tangentes réelles “apparu en rapprochant deux points d’un même ovale”?

REPONSE: je pense que oui, car Klein précise “isolierte reelle Doppeltangente”, comparez ma Quote 24.2[^36] page 205 de mon survey (si vous n’avez pas le volume 2 de Klein sous la main). Ainsi il me semble que votre interprétation moderne (Marin 1988) diffère un peu de l’original Kleinéen, en étant toutefois plus puissant que l’assertion d’origine.

PS J’espère que le déménagement se passe bien.

oui mais c’est long, à ce propos, vous trouverez sur

http://alexis.marin.free.fr/BIB/papier/

la liste des livres que j’ai en plusieurs exemplaires et (sauf ceux dont la colonne “héritier” est remplie (par Vinel et/ou Guillou)) qui sont à la disposition de qui (en particulier vous) les demande.

C’est une magnifique liste de trésor. Je voudrais bien les acquérir, mais je me demande si mon hygiène de vie (overwork) rend une telle acquisition raisonable...(Il faudrait que je passe à Grenoble avec la camionnette de mon oncle pour récupérer les “invendus”. Il est préférable en effet de trouver des preneurs plus compétents que moi. Si en dernier recours, vous ne trouvez pas de preneurs je pourrais récupérer les volumes restants en vrac...Merci infiniment pour cette généreuse proposition. Moi même je suis très marginal financièrement et spatialement, petit appartement à Genève partagé avec ma mère (avec environ 8 tonnes de littérature mathématique), mais dans le futur je pourrai peut être m’installer dans une ferme fribourgoise, où il reste de l’espace pour expandre la bibliothèque...)

PPPS: je me suis procuré une copie de l’article sur “il capo”,

Voulez vous dire “Le capo”? Oui, j’essayais d’improviser en italien, mais c’est une langue plus subtil que vous utilisez...

Cependant le gros quiz, c’est l’assertion de Teichmüller 1941, qui prétend que Klein 18XX? anticipe Ahlfors 1950, de 70 ans environ. Toute courbe séparante (ou surface de Riemann orthosymmétrique, pour reprendre le jargon kleinéen) admet un morphisme réel vers la droite dont les fibres au dessus des points réels sont toutes exclusivement formées de points réels.

Voulez- vous dire revêtement d’espace total l’ensemble des ovales? Il y a-t-il quelque chose de plus précis sur le degré et sa répartition parmis les ovales? Les références sont-elles dans votre article?

OUI, toute surface de Riemann à bord (=membrane compacte) s’exprime comme revêtement holomorphe ramifié du disque. C’est juste une version relative (à bord) du théorème d’existence de Riemann qui concrètise toute surface de Riemann close comme revetement conforme de la sphère (ronde). Il a fallut toutefois attendre la contribution d’Ahlfors 1950 qui donne en plus un contrôle sur le degré d’un tel revêtement conforme, à savoir r+2p, où r est le nombre d’“ovales” (mieux le nombre de contours de la membrane), et p son genre. La Thèse de moi-même (Gabard 2004, et l’article de 2006 au Commentarii Math. Helv.) donne un meilleur contrôle, à savoir $r+p$, en économisant donc une cartouche pour chaque anse. Les références précises sont dans le survey. L’énoncé d’Ahlfors était vachement anticipé dans le cas $p=0$ (membrane planaire ou schlichtartig pour reprendre la terminologie de Paul Koebe) par la grande lignée Riemann 1857 (Nachlass), Schottky 1875-77, Bieberbach 1925 et Grunsky 1937. Lorsqu’on passe au double de Schottky-Klein de la surface à bord on obtient (via Ahlfors) une courbe séparante avec un morphisme totalement réel vers la droite projective. Inversement toute courbe séparante est totalement réelle, puisqu’il suffit d’appliquer Ahlfors à une des moitiés orthosymétrique de Klein.

\[Gabard\] C’est cet énoncé fondamental qui me semble être sous-exploité! Evidemment comme la noté Viro, il implique la partie facile de l’assertion de Klein (1876): une courbe séparante ne peut gagner un ovale spontanément comme une bulle de champagne surgit du néant.

\[Marin\] Soyez plus précis pourquoi un tel morphisme admettrait-il une déformation le long de la modification d’adjonction d’un ovale?

\[Gabard\] Je pense que ça marche car lorsque la courbe est plongée dans le plan, le morphisme total d’Ahlfors admet une réalisation projective comme un pinceau de courbes planes dont tous les membres découpent seulement des points réels sur la courbe orthosymmétrique (=séparante). Par conséquent, en traçant la courbe du pinceau total qui passe par un point de l’oval spontanément créé, on obtient une contradiction avec Bézout. Donc Ahlfors 1950 implique Klein 1876, mais votre démonstration de 1988$-\varepsilon$ (votre preuve est déjà mentionnée dans Viro 1986) est surement plus intrinsèque et voisine de l’argument d’origine de Klein (s’il en avait un au delà de la pure contemplation empirique des quartiques notamment...)

Merci infiniment pour vos messages, et d’ici tout bientôt (3-4 jours) je vous enverrai une version mise-à-jour du survey qui clarifiera peut-être les assertions précédentes. Toutefois les grands problèmes et plein de détails m’échappent encore dans la pyramide Gudkovo-Rohlinienne. Quelle splendide pyramide qui joint à la perfection Klein et Hilbert! Un détail qui m’échappe, c’est le fait que le discriminant est de degré $3(m-1)^2=75$ pour $m=6$, tandis que que du point de vue des chirurgies “de Morse” il y a des cycles de longueur 4 dans la pyramide de Gudkov. Donc il y a un problème de parité si on déforme le long d’un pinceau générique (transverse au discriminant)...Désolé, de vous embêter avec ces détails que j’ai honte de ne pas réussir à clarifier depuis quelques jours.

Amitiés, et bon courage pour la suite du déménagement, Alex

$\bullet$ \[16h40 15.01.13\] Cher Alexis, Merci encore pour vos messages et vos remarques fascinantes que je dois encore bien digérer. De mon côté, j’ai fait de minimes progrès, et vous envoie malgré votre déménagement une version ajournée de mon survey. Il me semble que le truc fou dont nous parlions il y a quelques jours, que j’appele depuis “Klein-vache”, i.e. la possiblilité de faire naitre un noeud solitaire (à tangentes conjuguées) depuis n’importe quelle courbe diasymétrique est vrai pour les sextiques. Pour cela j’utilise un argument qui combine Rohlin 1978, Klein-Marin 1988, et Nikulin 1979 (classification isotopique) et un résultat relié de Itenberg 1994 (possibilité de contracter n’importe quel ovale vide, i.e. sans autre ovales dans son intérieur, sur un tel noeud isolé). Les détails de la preuve sont exposés dans la Prop.24.24\[meanwhile this is \[Klein-vache-deg-6:prop\]\], page 235 du fichier ci-joint. ? Evidemment, en principe “Klein-vache” n’a aucune chance d’être vrai en degré supérieur. Cependant la seule obstruction que je connaisse est ce résultat de Shustin 1985, dont je ne comprends toujours pas la logique de base (sans même parler du fait que c’est fondé sur la méthode de Viro, dissipation de singularités tacnodales..., une technologie que je n’ai jamais maitrisée). Mes objections naives à l’argument de Shustin se trouvent en page 248 (dans le paragraphe qui précède la Figure 94\[=meanwile Fig.\[Shustin:fig\]\]). Dans cette figure, je ne sais pas comment prohiber le $(M-1)$-schémas encadré par le carré vert (à mi-hauteur de la figure), et dans son article de 1985 Shustin n’est pas trés explicite. Mais bon, il s’agit la d’une question assez ennuyeuse et en fait je vais peut-être prendre l’initiative d’écrire un nouveau message collectif pour clarifier ce point d’ici quelques heures. Merci infiniment encore pour vos messages, et meilleurs voeux de courage pour la suite du déménagement, Amitiés, Alex PS: Pour l’instant j’ai inégré en vrac tous nos échanges e-mail dans le survey (p.219 et suivante), mais bien entendu dès que possible je censurerai les remarques plus confidentielles..., et masquerai les répététitions, voire l’intégralité de la discussion si je parviens bien à résumer votre apport malgré mon anglais catastrophique. Cependant en relisant vos remarques, elles apportent une prose substantielle que je ne saurais jamais reproduire en anglais, donc je trouverais très dommage de censurer vos souvenirs en vracs!!! Evidemment rien ne presse et je suis désolé de vous avoir dérangé durant cette délicate opération du déménagement inter-grenoblois. Amitiés, encore, et je vous tiens au courant d’éventuelles progrès...Je suis surtout curieux des réponses de Shustin (et Viro) s’ils parviennent à éclairer ma lanterne. PPS: Je joins une copie de la note de Shustin, si jamais, mais je ne veux pas vous distraire de votre tâche prioritaire...

$\bullet$ \[15.01.13–18h30\] Dear Evgenii, Ilia, Oleg and Alexis (and Felix Klein),

I was much fascinated those last days by Evgenii’s counterexample to (one part of) Rohlin’s maximality conjecture to the effect that a real scheme is of type I iff it is maximal in the hierarchy of all schemes. Quite interestingly this work of you (Shustin) also destroys an old (semi-)conjecture of Klein (1876) positing that any nondividing plane curve can acquire a solitary node by crossing only once the discriminant (the resulting Morse surgery then sembling like the formation a champagne bubble arising like a blue sky catastrophe of little green men’s coming with flying saucers).

Alas from Shustin’s note of 1985 (in its English translation), I was not quite able to understand your proof (compare optionally the attached file, on page 248, in the paragraph right before Figure 94\[=Fig.\[Shustin:fig\]\]). In fact I do not know how to prohibit the $(M-1)$-scheme $4/1 2/1 1/1 11$ enlarging Shustin’s (M-2)-scheme. Alas I am not an expert in the field and I feel quite shameful disturbing you with such a detail. Despite having myself full Leningradian origins (through my father), I do not master the Russian language so that it may well be the case that the original Russian text is more detailed than its translation. Of course it is much more likely that I missed something well-known, that you perhaps may not have made completely explicit in the note? (Incidentally I send you a copy of Shustin’s note for convenience!)

I apologize for this question of detail, yet it seems quite important to me for your result of 1985 is the only obstruction (I am aware of) to the naive desideratum of truth about Klein’s conjecture. Klein himself is extremely clever and quite ambiguous about stating this as a conjecture or as a result (compare optionally Klein’s original quote reproduced on page 206 of the attachment). Today I managed as a simple exercise to check the truth of Klein’s hypothesis in degree 6, via an armada of Russian results (especially Itenberg 1994 contraction principle for empty ovals), plus the Klein-Marin theorem (for the details of this exercise cf. optionally Prop.24.24\[=\[Klein-vache-deg-6:prop\]\] on page 235 of the attached text).

You, Oleg Viro, in the preface of that volume presenting Itenberg’s article (1994) advanced the (crazy?) conjecture that one might always be able to contract empty ovals!!! Do you know if there is meanwhile some counterexample (in high degrees)? Of course there is some vague parallelism between Itenberg’s contraction and the one required to implement Klein’s hypothesis (which must amount shrinking an anti-oval, i.e. an invariant circle acted upon antipodically by conj).

Sorry again for disturbing you with all these naive questions, and do not take the pain answering me properly if you are overwhelmed by other more important duties. Many thanks for all your attention. Sincerely yours, Alex (Gabard)

$\bullet$$\bullet$$\bullet$ \[16.01.13–02h57: Oleg Viro\]

Dear Alexandre,

I do not mind to pose crazy conjectures. I do not mind if my crazy conjecture would be disproved.

However, I suspect that my conjecture is not as crazy as possibility of shrinking of an anti-oval. The difference between the oval and an anti-oval is that the oval is assumed to exist and be empty, i.e., not linked with the complex curve in whatever sense, while the anti-oval apparently has none of these properties.

I am not aware about any counter-examples that you ask about. I do not bet that they do not exist, but find the question stimulating, and better motivated than the conjecture that was proven to be wrong.

Best, Oleg

$\bullet$$\bullet$$\bullet$ \[16.01.13–14h56: Stepan Orevkov\]

A small remark:

It is wrong that $11 U 1<1> U 1<2> U 1<4>$ is not a part of an $(M-1)$-scheme. It is[^37]. Moreover, there is no known example of $(M-2)$-curve of type II which cannot be obtained from an $(M-1)$-curve by removing an empty oval.

In contrary, there are $(M-1)$-curves of degree $8$ (which are necessarily of type II) which do not come from any $M$-curve. These are:

$3<6>$

$4 U 1<2> U 2<6>$

$8 U 2<2> U 1<6>$

$12 U 3<2>$

Constru\[r\]ction (inspired by Shustin’s construction of $4 U
3<5>$):

Consire\[der\] a tricuspidal quartic $Q_{sing}$ symmetric by a rotation $R$ by $120$ degree and perturb\[e\] is\[=it\] so that each cusp gives an oval (we assume that this perturbation is very small). Let $Q$ be the perturbed curve. Two flex points appear on $Q$ near each cusp of $Q_{sing}$. We chose flex points $p_0, p_1,
p_2$ (one flex point near each cusp) so that $R(p_0)=p_1,
R(p_1)=p_2, R(p2)=p_0$. We choose homogeneous coordinates $(x_0 :
x_1 : x_2)$ so that the line $x_i = 0$ is tangent to $Q$ at $p_i$ $(i = 0,1,2)$.

Let $C$ be the image of $Q$ under the Cremona transformation $(x_0
: x_1 : x_2) \mapsto (x_1x_2 : x_2x_0 : x_0x_1)$. Then $C$ has 3 singular points, each singular point has two irreducible local branches: a branch with $E6$ and a smooth branch which cuts it “transversally”. By a perturbation of $C$ we obtain all the four curves mentioned above.

The fact that these curves cannot be obtained from $M$-curves immediately follows from the fact that, for any $M$-curve of degree 8 of the form $b U 1<a_1> U 1<a_2> U 1<a_3>$, all the numbers $a_1$, $a_2$, $a_3$ are odd[^38].

Best regards Stepa O

$\bullet$ \[17.01.13 ca. 23h00\]

Dear Oleg and Stepa,

Many thanks for all your fascinating remarks and detailed answers. I look forward digesting them carefully tomorrow.

Sorry for my late reply as I have no internet at home and was quite busy trying to understand some basic facts, notably that one may have some “eversion” of a real scheme when the oval explodes at infinity undergoing a Morse surgery not affecting its connectedness. This implies that there is some hidden passages in the Gudkov-Rohlin pyramid of all sextics changing a Gudkov symbol $k/l \ell$ to its mirror $\ell/1 k$. The resulting combinatorics of this graph looks quite formidable and I wonder if it is known whether each of those secret edges corresponding to eversions (except those linking $M$-curves) can be explored algebraically. Perhaps the problem is related to Ilia’s shrinking process for empty ovals, but seems to involve yet another species of “anti-ovals”, namely those with two fixed points under conj, yet located on the same oval. All what I am saying is for sure well-known to you since time immemorial, yet I was very happy to understand this point which solved several paradoxes of mine, notably those related to the degree of the discriminant and the contiguity graph between chambers residual to the discriminant under elementary algebraic Morse surgeries, as Oleg says. Of course, I shall send you an updated version of my file, when I manage to reorganize slightly the exposition.

Many many thanks for all your excellent answers! All the best, Alex

$\bullet$ \[18.01.13 ca. 10h00, Viatcheslav Kharlamov\]

Dear Alex,

I followed rather attentively the discussions, but kept silence since had no much to add to the reaction of the others.

This “eversion”, as you call it, played some important role in the prehistory of the Gudkov conjecture. As you probably know, the first classification declared by Gudkov was wrong, and it is one of his “thesis referees”, Prof. Morosov, who had objected the first classification exactly because of a small irregularity with respect to “eversion” of the answer. Repairing this asymmetry Gudkov came to his final result, and, if my memory is correct, in particular, at this stage discovered the missing $M$-curve.

If honestly, I don’t remember did somebody ever before discussed seriously any conceptual explanation to this “eversion”. However, it was implicitly present in all results obtained through $K3$ and their lattices. Recently, studying the shadows of cubic surfaces with Sergey Finashin and having proven, to our own surprise, for them a very similar “symmetry”, which we have called “partners relation”, we have formalized it as follows.

First level of explanation is coming from lattices of double coverings: the partner relation consists indeed in transferring an $U$-summand (unimodular even lattice of rang $2$ and signature $0$) from one eighenlattice to another. Second level of explanation is coming from moduli in terms of periods: each partner in the partner pair can be deformed to a triple conic, near the triple conic the family looks as $Q^3+tbQ^2+t^2cQ+d=0$, and switching of the sign of $t$ (passing through the triple conic) replace curves of one deformation class by curves from the partners class: moreover, such degenerations are deformationally unique. Literally the same explanation (and with much easier proofs at the both levels) works for nonsingular sextics (the shadows are sextic curves with $6$ cusps on a conic; remarkably, in many respects they behave in a way more similar to that of nonsingular sextics, than other sextics with singularities).

Yours, Viatcheslav Kharlamov

$\bullet$ \[18.01.13, Kharlamov, title of message=Correction\] Writing the message a bit in a hurry I did not describe fully and appropriately the partner relation at the lattice language. The summand $U$ does play a crucial role, and it should be moved from one eighenlattice to another, but then additionally one should exchange the eighenlattices. In fact this $U$ contains indeed the $2$-polarization vector, $h¨2=2$, and thus the eighenlattice containing this distinguished U is aways $(-1)$-eighenlattice.

The existing exception to the partner relation (as I remember, in the nonsingular case, there is only one) is the case when the $(-1)$-eighenlattice does not contain such a pair $(U,h)$.

Sorry, for being in a hurry, but I should stop at this point. Hope that now it is more clear.

$\bullet$ \[18.01.13, Gabard, ca. 21h00\] Dear Viatcheslav, Oleg and Stepa (and all the others),

So many thanks for all the excellent comments, especially on Morosov. There was some allusion to this issue in Viro’s survey from 2006, in Japanese Journal of Math, as to the lack of symmetry in Gudkov’s initial answer. Yet Morosov was not mentioned if I remember well...

On my side I was quite stimulated by the last letter from Oleg, about the contraction conjecture, as looking indeed much more realist than Klein’s Ansatz on the champagne bubbling in any nondividing curve. I attempted today to imagine what sort of proof one could expect to find for this fascinating Itenberg-Viro contraction conjecture of empty ovals.

After some trials with orthogonal trajectories to the functional computing the area of the empty oval, I arrived at some sort of strategy (probably completely fantasist) consisting in using the Riemann mapping theorem as applied to the interior of the empty oval. Naively as the contour is algebraic so is the Riemann map and hence the concentric sublevels of it ought to be algebraic curves of the same degree!!??? This would give the shrinking.

I am sure that tomorrow while checking more carefully the details all this argument will crash down. Hence sorry for this premature message. Some more details about this and my naive understanding of “eversions” are in the attached file, especially Section 24.15 (p.251) and p.242 (Section 24.12 for eversions). Regarding eversions I wonder which edges in the Gudkov pyramid are actually realized algebro-geometrically? All, except those connecting the $M$-schemes is my naive guess, yet it is probably too optimistic...

Many thanks again for sharing all your knowledge on that fascinating topic, and all your exciting letters. All the best, Alex

\[21.01.13, ca. 20h00\]

Dear real geometers,

Thank you again, Oleg, Alexis, Stepa, and Viatcheslav, for all your messages which I have carefully integrated in my TeX-notes, and to which I frequently refer for citation in my text. Your messages suggested me several ideas I would never have explored without your precious hints.

On my side, I noticed of course that the cavalier Riemann mapping strategy toward the (Itenberg-Viro 1994) contraction conjecture (CC) of empty ovals fails blatantly (cf. Section 25.7(=\[CC-via-Riemann:sec\]), pages 255-258, roughly even if the Riemann map of an algebraic oval would be algebraic then its degree seems to be twice as big as it should, or better the polynomials arising as norms of algebraic Riemann maps are not the most general representatives of their degree!!!). Perhaps the Riemann method works for special ovals, but of course they are unlikely to be interspersed in all chambers of the discriminant! This failure drifted me toward another formulation of the contraction conjecture which I call CCC, for collective contraction conjecture. This posits that all empty ovals of a real algebraic curve can be contracted simultaneously toward solitary nodes (by a path having solely its end-point in the discriminant). This looks even more “crazy” than CC, but I found no counterexamples (in my pockets). I would much appreciate if you already thought about this natural variant, especially if you detected some counterexample (perhaps arising from the Viro-Itenberg patchworking method or the dissipation of higher singularities, with which I am alas still unfamiliar with, like in Shustin’s counterexample to Rohlin’s maximality conjecture).

Here are the trivialities I managed to prove. Via Brusotti 1921, it is plain that CCC implies the usual contraction conjecture (CC) (cf. details in Lemma 25.22(=\[CCviaCCC-Brusotti:lem\]) on page 263 of the attached file). On the other hand CCC implies (as a large deformation principle) several well-known prohibitions. E.g. a two-seconds proof of the Hilbert-Rohn-Petrovskii prohibition of the sextic $M$-scheme $11$ (eleven ovals without nesting), as well as Rohlin’s prohibition of the sextic scheme $5$ of type I (by the way causing the unique asymmetry in the Gudkov-Rohlin table of sextics). Under CCC, all these facts appear as trivial consequences of Bézout (compare Section 25.8(=\[CCC:sec\]) on pages 258-259). I found this simplicity quite exciting (even though it leads to nothing new as compared to Arnold-Rohlin). One can wonder if Hilbert already used this, at least as a heuristic tool??? Philosophically, I found also interesting that such large deformation conjectures produce prohibitions, in contradistinction to small perturbations as being primarily a method of construction (Harnack-Hilbert-Brusotti, etc.). There is accordingly some nice duality between Luigi Brusotti and Ilia-Oleg’s contraction conjecture. Of course you surely noted this issue a long time ago, yet for me it was a happy discovery (yesterday).

Perhaps CCC and CC are actually equivalent, yet this looks more hazardous but maybe not completely improbable... (One would just have to synchronize the death of all ovals posited by CC.) This is all the modest news I have collected during the week-end. Of course I still have some naive hope that CCC (hence CC) could be attacked via some gradient flow, but it looks quite difficult to locate the right functional (or Morse function). Looking at the area (or length) of all empty ovals is probably too naive...Perhaps some “degree of roughness” à la Gudkov could be projectively more intrinsic and useful...

Thank you so much for your attention and all your brilliant letters and answers, while apologizing me for sending you only easy doodlings. Best regards, Alex

\[26.01.13, ca 20h00\]

Dear Oleg, Stepa, Viatcheslav, Evgenii, Alexis, Thomas, etc.

I continued my naive investigations of real plane curves. What a beautiful story! I finally “understood” and studied in detail the marvellous construction of Gudkov $\frac{5}{1} 5$ (ca. 1971-73), as to understand the more tricky (but related) construction proposed by Stepa, which I attempted to depict on Fig. 111(=\[Orevkov2:fig\]) of the attached file. (I did not as yet assimilated the full details but feel on the good way. In fact I tried to use the dissipation of $Z_15$ in Viro’s survey from 1989/90 in Leningrad Math. J., which I hope is the same as the $E_6$ advocated by Stepa. Sorry for being very ignorant about singularities...)

Yesterday, I also finally understood the correctedness of Evgenii’s argument. (As helped by Stepa’s e-mail, the point which I missed is this obstruction of Viro extending that of Fiedler) for $M$-schemes of degree 8 as having necessarily “odd content”.

On the other hand, I was scared (since three days) by the fact that something which I subconsciously thought as evident (or rather which I was sure to have read somewhere) is perhaps not true. My (naive) question is whether two empty curves are necessarily rigid-isotopic? This looks at first between metaphysical nonsense and “triviality”? Maybe it is unknown, when $m$ is large enough. (m=6 follows from Nikulin 1979, and as far as I know there is not a simpler proof, say valid for all (even) degrees). So I am quite shameful asking you about this point: Is the empty chamber always connected?

I tried a dynamical approach (to this problem) in Section 25.12(\[rigidity-empty-scheme-via-dyna:sec\]), but it is not very convincing. On the other hand, if the empty room is connected, then maybe the space of all curves with one component is also connected? (Naively one would apply the Itenberg-Viro contraction conjecture, to reduce to the empty case, move there for a while to resurface at the other curve (the contraction thereof). Perturbing this path in the “visible world” would conclude the proof modulo some difficulties...) Again, you Oleg, in your wonderful survey of 2008 (in Japanese J. Math) lists as an open problem the question of deciding the rigid-isotopy of curves of odd degree having a unique real circuit. As you emphasize the word “odd degree”, I wondered if the case of even degree (again with only one oval) is already settled?

In Section 25.10(=\[CCCviaDynamics:sec\]), I have attempted a naive dynamical approach to the collective contraction conjecture(CCC). This states that we can shrink simultaneously all the empty ovals toward solitary nodes. This is a bit like a perfect landing in flight simulator where all wheels touch the ground simultaneously. My naive strategy is just to study the gradient lines of the functional measuring the total area of all empty ovals, but it is surely not serious. It would be exciting, in my opinion, to describe a counterexample to CCC if there is one.

Many thanks for the attention, all your patience about my naive reasonings, and above all for the brilliant answers you already gave me. Best regards, Alex

$\bullet$$\bullet$$\bullet$ samedi 26 janvier 2013 20:15:54, the prompt response of Eugenii Shustin:

Dear Alex,

The chamber of empty curves of a given (even) degree is indeed connected: two such curves can be defined by homogeneous polynomials, positive for any real not all zero variables, and their linear homotopy $(1-t)P+tQ$, $0\le t\le 1$, gives a path in the chamber of empty curves.

By the way, another (well) known connected chamber consists of hyperbolic curves (i.e. those which have totally real intersection with lines of certain pencil) - this is a consequence of Nuij W. A note on hyperbolic polynomials. Math. Scandinavica 23 (1968), no. 1, 69–72.

With best wishes, Eugenii

$\bullet$$\bullet$$\bullet$ samedi 26 janvier 2013 21:08:27, Oleg Viro:

Dear Alex,

The counterpart of the Rokhlin conjecture[^39] about rigid-isotopy of any two curves of odd degree with one component is the obvious observation described by Evgenii, about empty curves of even degree. The question about curves of even degree with a single oval is equivalent to the question about removing this single oval by an algebraic Morse modification. I don’t think it was ever discussed, but I could miss it.

$Z_15$ is not $E_6$. The easiest way to construct the Gudkov $M$-curve is by perturbing two $J_10$ singularities of the union of 3 non-singular conics tangent to each other at 2 points.[^40]

Best, Oleg

$\bullet$ \[28.01.13, lundi 28 janvier 2013 20:03:58\] Gabard wrote euphorically[^41] an e-mail titled “Some more metaphysical non-sense about the rigid-isotopy of empty curves?”:

Dear Eugenii, Oleg, Viatcheslav, Alexis, Stepa, etc.

So many thanks, Eugenii, for putting me again on the right track, and recalling me the argument which I shamefully forgot about. Yesterday, I was quite excited by trying to digest your argument (albeit it seems so simple). In fact the little detail that worried me is that I do not know why during the linear homotopy $(1-t)P+tQ$ the variable curve could not acquire (while staying of course empty if $P,Q$ have both the same sign) a pair of conjugate nodal singularities. This puzzled me for a while, and then using systematically your argument, I arrived at the somewhat opposite conclusion that the empty (smooth) chamber must be disconnected (for all even degrees $m\ge 4$)!!! This violates all what we know since Rohlin 1978 (and surely Gudkov as well??), while the former refers directly back to the argument of Klein 1876 based on Schläfli cubics surfaces $F_3$’s and Zeuthen correspondence between cubic surfaces and quartics (via the apparent contour). Klein’s proof is a bit tricky and uses as well his rigidification (Klein 1873) of Schläfli’s isotopic classification. Needless to say I could not follow Klein’s reasoning completely, as I just studied it today for ca. 2 hours. Marin informed me recently that he, in contrast, was able to digest all of those Kleinian works!

So using your method of linear homotopy, one sees quickly that the (cone) space $C^+$ of positive anisotropic (=not representing zero) forms is contractile (convex actually) hence simply-connected. Its projection in the space of curves is the invisible locus $I$ consisting of all empty curves. Since the latter is merely a quotient of $C^+$ (by positive homotheties) it follows that it is also simply-connected (via the exact homotopy sequence of a fibering). But the discriminant is visible inside this invisible locus $I$, since it is a simple matter via Brusotti (1921) to construct empty curves with a pair of conjugate nodes. Thus we see inside the simply-connected manifold $I$ a certain hypersurface (namely a portion of the discriminant), which by Jordan-Brouwer (or a slight extension thereof) should separate this manifold $I$ in pieces (at least so is my naive intuition). It follows that our empty chamber (consisting of smooth curves) is disconnected!!!! This is my proof in its broad lines (for more details, compare Section 25.13, page 282-283 of the attachement, Theorem 25.29 and its proof on page 283). This is just one page long...

Since this conclusion contradicts violently what is asserted by Klein 1876 (and approved by Rohlin 1978), it is of course very likely that my proof contains a serious flaw, or at least that I am confusing somehow the basic conceptions. However presently I do not see where is my mistake! Of course, my pseudo-theorem also violates the part of Nikulin 1979 concerned with the rigid-isotopy of the empty chamber of sextics.

Many thanks again for your attention, and sorry for overflowing your mail boxes with my naive questions (and dubious reasonings). Thank you again so much for all your excellent and detailed responses (especially on $E_6$ and $Z_15$).

Best wishes, Alex PS: I send you a copy of my TeX-file in case someone would like to work out a specific passage. At the occasion I would also be happy to send you my figures in zipped format so that one of you can continue the project in case I make a fatal bicycle accident (like Academician V.I. Arnold?)

$\bullet$ \[30.01.13, 18h10\]

Dear Oleg, Eugenii, and the other experts,

I think that I found the mistake in my “proof” of the disconnection of the empty locus (that you certainly noticed meanwhile in case my explanation is the correct one). The reason seems to be simply that the discriminant inside the invisible locus has only real codimension 2, hence cannot separate anything. I have attempted to explain this in Section 25.14 on page 288. If this is not wrong it seems that the next natural question is to decide which chambers residual to the principal stratum of the discriminant contains such smaller pieces of the discriminant shrunk to codimension 2. I think to have found a topological obstacle for $M$ and $(M-1)$-curves, and conjecture (very naively) this to be the sole obstruction. In more geometric terms, this amounts essentially to decide which smooth curves can acquire a pair of imaginary conjugate nodes.

Many thanks, Eugenii and Oleg, for your detailed answers. As you said, it seems that the (Itenberg-Viro) contraction conjecture of empty ovals implies the rigidity conjecture for even order curves with a unique oval. However I should probably still try to understand this implication in some more details. Perhaps it is somehow related to the previous codimension 2 phenomenon inside the “invisible” chamber.

Sorry for all my confusing messages, and many thanks again for all your kind efforts in trying to educate myself. All the best, Alex

$\bullet$ \[01.02.13, ca. 20h00\] Obstruction to rigid-isotopy (strictly) below height DEEP+2?

Dear Oleg, Eugenii, Alexis, Thomas, Stepa, etc.

Many thanks for all your brilliant messages and articles I am still slowly trying to assimilate properly. I hope not taking too much of your precious time. Albeit I met all of you only rarely, I remind very accurately your brilliant talks (in Geneva or Rennes), and so it is a special pleasure to remind each of yours while trying to explore this fantastic topic.

On my side I was those last two days fascinated by the conjecture that the one-oval scheme ought to be rigid, as Oleg or Rokhlin conjectures. (Let me say that a scheme is rigid, if all the curves representing it are rigid-isotopic.) Given a degree $m$, one may wonder what is the smallest height $r(m)$ at which there is a non-rigid scheme. (For me the height of a scheme just means its number of components.)

For any degree $m$, there is of course the deep nest with $r=[(m+1)/2]=:DEEP$ real branches. Two units above the latter’s height, it is easy to construct (for each $m$) curves having the same real scheme yet different types (I vs. II) hence not rigid-isotopic. (This is a simple iteration of Rohlin’s construction in degree $5$, cf. Figs. 102, 103 in my file). Using the Marin-Fiedler method of the lock it is even possible to exhibit at this height $DEEP+2$ curves of degree 7 or 9 having the same real scheme and the same type II, yet not rigid-isotopic (Figs. 105, 104). (Probably the method extends to all other odd degrees.) However, it seems much more tricky (and the lock-method seems ineffective) to detect obstruction below this height $DEEP+1$ (i.e. one unit above the height of the deep nest). Could it be the case that all schemes at or below this height are rigid? Of course this looks super-optimistic as we do not even know rigidity at height one, but I was unable to find a counterexample. I would be very happy if you know one? If there is a simple candidate, I hope to detect it alone during the next few days…. So do not take care answering me if my question is trivial. (As I just work on this since two days, I probably missed something accessible.)

Paraphrasing slightly, I found quite puzzling, that the very explicit function $r(m)$ measuring the smallest height of a non-rigid scheme is only subsumed to the large pinching $1 \le
r(m)\le [(m+1)/2]+2=DEEP+2$. Of course a better lower bound seems out of reach, but perhaps you know better upper bounds.

I also wondered if there is an extension of the Nuij-Dubrovin rigidity of the deep nest to, say, the totally real scheme of degree 8 consisting of 4 nests of depth 2. I should think more seriously on this at the occasion.

Many thanks for your attention, and sorry again for all my enthusiastic and naive e-mails. All the best, Alex

$\bullet$ \[written 08.02.13 and sent 09.01.13\]

Dear Oleg, Eugenii, Stepa, Viatcheslav, etc.

I still continued my trip through real plane curves and cannot say that my curiosity is starting to fade out. I tried for several days to find a counter-example to the conjecture (of mine so probably quite wrong) that all schemes below height $DEEP+2$ are rigid, where $DEEP=[(m+1)/2]$ is the number of branches of the deep nest of degree $m$. At least the method of the lock (Fiedler-Marin) seems quite inoperant to detect an obstruction to rigid-isotopy at such low altitudes. If true, the proof probably involves a geometric flow collapsing either the pseudoline to a line (by shortening its length like a systole) or improving the rotundity of some oval to a circle (via an isoperimetric functional?). If all this works, it would reduce the low-altitude rigidity conjecture to Nikulin’s theorem (or maybe even Klein’s on $C_4$) as the starting step of a big recursive process. Of course this seems still quite out reach (canary music) unless one feels very motivated!

Next I tried to corrupt the truth of Slava’s remarkable rigid-isotopic classification (Nikulin 1979) of sextics via the Marin-Fiedler locking argument using Bézout saturation. Of course I have nothing against Slava, but this was rather intended to test experimentally the power of Nikulin’s result. Specifically I looked at sextic schemes of the form $3/1 \ell$, and wondered if for some specific curves the distribution of the $\ell$ outer ovals away the fundamental triangle traced through the 3 inner ovals (those enveloped by the unique nonempty oval) could be different for different curves. On all examples I tested it seems that the outer ovals are never separated by the “deep” triangle. So we find no violation of Nikulin’s theorem, and the latter rather implies that as soon as we are able to visualize the distribution for a single curve it will be the same for all curves belonging to this scheme. The case most tricky to understand is the maximal permissible, namely $3/1 5$. I managed to construct it à la Harnack (as preconized in Gudkov 1974 or 1954). But being quite unable to decide from this model the distributional question of the outer ovals past the fundamental triangle, I decided to switch to Oleg’s method of construction via dissipation of the singularities of a triplet of coaxial ellipses. I played this game yesterday but could not decide the distributional question for this Viro curve (cf. especially Fig.126 on page 320 and the hypothetical Theorem 26.29 on page 322). In fact today I tried again to inspect directly Harnack construction and found Lemma 26.27 on page 319 whose proof seemed to me very transparent until I found the little warning, which I think is not fatal. In conclusion I believe now that there is no separation by the fundamental triangle!!!??

Of course I imagine that, if I am not completely wrong, what I am investigating must be quite familiar to you. I would much appreciate if you know if this hypothetical theorem (26.29 page 322) is true. It amounts essentially to check whether in Viro’s construction of $3/1 5$ the triangle through the 3 deep inner ovals does not separate the 5 outer ovals. I find this question quite attractive as it seems to require some understanding of the geometric location of the microscopic ovals arising in Viro’s method (optionally compare Fig. 127 (page 322) which shows a scenario with the two bottom micro-ovals aligned vertically in which case the fundamental triangle would separate the outer ovals). This scenario seems to me quite unlikely but it does not seem to be impeded by naive Bézout obstructions.

Many thanks for your attention, and sorry again for all my naive and confuse questions. Thank you very much again for your precious guidance and answers.

All the best, Alex

$\bullet$$\bullet$$\bullet$ (10.02.13) Bonjour Alexandre, Thomas m’a transmis ta question. La réponse est toute simple: soient $A$, $B$, $C$ trois ovales intérieurs et $D$, $E$ deux ovales exterieurs de ta sextique. Le triangle fondamental $ABC$ est entiérement contenu dans l’ovale non-vide. Si $D$ et $E$ sont dans deux triangles $ABC$ (non-fondamentaux) différents, alors la conique passant par $A$, $B$, $C$, $D$, $E$ coupe la sextique en $14$ points, contradiction. Avec des coniques, on montre plus généralement que: Les ovales vides de la sextique sont distribués dans deux chaines (int, ext), l’ordre cyclique est donné par les pinceaux de droites basés dans les ovales interieurs. Les ovales interieurs sont disposés en position convexe dans l’ovale non-vide. Bon dimanche,   Séverine

$\bullet$ \[12.02.13\] Is the Gudkov chamber simply-connected?

Dear Séverine, Viatcheslav, Ilia, Oleg, and all the other experts,

First many thanks, Séverine, for your excellent answer on my distribution question of ovals of sextic, and sorry for my late reply on it as I lack an Internet connection at home.

I tried today to understand when a dividing (plane) curve admits a transmutation, i.e. a rigid-isotopy permuting both halves of the curve. I also studied the weaker notion of mutation of when there is a linear automorphism of the plane permuting both halves. Using the Kharlamov-Itenberg calculation of the monodromy of sextics I think that I managed to get some obstruction to mutability, especially for the 3 dividing curves which have trivial monodromies (compare Lemma 26.6, page 288, which is hopefully correct). However I don’t know if the Gudkov chamber (or the 2 other related “antidromic” chambers, i.e. having trivial monodromies) is simply-connected. I hoped to detect some non simple-connectivity by looking at the monodromy induced on the halves instead of the ovals. At least this works of course for the deep-nest chamber which is not simply-connected since there is a symmetric model which can be mutated. So my (hopefully not too naive) question is the following: is it known whether or not the Gudkov chamber is simply-connected? (equivalently is the Gudkov curve transmutable?) The same question looks attractive for the other 2 antidromic curves, i.e. the left wing “Rohlin curve” $6/1 2$ and $4/1 4$ in type I.

Thank you so much for all your attention and patience, and in advance for your answer if it is known. Best regards, Alex

$\bullet$$\bullet$ $\bullet$ mercredi 13 février 2013 04:34:12

Dear Alexandre, If I understand correctly the question then the answer is not, if I state the question appropriately then the answer is yes. I mean the following precise statements. Let consider the part of the projective space of real sextics that is represented by maximal sextics of Gudkov’s type. Then the fundamental group of this part is $Z/2$. It becomes simply connected after taking quotient by the natural action of $SL(3,R)$. In fact, before factorization it is a fibration over contractible base with the fiber $SL(3,R)$. These results (and there analogs for other maximal sextics and certain curves of lower degree) are contained in my talk On monodromies of real plane algebraic curves at one of Petrovsky seminars in 80th, I guess (short summary should be found in Russian Surveys). The proof (in the case of sextics) is rather straightforward as soon as based on the $K3$ surfaces periods uniformization. As it happens rather often with this approach, to treat the maximal curves is extremely easy, since the corresponding eighenlattices become unimodular. In general the period domain, which is the product of two polyhedra in the real case, represents the studied sextics (or associated K3 surfaces) only up to codimension 2. Which makes laborious to treat the fundamental group. But, surprise, in the case of maximal curves there are no codimension 2 phenomena, since such holes appear only as traces of $(-2)$-cycles having nontrivial components in the both eighenspaces, which is impossible since in the maximal case the components are integral and the eighenlattices are even. I don’t remember by heart the final result for other maximal sextics. It should be pointed in the same summary and by the way easy to get following the same approach I have pointed. The key is that even if it is no more a pure fibration - it has special fibers which are quotients of $SL(3,R)$ by the corresponding monodromy group (which indeed coincides with the maximal possible group of symmetries for the given type of sextics) - its fundamental group is exactly the fundamental group of this special quotient. Yours VK

$\bullet$ mercredi 13 février 2013 11:46:20

Dear Colleagues, Am I alone who did not receive a copy of Severine’s letter? I would be happy to know its content :) Yours VK

$\bullet$ mercredi 13 février 2013 13:43:10

Dear colleagues, I had written only to Alexandre, sorry! My answer was this: let $A, B, C$ be three inner ovals, and $D, E$ be two outer ovals of the sextic. The fundamental triangle $ABC$ is entirely contained in the non-empty oval. If $D$ and $E$ are in two different (non-fundamental) triangles $ABC$, then the conic through $A, B, C, D, E$ cuts the sextic at 14 points, contradiction. Using conics, one proves more generally that there is a natural cyclic ordering of the empty ovals, given by the pencils of lines based at the inner ovals. The empty ovals are distributed in two consecutive chains (inner, outer). The inner ovals lie in convex position in the non-empty oval. Best regards, Séverine

$\bullet$ \[14.02.13\]

Dear Viatcheslav, Séverine and all the other colleagues,

Thank you very much for this beautiful answer on the Gudkov chamber. I look forward to digest properly all that incredible technology that you and Nikulin developed. Again many thanks also to Séverine for the clever argument which I digested yesterday with great pleasure, and integrated in my notes in Section 26.10(=\[LeTouze:sec\]) pages 332–334. This gave me yesterday some motivation again to attack the very first question of all our chat room, namely Rohlin’s claim that the pencil of cubics through the 8 deep basepoints located inside the 8 empty ovals of any sextic curve of type $6/1 2$ or its mirror $2/1 6$ is totally real, hence of type I (also called orthosymmetry by Klein ca. 1881-82 and his student Weichold 1883).

In fact I (naively) hoped to prove this Rohlin claim via Poincaré’s index theorem, yet the qualitative picture (Fig. 133 on page 337) rather inclined me to believe that the proof cannot reduce to mere combinatorial topology of foliations (i.e. Poincaré’s index formula of 1885). So I am still puzzled, but perhaps an argument like Séverine’s one do the job. At any rate I would be very excited if someone manages to reconstruct this proof asserted by Rohlin (1978) if it is not too tantalizing for the brain.

Otherwise I am also much frustrated by failing to visualize totally real pencil on the three $M$-sextics, whose existence is I think predicted by Ahlfors theorem of 1950 (or better the special zero-genus case thereof known to Riemann 1857, and reworked by Schottky 1875-77, or even Bieberbach 1925 and his more respectable student Grunsky 1937). Marin warned me recently that the transition from the abstract Riemann surface viewpoint to the planar context “of Hilbert’s 16th problem” may be not so easy as I always assumed subconsciously. (If necessary, all the correspondence I received from all the colleagues is gathered in Section 24.6, p.221). Overpassing this difficulty (which I hope is not fatal) there should be on all $M$-curves (more generally dividing curves) auxiliary pencils which are totally real. Alas for $M$-sextics (even $M$-quintics), I am completely unable to trace them and know nothing about the degree of the curves involved (in the pencil). I hope to be able to tackle such questions in the future, but perhaps you have better ideas (or motivations) than I do have.

Thank you very much again for all your brilliant answers, and kind messages. All the best, Alex

$\bullet\bullet\bullet$ samedi 16 février 2013 17:54:55

Dear Alexandre, dear other colleagues,

I have managed to prove that a pencil of cubics with eight base points distributed in the eight empty ovals of a sextic $2 \cup 1(6)$ is necessarily totally real. Details will follow soon in a paper. Yours,

Séverine

$\bullet$ \[16.02.13,19h41\]

Dear Séverine and colleagues,

Congratulations for this fantastic achievement. I am sure the proof must be very beautiful. On my side I tried to work out for all sextics of type I an optical recognition procedure of the type by some synthetical procedure akin to Rohlin’s claim, yet this is still much in embryo. In particular the case of $(M-4)$-sextics is quite puzzling as it seems to contradict the version of Ahlfors theorem due to myself (existence of a totally real map of degree the mean value the number of ovals and Harnack’s bound). I hope to send you more palatable material soon, but confess that the questions look quite hard and I seem much less efficient than Séverine. So I suppose that Rohlin’s claim is one among several other (less pure) total reality result. So I look forward with great interest to see Séverine’s article.

All the best, Alex

$\bullet$ \[19.02.13\] Dear colleagues,

Many congratulations again to Séverine for your fantastic achievement. Sorry to have been brief in my last letter, as I wrote (lacking an internet connection at home) from a friend of mine who had a romantic party with his girlfriend, and I do not wanted to disturb too long his romantic evening. Meanwhile I also tried hard to concentrate on a proof of the Rohlin-le Touzé’s theorem, which still overwhelms my intelligence. The last things that I have written are on pages 336–352 (Sections 27.1, 27.2, and 27.3), but this is poorly organized and supplies no serious proof of the Rohlin-Le Touzé’s theorem.

Some few days ago, I got Theorem 27.5\[=\[total-reality-of-plane-M-curves:thm\]\] (on page 346), which (if it is correct) answers one of the question I asked in my penultimate e-mail (as well as desideratum of Alexis), namely the question of estimating the order of curves involved in a total pencil on an $M$-curve. It seems that there is always such a pencil of order $(m-2)$, i.e. two units less than the given degree $m$ of the $M$-curve. In fact, the proof is a nearly trivial adaptation of the abstract argument going back to several peoples (in chronological order Riemann 1857, Schottky 1875, Enriques-Chisini 1915, Bieberbach 1925, Grunsky 1937, Courant 1939, Wirtinger 1942, Ahlfors 1947, 1950, a myriad of Japaneses, a myriad of Russians including Golusin 1953/57, etc....., up to Huisman 2000, and Gabard 2001/2006, who else?). The point is that total reality is trivial in the case of $M$-curves since we have one point circulating on each oval (such a group moves by Riemann-Roch!!!) and so we have like a train-track with only one train on each track, hence no collision can occur and total reality is automatic. If we work with plane curves we only need to take curves of order $(m-2)$ which have enough free parameters to pass through any given distribution of $M$ points (one on each oval), and this works by looking at the residual group of points (details in the proof on page 346). So this is quite interesting but probably only a first step toward deeper things. (One could dream to recover all the Gudkov-Rohlin/Arnold congruence via this method but that looks hard work...)

After this little discovery I focused again on the Rohlin-Séverine theorem, yet without any success. So I have not more to report for the moment.

Thanks a lot for the attention, and all my congratulations again to Séverine for your deep advance. Best wishes, Alex

$\bullet$ 19.02.13 Dear Alex,

Let me ask you a question from your previous field of interest. Do you know any example of a non-Hausdorff 1-manifold which does not admit a differential structure? I heard about existence and could easily construct examples of exotic, i.e., homeomorphic but not diffeomorphic non-Hausdorff 1-manifolds. See

http://www.map.mpim-bonn.mpg.de/1-manifolds

Sincerely, Oleg

20.02.13

Dear Oleg, David and Mathieu,

Many thanks, Oleg, for your lovely question, and best greetings to the other friends. Alas my memory is failing quite dramatically, so my answer will be of poor quality. If I remember well I asked myself the same question some 3-4 years ago, but I cannot record to have ever found an answer. Thus I forward your question to David and Mathieu, the leading experts of non-metric surfaces who perhaps will supply a better answer. On my side I hope to think more seriously to your question when I see clearer with Rohlin-Le Touzé’s sextics.

Maybe a first idea is that there ought to be a (non-canonical) “twistor construction” assigning to each non-Hausdorff curve a Hausdorff surface fibered by (real) lines. This construction should go back to Haefliger’s very first note in the colloque de topologie de Strasbourg ca. 1955-1956 (yet it is not very detailed). In substance it is like a train-track construction à la Penner-Thurston…(some intuition about this is given in my article ‘Ebullition and gravitational clumping, arXiv, 2011). Do not worry if you don’t understand me, as I myself remember only vague souvenirs and are not so convinced by what I am saying!!! In fact Haefliger (ca. 1956) claims this construction only for second countable curve (even with a proviso on the fundamental group), but when I was in touch with the subject I was fairly convinced that it must work universally.

OPTIONAL REMARK: Haefliger, and Haefliger-Reeb 1957 use this construction to prove that any simply-connected curve (second countable) arises as the leaf space of a foliation of the plane. (Sketch of proof: take the twistor of the given curve which is by the exact sequence of a fibering 1-connected and (by Poincaré-Volterra) second countable, hence it is the plane, q.e.d)

So the idea would be to descend a smooth structure on the surface to get one on the curve. Alas, it is a well-known open problem whether any (non-metric but Hausdorff) surface admits a smooth structure (Spivak 1971, Nyikos, etc.) However quite puzzlingly Siebenmann 2005 (Russian Math Surveys) claims (and even prove in some details) that a PL structure exists universally on all such surfaces, merely as a consequence of Schoenflies theorem. So perhaps Siebenmann argument work as well for DIFF structures, and the metaphysical problem of Spivak-Nyikos is cracked. If this works (ask maybe Siebenmann, or an Indian in the States(=Ramachandran) who albeit not an expert was fairly convinced that there should be no asymmetry between PL and DIFF in dimension 2), then there is perhaps some chance to get a smooth structure on all non-Hausdorff curves. Of course there is perhaps a more direct strategy without transiting through surfaces.

Otherwise, regarding exotic smooth structures on curves the original reference is Haefliger-Reeb 1957 article in L’Enseignement Math. Perhaps you could quote this in your brilliant web-page.

Sorry for this vague answer, but at the moment my brain is much concentrated on this Rohlin-Le Fiedler total reality claim which still puzzles me a lot!!!

Best greetings to all, as well as to Rachel and Chiara. All the best, Alex

$\bullet$ \[22.02.13\] Dear colleagues (especially Séverine),

I worked hard (but without success) on the Le Touzé’s theorem, at least for 8 basepoints assigned on the nonempty ovals of a sextic of type $6/1 2$. If I understood well Séverine’s announcement, you rather handle the case of $2/1 6$ and assign more generally the points in the insides of the empty ovals (but of course I suppose that your argument adapts to $6/1 2$). Even in my weaker form I am not really able to conclude but send you my last thinking on the question (Section 27.4, p.352–356, esp. Fig.141). Ultimately I found a method which I call “barrages”. A special rôle is played by nodal cubics of the pencil, and I try to get a corruption with Bézout by looking at nodal curves with a barrage, i.e. such that 4 arcs of some other cubic joins pairwise the 8 basepoints distributed on the loop of the original cubic. (By the loop of a nodal cubic, I mean the unique path from the node to itself which is null-homotopic in the plane $RP^2$.) Of course I am not sure that details can be decently completed, but for the moment it is the only reasonable strategy I could imagine. I am sure that Séverine’s argument is much more elegant and convincing. My reasoning is completely conditioned by Fig.141, and I am probably too naive in believing that it reflects the general situation.

Sorry for sending you this very coarse material, and of course do not take the pain to react to this message. Many thanks again a lot to all for sharing so generously your knowledge and for all your answers. Best regards, Alex

\[25.02.13\] Dear real geometers,

I was still much fascinated by the Rohlin-Le Touzé theorem (RLT) albeit still not able to prove it. Being frustrated by my failing attempts (probably due to a lack of stubbornness and competence in algebraic geometry) I decided to speculate a bit of why it is so important or at least to explore how the statement could generalize.

In its most elementary incarnation involving pencil of lines and conics, the phenomenon of total reality occurs along infinite series stable under the operation of satellite of a real scheme (of even order). Satellite just amounts to trace each oval with a certain multiplicity $k$ (jargon obviously borrowed from knot theory). So the unifolium scheme of degree 2 (allied to a conic) gives rise to the deep nests, and the quadrifolium scheme of degree 4 gives rise by taking its satellites to an infinite series of schemes of order multiples of 4 which are totally real under a pencil of conics (assigned to pass through the deepest ovals). It seems therefore natural to ask if the satellites (e.g. the second satellite) of the Rohlin’s scheme $6/1 2$ (or its partner $2/1 6$) are also totally real (and hence of type I) under the “same” pencil of cubics as posited by the Rohlin-Le Touzé theorem. Alas I was not even able to settle this question. (Of course this seems evident (granting RLT) for a small perturbation of the algebraic double (essentially $F \cup F+\epsilon$), since total reality forces transversality of the foliation induced by the pencil with the curve.)

Next, I tried to understand what are the higher order avatars of the RLT-theorem (in the hope that it is not an isolated phenomenon as vaguely suggested by Ahlfors theorem). I found using the Rohlin-Kharlamov-Marin congruence ensuring the type I-ness (=orthosymmetry) of some $(M-2)$-schemes an (obvious) infinite series of avatars of the Rohlin’s $(M-2)$-schemes of degree 6 . Those are also $(M-2)$-schemes and total reality seems to be possible for a pencil of curves of order $(m-3)$, exactly like for the Gürtelkurve of Zeuthen-Klein (bifolium quartic with 2 nested ovals totally flashed by a pencil of line through the deep nest) or for the Rohlin’s sextic (flashed by a pencil of cubics). So it seems that the theory of adjoint curves of order $(m-3)$ plays some special rôle in this question of Rohlin-Séverine. I would be very happy if one of you knows if it is reasonable to expect an extension the RLT total reality theorem to all this schemes whose type I ness is ensured by Rohlin-Kharlamov-Marin congruence (sorry if I am not hundred percent right in crediting as I could not extract the exact history of this subliminal result). Specifically I have Conjectures 27.17 and 27.18 (page 365 and 367 resp.) which list some candidate-schemes for total reality in degree 8 and 10. If the conjectures are right, it would be of great interest to know if Séverine’s proof adapts to them. Sorry if I am too naive about the real difficulty of such problems, but I found exciting to wonder if there is something more general behind the cryptical allusion of Rohlin. Of course I presume that he derived the synthetic result a posteriori from highbrow topology (or Kähler geometry in Kharlamov’s case?), but perhaps there is a simple explanation with (“basic”) algebraic geometry and total reality as Séverine was able to do? As Oleg knows my problem is that I wasted too much time with non-metric manifolds and so forgot all the little I ever knew about algebraic geometry.

During the way, I think to have found a counterexample to the conjecture of mine (inspired by the Itenberg-Viro contraction conjecture of empty ovals), and according to which all empty ovals could be contracted simultaneously to solitary nodes. This counter-example is Thm 27.16 on page 364 (which I hope is correct and sharp as far as the degree is concerned).

Thanks a lot for the attention, and sorry for all the modest news (you surely thought about in sharper form already). All the best, Alex PS: The material summarized in this message occupies page 357-367 (Sections 27.5, 27.6, 27.7), as usual I had not much time to polish, but I hope it is still readable.

\[27.02.13\] A census of 100 octic $(M-2)$-schemes of type I satisfying the RKM-congruence, plus a little addendum for Oleg’s non-Hausdorff curves

Dear colleagues,

I have pursued some preliminary study toward the total reality phenomenon, yet merely in its combinatorial aspect prompted by the modulo 8 RKM-congruence (for Rohlin-Kharlamov-Marin) ensuring the type I of $(M-2)$-schemes of degree $2k$ with $\chi = k^2+4 \pmod
8$. Accordingly, I call an RKM-scheme any $(M-2)$-scheme satisfying this congruence. While any RKM-scheme is of type I, I do not know alas whether the converse statement is true. If it is known I would be extremely grateful if someone can tell me (and our collective chat room) the answer. Further I noticed that the list given in my previous e-mail of RKM-schemes of degree 8 can be much enlarged. If I am not too bad in combinatorics, there are precisely 100 such schemes in degree 8, all of them being potentially subsumed to the phenomenon of total reality under a pencil of quintics akin to the Rohlin-Le Touzé theorem (for sextics flashed by cubics). This modest material is to be found in Section 27.8, p.368-373 (especially Fig. 146 page 370 and Lemma 27.24, p.372, plus all the 36 Gudkov symbols on page 372). I hope of course that I missed nobody in this catalogue. Extrapolating a bit using the (hypothetical) converse statement to RKM, I would say that there are precisely 100 schemes of type I which are $(M-2)$-schemes. Is this well-known and correct?

Actually, I do not really know if all these 100 schemes are realized algebraically, but presume that most of them (all?) are. Possibly I am much too naive. Of course it is quite amazing to see that the only two RKM-schemes of degree 6 (namely $6/1 2$ and $2/1
6$) demographically explodes to a menagerie of 100 such schemes in degree $8$, but that should be no surprise for you much acquainted with the higher cases of Hilbert’s 16th problem. It would be even more crazy if all those 100 schemes (or at least a good portion thereof) are subsumed to the phenomenon of total reality. If you have some ideas on those circle of ideas, I would be extremely thankful.

Many thanks again for the patience and attention, and I hope that what I am telling is nearly correct (not too surrealist). Very best regards, Alex

PS: For Oleg, regarding my loose answer on smooth structures on non-Hausdorff 1-manifolds, I would like to add another philosophical remark related to the method of Haefliger’s “twistor”. This is of course like a thickening along a normal bundle except that there is no ambient manifold (safe the ether) and so the construction must be intrinsic. To my knowledge it was never exposed in details (albeit Haefliger’s 1st article ca. 1955-56 in Colloque de Topologie de Strasbourg uses implicitly this construction). Now my point is that albeit the twistor method looks somewhat indirect, I think that it is fairly useful. For instance, I was since 2006-07 puzzled by the naive question if the fundamental group of a one-manifold is always a free group. (Of course such non-Hausdorff curves resemble somehow graphs, whence some intuition). For instance the line with 2 origins has $\pi_1=Z$ as follows quickly from Seifert-van Kampen (and if 3 origins or 2 doubled origins then $\pi_1=F_2$ is free of rank 2). Ultimately in 2011 I found a general answer to this “freeness” puzzle by using the Haefliger twistor construction, while showing first that all open (non-metric) surfaces have free fundamental groups. (This is actually a very modest extension of the metric case, which to my knowledge is first treated in Ahlfors-Sario book of 1960, albeit it may have belonged to the folklore much earlier, say Kerekjarto, H. Kneser, Rado, in the 1920’s, Papakyriakopoulos in the 1940’s???). This material is exposed in some details in my arXiv note of ca. 2011 (Ebullition in Foliated surfaces versus gravitational clumping). I hope that those results are nearly correct but they certainly require more professional treatments and exposition than I was able to do. I hope this little remark makes perhaps more plausible that the approach via the (Hausdorffizing) Haefliger twistor is also reasonable for your problem of DIFF structures.

$\bullet\bullet\bullet$ vendredi 1 mars 2013 19:07:12

Dear Alexandre, dear other colleagues, here is the note I had promised to send you. There are still many open questions, as Alexandre wrote. It would be also interesting to know whether one could find a totally real pencil with respect to the dividing $M-2$-sextics with real scheme of indefinite type. I will think about it when I have more time. Best regards,

Séverine

(01.03.13, 22h15)

Dear Séverine and the other colleagues,

So many thanks Séverine for sending us your splendid article. I am much excited to read the details tomorrow, as myself started today to doubt about the whole result (at least in the strong form that any points 8 points distributed on the empty ovals ensures total reality). (If I am not wrong the whole phenomenon depends upon the location of the 9th base point, namely the pencil is totally real iff the 9th base point lands in the inside of the nonempty oval.) So I was much depressed and lost in my poorly organized thoughts. So your sending arrives as a true deliverance for my brain.

Many congratulations again to Séverine for this fantastic work. Very best regards, Alex

\[02.03.13\] Can total reality fail for a distribution of 8 points on the empty ovals?

Dear Séverine and the other geometers (especially Professor Nikulin),

I enjoyed much a detailed look at your splendid article full of illuminating remarks. I will probably need much more time to digest the impressive technology you use, and need to print the material to make a deeper reading (especially of the former works upon which your argument seems to depend). So many thanks again for sending us your work in so rapid delay. I wrote some naive reactions in Section 27.11, where I mostly copied your sayings, and tried to add hopefully pertinent footnotes.

Regarding your question “Can conversely any dividing curve be endowed with some totally real pencil?”, I still wonder if a positive answer is not a trivial consequence of Ahlfors theorem (compare very optionally Gabard’s Thesis 2004, page 7). However since Marin warned me in January 2013 (cf. Section of e-mails) it may be the case that the transition from the abstract conception of Riemann-Schottky-Klein to the embedded viewpoints of Hilbert-Gudkov-Rohlin is not so easy. Yet I am still confident (or naive enough) to believe that it holds true. The point seems to be primarily a matter of projective algebraic geometry, namely the question if any abstract morphism on a concrete plane curve to the line $\PP^1$ is induced by a (linear) pencil of ambient curves. This is either trivially true or trivially wrong, but alas I do not know the answer due to my failing memory about the foundations of algebraic geometry.

Your article already helped much as I suffered under the misconception that your result states that any distribution of 8 points on the empty ovals induces a totally real pencil. Your statement is much more subtle, yet personally I do not know if this stronger (universal) form of total reality is wrong! If you know a counterexample foiling universal total reality I would be very happy. It could then still be the case that there is some special sextics for which universal total reality holds true, i.e. for all octuplets distributed on the empty ovals. (Perhaps reading more carefully your article, especially the aspect related to Nikulin-Kharlamov’s rigid-isotopic classification already answers those questions?)

(The newest material of mine (as usual confusing and poorly organized) occupies Section 27.9–27.11 on pages 373–378. Here I attempted a topological approach to the existence of octuplets inducing a totally real pencil, but alas was not able to conclude, presumably because I know too little on the predestination process creating the 9th basepoint as a function of the 8 assigned ones.)

Many congratulations again to Séverine for this breakthrough. Best regards, Alex

dimanche 3 mars 2013 18:07:57 a new version with small corrections?

Dear Alexandre, dear other colleagues, I owe you some apologies: the Theorem was slightly incorrect, as Alexandre pointed out. I let you discover this new version, where I have reformulated the Theorem, and added a few words in the end of the proof. Best regards, Séverine

(04.03.13) Dear Séverine and the other colleagues,

Many thanks for the new version. In fact, it seems that the main change is that you now assign the 8 basepoints ON the empty ovals instead of IN their insides. Rereading my previous message, I realize that I misstated your original statement and so it is pure chance that assignation on the ovals turned out to be “more correct”.

Your fascinating article gave me new forces to think about the problem, but alas still without success. For instance, I still do not know if there exist octuplets (on the empty ovals) failing to induce a totally real pencil. Of course assigning them in the insides gives more freedom, but presently it looks to me harder to ensure total reality. So despite your correction, it could still be the case (in my modest understanding) that the pencil is total for all octuplets chosen in the insides of the empty ovals. Perhaps you know a counterexample to this strongest form of the statement?

Many thanks again for the article, which guided much my thinkings. I hope to send you more exciting news soon, but the whole problem which you call “the lost proof of Rohlin” seems to me still much out of reach. All the best, Alex

$\bullet$$\bullet$$\bullet$ answer to Alexandre’s questions (mardi 5 mars 2013 13:30:42)

Dear Alexandre, dear other colleagues, let me try to answer the question with a new formulation.

Assume first that the base points are distributed [*inside*]{} of the empty ovals. Applying your nice “dextrogyration argument” to all nine ovals gives the following lemma:

[*The pencil is totally real iff 9 lies inside of the non-empty oval $O$ and outside of the empty ovals.*]{}

If 9 is outside of $O$, the bad cubics are as shown in Figure 2 of the paper. If 9 is inside of an empty oval $X$, the bad cubics have an oval passing through the two base points 9 and X only, and this oval is entirely contained in the empty oval denoted also $X$. To get rid of this latter possibility, it suffices to take the base points [*on*]{} the empty ovals.

In ii), I give an explicit description of the pencil, valuable for any generic choice of the eight base points [*inside*]{} of the eight empty ovals. (It turns out that the only possible non-generic situation is that of a pencil with a double base point $9=2$, this means that the points 1, ..8 lie on a nodal cubic with node at 2.) Recall that 2 is the base point chosen in the extreme inner oval forming a positive pair with $O$. For this pencil, the only possibly bad cubics are those with an oval passing through 9 and 2 only. To grant total reality, it suffices to choose the base point 2 [*on*]{} the corresponding empty oval, the other base points lie arbitrarily in the inside discs of the other empty ovals. Thus, your conjecture 27.29\[=\[SRLT:conj\]\] is true, and an even stronger result holds for the sextic with six inner ovals. Best regards, Séverine

\[07.03.13\] Little news from Alex, and so many thanks to Séverine for the answer

Dear Colleagues,

First many thanks to Séverine for your very detailed answer (which I will study in detail tomorrow). Sorry for being always a bit differed in time due to my lack of internet at home.

I added some material in my loose notes. In Section 28.1–28.2 (pp.384–392), I tried once more to explore the grand programme that Rohlin might have had in mind, namely total reality and its connection with his maximality conjecture. As I often said it seems to me that the missing link could be played by Ahlfors theorem, or perhaps Rohlin had a grand vision that he could arrange total reality by purely synthetical processes extending in all degree the already tricky theorem of Rohlin-Le Touzé in degree $m=6$. This idea when explored in full looks to me extremely vertiginous, but its net impact would be a sort of upper bound upon the complexity of Hilbert’s 16th problem, and in some sense subsume all prohibitions (à la Gudkov et cie.) to the paradigm of total reality. All this necessitates to be made much more precise, but I \[have\] attempted to make a psychoanalysis of what Rohlin may have had in the brain, without that he himself ventured to put it on the paper due to his own modesty and pragmatism.

Next I discovered the little Theorem 28.7\[=\[Thom-Ragsdale:thm\]\] (p.393), which is just a matter of making explicit the consequence of Thom’s conjecture (=Kronheimer-Mrowka theorem) as it pertains to Hilbert’s 16th problem. The result is the lovely estimate[^42] $\chi \le k^2$ for a curve of type I and degree $2k$. With this I realized that my former counterexample (with the scheme $20$ in degree 8) to CCC(=collective contraction conjecture) is actually killed by Thom, and realized (later only!!) that it is also killed by Rohlin’s formula. So CCC is again resuscitated but probably not for long!?

Then I tried to make a comparative study of Rohlin’s formula versus the Thom obstruction. It seems that the latter is often implied by Rohlin’s formula, but not always. More in Section 28.4 (p.393). It seems however that at least for degree $m \ge 10$ there is some cases where Thom really affords new information not covered by Russian congruences or Rohlin’s formula (cf. Thm 28.11, p.396). Finally using the Gudkov table in degree 10 (=Fig.148 on page 395), I got some naive hope to disprove the Rohlin maximality conjecture, but this quickly turned into disillusion (cf. Point 3 on p.396–397).

Sorry for all these messy remarks, yet I found the rôle of Thom quite pleasant. I am sure that this is not new, and that I read it somewhere, but again could not recover where precisely. (I thought it was in Degtyarev-Kharlamov 2000’s survey but apparently not, though Kronheimer-Mrowka is alluded to.) If you remember some anecdotes about the rôle of Thom’s conjecture in Hilbert’s 16th problem, and who puts it first into action as a such, I would be extremely happy to insert your remarks in my (messy) survey.

Thanks a lot for the attention, Best regards, Alex

Constructions
=============

\[29.03.13\] Construction of algebraic curves seems a syllogism since they are nearly God-given. Perhaps the word contemplation looks more appropriate, but clumsy. Despite existence of Gods, the art of tracing of algebraic (plane) curves goes back to time immemorial Newton, Plücker 1839 [@Plücker_1839], Zeuthen 1874 [@Zeuthen_1874], with the modern era usually identified by Harnack 1876 [@Harnack_1876], Klein 1876, Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege]). The game is especially interesting over $\RR$, else nearly everything follows from Riemann.

Much of the elementary aspects can be treated by the primitive method of small perturbation, which nearly gives a good picture of what happens in degree 6. This is how worked Plücker, Klein, Harnack, Hilbert, Ragsdale, Brusotti, etc. However already in degree 6 the classical method starts showing some limitation. Albeit the Gudkov curve can be distilled by small perturbation, it requires an extra twist by means of Cremona transformations (at first difficult to visualize). It took the community ca. 8 decades (including such masters as Hilbert, Ragsdale, Brusotti, Petrovskii, Gudkov first not an exception) until to discover the fairly trivial picture traced by Gudkov ca. 1972 (Fig.\[GudkovCampo-5-15:fig\]) exhibiting a curve with topology $\frac{5}{1}5$.

Ca. 1980 Viro described how to dissipates more complicated singularities, allowing experts to create more funny curves refuting most of the conjectures erected along the primitive method. For instance also Gudkov’s sextic appears fairly trivially when one knows how to smooth a triplets of ellipses tangent at 2 points, compare Fig.\[Viro3-15:fig\]c. A variant of Viro’s patchwork due to Itenberg (called the $T$-construction) is purely combinatorial and permitted to disprove severely the Ragsdale conjecture (cf. Fig.\[Itenberg:fig\]), as well as our naive Thom estimate $\chi\le k^2$ for dividing curves.

Constructing the two maximal $(M-2)$-schemes {#const-total-(M-2)-schemes:sec}
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\[05.01.13\] As to the existence of the two maximal $(M-2)$-schemes (namely $\frac{6}{1}2$ and $\frac{2}{1}6$), they can be constructed (as observed in Gudkov 1974/74 [@Gudkov_1974/74 p.42]) by a slight modification of Hilbert’s method. Let us reproduce his figure (Fig.4, p.16). This gives (after smoothing) the left-side of Fig.\[Gudkov-Hilbert-modified:fig\]. Alas, this is not the desired scheme. Is there a mistake in Gudkov at this place? Apparently not as it seems approved in A’Campo 1979 [@A'Campo_1979] (alas no detail).

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The right-part of Fig.\[Gudkov-Hilbert-modified:fig\] is just a variant inspired from Hilbert’s configuration. This has again the wrong real scheme. Since Gudkov does not seem to give exactly what he claims, we must rely on some do-it-yourself endeavor. A naive idea gives Fig.\[GudHilb2:fig\], failing again to have the correct scheme.

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Let us now work more systematically. The key is first to make Walt-Disney pictures of Hilbert’s method à la Gudkov. This involves a simplified art-form, far from geometrically realist, but topologically faithful and more malleable. This produces the following pictures (Fig.\[GudHilb3:fig\]). The trick of Hilbert’s method is to let oscillate an oval across an ellipse while smoothing their union (cf. left part of Fig.\[GudHilb3:fig\]). Such oscillations are Bézout compatible: each oscillating quartic intersects 8 times the ellipse. A posteriori it is a simple matter (Hilbert’s method) to realize such oscillation by rigid algebraic curves suitably perturbed by lines arrangements. Yet it is valuable first exploring the softer topological figures as to understand which oscillation is able producing a prescribed topology (e.g. $\frac{6}{1}2$ and its companion $\frac{2}{1} 6$).

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The variant of Hilbert’s construction involves letting oscillate various ovals across the ground ellipse. On the middle row of Fig.\[GudHilb3:fig\] we let oscillate thrice one oval and once the opposite oval with respect to some ground ellipse. Smoothing gives some $(M-2)$-curves not realizing the desired schemes. Choosing instead a triple oscillation of the upper oval of the quartic combined with a simple oscillation of the nearby oval gives the desired schemes (right row of Fig.\[GudHilb3:fig\]) with either 6 outer ovals (top) or 6 nested ovals (bottom). Gudkov was right albeit his discourse was not in perfect adequation with his picturing.

It remains to geometrize such oscillations à la Hilbert. This is an easy matter, except that realist pictures require judicious scalings to make things visible. The first mode of vibration leading to $\frac{2}{1}6$ is geometrized on Fig.\[GudHilb5:fig\] below.

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Some few comments on this figure: first one has the two blue ellipses forming a quartic $C_4$. Next one has 4 dashed lines (another quartic). Perturbing slightly the former along the other (within the pencil spanned by both) gives another $C_4$ traced in black. This has the virtue of oscillating across the circular ellipse (in blue). Finally, smoothing their union gives the sextic in red realizing the desired real scheme $\frac{2}{1}
6$ (i.e. 2 ovals captured in one and 6 outside).

The other scheme $\frac{6}{1}2$ is obtained similarly via the following system of oscillations (Fig.\[GudHilb6-12:fig\]) geometrizing the bottom-right part of Fig.\[GudHilb3:fig\]:

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Using either the schematic pictures or the more geometric one it is an easy matter to see that both curves just traced are dividing. This follows as usual (Fiedler’s law) by checking that all smoothings are compatible with complex orientations (cf. Fig.\[GudHilbdividing:fig\] below).

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Hence according to Ahlfors theorem there must be a total pencil of curves. Actually Rohlin claims much more that any sextic realizing those schemes is totally real under a pencil of cubics, but his argument has never been published[^43]. A bit like Hilbert in 1900, Rohlin 1978 says that his proof is too cumbersome to be written down. As we know Hilbert’s 2nd assertion that there is only two $M$-sextics was refuted by Gudkov some 7 decades latter, so it is not impossible that Rohlin’s claim is fallacious as well. Of course it can also be the case that Rohlin’s claim on the type I of the maximal $(M-2)$-schemes is correct, but that total reality via a pencil of cubics is erroneous. However as we noted cubics leads to a mapping-degree of $3 \cdot 6 -8=10$, in adequation with Gabard’s bound $r+p$ on the degree of circle maps.

At this stage the naivest thing-to-do is to convince that there is no trivial counterexample to Rohlin’s claim. So we trace more oscillations to get the following pictures (Fig.\[GudHilb8:fig\]). Some noteworthy species appear especially the remarkable scheme $\frac{4}{1}4$, occupying the central position of Gudkov’s table (Fig.\[Gudkov-Table3:fig\]). Our specimen is dividing and it looks hard to get the same scheme in the nondividing way (though Rohlin 1978 asserts its existence). Speculating that this scheme is of type I, while admitting the truth of Rohlin’s maximality conjecture, then all 3 sextics schemes dominating $\frac{4}{1}4$ would agonize along a blue sky catastrophe! Gudkov would be wrong and Hilbert right! Of course this seems a too apocalyptic scenario, yet up to now our text does not entail this option!

Also difficult to find are the schemes $\frac{5}{1}3$ and its mirror $\frac{3}{1}5$. Apart those exceptions, Hilbert’s method offers all possible $(M-2)$-schemes.

[*Insertion*]{} \[08.02.13\] For a Harnack method realization of $\frac{3}{1}5$, see Fig.\[HarnaGudkov3-15XXL:fig\] much below, while $\frac{5}{1}3$ truly requires the method of Gudkov (cf. Fig.\[GudkovCampo-5-15:fig\]).

If, via a small perturbation, one merges together $2$ small ovals on Harnack’s or Hilbert’s curve (cf. détail on Fig.\[GudHilb8:fig\]), then one gets the $(M-1)$-schemes $\frac{1}{1}8$ resp. $\frac{8}{1}1$. The other $(M-1)$-schemes of Gudkov’s table are somewhat harder to exhibit, except of course if one is aware of a large deformation able to extinct the inner oval (case of Harnack) or the outer oval (in Hilbert’s case). This contraction of empty ovals is actually possible via Itenberg 1994 [@Itenberg_1994]—using the apparatus of Nikulin’s (1979/80 [@Nikulin_1979/80]) (rigid-isotopy classification via K3 surfaces)—but of course this is surely not the most economical argument for our purpose (known to Gudkov 1969 or earlier).

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It seems clear that we have exhausted the faculty of Hilbert’s method (and its variation where the vibration is dissipated on several ovals). Some naive questions: what can be obtained by perturbing an arrangement of lines (in general position)?

To go further one is helped once more by Gudkov 1974/74 [@Gudkov_1974/74 p.42] asserting that the scheme $\frac{4}{1} 5$ can be gained by a modification of Harnack’s method. Alas, Gudkov makes no picture but was aware of this at least since 1954.

[*Insertion*]{} \[08.02.13\].—For a picture of this cf. Fig.\[HarnaGudkov4-15:fig\] much below.

\[07.01.13\] Of course it is also possible to apply Hilbert’s oscillations to a Gürtelkurve (or other quartics), cf. Fig.\[GudHilb9:fig\]. Alas the list of schemes so obtained is not very exciting (no new species over the previous vibrations).

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Constructing the indefinite types (Brusotti, Rohlin 1978, Fiedler 1978, Marin 1979, plus do-it-yourself) {#indefinite-types:sec}
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\[02.01.13\] Recall the definition (Rohlin 1978), a scheme is of [*indefinite type*]{} if it admits representatives of both types I and II, in the sense of Klein 1876, i.e. curves which are both dividing and not. This section aims to construct all schemes of indefinite type in degree 6 as to understand in full details Rohlin’s theorem (\[Rohlin-type:thm\]) enhancing Gudkov’s table by the data of Klein’s types. All the strategic information is tabulated in Fig.\[Gudkov-Table3:fig\], but each bit of coloring involves a little fight with the geometrical substratum. Again the ideas are purely those of Rohlin and his school, especially Fiedler. In Rohlin’s 1978 survey [@Rohlin_1978] detailed constructions are not given. After completion of this section, we noted that full details are given in Marin 1979 [@Marin_1979 p.57–58], whose constructions differ slightly from ours, but settling one case we failed to detect alone, namely the type $\frac{8}{1}_{II}$.

$\bullet$ First, consider the scheme $\frac{2}{1}2$. Smoothing positively a triad of conics gives the dividing curve on the left of Fig.\[R2-12:fig\].

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On the other hand, starting from a triangular configuration of ellipse (center part of Fig.\[R2-12:fig\]) one may by free-hand drawing (without taking care of orientation) arrange the real scheme to be the prescribed one. After reporting signs of our chosen smoothing we find them to be all negatives. At this stage I thought the curve to be dividing. However, right below one of the orientation is reversed but the smoothing effected left unchanged. Now it is of mixed signs, so the curve is in fact nondividing. On the right-top part of Fig.\[R2-12:fig\] is depicted another free-hand drawing realizing the given given real scheme. Mixture of signs implies the nondividing type of this curve.

$\bullet$ On smoothing positively the configuration on the center-bottom part of Fig.\[R2-12:fig\] we get the (dividing) curve on the left-bottom of Fig.\[R2-12:fig\] which belongs to the real scheme $\frac{1}{1}5$. Next starting from the 3 ellipses, we got the miniature figure on the bottom of Fig.\[R2-12:fig\] who alas had not the right number of ovals. We thus started anew for the “radioactive” (triangular) triad of ellipses to find the right-bottom curve on Fig.\[R2-12:fig\] which has the correct real scheme and is nondividing (as it involves mixed signs).

$\bullet$ On smoothing positively the “radioactive triad” of ellipses for the prescribed orientation gives the dividing curve on the left of Fig.\[R4-1:fig\]. This belongs to the scheme $\frac{4}{1}$ (i.e. 4 ovals nested in one big oval and nothing outside). It is easy to trace the same scheme using as template the “atomic triad” of ellipses, cf. middle-part of Fig.\[R4-1:fig\], and checking orientation one finds a mixture of signs imposing the nondividing character of this curve. Another option also yielding a nondividing curve is given on the right-part of Fig.\[R4-1:fig\]

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$\bullet$ Consider now a triad of ellipses with two ellipses invariant under rotation by $90$ degrees, plus one circle pinched in between. A positive smoothing creates the archipelago sextic on Fig.\[R8-1:fig\], which is dividing and of real scheme $\frac{8}{1}$ (i.e. $8$ ovals captured in a bigger one and nothing outside). It remains to find a nondividing realization of this scheme, cf. for this Marin’s picture=Fig.\[GudHilbMarin:fig\] below.

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$\bullet$ Dragging down the archipelago circle gives a configuration of ellipses smoothable positively to the scheme $\frac{3}{1}3$, cf. Fig.\[R3-13:fig\](left). After several infructuous attempts (depicted as miniatures) one finds the strange triad of conics on the right-part of Fig.\[R3-13:fig\] which admits a smoothing belonging to the same real scheme, but which is nondividing due to mixed signs.

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$\bullet$ After some patience and many trials (especially if one is tired) one finds another configuration of ellipses smoothable positively to the scheme $\frac{5}{1}1$, cf. left of Fig.\[R5-11:fig\]. Besides, one finds quickly the right-part of Fig.\[R5-11:fig\] belonging to the same real scheme, yet nondividing due to mixed signs.

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[**What remains to be constructed?**]{} At this stage we are nearly finished (compare the list of schemes we explored with those marked by rhombs on Fig.\[Gudkov-Table3:fig\]). It remains us to find the scheme $\frac{4}{1}4$. As we did not found it presently as a perturbation of 3 ellipses, and since this lies quite near (on Gudkov’s table=Fig.\[Gudkov-Table3:fig\]) to Gudkov’s $M$-sextic (notoriously difficult to construct) one is imbued of some suitable respect. Possibly it is impossible to exhibit as a deformation of (transverse) 3 ellipses. Notice yet that the curve $\frac{4}{1}4$ exists as shown by a variant of Hilbert’s method (cf. Fig.\[GudHilb8:fig\]). However presently this only realizes the scheme in the dividing way, whereas Rohlin claims this type to be indefinite.

\[12.01.13\] A somewhat mystical way to solve this question involves taking a curve lying just above the scheme $\frac{4}{1}4$, while contracting an empty oval via passage through a solitary node. (Remember this to be possible by Itenberg 1994 [@Itenberg_1994].) Reading the deformation backward it follows from Klein’s remark (1876)(=Marin’s theorem 1988 [@Marin_1988]) that the resulting curve has type II. However there is probably a more elementary proof by looking at the scheme $\frac{4}{1}5$, which according to Gudkov (1974 [@Gudkov_1974/74]) can be exhibited by a variant of Harnack’s method, while its mirror $\frac{5}{1}4$ is harder to construct (Gudkov 1954 [@Gudkov_1954] even claiming erroneously its non-existence).

\[07.02.13\] One elementary way to realize $\frac{4}{1}4$ in type II involves a modification of Harnack’s method depicted below (Fig.\[indef414:fig\]). (This is inspired by Gudkov’s text, but alas no picture there). This is the sort of bird hard to tackle down if one is tired. Beware also that in practicing Harnack’s method one never finds directly what one is seeking (I found this while searching $\frac{4}{1}5$.)

The trick here is that we leave much room between the vertical lines effecting Harnack’s oscillations. So we start with the 3 high vertical lines, and a slight perturbation of the circle union the horizontal line produces a cubic $C_3$ oscillating thrice about the horizontal line $L$. The reducible quartic $C_3\cup L$ is then perturbed by a quadruplet of lines, which again is much stretched so as to effect another intermediate vibration. Then we have an $M$-quartic $C_4$ oscillating 4 times across $L$. Via the same trick $C_4\cup L$ is perturbed by a quintuplet of vertical lines to produce a $C_5$ oscillating 5 times across $L$. Then using Brusotti’s theorem (that German workers used subconsciously it seems or used ad hoc tricks to complete their perturbations) we have two ways to smooth $C_5\cup L$ to get a smooth $C_6$. Taking caring of orientations, the first depicted choice leads a curve of type I, whereas the second involves a negative sign and therefore produces type II. It is easily checked that both curves belong to the scheme $\frac{4}{1}4$.

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In a similar way, we do not have yet constructed the type II incarnation of the scheme $\frac{8}{1}$. Again in somewhat sloppy fashion, one could argue by contracting successively two empty ovals in Hilbert’s $M$-curve (scheme $\frac{9}{1}1$), namely the one outside and one inside the nonempty oval. Granting such a deformation through two (successive) solitary nodes, Klein’s remark implies the resulting curve being of type II, and we are done. Yet I presume there must be a more elementary construction.

\[17.01.13\] Indeed one such is sketched in Marin 1979 [@Marin_1979 p.57, very bottom left of the table]. Let us reproduce Marin’s picture as Fig.\[GudHilbMarin:fig\]b:

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Marin’s trick here is to start from 2 ellipses tangent at one point but transverse elsewhere (Fig.a). Perturbing this by a suitable quadruplet of lines as in Hilbert’s method gives a quartic $C_4$ oscillating as depicted on Fig.a and with 3 ovals only. Hence the $C_4$ is nondividing (Klein’s congruence), and so is a fortiori the resulting sextic $C_6$ (as the nondividing character is dominant in the genetic sense), which realizes the $(M-1)$-scheme $\frac{9}{1}$ (Fig.a). A simple conjunction of two inner ovals yields the $(M-2)$-scheme $\frac{8}{1}$ (Fig.b), we were really interested in (and which is again of type II for the same genetical reason).

With this trick we can construct several other curves, depicted on the second row of Fig.\[GudHilbMarin:fig\], in particular we get the scheme $10$ as well as $\frac{8}{1}_{II}$ via a variant of Marin avoiding tacnodality. The little price to pay is that we concede two imaginary intersections between the ground ellipses so that the nondividing character of the $C_4$ (unnested) has to be derived by some ad hoc argument (e.g. Klein’s in (\[Klein-unnested-quartic-nondividing:lem\]), or Arnold’s congruence $2=\chi=p-n\equiv k^2 \pmod 4$, or Rohlin’s formula $0=2(\Pi^{+}-\Pi^{-})=r-k^2$ or even Bézout modulo the highbrow contraction conjecture CCC, cf. Sec.\[CCC:sec\]). Of course there must also be an elementary argument by noticing that the two imaginary intersections of both ellipses are “connecting” different halves, so that when smoothed as shown the resulting curve is nondividing. Once this $C_4$ is known to be nondividing the depicted $C_6$ is likewise by virtue of the genetical dominance of nondividingness. All this argument looks tricky but is in reality trivial (think-yourself, and compare optionally Rohlin 1978, Fiedler 1981 [@Fiedler_1981], Marin 1979 [@Marin_1979], and maybe Gabard 2000 [@Gabard_2000]).

As knowledge advances it will perhaps become as difficult to find new truths as to discover old mistakes. E.g., is Falting’s proof of Mordell correct? Is Freedman’s proof à la Bing reliable? Is Perelman’s proof of Poincaré really eclectic? If not should we retire him the million. No because because it was never accepted. Finding mistakes in those venerable implementations will perhaps be as challenging as claiming new truths? At any rate the game is always pleasant.

Gudkov’s sextic $\frac{5}{1}5$ (Gudkov 1969, 1973, etc.) {#Gudkov:sec}
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\[24.01.13\] Several constructions are available, but first some historical remarks.

$\bullet$ The very first treatment appears in D.A. Gudkov’s Doctor Thesis (1969 [@Gudkov_1969-Doctor's-Thesis]) under Petrovskii and the liberal supervision of Arnold (apparently none of the supervisors were able to digest the full swing of the candidate Dmitrii Andreevich). Upon this Polotovskii 1996 [@Polotovskii_1996-D-A-Gudkov] comments as follows: “It is interesting to remark that the first proof of this fact in \[18\](=1969 [@Gudkov_1969-Doctor's-Thesis]) was extraordinarily complicated. It takes up $28$ pages of text, is a “pure existence proof”, and was obtained by means of a combination of the Hilbert-Rohn method with quadratic transformations. Shortly after D.A. Gudkov suggested significantly simpler [*constructions*]{} of curves having this scheme, see \[19\](=1971 [@Gudkov_1971-const-new-ser-M-curv]), \[21\](=1973 [@Gudkov_1973-const-curve-deg-6-type-515]), \[23\](=1974/74 [@Gudkov_1974/74]).”

$\bullet$ This complicated proof was published in Gudkov-Utkin 1969/78 [@Gudkov-Utkin_1969/78] (English transl. issued in 1978).

$\bullet$ New simpler constructions, are due to Gudkov and to be found in Gudkov 1971 [@Gudkov_1971-const-new-ser-M-curv], or in [@Gudkov_1973-const-curve-deg-6-type-515]), reproduced in his survey Gudkov 1974 [@Gudkov_1974/74].

$\bullet$ This is also reexposed in A’Campo 1979 [@A'Campo_1979].

$\bullet$ Viro 1989/90 [@Viro_1989/90-Construction p.1076] also emphasizes Gudkov’s initial construction “was rather complicated” (an euphemism as compared to Polotovskii’s prose above). His second proof reduces “to the first stage of Brusotti’s construction, i.e., the classical small perturbation of the union of the curve and the line.” Yet the whole difficulty is to find a quintic oscillating 5 times across the line while enveloping 5 ovals in one “wave oscillation” while leaving one oval outside (cf. Viro’s figure 12 in , p.1077). According to Viro (, p.1076): “It was only in 1971 that Gudkov \[11\](=1971 [@Gudkov_1971-const-new-ser-M-curv]) found an auxiliary curve of degree 5 that did this.”

Hence Gudkov had two constructions of $\frac{5}{1}5$:

$\bullet$ A first very complicated one, published in Gudkov-Utkin 1969/78 [@Gudkov-Utkin_1969/78] (English transl. published in 1978) or Gudkov 1973 [@Gudkov_1973-const-curve-deg-6-type-515] (using Cremona), and

$\bullet$ a much more elementary one (à la Brusotti) given in Gudkov 1971 [@Gudkov_1971-const-new-ser-M-curv], by finding an $M$-quintic oscillating appropriately about a line, while smoothing their union.

(I don’t know upon which version is based A’Campo’s account, presumably the second simpler variant, yet A’Campo uses Cremona.)

Of course since Viro in the early 1980’s, Gudkov’s sextic may also be exhibited by Viro’s patchwork; or as a perturbation of three ellipses tangent at 2 points like Hawaiian earrings. This involves yet a deep understanding of how to dissipate such higher singularities. The interested reader can look at Fig.\[Viro3-15:fig\]c.

Now let us describe once more Gudkov’s trick (source used Gudkov 1974 [@Gudkov_1974/74 p.42–43] and some more détail in A’Campo 1979 [@A'Campo_1979 p.12–13]). This is artwork of the best stock (cf. Fig.\[GudkovCampo-5-15:fig\]).

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$\bullet$ The first step is easy and consists to perturb a line at one of the 8 flexes of the quartic $C_4$ with $r=4$ slightly so that it creates 3 nearby intersections. Look at the fourth intersection, and from here trace two secant intercepting some other oval as shown, while cutting 2 nearby point on the “large” oval such that the line through them cut a little teats on the large oval. That all this can be achieved is already clever and explained in detail, first in Gudkov 1971 [@Gudkov_1971-const-new-ser-M-curv] or in A’Campo 1979 [@A'Campo_1979].

$\bullet$ The 2nd picture right below is merely a qualitative redrawing of the first.

$\bullet$ The 3rd picture shows the transformation of the $C_4$ experimented under the Cremona transformation centered at the 3 points $4,5,6$, mutating it into a quintic $C_5$. One way to argue is via the birational invariance of the genus, keeping the value $g=3$ constant. Hence as the image curve has 3 nodes (arising as the intersection of the fundamental triangle through $4,5,6$ with $C_4$), it must be a quintic. Another way to argue is to remember the definition of Cremona as the projective (rational) map induced by the linear system of conics through the 3 basepoints (located on the large oval). Hence the pullback of a line is a member of the system, cutting the $C_4$ along 8 points, but 3 of them being assigned, we find $5$ for the degree of the image of $C_4$. Likewise the image of the diagonal line $L$ intersecting only the large oval $\alpha$ is of degree 1, hence a line. To understand the Cremona-map picture of Gudkov, one must keep in mind that Cremona contracts any edge of the triangle $4,5,6$ to the opposite edge of this triangle (and viceversa its explodes each basepoint to the opposite side of the triangle). The map being actually an involution (order 2). So the 4 chambers residual to the triangle are preserved. It is then fairly easy to check that Gudkov’s picture is realist, where tildes are images under Cremona. Life becomes easier if we number some few points on the $C_4$, while denoting by the same letters their images under Cremona (omitting the tilde for simplicity), compare Fig.\[GudkovCampo-5-15:fig\]. It remains to convince that the location of $\tilde{\gamma}$, $\tilde{\delta}$ is as depicted by Gudkov.

The line $L$ is imagined as invariant under Cremona. In fact if we remove the 3 fundamental lines it remains 4 open triangles (homeomorphic to a cell ${\Bbb R}^2$) which are preserved. An involution of the plane has necessarily a fixed point (Brouwer, Kerekjarto, Smith, etc.) in fact a line or a singleton of fixed-points depending on whether it reverse or preserve orientation. The usual formula for Cremona $$(x_0, x_1, x_2)\mapsto(x_0 x_1, x_1 x_2, x_0 x_2)$$ shows that $(1,1,1)$ is fixed, and solving the fixed point equation (outside the fundamental triangle whence all $x_i\neq
0$) gives $(1,1,1)=\lambda(x_1, x_2, x_0)$ as unique solution. So the fundamental triangle splits ${\Bbb R}P^2$ in 4 chambers preserved under Cremona. How are they permuted? If we normalize the sign of the first coordinate as positive then we have the following signs distribution corresponding to the chambers $$I=(+,+,+),\; II=(+,+,-),\; III=(+,-,+),\; VI=(+,-,-).$$ The first chamber is preserved by Cremona. The second mutates to the fourth. The third maps to the second, and the fourth maps to the second. This looks a bit anomalous for by functoriality we would have expected that an involution induces an involution on the set of components (functor $\pi_0$).

Changing the formula to the one written down in Gudkov 1974 [@Gudkov_1974/74 p.43] gives $$(x_0,x_1,x_2)\mapsto (x_1 x_2, x_0 x_2, x_0 x_1),$$ and we get $$I\mapsto I, II\mapsto(-,-,+)=II, III\mapsto (-,+,-)=III, \textrm{
and } VI\mapsto (+,-,-)=VI,$$ which is more pleasant. Thus after mutation the ovals $\gamma,
\delta$ stays in the same chamber. Yet this is not enough for if they would lye like $ \gamma, \delta$ (without tilde) then we would get the scheme $\frac{3}{1}7$, which is prohibited either by Hilbert-Rohn-Gudkov or by Rohlin 1972 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov] proof of Gudkov’s conjecture. (Remind Rohlin’s original proof to contain a little flow, repaired either by Rohlin via Atiyah-Singer or by Guillou-Marin!). So here we are quite close to adding another dramatic twist in the Hilbert-Gudkov saga.

However it is more realist that a more thorough examination of the Cremona map shows the location of $ \tilde\gamma, \tilde\delta$ to be the one depicted by Gudkov. Indeed the chamber (say $III$) containing $ \gamma, \delta$ is invariant (like any other). However the line $L$ is also invariant and divides the chamber $III$ in two pieces which have to be exchanged by the Cremona involution. It suffices indeed to use the topological classification of involutions in the plane ${\Bbb R}^2$ à la Brouwer, to notice that in all cases (orientation reversing or not) the involution is either a reflection about a line or about a point (rotation). In both cases the residual components of an invariant line are exchanged. Hence chamber $III$ splits in two halves ${III}_{+}$ and ${III}_{-}$, where the former contains $\gamma, \delta$. Their images have to lye in the other chamber ${III}_{-}$, and Gudkov’s depiction is verified. In fact looking at the image of the point $x$ as mapped to $6=7=8$ shows that Cremona restricted to $III$ acts as a rotation (having one fixed points). More algebraically, solving the fixed-point equation $(x_0,x_1,x_2)=\lambda (x_1x_2, x_0 x_2, x_0x_1)$ shows that $$x_0=\lambda x_1 x_2=\lambda^2 x_0 x_2^2$$ so that $x_2^2=1/\lambda^2$, and likewise—by repeating the calculation or anticipating it by symmetry—we find $x_0^2=1/\lambda^2$, and $x_1^2=1/\lambda^2$. Thus up to homothety we have $(x_0,x_1,x_2)=(1,1,1)$ modulo the 4 possible variations of signs $(+,+,+)$, $(+,+,-)$, $(+,-,+)$ and $(+,-,-)$. We conclude that Cremona has exactly 4 fixed points (one in the barycenter of each chamber). So in particular Cremona is orientation preserving (within each chamber).

$\bullet$ The fourth picture (of Fig.\[GudkovCampo-5-15:fig\]) contains also a little trick, namely the possibility to smooth the node (at $7=8$) of the trinodal quintic $C_5$ is such a way that its pseudoline penetrates slightly inside the line $\widetilde L$. Once this is done it suffices to smooth à la Brusotti $C_5 \cup
\widetilde{ L}$ to obtain the desired Gudkov sextic. And the miracle is full. Why did it took so long (ca. one century from Harnack up to Gudkov) to discover this curve? Why Hilbert missed it? Admittedly the construction is quite tricky, but completely elementary. Up to our knowledge there is not any further simplification in this second Gudkov proof, apart perhaps via Viro’s patchwork or dissipation method of higher singularities, which probably require more highbrow technologies making them didactically hard to compete with Gudkov’s construction[^44]. As a last (sentimental) outcome, look how the quintic $C_5$ of Fig.\[GudkovCampo-5-15:fig\] resemble a portrait of its happy discoverer, especially $\tilde \gamma, \tilde \delta$ are like the eyes, and $\tilde \beta$ the smiling mouth of Gudkov near to crack the centennial problem.

Finally it is plain from Gudkov’s curve to derive curves with less ovals, especially the $(M-1)$-curve $\frac{5}{1}4$ and the $(M-2)$-curve $\frac{5}{1}3$ (e.g. by changing the smoothing at the nodes $2$ and $1$). Those curves were notoriously hard to construct, and no construction independent of Gudkov’s is known. Using Fiedler’s signs-law it is plain that the curve $\frac{5}{1}3$ so constructed is of type II, as it should by virtue of say Arnold’s congruence.

If instead we change the smoothing in the inside of the oval along the smiling mouth $\tilde \beta$ of Gudkov, then we get the $(M-1)$-scheme $\frac{4}{1}5$, and the $(M-2)$-scheme $\frac{3}{1}5$. Those were however much easier to construct by a variant of Harnack’s method (as reported in Gudkov 1974); compare indeed our Fig.\[HarnaGudkov4-15:fig\] and \[HarnaGudkov3-15XXL:fig\].

Finally we note that we may also obtain the $(M-2)$-scheme $\frac{4}{1}4$ in type II by smoothing the Gudkov configuration $C_5\cup \widetilde{L}$. However there is surely a more elementary approach via $\frac{4}{1}5$ constructed by a variant of Harnack; yes indeed compare Fig.\[indef414:fig\].

Diophantine and probabilistic aspects {#Diophantine-and-proba:sec}
-------------------------------------

\[26.01.13\] Why did it took so long to discover Gudkov’s sextic? Is it only because it is the most secret part of the pyramid (Fig.\[Gudkov-Table3:fig\]), or because we have difficulty to visualize Cremona transformations? Is there some more intrinsic reason.

One boring algebro-arithmetic game is to think of curves as ternary forms $F(x_0,x_1,x_2)=\sum_{i,j,k:i+j+k=m} a_{i,j} x_0^i
x_1^j x_2^{k}$ with real coefficients. Up to rounding a bit the real coefficients randomly we may assume them rational numbers in ${\Bbb Q}$, and this can be done without affecting the topology nor the rigid-isotopy class. So we find nearby the given curve a smooth one defined over ${\Bbb Q}$, and we may put all coefficients in ${\Bbb Z}$ after scaling. As usual we may chase the common divisor of the equation to get a Diophantine equation with coefficients primes together $(\gcd (a_{i, j})=1)$. This we call the reduced equation of the rational curve (in the sense of Diophante as opposed to having genus $0$). It is unique up to sign. In particular there is a [*height*]{} defined as the largest coefficient of the equation.

Then there is a myriad of question. For instance, given an isotopy type of real curve (or even a rigid-isotopy class) what is the smallest height of a Diophantine equation realizing this type? To make this concrete imagine the case of sextics. The Fermat equation $x_0^m+x_1^m-x_2^m=0$ shows that the corresponding chamber (unifolium) has always height 1. Similar remark for the invisible curve $x_0^m+x_1^m+x_2^m=0$ (anti-folium) when $m$ is even (empty real locus). However it is unknown if the curve with $r=1$ real branches always corresponds to a unique chamber of the discriminant (cf. Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical]). What is the height of Gudkov’s curve? Can we write down (the best) explicit equation?

Another question is to look for some fixed integer $N$ (altitude) the set of all Diophantine equation $F(x_0,x_1,x_2)\in {\Bbb Z}
[x_0,x_1,x_2]$ of height $\le N$ and consider how they distribute between the chambers of the discriminant. If $m=6$ there is 64 many chambers by Klein-Rohlin-Kharlamov-Nikulin 1979 [@Nikulin_1979/80]) encoded by the chromatic Gudkov table of Rohlin (Fig.\[Gudkov-Table3:fig\]). Of course some sporadic equations may land on the discriminant. Now count the corresponding $65$ (or rather $64$, maybe I added one for the discriminant but this will tend to zero) frequencies and consider the corresponding probabilities $p_{i,N}$ (indexed by the Gudkov symbols $i=\frac{k}{1} \ell$ (plus $(1,1,1)$ deep nest) enhanced sometimes by the type as on Fig.\[Gudkov-Table3:fig\]). Is the probability assigned to Gudkov’s chamber $\frac{5}1 5$ particularly low, say as compared to Hilbert’s or Harnack chamber? Paraphrasing slightly, how long would it take to a stupid computer to discover Gudkov’s sextic by merely tracing with clever algorithms the real locus of an explicit Diophantine equation, while randomly trying one equation after the other.

In contrast one may expect that when $N \to \infty$ there is some equidistribution, with all probabilities tending to be equal. Perhaps some special rôle is played by the empty chamber which is connected by Nikulin 1979 [@Nikulin_1979/80], or better by the more elementary argument valid in all degrees, cf. (\[empty-chamber-connected-Shustin:lem\]). Of course a priori it is not even clear that the limiting probabilities converge as $N\to \infty$.

What about the height of Gudkov’s chamber, i.e. the least size of the coefficient of a defining equation. Idem for Harnack and Hilbert’s chambers. Are they lower? Can we estimate the heights from above using the classical constructions made effective over ${\Bbb Q}$?

Of course all these questions look perhaps a bit unnatural or somewhat out of reach. Also they depend on the height function (maximum coefficient), while there is perhaps other more natural ways to measure the complexity of an equation, e.g. by the Pythagorean distance (sum of all spares of the coefficient $\sum_{i,i} a_{i,j}^2$). This grows like a ball instead of like a cube, but perhaps the corresponding probabilities are independent of the exhaustion process? In that case there would be canonical probabilities and their estimation could be interesting.

All this seems out of reach even when $m=6$, e.g. because we lack serious algorithms to detect the type from the equation (compare e.g. Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical]). Of course the asymptotic probability as $N\to \infty$ of landing in the discriminant will tend to zero (being a hypersurface of Lebesgue measure zero). So we should really have a distribution between $64$ numbers $p_i\in [0,1]$ (some possibly zero? yet unlikely) weighting the Gudkov-Rohlin pyramid (Fig.\[Gudkov-Table3:fig\]) by real masses. Are those probabilities all equal (equidistribution), rational numbers, etc.? Is the empty chamber much more heavy than the other?

A crude intuition is that when coefficients get larger and larger, we get some thermodynamic excitation with all topological schemes (as complicated as they may be) fairly represented.

Another less arithmetical way to pose the question of the frequency (e.g. of curves as Gudkov’s) is just to put the natural(?) round elliptic volume element à la Riemann-Lebesgue on the space $\vert mH \vert\approx {\Bbb P}^N$ of all curves-coefficients dominated by the round (unit) sphere $S^N$. The latter is calibrated to volume $2$ as to arrange unity volume for its quotient ${\Bbb R}P^N$. Each of the 64 chambers (when $m=6$) has then a (natural) mass, which demands only to be explicitly determined. It would be again exciting to compare the mass of Gudkov’s $M$-chamber with those of Hilbert’s or Harnack’s. Now it is clear that the discriminant has measure zero being a hypersurface, whereas all other chambers are affected by positive masses.

How does a random equation (curve) look alike? Letting $p_i$ ($i=1,\dots, 64$) be the probabilities assigned to each of the (Rohlin-Kharlamov-Nikulin) chambers. Those are either all equal (equidistribution) which is quite unlikely, or some “curve” occurs more frequently? From zero-knowledge all what can be said is that some $p_i\ge 1/64$. What is the largest $p_i$? Maybe the empty chamber is the most massive?

Of course then there is also refined questions about the Riemannian geometry of those chambers. Assume for simplicity equidistribution of masses. Then the whole hotel $\vert mH
\vert-\frak D$ is shared by 64 families having chambers of the same volume, yet perhaps some are much more comfortable to live in. Annoying might be chambers highly contorted where there is little room to plug mobiliary inside. For instance we could look at the largest Riemannian ball expansible inside a given chamber, etc.

Perturbation of lines (Plücker 1839, Klein 1873, Finashin 1996) {#Line-perturbation:sec}
---------------------------------------------------------------

\[08.04.13\] This short section can be skipped. It was written at an early stage when we had not yet found all schemes asserted by Gudkov-Rohlin, primarily because we did not mastered sufficiently the Harnack method. So we attempted to realize schemes by perturbing lines. In this primitive context it could still be of interest to understand precisely what schemes are realized. If I remember well Felice Ronga (ca. 1999) once mentioned this problem as one challenging his imagination. Perhaps it is worth at the occasion trying to understand what can be said.

\[12.01.13\] As yet we missed several schemes whose existence is asserted in Rohlin 1978 [@Rohlin_1978]. This is one motivation for trying to look at what is obtainable by perturbing an arrangements of lines. Of course some more ancestral motivation like the work of Plücker 1839 [@Plücker_1839] as credited for by Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn] gives also such a motivation. In fact this section was motivated by a figure in Finashin 1996 [@Finashin_1996 Fig.10] which we shall now reproduce while trying to explore other choices. The general question could be which (typed) schemes of the Gudkov-Rohlin table (Fig.\[Gudkov-Table3:fig\]) can be realized by perturbing a line arrangement.

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Contraction conjectures (Klein 1876, Rohlin 1978, Shustin 1985, Itenberg 1994, Viro 1994)
=========================================================================================

“Klein-vache”: Nondividing implies champagne bubbling? (Klein 1876, disproof Shustin 1985)
------------------------------------------------------------------------------------------

\[14.01.13\] As early as 1876 [@Klein_1876], Klein asserted the firm conviction that curves of type I cannot gain an oval by crossing a solitary node. It required ca. 110 years until Marin 1988 took the pain to write down a proof of a somewhat stronger assertion (cf. Sec.\[Klein-Marin:sec\]). In the same paper, Klein (1876) speculated about a much more metaphysical converse allowing any nondividing curve to gain an oval after crossing a solitary node. This was never rigidly asserted by the cautious Felix Klein, but disproved 99 years later by Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]. Personally, we have not yet assimilated the full details of Shustin argument, as it uses much technology, but all experts (Shustin, Viro, Fiedler, Orevkov, etc.) have validated this disproof.

[(Klein’s hypothesis of 1876, abridged “Klein-vache” in the sequel, disproved in Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin])]{} \[Klein-1876:conj-noch-entwicklungsfahig\] Given any nondividing plane curve of arbitrary degree $m$, it is possible to let it cross the discriminant through a solitary node via a path of curves $(C_t)_{t\in [-1,+1]}$ traversing only once the discriminant. In other words any diasymmetric chamber bounds a solitary wall.

\[15.01.13\] The conjecture is nearly evident when $m=6$ in view of Rohlin’s enrichment of Gudkov’s table by types and the subsequent rigid-isotopic classification of Nikulin 1979/80 [@Nikulin_1979/80] (Theorem \[Nikulin:thm\]). With this data available one gets a bijection between chambers past the discriminant and Rohlin’s enriched schemes (cf. Fig.\[Gudkov-contigBIS:fig\] below).

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A moment contemplation of this table shows that all diasymmetric chambers admit at least one edge moving upwards in the hierarchy incrementing the number of ovals $r$ by one unit. Of course a priori such an increment does not necessarily correspond to the formation of a solitary node (isolated double point) but can also traduce the subdivision of an oval shrinking to a lemniscate. Hence more work is required, yet we believe strongly that “Klein-vache” holds true for sextics. In fact here is a complete proof:

\[Klein-vache-deg-6:prop\] [(Gabard 15.01.13, but a trivial corollary of Rohlin 1978, Nikulin 1979/80, Itenberg 1994 [@Itenberg_1994] and Klein-Marin 1876–1988)]{}.—The conjecture “Klein-vache” [(\[Klein-1876:conj-noch-entwicklungsfahig\])]{} holds true for $m=6$, i.e. any nondividing sextic can acquire a solitary double point by a rigid-isotopy crossing only once the discriminant transversally.

It is first a matter of paying attention to the combinatorics of Rohlin’s classification into types (Fig.\[Gudkov-contigBIS:fig\] above). The rest of the proof is then nearly self-explanatory. In slight contrast to Rohlin 1978 [@Rohlin_1978] we forbid the “hermaphrodite” [*indefinite schemes*]{} (allowing projective realizations of both types I/II) but rather imagine them as two superposed (but distinct) elements, with the dividing schemes (especially the blue rhombs on Fig.\[Gudkov-contigBIS:fig\]) levitating slightly above the sheet of paper of that figure. By Nikulin’s theorem 1979/80 [@Nikulin_1979/80] those combinatorial symbols (with levitating twins above the blue-rhombs) are in one-to-one correspondence with the chambers past the discriminant. Now imagine on Fig.\[Gudkov-contigBIS:fig\] a sort of random flow moving downwards along the red-edges of that figure.

Let us be more precise. By a result of Itenberg 1994 [@Itenberg_1994 Prop.2.1, p.196] (based upon techniques used by Nikulin ()) [*each empty oval of a sextic can be contracted to a solitary node before disappearing in the blue sky*]{}. (An oval is said to be [*empty*]{} if it contains no oval in its interior.) Pick a curve in each chamber and pick two contractions (given by Itenberg) shrinking either an outer oval or an inner oval, provided both are available on the real scheme. If only inner or outer ovals are available, pick only one contraction. This can be visualized as a “random” vector field moving downward along the diagrammatic of Fig.\[Gudkov-contigBIS:fig\]. Each Itenberg contraction necessarily lands in type II (diasymmetric) chambers. Else if landing in an orthosymmetric (=dividing) chamber, then reading the Morse surgery backwards corrupts the Klein-Marin theorem (even in its weak original formulation of Klein 1876 [@Klein_1876], though the latter gave no proof but see (\[Klein-via-Ahlfors(Viro-Gabard):lem\]), or (\[Klein-Marin:lem\]), or the 1st hand source Marin 1988 [@Marin_1988]). Hence our random vector field has its “trajectories” ending on the bottom sheet of paper (as we imagine orthosymmetric chambers levitating somewhat above the sheet of paper, see again the blue-rhombs on Fig.\[Gudkov-contigBIS:fig\]). It is plain now that all diasymmetric chambers (green squares on Fig.\[Gudkov-contigBIS:fig\], plus those lying behind the blue rhombs) do occur as extremities of our vector field encoding the varied Itenberg contractions chosen. Interpreting this process backward-in-time proves “Klein-vache” in degree $6$. The proof is complete.

[*Insertion*]{} \[30.03.13\] It should be noted that Itenberg’s contraction theorem affords in degree 6 another proof (independent of total reality) of Rohlin’s maximality principle (in degree 6), at least if we take for granted the RKM-congruence (\[Kharlamov-Marin-cong:thm\]). This prompts another strategy toward Rohlin’s maximality conjecture (independent of total reality) and perhaps worth exploring further. Of course the hearth of the problem seems to be the Itenberg-Viro contraction conjecture for any empty oval (\[Itenberg-Viro-contraction:conj\]), but this does not seem to imply Rohlin’s maximality conjecture. In contrast to “Klein-vache” the Itenberg-Viro contraction conjecture is still open and certainly worth investigating further. It is also worth noting that at the earth of the above proof (\[Klein-vache-deg-6:prop\]) we have Itenberg’s contraction theorem. Thus roughly Itenberg implies Klein-vache, yet this is not the sole ingredient for otherwise in degree 8 Shustin’s disproof of Klein-vache would refute the contraction principle (which is still open in degree 8). So the above proof really uses more than just the contraction principle. In some sense it uses results by Nikulin but only as a mean to get Itenberg contractions. What looks more pivotal is the role of the Gudkov-Rohlin table. One may thus wonder if in degree 8, we can get sufficient grasp on the Gudkov-Rohlin table as to infer the logical move from the contraction principle to Klein-vache. If feasible, then Shustin’s disproof (1985) would refute the Itenberg-Viro contraction conjecture (1994) in degree 8. This scenario looks a priori quite likely and requires perhaps just completing the full diagrammatic of Hilbert’s 16th in degree 8, plus the extra-data of types. (This is perhaps available within the next decade, if we appreciated correctly the optimism of experts). Factually, the above proof can be summarized by saying “Itenberg contraction+Gudkov-Rohlin diagrammatic$\Rightarrow$Klein-vache”, yet without that it is crucial to have a bijection between typed-schemes and rigid-isotopy classes à la Nikulin. This correspondence being disrupted in degree 7 (and so probably 8) by Marin 1979 (cf. Fig.\[Marin:fig\]). Hence it seems likely that a completion of the Gudkov-Rohlin table in degree 8, will imply a refutation of the Itenberg-Viro contraction conjecture.

The above proof of Klein-vache (in degree 6) is quite attractive, but to be really sublime it should extend to higher orders. Several obstacles arise. First Itenberg’s contraction principle becomes conjectural for $m>6$ (compare Viro’s preface in the same volume). Next our argument rests on the deep combinatorial classification of Rohlin 1978 [@Rohlin_1978], plus Nikulin’s rigid-isotopy classification via real schemes enriched by the type data (I/II). This ceases to be true for orders $m\ge 7$ (Marin 1979/80 [@Marin_1979], Fiedler 1982/83 [@Fiedler_1982/83-Pencil]). Thus the above proof looks jeopardized for higher orders. Of course, if one believes in Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]), then “Klein-vache” is actually false when $m=8$. Historiographically, it is of course quite improbable that Klein’s (weak) intuition about “Klein-vache” was based upon the above procedure (Torelli for K3’s being needed by Nikulin), yet it is also not completely impossible that a more elementary proof than the one above exists (cf. optionally Sec.\[Klein-vache-proof:sec\]). At any rate Klein’s power of prediction via geometric intuition is once more quite amazing. More modestly, it should be stressed that Klein, interpreted in the lowbrow fashion, merely asserts that there is no topological obstacle toward implementing “Klein-vache”, yet he is prudent enough in not claiming this as a theorem (compare again Klein’s original Quote \[Klein\_1876-niemals-isolierte:quote\] which is beautifully ambiguous).

\[11.01.13\] A first natural question is whether Klein-vache implies the direct sense of Rohlin’s 1978 conjecture (i.e. “type I implies maximal”). In fact Klein-vache shows rather that if a scheme is not of type I (so contains a nondividing representative) then it is non-maximal. Paraphrasing, “type I is implied by maximal”. This is however the part of Rohlin’s conjecture that was refuted by Shustin 1985/85 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]. So indirectly Shustin’s counterexample also destroys (the hard half of) Klein’s intuition (i.e. Klein-vache). Shustin’s result is somewhat stronger:

[(Shustin 1985)]{} \[Shustin:thm\] There exists a maximal scheme of degree $8$, which is of type II.

[(copied from the source)]{} Shustin proves first the following assertion.

There exists $(M-2)$-curves of degree $8$ with the schemes $10 \sqcup 1 \langle 1 \rangle
\sqcup 1 \langle 2 \rangle
\sqcup 1 \langle 4 \rangle$, and $6 \sqcup 1 \langle 2 \rangle \sqcup 1 \langle 4 \rangle \sqcup 1 \langle 5 \rangle$, in the notation of Viro (i.e. the notation $1 \langle k \rangle$ means one ovals enveloping directly $k$ empty ovals).

One starts with a certain quintic $C_5$ having controlled topology with respect to the $3$ axes (constructed in Polotovskii 1977 [@Polotovskii_1977/77]). Then applying a quadratic transformation gives a singular octic with “complicated” singularities. On dissipating such complicated singularities (Viro’s method 1980) one may create the 2 required schemes.

[*End of the proof of Theorem \[Shustin:thm\] (compare also Sec.\[Shustin-understood:sec\] for our slow assimilation of Shustin’s proof)*]{}. Applying a result of Viro 1983 [@Viro_1983/84-new-prohibitions], the $(M-2)$-schemes constructed above are of type II. It remains now to check that they are maximal.

[*Insertion*]{} \[31.03.13\].—The Euler-Ragsdale $\chi$ of the first scheme is $\chi=10+(1-1)+(1-2)+(1-4)=6$, while $k^2=16$. Hence Arnold’s congruence mod 4 (or the allied Rohlin’s formula) suffices to establish type II of the curve. For the second, $\chi
=6+(1-2)+(1-4)+(1-5)=-2$, and again Arnold/Rohlin suffices to show type II.

First Shustin says that the $(M-1)$-schemes obtained from them by addition of an oval (if they exist) are (always) of type II, referring to Rohlin 1978 [@Rohlin_1978 point 3.2]. Needless to say, this is actually a trivial consequence of Klein’s congruence (1876) $r\equiv g+1 \pmod 2$. Yet more seriously it seems to me (Gabard) that we do not need only to know these schemes being of type II, but rather that they do not exist at all!? So in my opinion there may be a trivial misconception here? In fact we can apply the Gudkov-Krakhnov-Kharlamov congruence (Theorem \[Gudkov-Krakhnov-Kharlamov-cong:thm\]) for $(M-1)$-curves to all possible enlargements (cf. Sec.\[Degree8:sec\] for details) yet this fails prohibiting a specimen. Shustin’s argument looks uncomplete at this stage, or presumably rests on stronger obstructions used subconsciously by the author!?) (\[24.01.13\] Compare again Sec.\[Shustin-understood:sec\] for our assimilation of Shustin’s proof; what is required is a prohibition of Viro.)

Next Shustin argues that the $M$-schemes obtained from the given ones by the addition of two ovals are forbidden by the extremal comparison in Rohlin 1978 [@Rohlin_1978 point 1.3], and Viro 1980 [@Viro_1980-degree-7-8-and-Ragsdale Theorem 4].

[*Conclusion.*]{}—Beside Polotovskii 1977, Shustin’s result relies massively on Viro’s revolutionary technique of construction via dissipation of complicated singularities (which came to be known as “patchworking”). Yet the basic logics of Shustin’s reasoning looks a bit elusive and perhaps flawed. (\[24.01.13\] Not all, cf. again Sec.\[Shustin-understood:sec\].) Hence it is not clear to me if it really destroys the hard-half of Klein’s intuition (i.e. Conjecture \[Klein-1876:conj-noch-entwicklungsfahig\]).

Let us repeat once more the crucial quote of Klein 1876: [*Z. B. kann bei den Kurven der ersten Art durch allmähliches Ändern der Konstanten niemals eine isolierte reelle Doppeltangente neu enstehen, um dann einen $(C+1)$-ten Kurvenzug zu liefern; während die Kurven der zweiten Art in dieser Richtung nicht beschränkt sind. Die Kurven der zweiten Art sind sozusagen noch entwicklungsfähig, während es die Kurven der ersten Art nicht sind. Doch soll hier auf diese Verhähltnisse noch nicht näher eingegangen werden.*]{}

Translated in English (while adhering to Russian notation and jargon) gives something like:

[*For instance, for curves of type I an isolated solitary node can never rise as to produce a new real circuit through progressive variations of the coefficients; whereas curves of type II are not restricted in this way. Curves of type II are so-to-speak still developable, while those of the first type are not.*]{}

This demonstrates that Klein only cautiously asserted that curves of type II are not obstructed to acquire a solitary node, yet not claiming something so radical as our Conjecture \[Klein-1876:conj-noch-entwicklungsfahig\], albeit his second sentence goes closer to suggesting this interpretation. \[24.01.13\] At any rate this Ansatz of Klein turns out to be corrupted by Shustin’s article, relying heavily on the new prohibition detected by Viro (cf. again Sec.\[Shustin-understood:sec\] for our ultimate assimilation of this).

Degree 8: the Grand pyramid of Gizeh {#Degree8:sec}
------------------------------------

\[12.01.13\] Can we picture out the Gudkov-Rohlin pyramid in order 8? Since $m=8$ we have $g=\frac{(m-1)(m-2)}{2}=\frac{7\cdot
6}{2}=7\cdot 3=21$. So $M=g+1=22$. It is first quite easy to extend upwards the Gudkov symbols as to build a larger pyramid (Fig.\[Degree8:fig\]). Yet this contains only schemes with 1 (or less) nonempty oval. One can easily report the modulo 8 prohibitions coming from Gudkov-Rohlin, etc., as discussed in Sec.\[Gudkov-hypothesis:sec\].

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[*Insertion*]{} \[02.04.13\].—A naive trick is to think of the whole pyramid as fibred over the depicted one (Fig.\[Degree8:fig\]) which shows the range of $(\chi, r)$. So a basic procedure is to start from the given elementary configuration with symbol $\frac{x}{1}y$ and to make a menagerie of transfer of ovals conserving $\chi$. This involves a Bonsai-cutting art-form of the Hilbert tree. Yet this does not really solve our puzzle of making a good chart of all possible schemes in degree 8.

The others schemes having $\ge 2$ nonempty ovals are a bit messy to report. In particular it seems unrealistic willing to report all schemes on a single table! Should we try several charts, but then how to track their interrelations and overlaps? Can we split in several classes? Let us try to use the number $N$ of nonempty ovals as a splitting recipe.

$\bullet$ if $N=0$ we have the schemes $\ell$ ($0\le\ell\le 22$), so 23 schemes.

$\bullet$ if $N=1$ we have the schemes $\frac{k}{1}\ell$.

$\bullet$ if $N=2$ we have $\frac{k}{1}\frac{\ell}{1}m$ (if $m=0$ this is still pictured as the left semi-triangle, yet for larger $m$’s one must imagine several layers lying above the sheet of paper). Of course it may be assumed $k\ge \ell$.

$\bullet$ if $N=3$ we have $\frac{k}{1}\frac{\ell}{1}\frac{m}{1}n$

$\bullet$ if $N=4$ we have $\frac{k}{1}\frac{\ell}{1}\frac{m}{1}\frac{n}{1}o$, but using a pencil of conics we see that $o=0$, and that $k,\ell, m, n \le 1$ and so we have unique such scheme, namely 4 nests of depth 2.

$\bullet$ Schemes with $N\ge 5$ are prohibited by Bézout with conics.

Okay but all this is a bit overwhelming to depict (except if one is able to visualize a pyramid in 4D!). Yet we could ask if there is a reasonable classification of all schemes according to their 3 types (as did Rohlin 1978 for $m=6$). Apart from the obvious schemes of type I, and the natural consequences of Arnold-Rohlin, etc. giving a complete answer looks again a herculean effort. Incidentally, it is an open problem as still some few cases are resisting to the experts of Hilbert’s 16th.

Yet we can ask more specific questions like (as did Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]) to corrupt one half of Rohlin’s maximality conjecture.

The trick is that under an enlargement of the scheme the number of nonempty ovals can only increase. So to see what lies above Shustin $(M-2)$-schemes with $N=3$, it is enough to contemplate the face of the pyramid with $N=3$, since $N=4$ is nearly empty.

Now writing one of Shustin’s scheme in Gudkov’s notation gives $\frac{4}{1}\frac{2}{1}\frac{1}{1}10$. Note that $\chi=p-n=(1-4)+(1-2)+(1-1)+(10)=+6=-2 \pmod 8$ so the Kharlamov-Marin congruence (\[Kharlamov-Marin-cong:thm\]) says nothing, but as observed above the more elementary Arnold congruence forces type II.

(Elementary B.A.-BA, hence skip).—To compute the value of $p-n$ (positive minus negative ovals also called even\[=pair in French\] and odds) one may use the trick of filling the ovals by an orientable membrane in ${\Bbb R}P^2$ bounding them in the obvious way, i.e. we take the interior of all the outer ovals, then remove the interior of the subsequent generation of ovals immediately nested inside, and aggregate again the inside of the next generation, etc. One has then the psychologically useful formula $p-n=\chi$, where $\chi$ is the Euler characteristic of this orientable planar membrane (which Möbius would call a reunion of binions, trinions, etc.)

Let us now examine the enlargements of Shustin’s scheme. First, we find four $(M-1)$-schemes ruling out those which are not Bézout permissible (cf. Fig.\[Shustin:fig\]). One of them $\frac{4}{1}\frac{2}{1}\frac{1}{1}11$ (framed on the figure) is not prohibited by the Gudkov-Krakhnov-Kharlamov-congruence $\chi=p-n\equiv k^2\pm 1 \pmod 8$ (Theorem \[Gudkov-Krakhnov-Kharlamov-cong:thm\]). Whether this scheme is actually realized is another question. If it is then Shustin’s result (1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]) would be erroneous. \[\[24.01.13\] No sorry this is a misconception of Gabard, cf. Sec.\[Shustin-understood:sec\] for a clarification, but detailed right now for the impatient reader. The point is that if this $(M-1)$-scheme is realized, then it will be the counterexample to Rohlin (hence to Klein) for there is nothing above it by a Viro prohibition (stating that $M$-schemes of degree 8 have an odd content trough out, cf. (\[Viro-Fiedler-prohibition:thm\])). Further if it does not exist then the $(M-2)$-scheme $\frac{4}{1}\frac{2}{1}\frac{1}{1}10$ will be a counter-example to Rohlin’s conjecture, since it would be maximal but of type II. So we get the promised disproof of one-half of Rohlin and of Klein-vache, without having to know precisely what happens above Shustin’s $(M-2)$-scheme. As a matter of fact it seems, the first alternative correspond to reality, i.e. the $(M-1)$-scheme $\frac{4}{1}\frac{2}{1}\frac{1}{1}11$ is realized.\]

On reading again Shustin argument () he does not explain convincingly why this $(M-1)$-scheme enlarging his one does not exist algebraically, but merely notice that all $(M-1)$-schemes must be of type II (a triviality of course since Klein’s congruence $r\equiv g+1 \pmod 2$). So again I cannot follow Shustin’s reasoning, and perhaps it is not a serious torpedo against Rohlin’s maximality conjecture! Is there a flaw in this Shustin note???

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Of course one can play the same game for the other scheme proposed by Shustin, namely $\frac{5}{1}\frac{4}{1}\frac{2}{1}6$, and we get the following Fig.\[Shustin2:fig\]. Alas again one larger $(M-1)$-scheme is not prohibited by Gudkov-Krakhnov-Kharlamov.

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Of course it can be the case that the $(M-1)$-scheme lying above Shustin’s scheme (we write in the singular as we fix attention to one of his scheme) can be prohibited by a stronger prohibition that Gudkov-Krakhnov-Kharlamov, which Shustin might have used subconsciously. Yet with what is written down in the article I could not verify his argument. So perhaps “Klein-vache” is still true (cf. Conjecture \[Klein-1876:conj-noch-entwicklungsfahig\]). In fact Shustin’s paper is the only (published) obstruction I am aware of against Klein’s intuition.

\[15.01.13\] Idea to explore (but skip as it leads nowhere \[02.04.13\]).—Maybe Shustin used subconsciously some inequalities stronger than the congruences, perhaps those à la Ragsdale-Petrovskii. (Those are discussed in Rohlin 1978 [@Rohlin_1978 p.87].) Let us try the Ragsdale conjecture. It states $$p\le \frac{3}{2} k(k-1)+1.$$ Rohlin asserts () that the first chance of refuting this inequality is the case $m=10$ (cf. indeed the breakthrough of Itenberg-Viro of Fig.\[Itenberg:fig\]). So Ragsdale should be true for $m=8$ (despite having been subsequently disproved by Itenberg via Viro’s patchwork of a variant thereof (again Fig.\[Itenberg:fig\]). Calculating on the first Shustin’s enlarged $(M-1)$-scheme we find $p=3+11=14$ versus $\frac{3}{2} k(k-1)+1=\frac{3}{2}
4(4-1)+1=2\cdot 9+1=19$. So Miss Ragsdale is far from violated.

Another idea would be to use a pencil of cubics through some deep ovals of the enlarged $(M-1)$-schemes. Yet some easy counting shows that we may force 22 real intersection by taking a connected cubics through 8 basepoints specified inside the deep ovals, but this not enough to overwhelm Bézout accepting $3\cdot 8=24$ intersections.

Finally understanding Shustin’s argument (with the help of Orevkov’s letter) {#Shustin-understood:sec}
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\[24.01.13\] Thanks to a letter of Stepa Orevkov, and a survey of Viro 1989/90 [@Viro_1989/90-Construction p.1126], we learned the presence of new Bézout-like prohibitions on $M$-schemes derived in Viro 1983 [@Viro_1983/84-new-prohibitions]. Those have the special feature of not being of topological origins, but rather algebro-geometric. (This also enabled Viro in 1979 to complete the (soft) isotopic classification of septics solving thereby the next case of Hilbert’s 16th, compare e.g. Viro 1989 [@Viro_1989/90-Construction p.1124].)

Here is the relevant Viro’s result of which we actually just need the first clause (transcribed in conservative Gudkov’s notation, having in our opinion a slight advantage of compactness when it comes to put the symbols on a pyramid):

[(Viro 1983 [@Viro_1983/84-new-prohibitions])]{} \[Viro-Fiedler-prohibition:thm\]

$\bullet$ ($M$)—If $
\frac{\alpha}{1}\frac{\beta}{1}\frac{\gamma}{1} \delta$ is the real scheme of an $M$-curve of degree $8$ with $\alpha, \beta$ and $\gamma$ nonzero, then $\alpha, \beta$ and $\gamma$ are odd.

$\bullet$ ($M-2$)—If $
\frac{\alpha}{1}\frac{\beta}{1}\frac{\gamma}{1} \delta$ is the real scheme of an $(M-2)$-curve of degree $8$ with $\alpha, \beta$ and $\gamma$ nonzero and with $\alpha+\beta+\gamma\equiv 0 \pmod
4$, then two of the numbers $\alpha, \beta, \gamma$ are odd and one is even.

—\[25.01.13\] We postpone the proof to a latter occasion, and merely reproduce now the remark to be found in Viro 1983 [@Viro_1983/84-new-prohibitions p.416]: “The special case of Theorem 2.2.E when $\delta=0$ and $\beta=1$ is due to Fiedler \[11\](=Fiedler 1982/83 [@Fiedler_1982/83-Pencil]). Theorem 2.2E was stated as a conjecture by A.B. Korchagin in connection with my results on realization of the real schemes of $M$-curves of degree $8$. The theorem rules out $40$ real schemes which are not ruled out by Theorems 2.2.A–2.2.D (of these forty, four are ruled out by the special case of Theorem 2.2.E which was proved by Fiedler).”

The proof is completed on p.422 of Viro’s text. It starts as follows: Let $C=C_8$ denote a smooth octic with real scheme $\la
\alpha \vc 1\la \beta\ra \vc 1 \la \gamma\ra \vc 1\la \delta \ra
\ra$, where $\beta, \gamma$ and $\delta$ are nonzero. The crucial result is Theorem 4.2 in Viro, which itself is based upon Fiedler. So the proof looks too technical to be reproduced here. A self-contained account encompassing Fiedler and Viro’s article would require several pages, and we postpone this to a future occasion.

One may wonder if the special case implemented by Fiedler does not suffice actually to corrupt “Klein-vache”, i.e. Klein’s Ansatz that nondividing curve can bubble out a new solitary node out of the blue sky. However Fiedler’s result prohibit only the scheme $
\la 1 \la 1\ra \vc 1 \la \alpha \ra \vc 1\la \beta\ra  \ra$ with even nonzero $\alpha$ and $\beta$ (cf. e.g. Viro 1983 [@Viro_1983/84-new-prohibitions p.420]), and a priori this is not enough to prohibit the enlargeability of some suitably chosen $(M-1)$-scheme (compare e.g. the constructions proposed by Orevkov in the next Sec.\[Orevkov:sec\]).

The first assertion prohibits the remaining $M$-scheme of Fig.\[Shustin:fig\].

As to the second clause of (\[Viro-Fiedler-prohibition:thm\]) pertaining to $(M-2)$-schemes, I do not know what to do with it. At this stage I read again Orevkov’s letter (cf. Sec. \[e-mail-Viro:sec\]), which I have some pain to interpret properly. Let us reproduce it right below for convenience, while adding some brackets of mine.

Before completing this reading, I finally understood Shustin’s argument. The point is that whether or not the $(M-1)$-scheme (framed on Fig.\[Shustin:fig\]) exists do not matter. Indeed if it does exist (algebraically) then it is of type II (by Klein’s trivial congruence) and maximal (by Viro’s prohibition in the above theorem), whereas if does not exist then Shustin’s $(M-2)$-scheme is maximal but of type II, by construction (or by Arnold). So in both cases Rohlin’s reverse implication “type I$\Leftarrow $ maximal” is foiled.

[*Insertion*]{} \[02.04.13\].—Of course it would be interesting to know if Shustin’s $(M-1)$-scheme enlargement do exist (algebraically), i.e. the scheme $\frac{4}{1}\frac{2}{1}\frac{1}{1}11$. If I interpret correctly the letter below of Orevkov (while removing a little misprint from it, namely trading the “11” for a “10”), it seems that the $(M-1)$-scheme written above is realized algebraically.

Stepa Orevkov’s letter {#Orevkov:sec}
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We now reproduce Orevkov’s letter (brackets=\[ \], are our additions):

$\bullet$$\bullet$$\bullet$ \[16.01.13–14h56: Stepa Orevkov\]

A small remark:

It is wrong that $11 \cup 1\langle 1\rangle  \cup 1\langle
2\rangle  \cup 1\langle 4\rangle $ is not a part of an $(M-1)$-scheme. It is. \[Not clear how to interpret this? Does it mean that Shustin’s claim is wrong, or simply that this scheme is an $(M-1)$-scheme. My question was whether this $(M-1)$-scheme is realized algebraically, of course. Yet, I admit that my question was a bit ill posed. In fact I wonder if Orevkov not intended to write a “10” instead of the above eleven.\] Moreover, there is no known example of $(M-2)$-curve of type II which cannot be obtained from an $(M-1)$-curve by removing an empty oval. \[So Klein looks still plausible for $(M-2)$-schemes, while Shustin looks wrong. No sorry, in fact I misunderstood Shustin for a long time, as he does not claim that the framed $(M-1)$-scheme does not exist.\]

In contrary, there are $(M-1)$-curves of degree $8$ (which are necessarily of type II) which do not come from any $M$-curve. These are[^45]:

$3\langle 6\rangle $[^46]

$4 \cup 1\langle 2\rangle  \cup 2\langle 6\rangle $

$8 \cup 2\langle 2\rangle  \cup 1\langle 6\rangle $

$12 \cup 3\langle 2\rangle $

Construction (inspired by Shustin’s construction of $4 \cup
3\langle 5\rangle $ \[should locate the reference\]):

Consider a tricuspidal quartic $Q_{sing}$ symmetric by a rotation $R$ by $120$ degree and perturb is\[=it\] so that each cusp gives an oval (we assume that this perturbation is very small). Let $Q$ be the perturbed curve. Two flex points appear on $Q$ near each cusp of $Q_{sing}$. We chose flex points $p_0, p_1, p_2$ (one flex point near each cusp) so that $R(p_0)=p_1, R(p_1)=p_2,
R(p_2)=p_0$. We choose homogeneous coordinates $(x_0 : x_1 : x_2)$ so that the line $x_i = 0$ is tangent to $Q$ at $p_i$ $(i =
0,1,2)$.

Let $C$ be the image of $Q$ under the Cremona transformation $(x_0
: x_1 : x_2) \mapsto (x_1x_2 : x_2x_0 : x_0x_1)$. Then $C$ has 3 singular points, each singular point has two irreducible local branches: a branch with $E6$ and a smooth branch which cuts it “transversally”. By a perturbation of $C$ we obtain all the four curves mentioned above.

The fact that these curves cannot be obtained from $M$-curves immediately follows from the fact that, for any $M$-curve of degree 8 of the form $b \cup 1\langle a_1\rangle  \cup 1\langle
a_2\rangle  \cup 1\langle a_3\rangle $, all the numbers $a_1$, $a_2$, $a_3$ are odd[^47].

Best regards Stepa O

This letter helped me much to understand finally Shustin’s proof, and is of course worth studying for its own (especially to make a picture of it). It gives another counterexample to Rohlin’s maximality conjecture, hence to Klein’s Ansatz of champagne bubbling nondividing curves.

\[25.01.13\] Now here is an attempt to vizualize Orevkov’s example. As he said we start with a tricuspidal quartic. This is known since time immemorial (maybe Euler 1745, Steiner 1857, cf. e.g. Briekorn-Knörrer 1981/86 [@Brieskorn-Knörrer_1981/1986 p.32] where it is described as a hypocycloid, cf. also Lawrence p.135, where it is called the Deltoid). This being given we smooth out the cusps to create some little ovals. I presume this can be done by hand, otherwise there is a theorem of Gudkov 1962 [@Gudkov_1962] extending to cusps that of Brusotti 1921. The more difficult task is to understand what happens under the Cremona transformation. Here I was much aided by the prototype of Gudkov’s example (cf. Sec.\[Gudkov:sec\]), which is the first place where Cremona maps were applied to topology of real varieties. Remind that Orevkov’s example, is inspired from Shustin, himself being a direct student of Gudkov.

So the first steps are fairly easy (say classical for Gudkov’s era), yet it took me some times to trace appropriately the Cremona transform of the $C_4$. I hope my picture is correct (ask Orevkov if needed)? It is imagined (I presume) that (like in Gudkov’s construction, cf. again Fig.\[GudkovCampo-5-15:fig\]) the flecnodal tangent is slightly perturbed to become transverse to the $C_4$. This implies then the funny behaviour “forth-back-and-forth” of the image $C_8$ at the place $1$, say. So there is an octic as depicted. To trace the picture it is useful to keep in mind that the Cremona map takes edges of the fundamental triangle to the opposite vertices of the triangle, while preserving the 4 residual component of the triangle.

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Now we arrive at the hard step, namely the dissipation of such singularities. Here comes the contribution of Viro (if we do not misunderstood history). As explained say in Viro 1989/90 [@Viro_1989/90-Construction p.1111–12], there is a myriad of high order singularity, and one would like to understand their dissipation. The singularity at hand in our case has 4 branches, while 3 of them have a second order contact (tangency) at the singular point. For this specific singularity, he quotes Korchagin (1988). So we have a table of possible template of dissipation, which may be locally glued in place of the singularity (exactly as in Brusotti’s method of small perturbation which amounts to the simplest singularity “$A_1$”).

Substituting one of this template, one may hope to find the schemes announced in Orevkov’s letter. Yet, it must be hoped that I use the right singularity (???, their naming being non-canonical apparently?), and as yet I failed

Worse if we take $\alpha=5$ and $\beta=1$ and the very first dissipation of Viro’s picture (right of Fig.\[Orevkov2:fig\]) (which is permissible in view of the congruence mod 4), then we get a curve with $18+3+3=24$ ovals violating frankly Harnack’s bound $M=g+1=(7\cdot 6/2)+1=21+1=22$. I got something wrong!!!

\[26.01.13\] Another explanation could be that for higher singularities there is no analog of Brusotti’s theorem on the independence of simplification. The latter is brilliantly explained in Gudkov 1974 [@Gudkov_1974/74], as reducing ultimately to Riemann-Roch, but also a theorem of Max Noether, and even special series. Note that in this passage of Gudkov, he seems to be not completely up-to-date with the problem of special series on curves, as was solved by Meis 1960 [@Meis_1960] in the special case of pencils, and by Kempf 1971 [@Kempf_1971], and Kleiman-Laksov 1972 [@Kleiman-Laksov_1972] independently in the early 1970’s.

Mistrusting Shustin 1985, while trying to prove “Klein-vache” 1876 (via Garsia-Rüedy or vanishing cycles à la Poincaré-Severi-Lefschetz-Deligne-Mumford, etc.) {#Klein-vache-proof:sec}
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\[06.03.13\] All of our (initial) mistrusting of Shustin’s proof is not really justified anymore, being in part clarified above (Sec.\[Shustin-understood:sec\]) modulo assimilation of the Viro-Fiedler advanced Bézout-style prohibition (\[Viro-Fiedler-prohibition:thm\]). Hence the sequel has to be read with suitable discernment, but was not completely censured as it may contains geometric ideas worth exploring further, and other issues of independent interest.

\[13.01.13\] Could it be that Shustin 1985 was wrong, while Klein 1876 is correct!? If so how to prove Klein-vache (\[Klein-1876:conj-noch-entwicklungsfahig\])? Of course this amounts to an amazing topological flexibility of Riemann surfaces as flying-saucers moving in the Plato cavern of plane projective geometry, where smooth curves are known to have “particular” moduli. More concretely one could imagine that this is always possible via pure Anschauung, namely the process dual to the subsequent Fig.\[Klein-Marin:fig\](left) read in reversed time. One would take a (globally) invariant cycle (=circle) traced on the diasymmetric surface which is however acted upon without fixed point by the symmetry (antipodal map on the circle). Such circles deserve a name:

\[antioval:def\] [An [*antioval*]{} of a symmetric (Riemann) surface is a topological circle traced on the surface invariant under the involution and acted upon antipodically by the symmetry.]{}

First, note as a trivial topological issue, the following.

\[antioval:lem\] Antiovals only exist on diasymmetric surfaces, all of them admitting one.

Assume the surface orthosymmetric (i.e. dividing) and containing an antioval. By definition an antioval lacks real points, being acted upon antipodically (by Galois). Take one point of the antioval and its conjugate (which is distinct) and look at an arc of the antioval linking $p$ to $p^\sigma$. This arc is in the imaginary locus, yet connects two conjugate points, violating the orthosymmetry assumption.

Conversely suppose given a diasymmetric surface $(S, \sigma)$, hence the quotient $S/\sigma$ is non-orientable. Choose a loop reversing the indicatrix (local orientation) and avoiding the boundary of $S/\sigma$. The counter-image of this circle in $S$ gives a circle $C$, since the orientation reversing loop lifts to an arc via the quotient map which is a genuine double cover outside the boundary (alias contour by analysts). Since the symmetric surface is recovered from the quotient via the orientation cover (Klein-Weichold yoga), the circle $C$ is the desired antioval.

By Klein 1876 (and Riemann), the number $r$ of ovals (better real circuits) is bounded by $r\le g+1$ (so-called Harnack bound, under the supervision of Klein who found a more intrinsic reason). It is natural asking about a similar bound for antiovals. The antipodal sphere $S^2$ shows that each great circle is an antioval, whence an infinity of such. Consider next an antipodal torus of revolution in 3-space invariant under rotation about the $z$-axis and acted upon by central symmetry $(x,y,z)\mapsto (-x,-y,-z)$. We see 2 evident antiovals by sectioning with the horizontal plane $z=0$ (Fig.\[Antioval-torus:fig\]a).

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Varying the slope of this plane gives an infinitude of antiovals until we reach the critical tangent plane (Fig.\[Antioval-torus:fig\]b) after which the 2 ovals are not nested anymore. The argument certainly generalizes to any genus upon placing some symmetric pretzels in 3-space. So any diasymmetric surface without fixed points has infinitely many antiovals. In fact the above proof (part 2) shows that infinitude is a general feature without resorting to a Euclidean realization of Klein’s symmetric surfaces.

After this topological triviality let us try to attack Conjecture \[Klein-1876:conj-noch-entwicklungsfahig\], i.e. “Klein-vache” (allusion to Lefschetz’ vache coined by Grothendieck? Weil? and used by Deligne, etc.) The idea we try to exploit (but we are unable to complete the argument) involves another crazy intuition of Klein validated by Garsia-Rüedy building over works by Teichmüller, namely the fact that any Riemann surface admits a Euclidean realization in 3-space.

Suppose given a diasymmetric (real) plane curve $C_m$ of (arbitrary) degree $m$. We may as usual look at the underlying Riemann surface. According to Klein’s intuition (validated by Garsia and Rüedy, building over a contribution of Teichmüller) we know that all closed Riemann surfaces admit a conformal model in Euclidean $3$-space. Let us dream that this adapts as well in some equivariant form for symmetric surfaces (with an anticonformal involution).

Note en passant: of course closed non-orientable surfaces do not embedded in $E^3$ (Euclidean $3$-space), but if bordered they do. So the only boring diasymmetric surfaces are the invisible ones (no fixed points) but those luckily enough admit a centrally symmetric model in $E^3$.

Choose a conformal model of $C_m({\Bbb C})$ in $E^3$ supplied by Garsia-Rüedy. By the lemma we know that there is an antioval on the diasymmetric surface. Shrink the latter to a point via a (plastical) deformation in $E^3$ akin to Fig.\[Klein-Marin:fig\](left) read in reverse sense.

Note at this stage that not all antiovals pinch to a “connected” surface (e.g. a pretzel of genus 2 with a belt dividing into two pieces). So even the topological aspect deserves to be precised, by looking at “good” (i.e. nondividing antiovals). Let us assume that those always exists.

Next look at our isotopic deformation in $E^3$ to a pinched pretzel, generating a one-parameter family of Riemann surfaces. The difficulty is to ensure that they stay planar (embeddable in ${\Bbb P}^2$) during the deformation. This looks a priori quite implausible, but we have not yet exploited the full punch of Garsia-Rüedy. Their result states that all Riemann surfaces arise in the tubular vicinity of any classical surface in $E^3$ via a normal deformation of arbitrarily small amplitude. Picturesquely, if you have any old woman(=Riemann) surface, but feel erotically bored by her due to an acute case of cellulitis just let vibrate her skin to get any girl you ever dreamed about. In the oldest lady hides any beautified young girl with taught epiderm, at least so conformally! So there is some chance that even if our initial plastical shrink deviate outside the realm of plane curves (seen as a stratum $\Pi$ in the moduli space ${\cal M}_g$, where $g=\frac{(m-1)(m-2)}{2}$) we can still rectify the trajectory so as to stay scotched along the planarity manifold $\Pi$ (for each time). We get so an “abstract isotopy”, i.e. a path in $\Pi$ the planarity manifold.

Next we have a canonical map $\vert mH \vert-\disc \to \Pi \subset
{\cal M}_g$ from the space of smooth curves of order $m$ to the moduli space ($\disc$ being the discriminant hypersurface). It should be easy to lift our abstract isotopy to $\vert m H \vert$ while having only the extremity ending in $\disc$ (necessarily at a solitary node by construction). Then one continues by letting emerge an oval. If all this is feasible (taking further better care of the involution) then Garsia-Rüedy implies “Klein-vache”.

Perhaps the above strategy requires to be adapted in $E^4$ to gain more flexibility and more care about the symmetry. Also if the Garsia-Rüedy trick in $E^3$ is not best suited to the problem at hand, a more direct approach could be to stay in ${\Bbb
C}P^2$. Recall indeed that Ko 2001 [@Ko_2001] has a fairly general extension of the theorem to any ambient Riemannian manifold. Alternatively more classic algebro-geometric methods (Severi’s Anhang F, etc., e.g. as modernized by Harris, etc.) are perhaps quite likely to imply “Klein-vache” if such methods of degeneration adapt to the reality context (equivariance w.r.t. Galois which is quite a rule when pure synthetic geometry is involved). But I must seriously refresh my memory on those works.

Another tactic toward “Klein-vache” via Itenberg-Viro suggesting a general evanescence principle
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\[14.01.13\] Is there a relation between “Klein-vache” and the natural Itenberg-Viro conjecture (cf. Itenberg 1994 [@Itenberg_1994], and the preface of that volume by Viro) positing that:

\[Itenberg-Viro-contraction:conj\] [(Itenberg-Viro 1994, abridged CC=contraction conjecture).]{}—Any empty oval of a (real, smooth) plane curve can be contracted to a point (solitary node) via a rigid-isotopy.

[*Historical note*]{} \[04.04.13\].—In Klein 1892 [@Klein_1892_Realitaet] (p.176 in the pagination of Ges. Math.Abh. 1922 [@Klein-Werke-II_1922]) there is discussed what he calls the “Doppelpunktsmethode” amounting essentially to contract any symmetry-line of the Riemann surface. This seems to anticipate the Itenberg-Viro contraction principle. It is not clear however that Klein ever formulated something as precise as the above conjecture (specific to plane curves). On p.176–177, Klein’s prose extracted from its context sounds a bit overoptimistic, namely: “Bei allen anderen Fällen hat die Durchführung des genannten Prozesses und damit die Zusammenziehung eines beliebiegen Ovals der Kurve zu einem isolierten Doppelpunkte keine Schwierigkeit.” This seems to trivialize the Itenberg-Viro conjecture but probably does not because Klein thinks really with abstract Riemann surfaces where there is much more flexibility than with plane curves. However it is not impossible that refining Klein’s argument/ideas could prove CC, but it is also quite likely that CC is false.

At first sight one may expect a direct logical subsuming of “Klein-vache” to “Itenberg-Viro’s contraction conjecture”. However some moment thought shows that there is no such direct “rapport de force”, i.e. “Klein-vache” is not implied, nor does it imply, the Itenberg-Viro contraction of empty ovals. However Prop. \[Klein-vache-deg-6:prop\] gives a logical subordination of Klein-vache to the contraction principle in presence of additional combinatorial knowledge available in degree $6$. Via Nikulin’s theorem (\[Nikulin:thm\]) on the rigid-isotopy classification of sextics it is nearly evident that Conjecture \[Itenberg-Viro-contraction:conj\] holds true for sextics. This is actually the object of Itenberg’s article just cited.

[*Insertion*]{} \[31.03.13\] In view of Viro’s isotopy classification in degree 7, and the philosophy that contraction plus combinatorial knowledge implies Klein-vache, one can also wonder if Klein-vache holds true in degree 7. Alas we lack a tool like K3’s in degree 7, and so the situation is somewhat obscure in degree 7. Possibly, Shustin’s disproof of Klein-vache descends from degree 8 to 7, and then maybe that the contraction principle is already disrupted in degree 7. Recall, that presently the contraction principle is wide open in degree 8, yet perhaps disprovable via Shustin (and a completed classification).

The true relationship between “Klein-vache” and Itenberg-Viro contraction hypothesis (\[Itenberg-Viro-contraction:conj\]) could be rather an analogy in the principle of proof that one might naively develop, namely the possibility of shrinking a cycle invariant under Galois(=complex conjugation). Indeed “Klein-vache” amounts essentially to shrink an antioval (cf. Def. \[antioval:def\]), whereas Itenberg-Viro amounts shrinking an empty oval.

Hence it may be suspected that there is a general strangulation principle specializing to both “Klein-vache” and “Itenberg-Viro” stipulating the following:

[(Shrinking principle)]{} \[shrinking-principle-vague:conj\] Any (Galois) invariant cycle(=circle) on a smooth plane curve of degree $m$ can be strangulated through a path in the hyperspace of curves crossing only once the discriminant at a smooth point of the latter (whenever there is no topological obstruction to do so).

The parenthetical proviso is required, for one cannot shrink a nonempty oval without shrinking all its inner ovals, creating thereby a singularity of higher complexity than nodal. The proof of (\[shrinking-principle-vague:conj\]) could be similar to the eclectic one sketched for “Klein-vache” in the previous section, i.e. either via Garsia-Rüedy (hence Teichmüller theoretic) or algebro-geometric via vanishing cycles à la Poincaré-Picard-Lefschetz-Severi, etc.

Let us examine the combinatorial possibilities for such a Galois-cycle. Being (by definition) invariant under complex conjugation $\sigma$, it can either be:

\(1) an oval (pointwise fixed by the Galois-Klein symmetry $\sigma$);

\(2) an antioval or dia-oval (acted upon antipodically by $\sigma$);

\(3) a pseudo-oval or ortho-oval (acted upon by $\sigma$ with two fixed points, hence like $(x,y)\mapsto (x,-y)$ on $S^1=\{x^2+y^2=+1\}$).

Our terminology ortho- and dia-oval is directly inspired by the figure of Klein 1892 (reproduced in our Quote \[quote:Klein-1891/92-ortho/dia\]), where given a circle and a point outside it one considers the involution of $S^1$ exchanging the 2 intersections of each line of the pencil. When the point lies inside the circle we get a diasymmetry (antipode like), while if it is outside an orthosymmetry (mirror with 2 fixed points).

Given a real curve (equivalently a symmetric Riemann surface in the sense of Klein), an oval exists except in the lowest diasymmetric case $r=0$ (of Klein’s classification). A dia-oval exists only in the diasymmetric case (Lemma \[antioval:lem\]). An ortho-oval can exist in both the dia- and orthosymmetric cases. An example of an ortho-oval is traced as the cycle $\beta$ on Fig.\[Klein-Marin:fig\].

Specializing the shrinking principle (\[shrinking-principle-vague:conj\]) to an oval implies the Itenberg-Viro contraction hypothesis (\[Itenberg-Viro-contraction:conj\]), to a dia-oval implies “Klein-vache” (\[Klein\_1876-niemals-isolierte:quote\]). Finally shrinking an ortho-oval leads to another natural:

Any two contiguous ovals can coalesce after crossing an ordinary node with real tangents.

[*Contiguous*]{} means here that both ovals can be joined in ${\Bbb R}P^2$ by an arc having only its extremities on the ovals. Two contiguous ovals can either be directly nested or unnested yet unseparated by a larger oval. One should not forget the possibility of a single oval subdividing himself. The latter operation is subsumed to no topological obstruction, except that one might enter in conflict with Bézout.

So we may dream of such an unifying principle explaining the perfect topological flexibility of “rigid-isotopies” permitted to traverse only once the discriminant transversally. In some sense (to be made precise) our shrinking conjecture asserts that any Galois-cycle shrinks provided there is no topological obstruction either in ${\Bbb R}P^2$ nor in the complex locus.

Alas our crude principle does not seem compatible with the:

\[Finashin-obstruction-to-coalesce-Harnack:lem\] [(Admitted, but not understood!, to whom is it due? Stated in Finashin 1996)]{} Harnack’s (sextic) scheme $\frac{1}{1}9$ can only degenerate toward the scheme $10$ by contraction of the inner oval, yet not by coalescence of the two nested ovals.

Cf. e.g. Finashin 1996 [@Finashin_1996 p.68, proof of Thm 6.2], who alas does not give a precise reference for this assertion.

So here we have a clear-cut example of a Galois-cycle (namely an ortho-oval) linking the inner oval with the nonempty one of Harnack’s curve (Fig.\[Finashin:fig\]), yet which cannot be shrunk.

Why is it so? Remember in contrast that the Gürtelkurve $C_4$ (quartic with 2 nested ovals) can see both its ovals coalesce (Fig.\[Finashin:fig\]). What is the difference between Zeuthen-Klein $C_4$ and Harnack’s $C_6$? If we take the pain of tracing the complex orientation (by Fiedler’s algorithm) we get the following pictures. It is seen that for the Gürtelkurve $C_4$ the complex orientation (in red-arrows) disagree from the orientation as the boundary of the annulus (grey-shaded), while for Harnack’s $C_6$ the ${\Bbb
C}$-orientation matches that as boundary of the ring. Could positive pairs of ovals be an obstruction to coalescence?

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\[16.01.13\] As we said we may also take an ortho-oval cutting twice the same oval. Shrinking this would effect a (cellular) subdivision of the oval. A first example is a hyperbola pinching to a pair of lines to become another hyperbola. Projectively we have permanently a conic with a single oval, so there is no naive minded subdivision like that of a cell in the naive organical sense. Incidentally 2 ovals for a conic corrupt either Bézout or Harnack, especially in the formulation of Klein. Likewise the unique oval of a cubic (if available) cannot be subdivided (without corrupting either Bézout or Harnack-Klein $r\le g+1$). However an oval of a quartic can sometimes subdivides (cf. Fig.\[Subdivide:fig\]). (If this figure is realist it is tempting to create an octic by small perturbation with $16$ unnested oval, yet let us not be sidetracked by this.)

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Visualizing the corresponding surgeries on the Riemann surface must be a pleasant exercise. If the curve is dividing (hence its Riemann surface orthosymmetric) then a subdivision is impossible without corrupting the strong form of Klein’s Ansatz proved by Marin 1988 [@Marin_1988].

It is still tempting to imagine an ortho-oval on a orthosymmetric curve, especially if it cuts only one oval. A myriad of choices are possible. Let us depict few of them. Fig.\[Orthoovals:fig\]a shows an ortho-oval dividing the surface. Hence when contracted we would get a reducible curve. As long as our naive picture (be it embedded or abstract) is respected this is incompatible with Bézout (or if you prefer the intersection theory of ${\Bbb P}^2({\Bbb C})$ whose generator $H$ of the second homology $H_2$ satisfy $H^2=+1$) unless both sides of the cycle $\beta$ have genus $0$. This proves the following lemma, whose significance is of course not confined to the case of real curves.

\[strangulation-impossible:lem\] A dividing cycle on a smooth plane curve $C_m$ of degree $m\ge 3$ cannot be strangulated by a rigid-isotopy crossing only once the discriminant.

By contradiction, assume strangulability possible along the given dividing cycle via a path of curves $(C_t)_{t\in [-1, +1]}$ starting from the given curve, i.e. $C_{-1}=C_m$ and so that only $C_0$ is singular and uninodal (smooth point of the discriminant). Denote by $S_t$ the corresponding Riemann surfaces, $S_t=C_t({\Bbb
C})$, where of course $S_0$ is mildly singular. Then the strangulated surface $S_0$ splits in two (smooth) orientable surfaces $S_1, S_2$ each porting a fundamental class $\sigma_i$ in $H_2({\Bbb C}P^2)\approx {\Bbb Z}$ ($i=0,1,2$). Hence we get in homology $\sigma_0=\sigma_1+\sigma_2$, and so taking respective degrees $m=m_0=m_1+m_2$, where $m_i=\deg \sigma_i$ (degree in the homological sense). The intersection $\sigma_1 \cdot \sigma_2$ computes as $m_1 \cdot m_2$, which have to be equal to $1$ (as the critical curve $C_0$ as just one normal crossing). It follows that $m_1=m_2=1$, violating the assumption $m\ge 3$.

The case $m=2$ is entirely different as a conic may degenerate to a pair of lines. The interesting option is to take a nondividing ortho-cycle $\beta$, as depicted on Fig.\[Orthoovals:fig\]b.

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Let us now shrink such a nondividing cycle $\beta$ to a point getting something like Fig.\[Orthoovals2:fig\]b, which is a nodal curve (still irreducible, because its Riemann surface is connected). After the critical level we could expect to find Fig.\[Orthoovals2:fig\]d, but this is impossible for the genus drops by one (remind that all smooth curves of some fixed degree have the same genus, since on the complexes the discriminant does not disconnect having real codimension $2$). In fact as soon as the handle is strangulated by the vanishing cycle it reappears instantaneously as depicted on Fig.\[Orthoovals2:fig\]c. On meditating slightly this occurs like a twisting, compare the miniature figures below depicting the pre- and post critical levels near the singularity. During the twisting one see that the north hemisphere of the orthosymmetric surface is suddenly connected with the south hemisphere forcing the diasymmetric nature of the post critical curve. (All this phenomenology is of course allied to the name of Picard-Lefschetz and Dehn.)

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Discovering eversions (Gabard 16.01.13, but surely in Möbius, von Staudt, Hilbert, Morosov, Gudkov, Kharlamov, Finashin, etc.) {#Eversion:sec}
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\[16.01.13\] How can this process (Fig.\[Orthoovals2:fig\]) occur at all if it is supposed to occur in the plane? In the naive Euclidean plane ${\Bbb R}^2$, any self-coalescence of a Jordan curve leads to a subdivision (compare the center of Fig.\[Klein-Marin:fig\] read backwardly) increasing the number of real circuits. However during our Riemann surface surgery (again Fig.\[Orthoovals2:fig\]) the number of real circuits is kept constant. Hence there seems to be a basic topological obstruction to our shrinking process, yet some more mature thinking shows this not to be the case. In reality we live in the projective plane ${\Bbb R} P^2$, so one oval may well expand “to infinity” to self-coalesce while keeping one component after having been “Morse surgered”. For varied depictions of this phenomenon, see Fig.\[Eversion:fig\] where as usual ${\Bbb R} P^2$ is depicted as a disc with contour antipodically identified.

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Thus an oval can be Morse surgered without splitting off a new oval. Let us call this process [*eversion*]{}[^48] of an oval. Note that after the eversion all the inside of the oval appears suddenly outside of it! This basic phenomenon resolves several misconceptions or paradoxes that foiled for ca. 6 days my understanding of that theory, especially when it comes to a parity anomaly between the degree of the discriminant for sextics $3(m-1)^2=75$ and the legal moves in the Gudkov pyramid (Fig.\[Gudkov-Table3:fig\]) encoding all real schemes combinatorially. If eversions are overlooked, the contiguity graph between chambers permits only closed circuits of even length, whereas by Bézout or Galois a generic pencil of sextics (defined of ${\Bbb R}$) has to cut an odd number of times the discriminant of degree 75. Nearly a contradiction in mathematics if eversion would not exist! Consequently,

\[eversion-deg-6-or-more-forced-by-loops:lem\] Any generic pencil of real curves of even order $m$ contains at least one eversion. At least this is clear for $m=4,6$, and hopefully correct in general.

(inserted \[01.04.13\], but contains a gap!) The discriminant has odd degree $3(m-1)^2$ when $m$ is even. Our pencil is generic in the sense of being transverse to the discriminant, hence induces a sequence of Morse surgeries. Those surgeries (if not “eversive”) can be visualized on the Gudkov table (Fig.\[Gudkov-Table3:fig\]) in degree 6 as moves along the lattice of red-rhombs which permits only closed pathes of even length, whence the assertion for $m=6$. In general we can introduce the invariants $(\chi, r)$ and notice that any Morse surgery which is not an eversion acts as one of the 4 transformations $(\chi, r)\mapsto ( \chi \pm 1,  r \pm 1)$ where signs can be chosen independently. (Not all of them being possible as shown for $m=6$.) Alas this does not seem to be enough to conclude, because those sole invariants $(\chi, r)$ amounts to a planar projection of the whole pyramid (which in general is not a “planar” object say for $m=8$). So one really needs to understand the crystallography of higher pyramids which hopefully still contain merely loops of even length when eversions are omitted. Hopefully our lemma is still true (cf. maybe a related argument in Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000]).

Of course an oval belonging to a certain real scheme can be (topologically) everted iff it is [*maximal*]{} (i.e. not included in any larger oval). Let us consider some examples. Suppose the given scheme to be $\frac{1}{1} 9$, i.e. Harnack’s $M$-sextic. Then there are 10 maximal ovals available. Everting the unique nonempty oval of Harnack’s scheme gives Hilbert’s scheme $\frac{9}{1} 1$ (cf. Fig.\[Eversion2:fig\]), while everting of of the 9 outer ovals leads to a configuration which is not Bézout-tolerable. The net consequence is that in the hyperspace of all curves two schemes (better chambers) may be in reality much closer than they look far apart on Gudkov’s pyramid (Fig.\[Gudkov-Table3:fig\]). There seems to be some secret passages permitting quick travelling in the pyramid. Of course whether Harnack’s chamber is really contiguous to that of Hilbert is another story! It would be fine so for them to sleep in good company, yet more mature thinking bring us back to the Riemann surface picture. The ortho-cycle effecting the strangulation is (since both are $M$-curves) necessarily dividing hence not strangulable by Lemma \[strangulation-impossible:lem\]. For instance one can imagine the top orthosymmetric surface with $r=g+1$ as the double of a planar domain $D$ (with $r$ contours). Suppose given on this an ortho-cycle $\beta$ meeting twice the same oval. The image of $\beta$ in one half (our plane domain) is an arc $\beta^{+}$ joining twice the same contour, and some moment thought shows that $\beta$ divides the surface. Indeed the arc $\beta^+$ can be completed to a Jordan curve in $D$ by aggregating an arc of the boundary and we apply Jordan. Better, argue that $\beta^+$ divides $D$ because we may shrink to a point the contour containing the extremities of $\beta^{+}$, and then apply Jordan separation. The separation effected by $\beta^+$ readily implies that by $\beta$. We have proven:

\[eversion-impossible-for-M-curves:lem\] An $M$-curve (of degree $m\ge 3$) cannot undergo an eversion (while crossing normally the discriminant). In particular Harnack’s chamber in the hyperspace of sextics is not contiguous to Hilbert’s.

If it could be everted, the corresponding path of curves would be materialized by the evanescence (strangulation) of an ortho-cycle $\beta$ cutting twice the same oval. But $M$-curves correspond to the top-orthosymmetric case with planar half. Thus by Jordan separation our cycle $\beta$ divides and therefore cannot be strangulated (by Lemma \[strangulation-impossible:lem\]).

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One can imagine more complicated eversions of Harnack’s scheme (cf. bottom of Fig\[Eversion2:fig\]), yet the result is still the same. Everything depends merely on the oval being everted, for what was inside becomes outside and conversely. For instance Gudkov’s scheme $\frac{5}{1}5$ turns into itself under eversion of the nonempty oval, whereas everting a nonempty oval leads to a scheme enlarging $(1,1,1)$, the deep nest of depth 3, hence Bézout incompatible.

Is the Gudkov chamber self-contiguous to itself via an eversion? Again this is merely a topological possibility, but it requires a deeper investigation to see if it is really so. This would imply Gudkov’s chamber to be highly contorted like a banana-shaped, and it is quite likely that its closure is not simply-connected. However the lemma above (\[eversion-impossible-for-M-curves:lem\]) precludes a self-contiguity of the Gudkov chamber to itself.

The real option however is that there are two (non-maximal) chambers past the discriminant related by eversion, and actually we know this phenomenon to exist a priori in view of the degree argument of Lemma \[eversion-deg-6-or-more-forced-by-loops:lem\].

It seems of interest to understand the secrete passages between Gudkov symbols of Gudkov’s pyramid, at least those topologically permissible under eversion. For sextics we get the following enhancement of the Gudkov-Rohlin pyramid with curvilinear-edges amounting to the varied eversion (Fig.\[Gudkov-eversion:fig\]). Note that we may only evert the nonempty oval without corrupting Bézout (maximality of the deep nest $(1,1,1)$). The sole exception arise with the unnested schemes, plus the scheme $\frac{1}{1} {1}$ whose empty-oval eversion is precisely the deep nest, whereas the nonempty-oval eversion flips back the scheme to itself. We get something like the following messy picture (Fig.\[Gudkov-eversion:fig\]) attempting to keep track of all logically possible eversions.

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The question is to decide which among those are effectively realized algebraically. We already know that those interconnecting $M$-schemes cannot be realized so due to a topological obstruction (Lemma \[eversion-impossible-for-M-curves:lem\]).

As usual blue-rhombs have to be duplicated according to their types and really correspond to 2 distinct chambers of the discriminant.

If a dividing curve (of degree $m\ge 3$) undergoes an eversion then the post-critical curve is nondividing.

Fig.\[Orthoovals2:fig\] nearly proves this via the occurrence of a Dehn-twist in the Picard-Lefschetz transformation. Again the proviso $m\ge 3$ is evident since Dehn twisting an equatorial sphere leads to the same equatorial sphere which is still orthosymmetric.

Incidentally this gives another proof of the impossibility of everting $M$-curves (Lemma \[eversion-impossible-for-M-curves:lem\]) since the latter are necessarily of type I (by Klein 1876 [@Klein_1876]).

[*Insertion*]{} \[01.04.13\] Moreover the lemma implies that both Rohlin’s chambers $\frac{6}{1}2$ and $\frac{2}{1}6$ are not connected by an eversion, since those schemes are of type I (either by the RKM-congruence (\[Kharlamov-Marin-cong:thm\]) or via Rohlin’s (lost) proof of total reality via a pencil of cubics). Note here that the newly discovered version of Le Touzé is not strong enough for this purpose (as it uses RKM).

Thus if we imagine the type I chambers levitating somewhat higher than the sheet of paper, the eversion starting from a dividing chamber always moves down to the ground floor of the diagram. Can we conclude that conversely the diasymmetric type always rises up to orthosymmetric via eversion?

As already noticed, eversions are impossible for $M$-curves (except of course if $m=2$, i.e. conics). Thus the 2 top $M$-curves eversions are actually impracticable. Looking one stage lower at $(M-1)$-curves we see 3 eversions. Examining the corresponding Riemann surface, we can imagine something like Fig.\[Orthoovals2:fig\], i.e. an orthocycle cutting only one real circuit while Dehn twisting the handle it is strangulating. So the real picture is exactly the same as Fig.\[Orthoovals2:fig\] safe that one real circuit has to be imagined missing. It seems plain that the eversion will conserve the diasymmetric character. At this stage the problem becomes quite fascinating: for instance we could via eversion travel from $5_{II}$ to $\frac{4}{1}_{II}$ but not to $\frac{4}{1}_{I}$. Hence $\frac{4}{1}_{I}$ could not be everted to $5_{II}$. This anti-commutativity looks a bit puzzling, since any path can be travelled backwardly or forwardly. All this properly understood could help unravel the mystery allied to the break of symmetry prompted by Rohlin’s complex orientation formula, forbidding the scheme $5_{I}$. Despite this and other intricacies, it seems reasonable to put forward the:

\[eversion-and-other surgeries:conj\] All the red-edges of Fig.\[Gudkov-eversion:fig\] (except the top curvilinear ones linking $M$-curves, and the one linking the Rohlin’s $(M-2)$-schemes $\frac{6}{1}2$ and $\frac{2}{1}6$) can be realized by an eversion (crossing only once transversally the discriminant).

This would give a complete picture of the contiguity graph between chambers residual to the discriminant via elementary algebraic Morse surgeries. Nothing forbids that some edges actually correspond to various Morse surgeries, hence different wall crossings (e.g. coalescing two inner ovals amounts to coalesce one inner oval with the nonempty oval). Hence it is clear that our conjecture is only a crude approximation, for one might really want to catalogue all walls between chambers (including self-contiguous one) while describing the corresponding Morse surgeries. This number of walls is very likely to be much bigger than the number of edges depicted on Fig.\[Gudkov-eversion:fig\].

Modulo little adjustments about the combinatorics, such a spectacular result is perhaps not completely out of reach, as its proof could be akin to that implemented in Itenberg 1994 [@Itenberg_1994] for contracting empty ovals of sextics. (The latter implies our conjecture (\[eversion-and-other surgeries:conj\]) for all the [*straight*]{} edges.) The rough philosophy (at least for sextics) is that when there is no topological obstruction to shrink, one can strangulate in a rigid-isotopic way. However it is quite evident that there must be some extra obstruction, at least if Finashin’s claim (\[Finashin-obstruction-to-coalesce-Harnack:lem\]) that the schematic move $\frac{1}{1}9 \mapsto 10$ (from Harnack’s scheme to the configuration with 10 unnested ovals) can only be accomplished via contraction of the nested oval but not by its coalescence with the nonempty oval. Perhaps Finashin’s claim is merely subsumed to a topological obstruction, which we did not yet understood properly.

Strangulation principle (infarctus, etc.)
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\[27.03.13\] Infarctus=hearth-attack seems to be the generic mortality cause by geometers too much in love with their topis (Dirichlet, Gudkov, Rohlin, etc.)

\[18.01.13\] A message from Viro (16.01.13, cf. Sec.\[e-mail-Viro:sec\]) suggested us the following naive remark. Assume an oval (or a priori just a real circuit) to be contracted to a solitary node via a rigid-isotopy $C_t \in \vert m H \vert$ (having only one extremity in the smooth locus of the discriminant parametrized by uninodal curves). Call such a path a [*pseudo-isotopy*]{} for short. Of course our contraction supplies a homotopy shrinking the oval to a point, hence our circuit is null-homotopic in ${\Bbb R}P^2$ and so forced to be an oval. (By an extension of the theorem of Schoenflies ca. 1906 a Jordan curve in a surface is null-homotopic iff it bounds a disc; compare e.g. R. Baer 1928, Epstein 1966, or Gabard-Gauld 2011 [@Gabard-Gauld_2010-Jordan-and-Schoenflies], etc.)

The presence of this canonical bounding disc gives some evidence to the Itenberg-Viro contraction conjecture (IVO) of empty ovals, which supplies some membrane for the strangulation to occur.

[*Added in proof*]{} \[01.04.13\] If pessimistic, it may be also the case that the contraction conjecture is violently false (and perhaps deducible via Shustin 1985), along the line our Prop.\[Klein-vache-deg-6:prop\] which shows crudely that contraction plus classification implies Klein-vache.

Note at this stage that if there were some dividing plane curve with only one oval and of degree $m\ge 3$ then strangulating this oval would be impossible by Lemma \[strangulation-impossible:lem\]. Quite fortunately such curves do not exist by Rohlin’s inequality $r\ge m/2$ (\[Rohlin’s-inequality:cor\]).

So the contraction conjecture for empty ovals (CCEO) suggests perhaps having some bounding disc for a cycle to be strangulable, as to refine slightly the statement of the Shrinking principle (\[shrinking-principle-vague:conj\]):

[(Strangulation principle)]{} \[shrinking-principle:conj\] Any (Galois) invariant cycle (=circle) $\beta$ on a smooth plane curve $C$ of degree $m$ can be strangulated (through a path in the hyperspace of curves crossing only once the discriminant at a smooth point of the latter), whenever there exist in ${\Bbb C}P^2$ a smooth disc $D$ bounding $\beta$ which is invariant under complex conjugation, and intersects the complexification only along $\beta$ (i.e., $D\cap C({\Bbb
C})=\beta$). Say in this case that $\beta$ is fillable.

This is of course a true extension of (CCEO) perhaps susceptible to imply “Klein-vache”, i.e. any nondividing curve can acquire a solitary node by a pseudo-isotopy. Of course any nondividing curve admits an anti-oval (just lift an orientation reversing loop from the non-orientable quotient $C(\Bbb C)/\sigma$), but is another story to find one which is fillable. If so and if the above conjecture extending Itenberg-Viro’s conjecture is right we could deduce “Klein-vache”, which is however disproved in Shustin 1985 [@Shustin_1985].

If there is a filling disc $D$ then as it is invariant under conj=$\sigma$, we have an involution on the disc (so with a fixed point by Brouwer). In the case of an anti-oval acted upon by antipody, it seems that the involution has to act as an antipody on the whole disc. In general involutions on the disc are of 3 types (identity of order 1, orthosymmetry fixing a diameter, and antipody fixing the center). (This must be ex/implicit in work by Brouwer, Kerékjártó ca. 1914-1922.)

As argued by Viro’s e-mail, CCEO looks more natural that “Klein-vache” since the filling is virtually God-given, just taking the (sealed) inside of the oval. Yet perhaps this has some analog for an anti-oval in term of differential-geometric fillings, e.g. minimal surfaces in ${\Bbb C}P^2$ endowed with its “round” Fubini-Study metric coming from $S^5\to {\Bbb C}P^2$ (Hopf fibering).

One problem with anti-ovals is that there are plenty of them (not just finitely many like for ovals), and so one’s idea could be to select some preferred one, maybe as “the” systole. In the case of the diasymmetric sphere this is not enough to ensure finiteness, but perhaps suffices to single out some natural class of anti-ovals. Recall that systoles are geodesics, and so are usual ovals.

A recipe could be as follows: given a nondividing curve $C_m$ of some degree. Endow it with the natural Fubini-Study metric of ${\Bbb C}P^2$ to get a Riemannian metric on the Riemann surface $C_m(\Bbb C)$. Since the curve is diasymmetric it contains an antioval (invariant circle acted upon by antipody by $\sigma$). Hence by compactness there is also a such of minimal length, the so-called systole, not perfectly unique of course, but choose one such systolic antioval. Consider the latter as a circuit in the ambient ${\Bbb C}P^2$ and solve the Plateau problem for that contour, in its classical setting of soap films diffeomorphic to the disc. Plateau makes also sense for membranes of higher topological structure, but ignore them to stay closest to the Itenberg-Viro conjecture. Plateau is always soluble but the notorious difficulty is to ensure embeddedness of the solution. Perhaps this is true in $E^3$ and also in ${\Bbb C}P^2$ due to some simple-connectivity, or perhaps the special systolic properties of the boundary data. (\[21.01.13\] Beware that a minimal surface has vanishing mean curvature, while the natural Itenberg-Viro “reality” membrane is positively curved. But the former assertion is specific to $E^3$…)

2[${\Bbb C} P^2$]{}

As the given contour is invariant under $\sigma$ (an isometry of ${\Bbb C} P^2$) it is likely that Plateau’s solution enjoys a similar invariance, and we would be essentially finished (modulo the difficulties enumerated).

At this stage we would have a perfect analog of the bounding disc of Itenberg-Viro’s empty oval, via our Plateau filling of “the” systole realizing the anti-oval of shortest length. For the analogy to be perfect one should ensure that the Plateau film intersects the Riemann surface $C_m({\Bbb C})$ only along the contour (systolic anti-oval). This looks either hard or trivial. For instance recall (from Wirtinger, cf. also Mumford’s book [@Mumford_197X-BOOK-complex-alg-var]) that algebraic subvarieties of ${\Bbb C}P^n$ endowed with Fubini-Study are (precisely?) minimal surfaces. So there is perhaps some chance to prove disjointness. (If they intersect interiorly then try to build a canal surface by surgering a piece of $C_m({\Bbb C})$ to the Plateau film, trying so to violate its area minimization ….)

Maybe some interesting twist of Plateau’s problem is that one may be able to reconstruct the whole complex locus via Plateau if we are given only the real locus of the curve. Of course as there is now handles (except for $M$-curves) and several contours this will necessarily involve the so-called Plateau-Douglas problem permitting membranes of higher topological structure (than the disc). As hazardous as it is, this claim would perhaps only work for dividing curves.

Assume all this to work then we have a perfect analogy with Itenberg-Viro, but it is still not explained why the empty oval or our anti-ovals are strangulable via a pseudo-isotopy of algebraic character.

Naively from the given data consisting of minimal film bounding the cycle $\beta$ we can hope to shrink concentrically the disc (put via the Riemann mapping in conformal equivalence with the round disc) to its center. Solving the higher Plateau-Douglas problem for this shrinking contour gives a minimal surface (which by the converse of Wirtinger) would be an algebraic curve realizing the deformation we are looking for. Since this concentric shrinking respect the symmetry (of the round disc whatever its type, i.e. identity, antipodal diasymmetry or orthosymmetric mirror like $z\mapsto \bar z$), the given smaller contours are invariant under $\sigma$ in 2 and so we get real curves by solving Plateau-Douglas, and therefore the desired pseudo-isotopy from $C_m$ to a nodal curve (with a solitary point).

All this is a bit reminiscent of Riemann’s spirit (except for being of lesser vintage) yet dreaming like a canary is quite pleasant. The above strategy (with all its gaps) suggests even that in the Itenberg-Viro shrinking of an empty oval it could be arranged that the subsequent curves all have their ovals progressively shrinking inside the initial one. Whether this stronger conjecture has some chance to hold true is unknown to me.

The above argument suggests that a real algebraic curve should be reconstructible from a single oval. This is certainly true via something like the Nullstellensatz, yet the assertion that all this algebra can be supplanted by differential geometry à la Plateau looks a bit doubtful for we are not controlling the full contours of the membrane. Actually the dividing case looks psychologically more comfortable to a direct appeal of Plateau.

As “Klein-vache” is probably false (cf. Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin] and our partial discussion in Sec.\[Shustin-understood:sec\]), we shall from now on concentrate on the Itenberg-Viro contraction conjecture for empty ovals, which is still non refuted (hence more likely to be true) and easier technically due to the canonicalness of the film bounding the vanishing cycle.

So suppose given any empty oval of a smooth plane real algebraic curve. Goal: strangulate it algebraically via a pseudo-isotopy, i.e. a rigid-isotopy except for its extremity which is a solitary nodal curve. To the disc bounding the (marked) empty oval, apply the Riemann mapping theorem to take it conformally to the unit disc $\Delta:=\{z\in {\Bbb C}: \vert z\vert \le 1\}$ via a map $f\colon D\to\Delta$. This mapping is canonical up to conformal automorphism of the disc, hence unique once a center and a boundary point are chosen (variant choose $3$ boundary points). Consider the pullback $C_\rho:=f^{-1}(\Gamma_\rho)$ of the circles $\Gamma_\rho: \vert z\vert=\rho$ of radii $\rho$ ($0\le \rho\le
1$). Question: are those still algebraic ovals when $\rho<1$? In other words are the Riemann levels $C_\rho$ of an algebraic oval (which is empty) still algebraic curves (at least part thereof)? The truth of this assertion is of course a necessary condition for our above strategy of constructing the pseudo-isotopy via Plateau.

If the initial oval is a circle (or even an ellipse), algebraicity of the Riemann levels looks evident (at least classical I think). For the ellipse it could involve Schwarz’s explicit solution to the Riemann mapping.

Another idea: it would we nice if there is some flow effecting the contraction conjectured by Itenberg-Viro. One idea could be to take any empty oval, and look at its normal curvature flow à la Huisken. Usually this is presented in $E^2$ but there is surely a variant in $S^2$ the double cover of ${\Bbb R}P^2$ on which Fubini-Study induces the round metric (I think). Is it true that the normal curvature flow preserves algebraicity of ovals? If so, the flow would shrink one of the empty oval to a point (yet not necessarily one oval chosen in advance like by Itenberg-Viro), and perhaps it will shrink all ovals in some succession it is alone able to decide until all get shrunk. This could prove a weak form of the contraction conjecture.

Toward a naive dynamical treatment of the Itenberg-Viro conjecture {#CC-via-dynamics:sec}
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\[18.01.13\] How large can an oval be? If we imagine a real projective curve as traced on the sphere $S^2$ we can wonder what the area or length of an oval can be. In degree 2 a quadratic cone can be as large as we please and so the inside area of the oval can be as close to $2\pi$ as we please, but of course not larger as its “twin” occupy the antipodal area of the sphere. If we restrict to even degrees then we have the Ragsdale orientable membrane bounding the ovals (that one with $\chi=p-n$). How large can its area be when lifted to $S^2$? For quartics and when $r=1$ we can enlarge the Fermat equation $x^4+y^4=+1$ as much as we want by taking $x^4+y^4=R^4$ which conserve the same shape (homothety), while covering more and more space of the sphere. So here the upper bound is again $4\pi$ (the full sphere). (We count now the full lift to $S^2$). But what about other quartics, e.g. the Gürtelkurve. Again we may imagine the latter just as a perturbation of two concentric circles. While the outer circle enlarges the inner contracts and so we get the Ragsdale membrane of nearly full area. It seems that there is little chance to find nontrivial upper bound. But what about an $M$-quartic with 4 ovals. How large can the area of its interior be? Again we can imagine a configuration of 2 transverses ellipses one very large but of small eccentricity and one fairly small but orthogonal and smooth it in the usual way to get 4 ovals; one very large and all 3 remaining fairly small. Expanding this at infinity shows that the Ragsdale membrane can cover nearly all the sphere and the upper bound is again $4\pi$. Such consideration could extend to all curves constructed say via Hilbert’s method. But how to treat the general case? At any rate to each rigid-isotopy class of curves we can consider the supremum of all areas of the corresponding Ragsdale membrane. Is this always equal to $4\pi$? Looking at the area of the Ragsdale membrane assigns to each curve a numerical value in $]0, 4\pi[$. Perhaps it is interesting to look at the orthogonal trajectories of this Ragsdale function? The allied gradient flow could provide a dynamical flow shrinking some empty ovals. There is plenty of other functionals perhaps better suited to a dynamical treatment of the Itenberg-Viro conjecture. For instance instead of Ragsdale area we could look at the [*empty area*]{} defined as the cumulated area of all empty ovals. Looking at the descending orthogonal trajectories of this function is likely to shrink ovals. Another choice is the function looking at the area of the smallest oval. Perhaps this has the drawback of lacking smoothness in case two ovals enter in competition for the infimum? Another strategy more suited to the Itenberg-Viro contraction problem is that we are given an empty oval, and during a rigid-isotopy we can follow him continuously. Hence given a curve with a marked empty oval we can define in the whole chamber residual to the discriminant (alias the rigid-isotopy class of the curve) the functional ascribing the area of the inside of this marked but moving oval. Of course when dragging the curve around a loop in its chamber the oval can be to another oval, so the function looks multivalued. Yet we get it single-valued on the space of curves with a marked oval. So the space of curves with a marked oval is actually an $r$ sheeted cover of the usual space of smooth curves (with $r$ variable on the different chambers of course).

As long as we keep the marked oval into view there is a way of steepest descent diminishing maximally the area of the oval. For this to make good sense we require orthogonal trajectories and so a metric on the space of all curves. The canonical choice seems to be the elliptic geometry on $\vert m H \vert \approx {\Bbb
P}^N({\Bbb R})$ the space of coefficients double covered by the round sphere $S^N$.

Now follow the corresponding trajectory of steepest descent. What can happen? By construction our marked oval will decrease in area, but will it docilely shrink to a point? Here are some evident difficulties (D.$n$, $n=1,2,\dots$)

(D.1) [*Wrong attractor (stable equilibrium).*]{} First one can imagine that our function as a sink trapping us into some “depression” like the basin of a lake yet not at zero altitude (e.g. lake Baikal). Then our motion stops and the goal fails blatantly, having only reached an algebraic curve realizing a local minimum of the area yet still positive. Perhaps some clever argument precludes such depression (e.g. if our functional turned out to be harmonic by some miracle?)

(D.2) [*Saddles points (unstable equilibrium).*]{} We may of course also reach something like a saddle point, where we need then to choose quite randomly one of the two (or more if not Morse) way of steepest descent. Generically up to perturbing the initial curve, we can avoid such accidents.

(D.3) [*Controlling the limit.*]{} Hence let us assume that the area shrinks to zero (assuming (D.1) to have been overcome). Naively one can imagine the oval shrinking to a complicated dendrite (though Bézout unlikely) or to a segment (again algebro-geometrically improbable). The sole possible limit seems to be a point. (This seems an easy task via implicit function theorem, Bézout, etc.)

(D.4) [*Choosing the right functional (i.e. arranging a “convex” or “harmonic” landscape).*]{} We have as yet only considered the area functional yet the length functional looks nearly as appropriate or better? Or even one could use a mixture of both like the isoperimetric ratio. Note that nearby a solitary node the behavior become nearly circular or at least elliptical like in the local model $x^2+y^2=\varepsilon^2$ or $ax^2+by^2 =
\varepsilon^2$ with $\varepsilon \to 0$.

Among all difficulties the most stringent seems to be (D.1) which of course as to be settled by playing with (D.4), i.e. choice of the functional. To settle (D.1) it is enough showing that nearby all curves there is one of smaller “energy”. For the area functional one could imagine an oscillation by perturbing slightly like in the Harnack-Hilbert method our marked oval by a collection of $m$ lines. Alas it result a vibration of the oval slaloming across its initial position so it is not obvious how to decrease area. So to impede getting blocked by (D.1) we are reduced to the “local” problem of finding an appropriate function which can always be decreased by small (algebraic) perturbations. What about the total curvature of the oval (or the inverse thereof as to go to zero for a shrinking circle), etc. If we work with area functional and if the oval is nearly circular, we can plug in it a smaller circle and taking this equation $k$ times (assume $m=2k$ even for simplicity). Perturbing the curve along this multiple circle may decrease area. This is of course just a very special case. It seems to apply to the case when the curve is a deep nest in which case the innermost (=empty) oval as to be “convex” (else Bézout is violated, compare Zeuthen 1874 [@Zeuthen_1874]).

The whole problem seems reduced to that of finding a good functional $\varphi$ which has no local minimum. Above it is not fundamental that the oval is convex, to plug a circle inside it. This just uses the fact that the interior of the oval is open, and tracing a little circle inside the oval while taking its $k$th multiple gives another curve $k\cdot E_2$, along which to deform inside the spanned linear pencil $\lambda C_m +\mu (k\cdot E_2)$. Alas nothing ensure the oval to stretch within its interior.

Another idea is to apply the Riemann mapping theorem and shrink the radius of the representing circle, while hoping that this new smaller Riemann level is still an algebraic curve of the [*same*]{} degree. If this work we are able to decrease the area functional.

This reminds me some work of Bell and Aharonov-Shapiro [@Aharonov-Shapiro_1976] to the effect that the Riemann (and more generally Ahlfors) map of a quadrature domain is algebraic, and that quadrature domains are dense in the space of all domains so that virtually any Riemann map is algebraic.

But in our context we have an algebraic contour(=oval) and the following would simplify life:

\[Riemann’s-level-algebraic:conj\] Given a nonempty oval of a real plane algebraic curve of degree $m$ and suppose the corresponding spherical calotte conformally mapped (via Riemann) to the unit disc $\{z: \vert z\vert \le 1
\}$. Then the pullbacks of the smaller circumferences $\vert
z\vert =r $ are still algebraic curves of the same degree $m$!(???)

If so then we can decrease area thus solving difficulty (D.1), and perhaps the whole conjecture of Itenberg-Viro. Of course this looks a bit optimistic (due to the a priori highly transcendental nature of the Riemann map), but at least the dynamical strategy looks quite stimulating. Needless to say we have not proved the Itenberg-Viro conjecture, but in case it is true, perhaps the above vague ideas are quite close (at least in broad lines) to its ultimate technical solution. We Summarize the discussion as follows:

Let $\vert m H\vert$ be the space of all real algebraic curves of fixed degree $m\ge 1$ and $\frak D$ be the corresponding discriminant parametrizing singular curves. The complement $S_m=\vert mH\vert - \frak D$ is the space of smooth curves. Suppose given some real positive-valued smooth functional $\alpha$ on the space MEO of all curves with a marked empty oval which has

\(H) no stable equilibrium (local minimum) and such that if $\alpha(C)\to 0$ then the curve $C\in S_m$ tends to a solitary nodal curve.

Then the trajectory of steepest descent (gradient flow) always converges toward a curve with a solitary node, after possibly slight perturbation of the initial data (permissible as we work up to rigid-isotopy). In particular the Itenberg-Viro contraction conjecture holds true.

Of course this is just the formal aspect of the story (i.e. imputable to the theory of ordinary differential equations). Yet the real problem is to find a functional $\alpha $ suiting hypothesis (H). A candidate is to take $\alpha$ the area of the marked oval, and then hypothesis (H) could follow from the optimistic Conjecture \[Riemann’s-level-algebraic:conj\] on the algebraicity of Riemann’s level.

Call for an attack via the Riemann mapping (yet another Irrweg=aberration?) {#CC-via-Riemann:sec}
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\[18.01.13\] Let us do some experiments. Suppose given some real algebraic curve and take an empty oval on it. Mark an interior point and consider the Riemann map $f$ taking the domain $D$ interior of the oval to the unit disc $\Delta$. Pull-backing polar coordinates on the disc gives an isothermic system of coordinates on $D$. Fig.\[ItenbergViroRiem:fig\] gives some qualitative pictures for an ellipse or with ovals of a cubic or even of some quartics.

-5pt0

-5pt0

Naively the levels $\vert f \vert = \rho$ of the Riemann mapping look again like curves of the same degree. This is especially striking for the contorted quartic with one oval of Fig.\[ItenbergViroRiem:fig\]c. It is like a volcano spreading its lava on the whole territory available in the island interior to the oval of this $C_4$. If algebraicity is true and degree conserved, then the levels of the Riemann map gives directly the contraction of the Itenberg-Viro conjecture (\[Itenberg-Viro-contraction:conj\]). Is there such a miracle? Maybe giving to Riemann an algebraic contour, the Riemann map itself is algebraic of the same degree and so are all its sublevels $\vert f\vert = \rho= const$.

Let us first consider the more basic converse assertion that if the Riemann map is algebraic then so is its contour. So let $f\colon D \to \Delta$(=unit disc) be a Riemann map which is algebraic, i.e. the power series of this analytic function is finite, i.e. a polynomial of finite degree $f(z)=c_0+c_1 z+c_2
z^2+ \dots +c_n z^n$. Then as $f^{-1}(\partial
\Delta=S^1)=\partial D$ we see that $$f(z)\overline{f(z)}=1$$ identically on the contour $\partial D=:\Gamma$. Doing the usual splitting in real and imaginary parts by letting $z=x+iy$ and $c_n=a_n+ib_n$ one sees that the product $f(z)\overline{f(z)}$ becomes a polynomial $P(x,y)+iQ(x,y)$, where $P(x,y)$ has degree $n$ \[[*Warning*]{}.—this is disproved in the sequel!\] while $Q(x,y)$ is identically zero. It follows that $P(x,y)-1=0$ identically on the contour $\Gamma$ which is therefore algebraic. Of course the same holds true for any sublevels of the Riemann map, i.e. sets $f(z)\overline{f(z)}=\rho$ are also algebraic. This proves the

\[Riemann-level-algebraic\] If the Riemann map $f$ is algebraic then the contour is a real algebraic curve and so are all sublevels. Further the degree of all this Riemann levels $f \bar f =\rho$ ($0\le \rho \le 1$) have the same degree as $f$.

This is the trivial sense. What about the converse? Suppose given a contour which is an oval of some algebraic curve can we conclude that the Riemann map is algebraic? This is what Arnold would call pure Riemannian predestination. We think this to be true as follows:

\[Riemann-map-algebraic:thm\] Suppose given an (empty) oval $\Gamma$ of a real algebraic curve $C_m$ of degree $m$. Mark any point $p$ inside this oval and let $D$ be the sealed interior of the oval. Let $f\colon D \to \Delta$ be the Riemann mapping taking $p$ to the origin $0\in \Delta$ (unique up to rotation). Then the Riemann map is a polynomial of degree $m$.

(vague sketch) The idea is simply to look at the function $f \bar f$ which is analytic and vanishes on an algebraic locus, namely $\Gamma$. So by an appropriate Nullstellensatz (Arnold’s predestination? or Bloch’s slogan “Nihil est in finito quod non prius fuerit in finito”) it must follow that the function $f \bar f$ is itself algebraic. The proof is complete. (???)

At this stage we would have deduced Itenberg-Viro’s contraction conjecture (as a simple corollary of the Riemann mapping theorem!!!):

\[Itenberg-Viro-via-Riemann:cor\] Any nonempty oval of a plane real algebraic smooth curve (of degree $m$) can be contracted to a solitary node.

Apply Theorem \[Riemann-map-algebraic:thm\] to the nonempty oval under consideration, to obtain a Riemann map $f$ which is algebraic. By Lemma \[Riemann-level-algebraic\] all the levels $\vert f\vert=\rho$ are algebraic curves of degree $m$, while shrinking the radius to $\rho=0$ the curve acquires a solitary node.

Some objections to the method (or details to be filled):

(DET.1) How to ensure that the nodal curve so obtained has only this solitary node as sole singularity?

(DET.2) We have worked as if the curve were affine and not projective. This is usually a harmless nuance via the usual yoga, (des)homogenization of the equations. More severe is our supposition that our oval can be put in some affine chart! As I learned from (the late) Felice Ronga (ca. 1999–02), there is a sextic with one oval only, such that any line cuts it in at least two (real) points. In other words no line avoids this sextic. This is simple to construct by perturbing à la Plücker-Brusotti a configuration consisting of two concentric circles, plus the two axes of coordinates. Smoothing this sextic arrangement quite randomly (slalom as much as you can), one gets easily the required curve (Fig.\[ItenbergViroRiem:fig\]d). \[Ronga’s original picture is to be found as the front cover of his book “Analyse réelle post-élémentaire, 1999 après J.Christ.” [@Ronga_1999-BOOK].\]

Such a “Ronga curve” causes some trouble to our procedure, which requires putting the oval in an affine chart which we identify to the complex plane. Perhaps this is not fatal as the Riemann mapping theorem has some more intrinsic character, namely any Riemannian membrane topologically equivalent to the disc is conformal to the unit disc. So maybe one should use this more general version, and adapt the above affine argument in this more global setting. Algebraically this would amount to work always with homogeneous coordinates and think with cones in ${\Bbb R}^3$. We may then apply the Riemann mapping theorem to the spherical calotte bounding the oval (there is two of them, but choose one), and use as cut function homogeneous polynomials. A variant is perhaps just to pass from the sphere covering ${\Bbb R}P^2$ to the complex plane via stereographic projection (from a point outside the oval). This projection is conformal, but does it preserve the degree of polynomials?

\[19.01.13\] Here is a more fatal destruction of the above pseudo-proof of the contraction conjecture via the Riemann mapping theorem.

The key is a little computation of $f(z)\overline{f(z)}$, where $$f(z)=c_0+c_1z+c_2 z^2+\dots +c_n z^n,$$ $$\overline{f(z)}=\overline{c_0}+\overline{c_1}\bar z+\overline{c_2}
{\bar z}^2+\dots +\overline{c_n} {\bar z}^n.$$ When expanding this product it is useful to write it as a “diamond”: $$\begin{aligned}
c_0 &\overline{c_0} \cr
c_0 \overline{c_1} \bar z  + & c_1 \overline{c_0}  z \cr
c_0 \overline{c_2} {\bar z}^2 +c_1 \overline{c_1}&  z \bar z+ c_2
\overline{c_0} {z}^2 \cr
c_1 \overline{c_2} z {\bar z}^2  + & c_2 \overline{c_1}  z^2 \bar
z \cr
c_2 \overline{c_2}& z^2 {\bar z}^2.\end{aligned}$$ This is the end result when $n=2$, but otherwise it will expand to larger rhombs. At any rate, the weightiest term is $c_n
\overline{c_n} z^n\bar{z}^n=\vert c_n\vert^2 (x^2+y^2)^n$, and so we get indeed a polynomial, but one of degree twice as big as that of $f$.

Now suppose the given contour to be an ellipse (which is not a circle), then even if the qualitative part of Theorem \[Riemann-map-algebraic:thm\] ought to be true, i.e., if the Riemann map $f$ of an algebraic contour is algebraic, then the degree of $f$ cannot be $1$ (for then $f$ is a similitude which preserves circles). Hence the degree of the Riemann map $f$ of an ellipse is at least $2$, hence by the above computation the degree of $f\bar f$ is at least $4$, and so the sublevels of the Riemann map are at least quartics, and not ellipses as we initially imagined (compare e.g. the misleading Fig.\[ItenbergViroRiem:fig\]a). Hence even granting algebraicity of the Riemann mapping of an algebraic contour, the corresponding levels would not be of the same degree.

Actually the above argument shows that our strategy fails for a conic, but is it really a disproof in general? Assume the initial curve $C_m$ to be a quartic, then the degree of the Riemann map could be $n=2$, and thus the degree of $f\bar f$ is four and so the sublevels would stay a deformation within quartics. More generally this numerology makes sense for any $C_{2k}$ curve of even degree $2k\ge 4$, by taking $n=k$.

A more severe objection is surely, as already apparent by looking at the highest power $c_n \overline{c_n} z^n\bar{z}^n=\vert
c_n\vert^2 (x^2+y^2)^n$, the fact that $f\bar f-1=0$ is not the most general curve of degree $m=2n$ (e.g. the monomial $x^{2n-1}y$ is missing). Hence there is no chance the Riemann map of any curve $C_{m=2k}$ being algebraic of degree $k$, and our strategy is definitively foiled.

As a modest consolation one can apply the above method to some few ad hoc curves with equations of the shape $f\bar f-1=0$ where $f$ is some algebraic Riemann map. The corresponding curves (say “Riemann curves”) could be contracted by the above recipe. This is of course far from settling the initial desideratum of Itenberg-Viro.

What can be retained from this attempt? Let us start with a polynomial $f(z)\in {\Bbb C}[z]$ of degree $n$. This induces a holomorphic map ${\Bbb C}\to {\Bbb C}$ of degree $n$ (Gauss–D’Alembert fundamental theorem of algebra) which is a branched covering. Sometimes it turns out that the unit disc is a trivializing open set for this covering. In the language of complex analysts (Bloch, Landau, Ahlfors, etc.) this is also what they would call a schlicht unit disc. Taking one among the $n$ many sheet lying over $\Delta$ gives a simply connected domain which by $f$ is conformally mapped to the disc, and so we recover a Riemann map. For all those domains (which are algebraic of degree $2n$ via the equation $f\bar f-1=0$ but with $n$ unnested ovals), we may a priori implement our contraction algorithm. (Overlooking the unnested condition, we could hope that such Riemann curves are spread in all chambers of the discriminant and hope a general attack.) Now back to our setting it must be noticed that during the shrinking $f \bar f=\rho$, with $\rho \to 0$ our algebraic curve sees all its $n$ ovals being simultaneously shrunk. Hence in this very favorable setting, the solitary node condition fails blatantly. Conclusion: our strategy via Riemann leads nowhere, if it pertains to implement the contraction conjecture of empty ovals.

CCC: collective contraction conjecture, as an avatar of Itenberg-Viro (Gabard 2013)
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[*Insertion*]{} \[04.04.13\].—As we noticed only today, in substance it seems that the conjecture posited below bears some analogy with Klein 1892 [@Klein_1892_Realitaet] (p.177 of Ges.Math.Abhd.) who wrote: “Wir könnten z.B. mehrere Züge unserer Kurve gleichzeitig in isolierte Doppelpunkte überführen”. More generally Klein 1892 discusses at this place (p.176) what he calls the “Doppelpunktsmethode” amounting essentially to contract any symmetry-line of the Riemann surface, and this of course seems to anticipate what we called before the Itenberg-Viro contraction principle. It is not clear however that Klein ever formulated something as precise as the Itenberg-Viro contraction conjecture (specific to plane curves). On p.176–177, Klein writes something which taken out from its context looks a bit overoptimistic namely: “Bei allen anderen Fällen hat die Durchführung des genannten Prozesses und damit die Zusammenziehung eines beliebiegen Ovals der Kurve zu einem isolierten Doppelpunkte keine Schwierigkeit.”

Failing with Riemann suggests a variant of Itenberg-Viro, viz. CCC=collective contraction conjecture: deformation in the large as a method of prohibition {#CCC:sec}
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\[19.01.13\] The above discussion suggests the following variant of the contraction conjecture, which has maybe some spontaneous appeal and independent interest. Suppose given a projective smooth plane real curve $C_m$. Look at all empty ovals simultaneously. Is it possible to shrink all of them in one single stroke toward solitary nodes (via a deformation of smooth curves sole for its end-point being in the discriminant)?

It seems likely that the contraction conjecture (\[Itenberg-Viro-contraction:conj\]) implies this, roughly by shrinking one oval, and then the second, etc. Of course one then needs to arrange a bit things so that prior to extinct one oval completely, one waits until the second empty oval becomes “small” enough, etc. Finally one synchronizes the ultimate “coup de grâce” to kill all the empty ovals at the same time (“time” being just the parameter of the path $[0,1]
\to \vert mH \vert$ in the space of all curves of order $m$).

[*Insertion*]{} \[02.04.13\].—It may help reading the sequel to remarked first that the reverse process, of deducing an individual (solitary) contraction (à la Itenberg-Viro) from our collective one, is much easier and a trivial consequence of Brusotti, if we did no mistake, cf. Lemma \[CCviaCCC-Brusotti:lem\] below. So the conjecture posited right below is stronger than the one of Itenberg-Viro (yet perhaps equivalent, or at least easier to disprove).

So let us (somewhat cavalier) formulate the:

[(Collective contraction conjecture=CCC, \[19.01.13, 22h40\])]{}\[CCC:conj\] Given any smooth real curve $C_m$ of degree $m$, it is possible to shrink all the empty ovals simultaneously toward solitary nodes. (Solitary but synchronized death of all ovals.)

This is obviously true for $m=2,3$ (being actually equivalent to the individual contraction principle) since there is at most one empty oval available.

The case $m=4$ is already more tricky, yet still compatible with Bézout. If $r=4$ ($M$-quartic), we would have a quartic with four isolated (solitary) nodes. This exists just take an imaginary conic $C$, and aggregate it with its conjugate $C \cdot C^\sigma$ (this is real but a priori the four intersections need not all be real). More simply take two transverse conics, look at signs and arrange a level so that there are 4 isolated points by making a naive picture of the graph of $E_2 \cdot F_2$. Since the real scheme encodes completely (in degree $m=4$) the rigid-isotopy class (Klein 1876, etc.) it follows that CCC holds true in degree $m=4$. The case $r=2,3$ are treated similarly by looking at the graph of a special equation and passing a plane tangent to the 3 (or lesser) hills.

Now what about degree 6? Deciding the truth of the above conjecture in degree 6 (CCC6), is already more tricky. Perhaps this follows from Itenberg’s CC6, if not formally by the method used therein, i.e. Nikulin’s theory with $K3$-surfaces. As said at the start it could be that CC implies CCC in all generality. In the sequel we assume CCC as granted and look what can be derived from it.

Let us first suppose that there is an $M$-sextic $C_6$ with 11 unnested ovals (what Hilbert, Rohn, Petrovskii, Gudkov, Arnold, etc. were fighting hard against). Shrink all of them to a point according to CCC (\[CCC:conj\]). Then the Riemann surface is strangulated along all its oval in two (algebraic) pieces which are topological spheres. Since the nodes are supposed to be solitary these two pieces are smooth curves of genus 0 intersecting transversally. Therefore (via the genus formula $p=\frac{(d-1)(d-2)}{2}$) they are of degree $1$ or $2$, but have to intersect in 11 points. Bézout is overwhelmed! (Alternatively the genus formula is corrupted, since we have a degeneration of $C_6$ toward two cubics $C_3$ and its conjugate $C_3^\sigma$!) This gives a new “proof” of Hilbert-Rohn-Petrovskii via CCC. Of course it is quite tempting to wonder if Hilbert (or subsequent workers) did not knew about this argument at least as a heuristic tool.

More generally:

[(like Hilbert 1891)]{} Under axiom CCC, a smooth $M$-curve (of even degree) cannot have all its ovals unnested unless its degree $m$ is less than four ($m\le 4$). In particular an $M$-sextic cannot have all its $11$ oval unnested (which is Hilbert’s original claim as early as Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege].)

Shrinking collectively all the empty ovals of $C_m$ (via CCC) gives a splitting $C_m \to C_d \cup C_d^\sigma$ in two algebraic curves of degree $d=m/2$ intersecting transversally in $r$ points. So by Bézout $d^2=r$. Since both strangulated halves have genus $p=0$ (for we started from an $M$-curve), their common degree $d$ can only be $1$ or $2$. Hence $d\le 2$, and so $r=d^2\le 4$. Since $r=g+1$ ($M$-curve assumption) it follows $g\le 3$ and so $m\le
4$. (Variant: conclude more directly via $d=m/2$.)

As a matter of philosophical dilettantism (?), it may be wondered, whether Hilbert himself used the above argument, at least as a heuristic tool. To my knowledge there is no record in print along this sense. Yet, Hilbert, say unlike Poincaré was a formalist, in particular never writing down crazy ideas. Thus, it may be not be impossible (our subjective speculation) that Hilbert may have argued along this route. In fact it is probably more realist that Hilbert argued along another idea, cf. e.g. the passage of Gudkov 1974 [@Gudkov_1974/74], where Hilbert’s method is described as implemented by his students Kahn and Löbenstein.

Even more generally:

Under CCC, a smooth dividing curve (of even degree $m=2k$) cannot have all its ovals empty (equivalently unnested) unless:

[(1)]{} its number of ovals $r$ is a square ($1,4,9,\dots$), and actually the square of its semi-degree $k$[^49];

[(2)]{} some stringent arithmetical conditions (say predestination or coincidence) are verified, namely all the displayed formulas in the proof below have to be satisfied.

By CCC shrink all empty ovals of the curve $C_m$. The Riemann surface $C_m({\Bbb C})$ (“complexification”) has genus $$g=(m-1)(m-2)/2.$$ Once strangulated, it splits in two Riemann surfaces of genus $$p=[g-(r-1)]/2$$ (since $g=(r-1)+2p$ by visualizing the orthosymmetric surface). Both halves are algebraic smooth curves intersecting in $r$ points. Being actually interchanged by conjugation, they have some common degree, say $d$, verifying $$p=(d-1)(d-2)/2.$$ So by Bézout (or homological intersection theory) we infer $$d^2=r.$$ Hence $r$ must be a square. In fact a sharper argument based on the degeneration $C_{2k}\to
C_k\cup C_k^\sigma$ shows that $r=k^2$ directly by applying Bézout to both halves of the limiting curve of the collective contraction.

A this stage “Eureka” \[23h41\] we have already proved that the sextic scheme $5$ is necessarily of type II (as followed first from Rohlin’s complex orientation formula).

Likewise the sextic schemes $\ell$ ($\ell = 9$ excepted) cannot admit a type I incarnation (though this was already implied by Klein’s congruence modulo 2, and Arnold’s congruence mod 4), safe for $\ell =1$ where either Rohlin or our suggestive geometric argument do instead the job.

Indeed if $r=1$ and $m=6$, then both strangulated parts have genus $p=5$ (imagine the Riemann surface of genus $g=10$ split by the one oval), but this is not even the genus of a smooth curve. Hence strangulation is impossible violating axiom CCC. Of course the philosophy behind CCC is quite akin to the filling trick of Arnold-Rohlin safe that the closing is God given by some postulated (but hypothetical) shrinking procedure in the rigid algebraic category.

In general it remains the boring task of extracting the exact arithmetical consequences of CCC, while checking if it is really compatible with factual data. In degree $6$, CCC seems to live in perfect harmony with Rohlin’s enhancement of Gudkov’s table (Fig.\[Gudkov-Table3:fig\]). Since the usual (individual) contraction conjecture CC holds true in degree $6$ (by Itenberg 1994 [@Itenberg_1994]), it is likely that the collective variant CCC holds good as well. Of course all the arithmetical relations are in reality less stringent that they look at first glance, since they are all coming from the genus formula which itself may be interpreted as a surgical process regulated by Bézout. (Recall the simple proof of the genus formula based on the morphogenesis of lines getting smoothed under surgeries.)

Applying CCC to Harnack’s sextic configuration leads nowhere since the Riemann surface keeps connected.

Another exercise: assume there is a dividing quartic with two unnested ovals. Apply CCC to both ovals. Then the Riemann surface of genus $g=3$ is strangulated in two surfaces of genus $p=1$, hence of degree $d=3$. But the latter cut themselves in 9 points and not two. This contradiction reproves (modulo CCC) the well-known fact (\[Klein-unnested-quartic-nondividing:lem\]) due to Klein 1876, Arnold 1971, Rohlin 1972–1978, Wilson 1978, Marin 1979, Gross-Harris 1981, etc., that a quartic with two unnested ovals is necessarily nondividing. (Variant of the argument: $r=2$ is not a square.)

The principle emerging is that large deformations prompted by contraction conjectures affords a puissant method of prohibition, as opposed to the method of small perturbations which is merely a toolkit for construction.

\[20.01.13\] At this stage, the method CCC looks quite powerful, at least as a heuristic tool, reducing to Bézout several deep assertions and results of Klein, Hilbert, Rohlin, etc. However as yet the method is quite limited to the case where all the ovals are empty so that the strangulation really implies an algebraic splitting of the dividing Riemann surface.

Perhaps the method can be extended beyond this proviso. For instance after shrinking the first generation of all empty ovals and making them effectively disappear from the real locus, a second generation of empty ovals appears, which would be contracted in turn to solitary nodes, etc. Iterating so collective contractions would contract all ovals and so achieve a splitting of the dividing Riemann surface. For the method to be effective it seems that the solitary nodes of the first generation ought to resurface as such right after the contraction of the second generation of empty ovals. All this looks very dubious, yet some more clever intelligence can perhaps extract something from this procedure.

Consider a specific example to make the difficulty more concrete. Consider the sextic scheme $\frac{10}{1}$, and let us call it (improvising terminology) the Rohn scheme (for Rohn 1911–13 [@Rohn_1913]) was the first attempting to disprove its existence via a substantiation of Hilbert’s method. Let us contract all empty ovals. The resulting Riemann surface is still connected, and we lack a splitting suited to an application of Bézout. The obvious idea is to make first an eversion of the nonempty oval so as to reduce to the unnested scheme $11$. (For the definition of “eversion” cf. Sec.\[Eversion:sec\].) So we need another highbrow large deformation principle, dual to the contraction principle stating that any maximal oval can be everted provided the resulting scheme is not prohibited by Bézout. Alas, eversions are not permitted for $M$-curves by virtue of Lemma \[eversion-impossible-for-M-curves:lem\], and this stratagem looks jeopardized.

[*Insertion*]{} \[01.04.13\].—After the collective strangulation of all empty ovals of a Rohn curve of scheme $\frac{10}{1}$, we would get a Riemann surface of genus 0 and degree 6, but with 10 nodes. Alas this is still permissible! Of course our argument being merely abstract (i.e. using the abstract topology) would equally well apply to the veritable $M$-curves, and this explains that.

Consider next the sextic $M$-scheme $\frac{2}{1}8$. Shrink all empty ovals to get a curve of genus $p=0$ with one oval (once it is desingularized). So it is really just a rational curve of the dividing type (like our orthosymmetric equatorial planet Earth). By a trivial case of Ahlfors (actually the Riemann mapping theorem) this has a unique conformal structure. So there is a total map of degree $1$. This in turn gives a total pencil of curves, of order say $k$. It seems clear that all solitary nodes of our (contracted) $C_6$ must be in the base locus of the pencil. Indeed else the pencil is sweeping out some node, and so the curve through it has one intersection (counting double) but zero nearby whence a disappearance in the imaginary locus (violating total reality). So our pencil must exhibit at least $10$ basepoints. Naively the disc inside the nonempty oval looks foliated by two foyers (index$=+1$) violating Poincaré’s index formula, but this looks too naive because it would kill as well Hilbert’s or Gudkov’s sextics. So we are again confronted to some complicated foliation argument which we know to be quite difficult to implement. We could dispense using CCC, by applying instead directly Bieberbach-Grunsky to the smooth curve $C_6$ (i.e. the genus zero case of Ahlfors 1950 [@Ahlfors_1950]). All this looks difficult and let us abort here shamefully.

\[23.01.13\] Let us insert here an optional side remark. It is tempting to wonder what follows from Thom’s conjecture (Kronheimer-Mrowka theorem meanwhile of 1994) to the effect that algebraic curves minimize the genus among smooth orientable surfaces embedded in ${\Bbb C}P^2$ realizing the same fundamental homology class. If one fills by the half of a hypothetical real $M$-sextic of real scheme $11$ (eleven unnested ovals) by the interior of all ovals, one obtains a smooth surface of genus $0$ (round the corners) of degree $6/2=3$. This violates Thom-Kronheimer-Mrowka, which therefore implies again the Hilbert-Rohn-Petrovskii prohibition.

[*Insertion*]{} \[01.04.13\].—The degree 3 case of Thom is really due to Kervaire-Milnor 1961 [@Kervaire-Milnor_1961], based on deep works by Rohlin, ca. 1952.

Some thinking shows however that Thom’s conjecture does not imply much more, for the filled membrane has then genus $\ge 1$.

++[C++]{}

Do iterated contractions (++) imply Rohlin’s formula?
-----------------------------------------------------

\[02.04.13\] We recommend to skip this section which is neither exciting nor seriously written.

\[22.01.13\] One may wonder if there is not a stronger mode of degeneration (alias contraction) of a real smooth plane curve $C_m=C_{2k}$ than CCC yielding the Rohlin complex orientation formula $2(\Pi^{+}-\Pi^{-})=r-k^2$ as a corollary of Bézout. In the case of no nesting this is precisely what did the previous section. Indeed the solitary node degeneration of CCC implies symbolically $C_{2k}\to C_k \cup C_k^\sigma$ (a topologically dividing curve divides algebraically!), whence by Bézout $r=k^2$. This coincides with Rohlin’s formula since there is no nesting.

Let us call such a hypothetical mode of degeneration C++ (like Turbo Pascal?). If this exists this would be a geometrization of Rohlin’s formula, in the sense that topology (homological intersection theory à la Poincaré-Lefschetz-Weyl(1923)-Pontrjagin-de Rham (1930), who else?) would be subsumed to Monsieur Étienne Bézout (ca. 1768) alone. This would also posit a wide extension of the Itenberg-Viro conjecture. It seems evident that such a contraction C++ should exist. It remains only to be not overwhelmed by the combinatorics.

So suppose given a dividing curve $C_{2k}$ in the plane. The idea is to contract all its ovals so as to split the curve in two algebraic pieces exchanged by Galois(=conj), all this being just caused by a strangulation of the underlying Riemann surface.

Of course we apply first CCC to contract all empty ovals toward solitary nodes. Then it appears a second generation of nearly empty ovals (those which formerly were at height 1 in the tree of the nesting structure). We may hope to shrink those in turn while necessarily coalescing together all the solitary nodes inside this oval. Perhaps we can do this while keeping the tangents distinct at those solitary nodes gravitationally clumped together. One continues this big crunch process and once all ovals have been contracted one get a splitting $C_{2k}=C_k\cup C_k^{\sigma}$. Counting properly intersections with Bézout should give Rohlin’s formula $$k^2=r-2(\Pi^+ - \Pi^{-}).$$

Of course we need to be much more explicit (as if Rohlin would not have influenced us). Recall that $\Pi^{+}$ is the number of positive (injective) pairs of ovals that is with complex orientation matching that of the bounding annulus of ${\Bbb
R}P^2$, and likewise $\Pi^{-}$ being the number of pairs with disagreeing orientation when induced from the complexes versus the real bounding annulus. Recall that all pairs are taken into consideration not just oval succeeding themselves immediately.

On applying first CCC we can shrink all empty ovals to solitary nodes. Naively one would then like to shrink all the nearly empty ovals containing only solitary nodes, and so on. So we would have a degeneration $C_{2k}\to C_k\cup C_k^\sigma$. Computing the intersection $C_k\cap C_k^\sigma$ with Bézout gives $k^2$ algebraically. Geometrically, as each oval is shrunk to a pair of conjugate lines, and this explains the presence of the term $r$ on the RHS of Rohlin’s formula (as displayed above). Further the Riemann surface of the reduced curve can be naively imagined in 3-space as a pair of paraboloid of revolution together with their orthosymmetric replicas. Each branches intersect its conjugate in one point, but those contribution where already taken into account. So it remains to count the intersection of the top small paraboloid with the bottom large paraboloid, and vice-versa the large top with the bottom small. So we get 2 additional intersections, and this explains the term $+2\Pi^{-}$ of Rohlin’s formula. (Alas the term $-2\Pi^+$ looks much harder to explain.)

More clarification is required. A negative pair of ovals can be shrunk simultaneously. An example is provided by the Gürtelkurve, quartic $C_4$ with two nested ovals. Either via Fiedler or by Ahlfors it is plain that this $C_4$ has a negative pair of ovals. Looking at an equation like two concentric circles and perturbing slightly to get away from the reducible locus (and the discriminant) we have $$(x^2+y^2-\rho^2) (x^2+y^2-R^2)=0,$$ and if $0\le \rho<R$ then we can shrink $R\to 0$ and obtain the required multi-contraction. This example obviously extends to deep nest in any (even) degree, as to shrink negative towers of ovals.

Our guess is in contrast that positive pair of ovals resist simultaneous shrinking. Probably there is an evident topological obstruction which I missed to notice as yet.

If so then there is no possibility to reduce Rohlin’s formula to Bézout via a super strong contraction principle C++ reducing the whole curve to a microcosm of solitary nodes. If so, the whole curve under the action of some gravitational clumping would truly reduce to a constellation of isolated points with real scheme condensed at the atomic scale. (Imagine points and then infinitesimal circle surrounding the first generation of point, etc.) It seems more likely that there is an obstruction to shrink everything (algebraically), and so Rohlin’s proof is surely the best one can implement. Yet there could still be some geometry behind it suggested by the contraction principle.

[*Insertion*]{} \[01.04.13\] The latter is true in degree 6, and actually stronger than Rohlin’s formula, when combined with RKM (\[Kharlamov-Marin-cong:thm\]) since it rules out all schemes lying above the $(M-2)$-schemes of type I (e.g. $\frac{7}{1}3$), what Rohlin’s formula is unable to do alone.

Failing to reduce Rohlin to Bézout suggests again a dynamical approach {#CCCviaDynamics:sec}
----------------------------------------------------------------------

\[22.01.13\] What is this geometry and is it worth paying attention at? Before trying answering this, note that even if a positive pair of ovals resists to shrinking it could undergo another type of Morse surgery, namely coalescence to a figure eight (lemniscate looking like a “sweetheart”, i.e. with one branch lying inside the other). Yet this operation corresponds to the contraction of an ortho-cycle without disconnecting the Riemann surface so as to produce an algebraic splitting $C_{2k}\to C_k \cup C_k^\sigma$.

Let us turn to the geometric aspect. Our goal is essentially to shrink the ovals at least those which are empty (CCC), and then eventually push further the deformation as to shrink the negative pairs (memno-technic trick imagine negative=depressive=shrinkable). The other positive pairs may offer some resistance (due to a topological obstruction, which we should still understand better).

To achieve such a shrinking it looks natural to look at the length-functional of [*all*]{} ovals (not just the empty ones). Consider the round metric on the unit sphere $S^2$ lying above ${\Bbb R}P^2$, and measure lengths in this metric. Given $C_m$ a curve (i.e. a homogeneous ternary form $F_m(x_0,x_1,x_2)$ with real coefficients up to homothety), look at the set $(F_m=0)\cap
S^2$ which is obviously rectifiable (Lebesgue, Jordan, Riemann, Gauss, Archimedes, etc.). Denote its length by $\lambda (C_m)$. This is zero iff $C_m({\Bbb R})$ is empty or contains merely isolated points. Further there is an obvious way to take into account the multiplicity of branches; e.g. a conic (degree 2) consisting of a double line has length not just $2\pi $ but twice that quantity. This is crucial to ensure continuity of $\lambda$ on the parameter space of all $m$-tics. Since the latter space is compact (actually an ${\Bbb R}P^N$, $N=\binom{m+2}{2}-1$) the length functional $\lambda$ reaches a maximum.

How long can an $m$-tics be? By the above compactness argument there is some universal constant $L_m$ bounding the length of all curves $C_m$ of some fixed degree $m$: $\lambda(C_m) \le L_m$, and the maximum is actually realized (a priori not by smooth curve). A configuration of $m$ lines produces the lower estimate $2\pi m \le L_m$, and by Brusotti 1921 [@Brusotti_1921] there are smooth curves $C_m$ of length as close as we please to $2\pi
m$, but slightly longer. Imagine a crossing getting smoothed then the geodesic of $S^2$ are entailed by curvilinear arcs which are longer (triangle inequality or Pythagoras in the small). So there certainly exist longer curves! But how long can an $m$-tic be? This is probably very difficult to answer.

When $m=2$, the above argument via Brusotti still makes sense. If we imagine quadratic cones in 3-space $E^3$ (say with elliptical affine cross-section at $x_2=1$), then they may cut strange ovals on $S^2$ possibly longer that $2\pi \cdot 2$???

When $m$ is odd then there is a pseudoline (or at least a circuit possibly singular) not null-homotopic in ${\Bbb R}P^2$. Obviously its length is at least $2\pi$, which is the lower bound of the functional $\lambda$ when $m=2k+1$ is odd.

The estimation of $L_m$ is surely an attractive problem, but let us try to be not sidetracked by this. Our goal would be rather to study the gradient flow of $\lambda$ as a dynamical process susceptible to implement the collective contraction conjecture CCC, or more elaborated versions thereof if feasible (e.g. some iterated contractions like C++).

Assume first $m$ even. Then $\lambda$ vanishes identically on the chamber corresponding to empty curves, as well as on its adherence which consists of curves with isolated real points (either solitary double points or of higher multiplicity necessarily even). Thus $\lambda $ cannot be analytic, but seems rather being $C^\infty$. (As usual with distance functions (e.g. $\vert x
\vert$) they sometimes lack even smoothness until tacking their squares. So perhaps take $\lambda^2$ squared.)

Consider the gradient flow of this functional $\lambda$, while hoping that the corresponding trajectories materialize the collective contraction conjecture (CCC).

Usually ovals fails severely to be geodesics on $S^2$, but are perhaps so when we look them in the Riemann surface $C_m({\Bbb
C})$ endowed with the Fubini-Study(=FS) metric on ${\Bbb C}P^2$. Is the corresponding length of the ovals the same, in other words does FS induce the round elliptic metric on $S^2$? It is (always) tempting to regard ${\Bbb C}P^2$ as the variety of groups of two points on the Riemann (round) sphere. In this model how to describe the FS-metric?

Another idea, at least if we restrict to smooth curves, is to take the uniformizing hyperbolic metric (of Schwarz-Klein-Poincaré-Koebe) on $C_m({\Bbb C})$ with curvature $K\equiv -1$, when $m\ge 4$ (so $g\ge 3$). Then we get another measure of length of the ovals, which we shall denote $h$. The problem here is that this length functional is not a priori defined on the full hyperspace of curves $\vert m H\vert$.

As soon as we look also in the complex domain, there is a myriad of other functionals like the systole of the Riemann surface, the area of one half in the dividing case, etc. We just remark that from the systolic viewpoint there might by an ortho-cycle of much shorter length than the real ovals, and which dynamically might be advantageous being first contracted.

Of course the technical difficulties look immense, but the problem involves a mixture of Poincaré-Morse versus Hilbert-Petrovskii, i.e. a synthesis thereof. So the game is certainly worth paying attention at. What seems called upon is a dynamical study of algebraic equations governed by motions regulated by (natural) geometric functionals on the corresponding varieties (zero loci). In particular find appropriate functionals whose trajectories converge (generically) to curves with solitary nodes as to implement the Itenberg-Viro contraction conjecture or its collective variant CCC.

To shrink all empty ovals simultaneously it seems not so fruitful to shrink the shortest oval. More collective optimization is asked as if one had to bring fastest to the harbor a convey of ships each carrying rough materials involved in the manufacture of some complex end-product (Polyà’s metaphor for Rayleigh eigenvalues). Here we are in a similar situation. If all empty ovals have to dye simultaneously (scenario posited by CCC), it is important to shrink faster the longer ovals. Perhaps this suggests looking and $\lambda^2$ the squared length functional penalizing longer ovals.

It is also tempting to look at the area $\alpha$ (of the interior of all empty ovals measured on $S^2$), and to play perhaps with the isoperimetric inequality. For instance the functional $\lambda^2/\alpha$ looks natural, and is bounded from below in the small by $(2\pi \rho)^2/(\pi \rho^2)=4\pi$, so it admits a finite limit when it shrinks. The isoperimetric functional $\iota=\lambda^2/ \alpha$ intuitively forces ovals to dye in a round manner, penalizing agonies along eccentric ellipses.

If optimistic, integrating the gradient flow of either $\lambda$ or $\alpha$, length resp. area of the empty ovals directly leads to a solution of CCC.

The serious obstacle is that there may be a sink, i.e. a local minimum of the functional preventing convergence to a curve with solitary nodes arising as contractions of the $r_0$ empty ovals.

One naive idea is to let vibrate the ovals via a configuration of lines (as in the Harnack-Hilbert method), hoping to decrease area through this perturbation. The oval then oscillates inside and outside itself but on a larger portion it would move inside himself, hence area decreases (cf. Fig.\[Vibrate:fig\]a). The similar assertion for the length functional looks even more fantasist. Hence the area functional looks better suited to the problem.

[*Insertion*]{} \[01.04.13\] One problem is that if we have several ovals it is not clear that decreasing the area of one will not enlarge area of the other empty ovals. This problem dissipates somewhat if we look only at the usual Itenberg-Viro conjecture, but of course also the latter is subject to doubts, e.g. those allied with Shustin’s disproof of Klein-vache.

-5pt0

-5pt0

A more naive idea is to look at some sublevel of the equation. For simplicity assume the given curve completely inside some affine chart (cf. however Ronga’s counter-example on Fig.\[ItenbergViroRiem:fig\]) with equation $f(x,y)=0$, and w.l.o.g. positive on the outside unbounded (in ${\Bbb R}P^2$ nonorientable) component residual to the curve. Then the sublevel $f(x,y)=-\varepsilon<0$ (small negative constant) ought to have a smaller area $\alpha$ (compare Fig.\[Vibrate:fig\]b).

Let us be more precise. Look at all empty ovals of the curve $C_m$. Then following Ragsdale-Petrovskii, some are positive and some negative (or, even and odd depending on the parity of the number of ovals surrounding it). Maximal ovals are even (being surrounded by zero ovals), those immediately inside them are odd, etc. Under our sign convention for the equation $f$ (positive outside) we see that even ovals decrease in area when considering the sublevel $f=-\varepsilon$, while odd ovals increase in area (compare Fig.\[Vibrate:fig\]b). The net bilan is hard to quantify, but on the situation of the picture where the even empty oval is much larger than the other empty ovals there is some chance to decrease the functional $\alpha$.

Is there some chance to deduce a general argument from our naive picture? Split all empty ovals in even and odd ones (denoted resp. 0 and 1 depending on their class modulo 2). Look which of both collections has more massive total area, i.e. compare $\alpha_0$ vs. $\alpha_1$. If $\alpha_0>\alpha_1$ then take $\varepsilon$ positive (and vice-versa if $\alpha_0<\alpha_1$ then take $\varepsilon$ negative). Of course if unlucky both magnitudes $\alpha_i$ are equal, in which case we are a bit lost (perhaps avoidable by genericity, as we work up to isotopy, hence can always perturb slightly the data).

Let us assume to be in first case $\alpha_0>\alpha_1$ (as on the picture). By a classical continuity lemma (cf. e.g. Gudkov 1974 [@Gudkov_1974/74]) we can preassign a tubular neighborhood of the curve in which the perturbation will stay confined. Thus we can probably control from above the area expansion of the odd ovals. It seems indeed that the tubular expansion is maximum for a circle as follows from the isoperimetric inequality. However it is not clear that the deflation of area of the even ovals ought to supersede this as it may be a very thin penetration (imagine the big south island as very mountainous hence poorly affected by a raise of the ocean level).

This leads to the idea of looking at the normal derivative of the defining (polynomial) function $f(x,y)$ across the sea level ($f=0$). The inflation of area resp. deflation of area of the empty ovals ought to be proportional to this normal slope and the length of those ovals via some explicit formula given by differential calculus. So we compare both quantities for even an odd ovals, and choose the right sign for $\varepsilon$ in order to create a deflation of area while keeping the degree constant. Of course if both quantities coincide we are a bit disturbed, but perhaps avoidable by genericity (two random real numbers are generically distinct.)

If this trick works there is some hope to show that the area functional $\alpha$ lacks local minima, and the corresponding orthogonal trajectories of steepest descent ought to converge toward curves with solitary nodes.

Note another phenomenon: imagine one empty oval shrinking prematurely before the others. Soon after this death another oval (formerly nonempty) may become suddenly empty implying a large jump of the area functional $\alpha$ which looks therefore discontinuous. The only reasonable parade against this catastrophe is that the trajectory of steepest descent will not kill abruptly the small oval as it is much more profitable to shrink first the voluminous ovals. Intuitively, the $\alpha$-flow would promote a collective contraction. Nonetheless the $\alpha$-functional can be discontinuous with big jumps across walls of the discriminant, as caused by the fact that we measure only empty ovals, which in contrast to all ovals, is subsumed to violent fluctuation.

Have we proved something? Maybe yes if quite sloppy. Let us resume some of the difficulties:

(D.1) First Ronga’s example of a $C_6$ not confined to an affine chart is presumably not a severe obstacle. For even degrees the sign of the projective equation $F(x_0,x_1,x_2)$ is always well-defined so that there is a variation $C_m^{\varepsilon}$ with disjoint real locus ($C_m\cap C_m^{\varepsilon}=\varnothing$). (For odd degrees nothing similar can be done so easily, but we confine attention to even degrees. That is already hard enough.)

(D.2) We look at the penetration index under a small perturbation as measured by the normal derivative of the landscape pondered against the length element of the oval. More precise, calculate along each point of $C_m$ the normal derivative $
\frac{\partial f}{\partial n},
$ and then integrate this against the length element $ds$ of some fixed oval $O$ to get $$\int_{O=an oval} \frac{\partial f}{\partial n} ds=: \pi (O).$$ This real number measures the rate of area change under a flood (variation of $\varepsilon$). Of course the normal derivative above can be interpreted as the gradient of $f$ on the coast line $f=0$. (If it is big in norm then the slope of the coast is low hence the territory much affected by floods, while if small then the coast slope is steep and the island has little to fear from inundations).

Call the above quantity $\pi(O)$ the “piaf” (protection index against floods). It is well-defined for any oval (up to sign and anodyne choices effecting a collective change). It seems to make also good sense in the projective context.

Next look at all the empty ovals $O_1, \dots, O_{r_0}$ of some smooth curve $C_m$, splits them in even and odd, and look which of both collections have the highest piaf. Depending on this knowledge, an appropriate choice of $\varepsilon$ create a variation of the curve with smaller inner area $\alpha$ .

(D.3) What to do exactly when both (even and odd) piafs are equal? Can we avoid this just by slight perturbation, i.e. is there always a small perturbation making them different? (Perhaps Petrovskii thought about such questions…)

Assume now that (D.3) can be overcome. Then the functional $\alpha$ has no stable equilibrium (local minimum) and we interpret it as a Morse function on the space $\vert mH \vert$ of all curves. In this generality it may rather look like a Grand canyon with big ravine when one cross a solitary node due to the brusque change of empty oval. Two attitudes are possible:

(A.1)—either we localize $\alpha$ to one chamber of the discriminant[^50] or

(A.2)—we look at the whole space of curves hoping that the trajectories dictated by the functional never cross such ridges (=ravine of $\alpha$).

After having introduced a natural metric on $\vert m H \vert$, e.g. the elliptic geometry available on each projective space, we look at the trajectories of steepest descents w.r.t. the function $\alpha$ (area of the nonempty ovals). This is as usual obtained by integrating the vector field ${\rm grad} \alpha $. By the above (D.2), for almost every initial condition $C_m$ it decreases endlessly up to reach level $\alpha=0$, which must necessarily be a curve with solitary nodes. Could the trajectory starts oscillating like a $\sin (1/x)$ curve without reasonable convergence? Looks unlikely due to the algebraic nature of our problem, but requires perhaps an argument. If the trajectory of $C_m$ converges to a saddle point (unstable critical point of $\alpha$) it suffices to perturb slightly the initial condition $C_m$ (which is allowable up to small rigid-isotopic perturbation). (In fact it is likely that such exceptional saddles correspond precisely to curves having the same even and odd piafs, especially if we have a rigorous proof that there is no local minimum for $\alpha$, as we tried to argue in Step D.2.)

At this stage we believe the proof would be completed (no additional difficulties) and we would conclude:

(Hypothetical!!!) Given any (non-void) curve $C_m$ (of even degree $m$ for simplicity), the trajectory of the gradient flow of the empty-ovals area $\alpha$ generically converges to a curve with solitary nodes in finite time, while the empty ovals themselves converges to the solitary nodes. If not then it finishes its trajectory to an unstable equilibrium and it suffices to perturb slightly to ensure convergence toward a solitary nodal curve (soliton for short, as compression of solitary and singleton). In particular, CCC holds true, i.e. there is a path in the space of curves such that all empty ovals contract to solitary nodes.

That the extinction of all the empty ovals occurs in finite time merely follows from the fact that the time parameter of any gradient flow is just the “height” function, here the functional $\alpha\colon \vert mH \vert \to {\Bbb R}$ but taking value in $[0, 4\pi]$ (where $4\pi$ is the area of the full sphere or $2\pi$ if you count this area divided by two).

Here we have looked at the empty-oval area functional $\alpha$. What happens if we look the same functional for all ovals. A priori the functional looks more continuous but be careful with eversions.

\[23.01.13\] Metaphor.—Problems of rigid-isotopy (or large deformations of curves) are like a video game in the sense that there is a joystick upon which one may act by freewill by varying the coefficients while there is in reaction a canonical picture emerging on the screen (the corresponding real locus of the algebraic curve conceived as an optical object). In some sense it is like a flight simulator (you move the “manche à balais” and the aircraft responds accordingly). The contraction conjecture CCC says that using the full freedom of the joystick one can always shrink the empty ovals simultaneously. The above theorem states roughly that there is some predestination, i.e. that a very sleepy autopilot or video game player suffices to land safely the aircraft, while performing actually a perfect landing (all wheels touch the ground simultaneously!). Of course in reality the autopilot in question is very well programmed for its action is governed by a principle of least action. The only little impulse required is when the aircraft arrives at critical points (global maximum of $\alpha$ or its saddle critical points), where some jiggling is required to perturb the initial condition.

Some few other applications of CCC {#application-of-CCC:sec}
----------------------------------

\[23.01.13\] What can be deduced from CCC? Quite a lot and alas no so much, compare the case of $M$-sextics. Applying it to Gudkov’s $C_6$ gives a rational (genus 0) real sextic with 10 solitary nodes equidistributed as $5$ inside and $5$ outside the unique oval. Of course there is no obstruction given by the genus formula to such a eventuality.

In general, given any curve $C_m$ with say $r$ ovals, then can be split as $r_0\le r$ empty ovals, which can be contracted to solitary nodes. Then applying Brusotti 1921 [@Brusotti_1921] we can let them disappear all, and so appears a new generation of empty ovals, to which the collective contraction process can be applied again, and so on. So we can reach the empty curves (or a pseudoline if $m$ is odd) after some few iterated contraction (as many as the height of the oval-graph encoding the nested structure of the original $C_m$). This we call the height of the curve $C_m$, denoted $h(C_m)$

This implies (as a crude estimate) that any two curves $C_m, D_m$ can be related by a rigid-isotopy crossing only $h(C_m)+h(D_m)$ times the discriminant. Of course it is not a transverse crossing in general for our critical curves have several solitary nodes. However by perturbing slightly we may cross the discriminant transversally, and each initial crossing through a multi-solitary curves implies as many intersection with $\frak D$ as there are nodes.

So counting properly we deduce:

[(modulo CCC and the connectedness of invisible curves=CIC)]{} Any two smooth curves $C_m, D_m$ of even degree $m$ can be joined by a path in the hyperspace of curves transverse to the discriminant while crossing it exactly $r(C_m)+r(D_m)$ times, where $r$ is the number of ovals (composing the real locus).

Applies iteratively CCC (conjointly with Brusotti) to both curves, to derive two curves with empty real locus. The latter are known to form a unique chamber of the discriminant, in other words to be rigid-isotopic by Lemma \[empty-chamber-connected-Shustin:lem\].

A similar assertion holds perhaps true in case of odd degrees, however it is still unknown whether two curves of the same odd degree are rigid-isotopic provided their real loci reduce to a pseudoline (compare Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical p.199]).

This goes in the sense of showing that the contiguity graph of chambers residual to the discriminant is a “small world”, in the sense that is has high connectivity and much “consanguinity”.

How good is the above estimate? More precisely the distance $\delta$ (or Erdös number) in the contiguity graph (of chambers) is majored by $\delta(C,D) \le r(C)+r(D)$. This is fairly good as compared to the estimate coming from the degree of the discriminant $3(m-1)^2$, which implies $\delta(C,D)\le
3/2(m-1)^2$, or the integral part thereof, as we may always choose the one side of the circle hitting less many times the discriminant.

For $m=6$, the discriminant estimate gives $\delta\le
[75/2]=[37.5]=37$, while the CCC estimate gives $\delta\le
11+11=22$. We can be more economical by not going down to the empty chamber but that having only one oval, which form already a unique rigid-isotopy class by Nikulin 1979 [@Nikulin_1979/80].

This raises the following question: we know either by Rohlin’s formula (or less rigourously by CCC) that any curve with one oval (hence of even degree) is nondividing provided $m=2k\ge 4$. Indeed if dividing apply CCC to get a splitting $C_{2k}\to C_k\cup
C_k^\sigma$, whence by Bézout $C_k \cap C_k^\sigma=k^2=r$, where $r=1$ whereas $k^2\ge 4$. Thus there is no obstruction to rigid-isotopy given by the Klein’s type between any two curves having only one oval, and extrapolating (violently) we arrive at the:

\[OOPS:one-oval-rigid-isotopic:conj\] [(OOPS=One oval postulation)]{} Any two smooth curves having only one oval are rigid-isotopic. (“Oval” is interpreted here in the strong sense of a Jordan curve which is null-homotopic, hence our curves are of even degree.)

(Perhaps there is an obstruction à la Fiedler-Marin, but unlikely as it seems to require a splitting of ovals, cf. Marin’s argument exposed below.) Remind also from Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical] that replacing above “oval” by pseudoline is still an open problem. Further Viro in his e-mail (dated 26.01.13 in Sec.\[e-mail-Viro:sec\]) confirmed me that this is still an open problem and goes back to Rohlin.

[*Insertion*]{} \[02.02.13\].—A naive approach to this would be to assert that any $C_{2k+1}$ may degenerate by a large deformation to $L_1\cup C_{2k}$ a line $L_1$ union an invisible curve $C_{2k}$ of even degree. Perhaps one can demand that both curves in the limit are transverse. Call this process a rectification of the pseudoline. Now given two smooth curves whose real scheme consist of a unique pseudoline, one may apply twice rectification, and then isotope both corresponding empty curves. This could imply the Rohlin-Viro conjecture.

In degree 6 the Erdös number of the graph (supremum of $\delta$) is probably much smaller in view of Fig.\[Gudkov-eversion:fig\], i.e. Gudkov’s table with eversions. Of course this figure posits that all logically possible eversions are realized geometrically (safe those linking $M$-curves and both $(M-2)$-schemes of type I). Under this circumstance the Erdös number looks hardly greater than 8, i.e. $\delta \le 8$ universally. It seems indeed that the maximal distance is realized by Hilbert vs. Gudkov or Gudkov vs. Harnack. Naively on the table (without eversions) Hilbert’s scheme and Harnack’s looks far apart, but using the eversion $\frac{8}{1}1
\to \frac{1}{1}8$ shows that their real distance is only 3, i.e. $\delta (Hilbert, Harnack)=3$. This is the answer if we confine attention to the top of the table, but of course the most distant chambers are the $M$-curves as separated from the empty scheme $0$. Those are at distance $11$ apart. So the correct Erdös number is $\delta=11$, with this maximal distance being realized thrice (empty vs. Hilbert, Gudkov, Harnack respectively). Morally eversions do not shorten the vertical distance, and we have proven the following (modulo Conjecture \[eversion-and-other surgeries:conj\]).

[(Semi-conjectural)]{} \[Erdos-number-of-sextics=11:prop\] The Erdös number of the contiguity graph of sextics is actually equal to the Harnack bound $M=11$.

It is tempting to wonder if it so in general. Perhaps there is some little chance to answer this without having to work out the exact rigid-isotopy classification in each degree (an insurmountable task!?). The trick would be that eversions collapse sufficiently horizontal distances, so as to make only the vertical chain the only plausible candidate for maximizing $\delta$.

As to the above OOPS conjecture (\[OOPS:one-oval-rigid-isotopic:conj\]), one could of course also imagine a dynamical proof. The whole task reduces to finding the right functional. Heuristically the obvious attractor ought to be a circle (possibly multiple). This inclines to look at the isoperimetric functional looking the length squared divided by the area of the unique oval.

More pragmatically,

[(but apparently validated by Viro, cf. comments right-after the proof)]{}.—It seems clear that CCC (and of course the weaker formulation CC) implies the one oval postulate (OOPS).

(pseudo, of course!) Indeed take any two curves having only one oval. By CCC (CC suffices) each of them can be shrunk to a solitary node. By Brusotti there is slight perturbations to empty curves. The latter can be linked by a path, by virtue of the connectivity of the empty chamber (Lemma \[empty-chamber-connected-Shustin:lem\]). Since the empty chamber of the discriminant is a manifold with corners (like probably any other chamber by the way) which topologically is a manifold with boundary there is such a path staying close to the boundary. (Actually the boundary of the empty chamber has faces consisting of uninodal solitary curves with one isolated real point.) Pushing this path slightly outside the empty chamber would give the required isotopy. However there is a serious difficulty, if our path meets another wall of ${\frak D}$ outside the empty chamber. However by genericity of this path (as avoiding sets of codimension 2) we may assume this crossing to be a transverse one of a wall which must keep $r=1$ constant since we stay in the vicinity of the empty chamber with $r=0$. Note of course by CCC as applied to quartics for instance that chambers with higher $r$’s are also contiguous to the empty chamber, yet the are like cubes hitting the empty chamber imagined as a cube at some vertices of codimension 2. So we have some wall crossing keeping $r=1$ constant, which as a Morse surgery must correspond to an eversion. But this means that there is an eversive wall falling down to the boundary of the empty chamber like a tripod. This looks incompatible with the local structure of algebraic sets? Maybe there is a more direct argument in ${\Bbb R}P^2$. Another obstruction comes also from Brusotti’s description of the discriminant as branches with normal crossing. All this is very confuse, we confess.

\[30.01.13\] As kindly informed by Viro (cf. his e-mail, dated 26.01.13 in Sec.\[e-mail-Viro:sec\]), it seems that the implication CC$\Rightarrow$OOPS causes no problem. It would be nice to write down complete details. Perhaps this simply follows from the fact that inside the empty locus the discriminant has real codimension 2 (cf. Lemma \[invisible-discriminant-codim-2:lem\]). Hence there is no wall inside it, hence no wall outside it by the “implicit function theorem”.

[*Insertion.*]{} \[01.04.13\] If CC$\Rightarrow$OOPS causes no troubles it would be interesting to extend the method to higher schemes (having more ovals than one).

\[URS:conj\] (URS)=(Unnested rigidity speculation).—The unnested configurations (which are of type II provided $r<k^2$ or even $r\neq k^2$, as shown by Rohlin’s formula) are always rigid (i.e. any $2$ curves representing the unnested scheme are rigid-isotopic provided $r\neq k^2$). When $r\neq k^2$ it could be that the type is the sole obstacle to rigid isotopy.

This holds true in degree 6 by Nikulin’s theorem (\[Nikulin:thm\]). Further it looks hard to disprove this by the Fiedler-Marin method as we lack a canonical choice for the fundamental triangle. Finally, it could be that the same argument as above shows that CCC implies URS.

[*Sketch of proof that CCC$\Rightarrow$URS*]{}.—Contract all ovals simultaneously to land in the connected empty chamber of curves without real points. So given 2 curves connect them by respective contractions to the empty locus, and therein by a path of invisible curves. The hard part is then to push this path by a small deformation again in the visible locus, and this in such a way that we never meet the discriminant. This looks feasible as the (closured) empty locus seems to be a bordered manifold, with connected boundary (being essentially fibred over $\RR P^2$ via assignment of the unique solitary node). So the joining-path in the empty locus can be pushed on the boundary and then further inside the chamber. Of course it remains the difficulty of ensuring that we do not meet other nappes of the discriminant. As above the loose argument is that since the discriminant as codimension 2 inside the empty locus, it will appear outside along the same dimension, and not effect any separation. This would supply the required rigid-isotopy between our pair of unnested curves. Of course it is essential to assume $r\neq k^2$ since otherwise there is curves of both types, yet this condition did not as yet appeared frankly in our argument, which is far from a serious proof. Of course one could expect that for $r=k^2$ the type is sole additional obstruction.

Looking around (in vain?) for counterexamples to CCC (=collective contraction conjecture)
-----------------------------------------------------------------------------------------

\[20.01.13\] An a priori easier game is to test if CCC (\[CCC:conj\]) has really some chance to be true. As usual experimentation is required. We may first consider a curve $C_8$ arising through perturbation of 4 ellipses rotated by $180/4=45$ degrees (cf. Fig.\[CCCRoses:fig\]b).

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The smoothing being compatible with orientations, this curve $C_8$ is dividing (Fiedler). It has $r=16$ unnested ovals (cf. Fig.\[CCCRoses:fig\]b). Assume it shrinkable via CCC, it will degenerate and decomposes as $C_8 \to C_d \cup C_d^{\sigma}$, where $C_d$ has genus $p=\frac{g-(r-1)}{2}=\frac{21-(16-1)}{2}=3$, hence of degree $d=4$ (this can be inferred more directly via the degree of the degeneration). The relation $d^2=r$ is verified and there is no numerical obstruction to CCC. Note in contrast that in the same construction for 3 or 5 ellipses, we cannot arrange all ovals unnested while smoothing in a sense preserving way. Look and see! (Figs.\[CCCRoses:fig\]a and c).

The construction of our $C_8$ generalizes whenever the degree $m=4\ell$ is a multiple of four. Indeed rotate an ellipse by $\pi/
\ell$. Orient the ellipses “alternatively” and smooth in a sense preserving way. The resulting curve has $r=4 \ell^2$ unnested ovals (as easily counted by extrapolating the figures $C_8$ and $C_{12}$ of Fig.\[CCCRoses:fig\], while noting that the ovals in $\ell=m/4$ couches containing each $m=4\ell$ ovals). Shrinking $C_m$ via CCC gives two curves of genus $p=\frac{g-(r-1)}{2}$, where $g=\frac{(m-1)(m-2)}{2}=(4\ell-1)(2 \ell-1)$. Hence $$\begin{aligned}
p
=[(4\ell-1)(2 \ell-1)-(4 \ell^2-1)]/2
&=[(2\ell-1) [(4\ell-1)-(2\ell+1) ]]/2 \cr
&=[(2 \ell-1) (2\ell-2)]/2,
$$ so that the half has degree $2\ell$, and intersects its conjugate in $4\ell^2$ points, which is precisely $r$. This little numerical miracle implies an absence of numerical obstruction to CCC via our rosewindows constructions. Of course, all the above computation can be shortcuted by noticing a degeneration $C_{m=4\ell} \to C_d \cup
C_d^{\sigma}$, where $d=2\ell$ necessarily (since the conjugation $\sigma$ preserves the degree of equations as it just acts upon the coefficients).

As yet the center of rotation was chosen inside the ground ellipse. Another series of picture arise when choosing it outside instead (but sorry this is probably not the right thing to do). Let us instead manufacture the dividing $C_6$ with $r=9$ unnested ovals as on Fig.\[CCCRoses2:fig\]a. Next we tried to construct a $C_{10}$ with 25 unnested ovals but failed somewhat. Note that this is not obstructed by Arnold’s congruence $\chi=p-n=k^2 \pmod
4$, as both sides are 25.

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Assume there is such a $C_{10}$ with real scheme $25$ (Gudkov’s notation) which is dividing. On applying CCC we find a splitting $C_{10} \to C_5 \cup C_5^{\sigma}$. The genus $g=9\cdot 8/2=9\cdot
4 =36$, hence $p=[g-(r-1)]/2=(36-24)/2=6$ which is indeed the genus of a quintic. So no obstruction on this side. Note also that Rohlin’s formula $2(\Pi^+ -\Pi^-)=r-k^2$ implies no obstruction, since $\Pi^{\pm}=0$ (no nesting) and so $r=k^2$ with $k=5$.

After our stupid trials we find ultimately the right ground configuration of ellipse as Fig.\[CCCRoses2:fig\]y, which is dividing with $25$ ovals. It is clear now how to extend this in an infinite series of curves $C_{4 \ell +2=2( \ell +1)}$ with $(\ell+1)^2$ ovals lying outside each other, i.e. the scheme $(\ell+1)^2$ in type I. Fig.\[CCCRoses2:fig\]z gives the case of a $C_{14}$.

All this pictures are pleasant, yet they do not help at all to corrupt the conjecture CCC. Of course the latter has some chance to be true especially if the Itenberg-Viro conjecture (\[Itenberg-Viro-contraction:conj\]) is true. A dividing curve without nesting has to satisfy $r=k^2$ by Rohlin’s formula, and therefore applying CCC gives a degeneration $C_{2k}\to C_{k} \cup C_{k}^{\sigma}$ yielding no chance to corrupt Bézout. Also there is no chance to corrupt Klein $p=[g-(r-1)]/2$. Indeed $g=(2k-1)(2k-2)/2$, hence $$\begin{aligned}
p
=[(2k-1)( k-1)-(k^2-1)]/2
&=[(k-1) [(2k-1)-(k+1) ]]/2 \cr
&=[(k-1) (k-2)]/2,
$$ so that the half has degree $k$, as it should.

CCC versus Brusotti: large deformations vs. small perturbations
---------------------------------------------------------------

\[20.01.12\] Philosophically, it seems that such contraction conjectures (Itenberg-Viro (\[Itenberg-Viro-contraction:conj\]) or our collective version thereof (\[CCC:conj\]))—if they turn out to be true by some lucky stroke—incarnate a sort of large deformation principle illustrating once more the perfect graphical flexibility of algebraic curves despite their intrinsic rigidity. In some sense this is an avatar in the large of the small perturbation method à la Plücker-Klein-Harnack-Hilbert-Brusotti-Viro. Hence it seems indeed (as Viro advocated) being of some primary interest to establish the contraction conjectures. Flexibility in the small gives rise to the perturbation method which is primarily a method of construction, whereas flexibility in the large (contraction principles) implies as a byproduct prohibitions (as we superficially experimented at the beginning of Sec.\[CCC:sec\] and more convincingly because Itenberg’s contraction theorem for sextics re-explain all Gudkov-style prohibitions by reduction to the RKM-congruence). At this stage we feel some big duality: local versus global and constructions versus prohibitions (to be or not to be). What would be the net impact of the contraction principle for Hilbert’s 16th problem (in the extended sense of high degrees)? Somewhat optimistically it would reduce the whole task (or rather adventure) to a combinatorial video game best suited for machines. So exaggerating slightly, the contraction conjecture seems quite close to reveal the ultimate secret of the whole problem. Of course some supplementary large deformation principles ought also to complete the picture, e.g. certain permissible eversions compatible with Bézout, and more generally the full morphogenesis of all algebraic Morse surgeries. If all this is available, the video game solving Hilbert’s 16th problem would show in real time all the possible perestroikas which the real loci of projective curves of some fixed degree can undergo, while dragging at free will the joystick in the parameter space.

As sketched in the previous Sections \[CC-via-dynamics:sec\] and \[CC-via-Riemann:sec\]), in order to prove the contraction conjectures (CC or CCC), we could either imagine a dynamical proof via orthogonal trajectories, hence akin to Morse theory, or a direct intervention of conformal geometry à la Riemann (albeit our implementation failed seriously). Whatsoever the exact details it is quite likely that the proof of CC or CCC will employ the calculus of variation in the large over which practically every deep geometrical theorem is based upon (from the brachystochrone, to the Dirichlet principle, via the Riemann mapping theorem up the recent solution of the Poincaré conjecture via the Ricci flow.)

Note finally another very modest piece of evidence in favor of CCC. Remember that for small perturbations à la Brusotti, there is complete freedom to smooth the nodes of a plane curve with normal crossings (compare e.g. Gudkov 1980 [@Gudkov_1980/80-Brusotti]) in the sense that all crossings may be smoothed away or some may be conserved. By analogy CCC is just the case where all empty ovals are contracted simultaneously, while Itenberg-Viro’s CC is just the contraction of a single empty oval. Of course there ought to be the full panoply of intermediate contractions.

CCC implies CC (i.e. Gabard stronger than Itenberg-Viro)
--------------------------------------------------------

\[21.01.13\] Let us now observe that CCC implies CC, just via Brusotti’s theorem (1921 [@Brusotti_1921]):

\[CCviaCCC-Brusotti:lem\] Suppose given a collective contraction of all the empty ovals of a smooth real curve $C_m$, then it is possible to construct all partial contractions via Brusotti. In particular if CCC holds true then so does the Itenberg-Viro contraction conjecture [(\[Itenberg-Viro-contraction:conj\])]{}.

Let $C_m$ be a smooth real curve. W.l.o.g. let us assume it having some empty ovals, say $r_0\ge 1$ many (otherwise the curve just reduces to a pseudoline or to empty real locus). By CCC there is a path $c\colon [0,1] \to \vert mH \vert$ such that $c(0)=C_m$ and $c(t)\in \frak D$(=the discriminant) only for $t=1$ where $c(1)$ is a nodal curve with solitary nodes only ($r_0$ many). By Brusotti’s theorem the neighborhood of the nodal curve $c(1)$ consists of $r_0$ “falde analytiche” (=analytic branches or better [*nappes*]{}) meeting transversally at $c(1)$. Further each of those nappes corresponds to the conservation of some node in the vicinity. In other words the chamber of $C_m$ looks like manifold-with-corner near the nodal curve $c(1)$, locally diffeomorphic to ${\Bbb R}^N$ with $r_0$ many distinguished hyperplanes of coordinates. It is now plain how to construct all other contractions, in particular all the ones of Itenberg-Viro CC contracting just a single empty oval (compare Fig.\[CCCBrusotti:fig\]).

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So if optimistic about the truth of CC, it may even seem that CCC is a fairly reasonable angle of attack. Of course one then needs a good functional (e.g. the area of all empty ovals).

Alas, it is less evident that CC implies CCC, as we suggested at the beginning of the investigation. This could be slightly easier if there is a contraction principle extended to solitary nodal curves while keeping the “solitons” in place.

Naively, one may dream of a contraction principle effecting a retraction of a whole chamber (past the discriminant) to its boundary. Yet compact bordered manifolds never retract to their boundaries (as shown by homology mod 2, as we learned from J.-C. Hausmann). This is evidently no obstacle against the contraction conjectures, for the retraction may be undefined on small loci, as since we work up to isotopy such equilibrium points can be avoided. More lucidly nobody ever asserted that the contractions should depend continuously on their initial point(=curve). Imagine as a very naive picture, the chamber as being a disc with a radial projection upon the boundary. Then it is undefined on the center of the disc but this is not a problem for perturbing it slightly it will get mapped somewhere. The whole analogy with retraction of bordered manifolds is not extremely pertinent as in general the chamber will have a boundary consisting not merely of faces touching the empty chamber. Hence under a retraction a curve close to the discriminant could first coalescence 2 ovals instead of shrinking one empty oval.

To show CC or even CCC we could employ a dynamical system (continuous flow) spreading nearly all curves toward the boundary of this chamber at curves having solitary nodes. This could occur as the orthogonal trajectories of some functional. (Another idea would be to look at the Green function of the chamber yet the Green’s lines, streamlines of the flow, would often finish at curves with non-solitary nodes.) Our flow should be strongly attracted by the multi-solitary nodal curves manifold, whereas all other walls corresponding to non-isolated singularities have to repulse the flow. A qualitative picture is given on Fig.\[CCCflow:fig\].

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This picture proves nothing, safe maybe the absence of topological obstruction (a priori) to find such a flow (especially if it is allowed some equilibriums when the chamber has complicated topology). Again to get a proof it is likely that one should consider the gradient flow of some real-valued functional on the chamber. This could be the area or length of all empty ovals (as measured on the round metric of $S^2$ double covering ${\Bbb
R}P^2$). Note that CCC implies a little technical simplification over our previous pseudo-proof of CC, where we were troubled by marking one oval.

As already discussed, the main obstacle occurs if our function has some global attracting basin inside the chamber preventing us to reach the desired multi-solitary nodal curve shrinking all empty ovals. This is basically the sole difficulty yet it looks quite insurmountable.

At least for the area or length functional, we saw no obvious way to produce small variations diminishing the “energy”. Perhaps there is some more clever (projective) invariants like degree of roughness à la Gudkov (cf. e.g. Gudkov 1974 [@Gudkov_1974/74]). For instance one could look at the largest (quadratic) cones in ${\Bbb R}^3$ which can be nested inside the ovals, and the corresponding area intercepted on the unit sphere. This is another functional measuring the conical area of the empty ovals. Can we show that this functional is “good”, i.e. no sink inside the chamber? Naive idea: trace inside each oval some maximal ellipse and try to deform the $C_m$ along suitable multiples of those ellipses.

Now the problem looks reduced to a fantastic game probably only soluble by such authorities as Andronov, Leontovich, Gudkov, etc. mixing the qualitative theory of differential equations with that of algebraic curves. So it is truly a Poincaré-Hilbert Verschmelzung(=fusion in Klein’s prose) which seems demanded to settle CCC (or its avatar CC).

\[03.04.13\] Of course, it also safe to say pessimistic and expect maybe that Shustin’s disproof of Klein-vache in degree 8 also implies a disproof of CC. We do not repeat our vague strategy for this, but refer to Sec.\[Challenging-open-prob:sec\].

Problems of rigid-isotopy
=========================

\[09.02.13\] Two (real, plane) curves are said to be [*rigid-isotopic*]{} if one can pass from one to the other by continuous deformation of the coefficients of the defining equations which avoids the discriminant. This involves again the paradigm of [*large deformations*]{} like the contraction conjectures discussed in the previous section. In fact there should be some direct connections between both topics.

As a rule very little is known about the phenomenon of rigidity. All what is trivial is that any topological characteristic persists during a rigid-isotopy, so for instance the [*real scheme*]{} (i.e. the isotopy class of $C_m(\RR)\subset \RR P^2$) as well as Klein’s type I, II measuring the situation of the curve in its complexification. Such invariance were intuitively clear since the era of Schläfli, Zeuthen, Klein 1876 [@Klein_1876_Verlauf], but requires perhaps Ehresmann’s lemma that a locally trivial fibering over a contractible (paracompact$\approx$metric) base is globally trivial. Paracompactness is essential as shown by the simply-connected (indeed contractible) [*Prüfer surface*]{} (cf. works by Prüfer 1922, Radó 1925 [@Rado_1925], Calabi-Rosenlicht 1953, Spivak’s book on Diff. Geom., Vol. I, Appendix, Baillif).

Up to degree $\le 4$ the real scheme (or Klein’s types I/II) suffices to encode the rigid-isotopy class as knew Klein 1876 [@Klein_1876_Verlauf], building over Schläfli and Zeuthen’s works. In degree 5, and 6, the same real scheme plus Klein’s type suffices to ensure a rigid-isotopy. This spectacular result is joint work of Nikulin, with the collaboration of Kharlamov building over two pillars, namely:

\(1) the Gudkov-Rohlin census solving Hilbert’s 16th problem (for sextics) while revitalizing the earlier Riemannian conceptions of Klein about the complexification and,

\(2) the theory of K3 surfaces (Torelli, etc.).

The situation changes drastically from degree 7 upwards as shown by the Fiedler-Marin trick using a locking triangle which consists of $3$ Bézout-saturated lines, hence which cannot be crossed by ovals during a rigid-isotopy. Here the basic idea is that if one can associate to a curve in some canonical manner an auxiliary curve called the lock then the distribution of ovals past the lock is rigid-isotopically invariant. A typical example in degree 7 is the lock consisting of 3 lines through the inner ovals of a curve having the scheme $\frac{3}{1}\ell J$ for some $\ell$. When $\ell
2$ the fundamental triangle can separate in different ways the $\ell$ outer ovals, and curves having different splittings past the lock will not be rigid-isotopic.

This obstruction to rigid-isotopy requires several ovals, and does not seem suited to curves with few ovals where the rigidity problem looks fully open. It is presently very unclear which sort of scenario is to be expected. For instance it is still undecided for $m\ge 7$ whether curves having only one real circuit are always rigid-isotopic. As informed by Viro, this is at least for odd degrees a (confidential) conjecture due to Rohlin.

More generally we posit the following speculation (not that we strongly believe in it, but just as a way to confess our ignorance):

\[LARS:conj\] (LARS).—Curves of degree $m$ with less than $DEEP+2$ real branches, where $\Delta(m)=DEEP(m)=[(m+1)/2]$ is the number of components of the deep nest of degree $m$, are always rigid-isotopic provided they have the same real scheme.

The basic motivation for this conjecture is that below altitude $r\le \Delta+1$ all schemes are of type II except the deep nest which is of type I (by total reality under a pencil of lines). This follows from Rohlin’s formula (\[Rohlin-formula:thm\]), especially its corollary known as Rohlin’s inequality (\[Rohlin’s-inequality:cor\]), as well as Klein’s congruence $r\equiv_2 g+1$ forcing dividing curves to have their numbers of real circuits $r$ jumping by quanta of 2 units, hence we gain one type II level right above the deep nest. Further the deep nest is known to be rigid by Nuij 1968 [@Nuij_1968].

Apart form those elementary facts, we have little evidence for this “low-altitude rigidity speculation” (LARS), except suspecting that if the assertion is true it will use a geometric flow permitting a degeneration to curves of lower orders after splitting off a line or a conic. For instance given a curve of odd degree it is tempting to look at the flow shortening the length of the unique pseudoline. Orthogonal trajectories of this functional should abut to a curve splitting off a line, which after all is the shortest pseudoline. This could be the basis of a grand inductive process reducing the rigidity of curves with few branches ($r\le \Delta(m)+1$) to that of curves of lower orders.

It seems that much can be explored along this line of geometric flows, somewhat reminiscent of say Möbius 1863, Poincaré, Morse, etc., up to Perelman’s proof of Poincaré’s conjecture, except that in our case the dynamics lives merely on a finite-dimensional manifold (the hyperspace of all algebraic curves of some fixed degree).

Another source of rigidity comes from the empty scheme, which is rigid. In fact I started to doubt about this issue, until Shustin kindly remembered me the following simple argument. If we have two curves with empty real locus ([*invisible curves*]{} for short), then after choosing equations of the same sign, the linear deformation $(1-t)P+tQ$ will connect both curves while conserving the same sign (provided $0\le t\le 1$), hence producing a path of invisible curves. However it is not a priori (nor a posteriori!?) evident that our path avoids curves with singularities, which could occur in imaginary conjugate pair. Hence the complete proof seems to use the fact that empty curves with singularities form a locus of codimension 2, since there are two conjugate nodes generically.

Once rigidity of the empty scheme is known, rigidity of curves with one oval should follow simply by contracting the one oval and letting it then disappear. This gives the basic connection with the former section (which is primarily a remark of Viro). Also as we remarked earlier, the stronger version CCC of the contraction conjecture could accomplish stronger rigidity results like URS, cf. (\[URS:conj\]).

[*Insertion*]{} \[03.04.13\].—Conversely if the one-oval scheme (unifolium) is rigid then it suffices to contract the Fermat curve (of even degree) to establish CC, but alas only for this unifolium chamber. Incidentally the validity of CC even for the unifolium scheme is not completely obvious, for taking the linear pencil between such a curve and an empty one leads (after assuming general position w.r.t. the discriminant, i.e. transversality) to a sequence of Morse surgeries a priori much more complex than just the death of the oval.

\[10.02.13\] Another question of didactic interest is to study the interplay between Fiedler-Marin locking method and Nikulin’s rigid classification. For sextic schemes of the form $\frac{3}{1}\ell$ we have an obvious lock given by the 3 lines through pairs among the 3 deep inner ovals. Each such line cuts twice the ovals it visits and twice the (nonempty) surrounding oval, hence is Bézout-saturated. Further this fundamental triangle is canonically assigned to the configuration in the sense that the position of 3 points in the insides of the ovals is parametrized by the 3rd symmetric power of a cell which is a contractible space. Hence the distribution of the outer ovals past the deep (Bermudian) triangle is invariant under rigid-isotopy. This adumbrates a strategy toward corrupting Nikulin’s theorem, but in reality the latter rather implies an invariance of this Bermudian distribution of outer ovals for all curves having the same real scheme (and the same type in Klein’s sense). It is therefore of interest to determine this outer distribution past the fundamental triangle for some specific curves as it will imply the same for all isotopic curves. This question is elaborated in Sec.\[Nikulin-corruption:sec\].

[*Insertion*]{} \[03.04.13\].—All this problematic went in decrepitude after an illuminating message of Le Touzé (cf. Sec.\[LeTouze:sec\]) yielding a conceptual explanation of why it is impossible to corrupt Nikulin via Fiedler-Marin. The reason is a simple chromatic law for conics passing through $5=3+2$ points with 3 of them black-colored (situated or defining a triangle), while the location of the 2 remaining points (white colored) past the triangle will determine how the sequence of 5 points distributes on the conic interpolating them. When the 2 white-points belongs to different component of the (black) triangle the distribution will be dichromatic in the sense that the 2 white points are not standing nearby, but separated by black points ($1$ or $2$ depending on the path chosen on the topological circle underlying the conic). Applying this lemma on conics to the above setting, shows that all ovals of a sextic enlarging the scheme $\frac{3}{1}$ are necessarily not separated by the deep triangle, for otherwise we can trace a conic with 4 transitions black-and-white (i.e. inside-vs.-outside of the nonempty oval) with therefore $5\cdot 2+4=14>12=2\cdot 6$ real intersections violating Bézout.

A last phenomenon is the rigidity of the deep nest established in Nuij 1968 [@Nuij_1968] or Dubrovin 1983 [@Dubrovin_1983/85]. This seems connected with Ahlfors total reality, since the deep nest is totally real under a pencil of lines. Extrapolating the Nuij-Dubrovin rigidity one can speculate that curves (or schemes) totally real under other pencils are likewise rigid. (A real scheme is [*rigid*]{} if all curves belonging to it are rigid-isotopic.) For instance the scheme of degree 8 consisting of 4 nests of depth 2 is totally really under a pencil of conics and thus could be rigid. Note however that total reality in the abstract sense which is actually (by Ahlfors theorem) synonymous to “type I” is not sufficient to ensure rigidity as exemplified by Marin’s construction of two isotopic $M$-septics, yet not rigid-isotopic (cf. Fig.\[Marin:fig\] below). Therefore if there is any connection between total reality and rigidity it must be a more subtle one. This theme is explored in Sec.\[Nuij-Dubrovin-extended:sec\], but we lack any serious result presently.

Problems of rigid-isotopy amount studying the residual components past the discriminant $\disc$ which is a hypersurface of degree $3(m-1)^2$ in the hyperspace of all curves of degree $m$. Call such components, [*chambers*]{} “of” the discriminant. When $m\le 6$ virtually everything is known, e.g. there are precisely 64 chambers of sextics by Nikulin’s theorem built upon the Gudkov-Rohlin census (cf. Fig.\[Gudkov-Table3:fig\]). A myriad of questions occur which are hard to handle systematically. For instance how many chambers in function of $m$? Is there an universal upper bound on the number of chambers residual to a hypersurface in function of its degree and dimension. This seems perhaps accessible via a conjunction of Harnack-Klein-Smith-Thom-Milnor and Jordan-Brouwer separation (plus Phragmen?). Asymptotic results in this sense were studied by Kharlamov-Orevkov.

One would like to describe the contiguity graph between chambers where edges label Morse surgeries while crossing the discriminant transversally along a principal stratum of codimension 1 (so-called [*walls*]{}). One can also investigate the topology of the varied chambers. Here one tool is the monodromy representation encoding how ovals permute when the curve is travelled along a loop in the given chamber. This and other issues is the object of next section, which is probably not extremely relevant to our main topic of the Ahlfors map, yet pleasant for its own. It can be left with loss of continuity.

The topology of chambers, symmetry, monodromy and transmutation (Kharlamov 1980, Itenberg 1994)
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\[10.02.13\] To each real scheme is attached a (Zeuthen)-Hilbert (multi-)tree (“forest”) with vertices the ovals and with edges whenever there is a nesting. Since any oval is immediately enveloped in at most one other oval this forest looks like a forest of pines (or a mushroom if you prefer). Hence, it is a directed set branching only downwards. The monodromy acts on this tree respecting its combinatorial structure. So for instance the deep nest is a “naked” tree having only a trunk but no branches. The automorphism group of this trivial tree is trivial, and so must be the monodromy representation. There is no obstruction to the deep-nest chamber having a simply-connected topology, and we can conjecture it to be simply-connected.

What about the empty chamber $E$? Define the [*invisible locus*]{} $I$ as the set of all curves with empty real locus. The empty chamber is $E=I-\disc$. A simple argument (detailed in the sequel) shows that $I$ is simply-connected and even contractible. Another simple argument based on Brusotti shows that $\disc \cap
I$ is nonempty (when $m\ge 4$). Further $I\cap \disc$ has real codimension 2 in the hyperspace of curves $\mH$ (or in $I$) and so is like a knot (possibly with singularity). In any case, it seems to follow ($I$ being noncompact) that the fundamental group $\pi_1(E)$ is non-trivial despite triviality of the monodromy. Perhaps $E$ is an Eilenberg-MacLane space $K(\pi, 1)$, i.e. aspheric. Can we compute $\pi_1(E)$ as a function of $m$? What about $m=6$ or even $m=4$?

Likewise for the deep chamber $D$ it seems hasty to expect simple-connectivity from trivialness of the monodromy. Consider for instance the Gürtelkurve $C_4$ quartic (with 2 nested ovals) and assume it a very symmetric perturbation of 2 transverse ellipses rotated by $90$ degrees. Assume further the existence of say a symmetry $\tau$ about the vertical axis. Since the group $G=PGL(3, \RR)$ is connected we can connected the identity to $\tau$ by a continuous path $c$. This $c$ induces a loop $\gamma$ in the space of quartics from $C_4$ to itself. As $C_4$ belongs to the deep chamber $D$ upon which the group $G$ acts, it makes sense to ask whether $\gamma $ is trivial or not in $\pi_1(D)$. For $c$ we may choose the path in $SO(3)$ given by $180$ degrees gyration about the vectorial line of $\RR^3\ni (x,y,z)$ parallel to the axis of symmetry of $C_4$ (viewed in the affine chart $z=1$). (Warning actually it seems that the line orthogonal to that is required!)

Actually choosing any path $c$ from $id$ to $\tau$ in $G=PGL$, its image in $D\subset \mH$ is a loop $\gamma$ likely to be not null-homotopic. Alas the map from $G$ to the orbit of $C_4$ is not really a covering, the argument looks a bit sloppy. A more convenient way to argue is to consider the double cover of the deep chamber $D$ by polarized curves, i.e. with a preferred half of the underlying orthosymmetric Riemann surface. Polarizing amounts specifying a complex orientation à la Rohlin (by taking the oriented boundary of the preferred half w.r.t. the canonical orientation induced by the complex structure). The loop $\gamma$ based at $C_4$ lifts—w.r.t. the polarized cover—to a [*non-closed*]{} path, since $\tau$ exchanges both halves of the complexification. Imagine indeed the symmetric surface underlying the Gürtelkurve as a pretzel of genus 3 with 2 ovals acted upon by an involution ($\tau$) with 4 fixed points then it must necessarily be a rotation by a half-twist about a line in 3-space perforating the ovals in 4 points.

It is clear that this argument extends to all deep nests and we obtain the:

For any integer $m$ (odd or even do not matter) the chamber of the deep nest (alias deep chamber) is not simply-connected.

Any curve in the deep chamber $D$ is of type I (Klein’s orthosymmetry) since there is a totally real pencil of lines. (This is the trivial sense of Ahlfors theorem so-to-speak.) We consider for each plane orthosymmetric curve the two possible ways to paint one half of the curve in black, and call the corresponding painted object a polarized curve (or Riemann surface). If $O$ is the union of all orthosymmetric chambers, we have a natural way to topologize the space $O_2$ of all polarized curves to turn it in a double cover of $O$, the orthosymmetric locus. In particular we have a double cover of the deep chamber $D_2\to D$.

Any member of the deep chamber admits a representative $C_m$ with a mirror involution $\tau$ given as $(x,y)\mapsto (-x,y)$ in affine coordinates. It suffices indeed to define $C_m$ as a small perturbation of an union of concentric circles (plus a horizontal line outside them when $m$ is odd). Either by inspecting the Riemann surface or just by noticing that $\tau$ reverses orientation of the ovals (and the pseudoline if $m$ odd) we infer that $\tau$ takes the polarized curve to its opposite (where the other half is preferred).

Using connectedness of $G=PGL(3, \RR)$ (more generally $PGL(n,
\RR)$ is connected whenever $n$ is odd because then both components of $GL(n, \RR)$ given by the sign of the determinant coalesce together since the identity matrix $I_n$ and its opposite $-I_n$ are homothetic yet of opposite determinants), we infer existence of a path $c$ in the Lie group $G$ connecting $id$ to the symmetry $\tau$. Applying this path to $C_m$ gives a loop $\gamma$ in the deep chamber $D$ based at $C_m$. Lifting this loop to the $O_2$ cover continuously amounts tautologically to apply the path $c$ to the polarized curve, whose end-point $c(1)$ is the opposite polarization as the one we started with. Hence the lift of the loop $\gamma$ is not a loop, and covering theory tell us that $\gamma$ is not null-homotopic.

If now $O$ denotes a specific orthosymmetric chamber we also have the double cover $O_2\to O$ (by polarized Riemann surfaces) and it is likely that the above argument extends to all or at least some orthosymmetric chambers having a representative with a mirror symmetry $\tau$. Each such chamber would not be simply-connected.

Abstractly an orthosymmetric surface can always be rotated by an half-twist permuting both halves. However it is not evident that this can be done in the plane, at least we know about no general argument. Thus we retract to examples in degree $m=6$, where due to the combined efforts of Harnack-Hilbert-Gudkov-Rohlin we know exactly what happens (cf. the Gudkov table=Fig.\[Gudkov-Table3:fig\]). In each orthosymmetric chamber we look for a symmetric representant under an involution fixing a line.

Consulting this table and gathering earlier constructions on a single plate (Fig.\[Symmetry:fig\] below) gives the following symmetric realizations of dividing sextics:

$\bullet$ the $M$-schemes of Hilbert and Harnack can both be given a symmetric realization as evidenced by the picture below. In both cases the invariant line intercepts 3 ovals.

$\bullet$ for Gudkov’s scheme $\frac{5}{1}5$ the existence of a symmetry is less obvious in Gudkov’s original construction (cf. Fig.\[GudkovCampo-5-15:fig\]). The situation appears more pleasant on Viro’s construction of the latter (compare Fig.\[Viro3-15:fig\]c), but alas since we cannot choose $\alpha=\beta=2$ both in V1 and V2 (please refer to the notation of that figure) we cannot conclude the existence of a global symmetry. Should we conjecture that Gudkov is somehow asymmetric?

$\bullet$ $\frac{8}{1}$ admits a symmetric realization as shown by a variant of Hilbert’s method (cf. figure below). Notice also the model with double (dihedral) symmetry. Again 3 ovals are intercepted by the (vertical) axis of symmetry.

$\bullet$ $\frac{6}{1}2$ in Hilbert’s realization below severely lacks symmetry. Appealing to Viro’s method (Fig.\[Viro3-15:fig\]c or below) does not aid (V5 twice looks promising yet do not confuse the values of $\alpha,\beta$!).

$\bullet$ $\frac{4}{1}4$ along Hilbert’s realization again lacks symmetry, and Viro’s method does not seem to help.

$\bullet$ $\frac{2}{1}6$ in Hilbert’s realization is asymmetric. Via Viro’s method this is realized by taking in V1 bottom $(\alpha, \beta)=(1,1)$ and in V2 top $(\alpha, \beta)=(0,4)$. Alas this is highly asymmetric.

$\bullet$ $9$ is symmetric under Hilbert’s construction, or a more elementary (Plücker-style) deformation of 3 ellipses, which is even more symmetrical.

$\bullet$ $\frac{5}{1}1$ is symmetric as shown below via a primitive perturbation of ellipses à la Plücker-Klein (pre Harnack-Hilbert oscillation trick). Again 3 ovals are intercepted by the symmetry-axis.

$\bullet$ $\frac{3}{1}3$ is symmetric by a perturbation of ellipses depicted below.

$\bullet$ $\frac{1}{1}5$ is likewise symmetric as shown by the depiction below.

$\bullet$ $\frac{4}{1}$ is highly symmetric as shown by a perturbation of ellipses below.

$\bullet$ $\frac{2}{1}2$ is symmetric as shown by the perturbation of ellipses below (2 models).

This is the exhaustive list of sextics of type I, modulo the omission of the deep nest (which is certainly symmetric). When taking 3 concentric circles one gets the impression of a continuous Lie group of symmetries, yet any perturbed curve will be more rigid (recall finiteness of automorphisms due to Schwarz-Klein-Poincaré-Hurwitz and the bound $84(g-1)$).

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From this investigation it follows the:

All orthosymmetric chambers of sextics are not simply-connected except perhaps the $4$ “antechambers” nearby Gudkov’s scheme that is $\frac{5}{1}5$, $\frac{6}{1}2$, $\frac{4}{1}4$, and $\frac{2}{1}6$. In fact all sextic orthosymmetric chambers $O$ have nontrivial polarization covering $O_2\to O$, safe perhaps the $4$ above asymmetrical schemes.

\[11.02.13\] Perhaps there is an obstruction for those 4 schemes to admit a symmetry about a line. In case of a Gudkov curve (of type $\frac{5}{1}5$), the symmetry has to leave invariant 3 ovals for the unique nonempty oval has to be preserved while the number of inner and outer ovals are odd. At this stage (or earlier) it is pleasant to visualize the Riemann surface in 3-space. Besides the horizontal orthosymmetry imagine a rotational symmetry under half-twist (180 degrees) leaving 3 ovals invariant while the 8 remaining one are pairwise exchanged. Of course per se this is no obstruction since Hilbert or Harnack have such a symmetry. So the obstruction is necessarily a subtle one if it exists perhaps say à la Arnold-Rohlin. Another idea is to smooth the Gudkov curve along the axis of symmetry and hope to get a septic violating Bézout (Fig.\[Symmetry:fig\]d), but looks improbable.

Let us look at Rohlin’s formula $2(\Pi^+ -\Pi^-)=r-k^2$ (see (\[Rohlin-formula:thm\])). Applying it to a Gudkov curve we have $r=11$ and $k^2=9$, hence $(\Pi^+ -\Pi^-)=1$. But on Gudkov’s curve we have $5$ injective pairs of ovals, i.e. $5=\Pi=\Pi^{+}+\Pi^{-}$, and it follows $2\Pi^+=5+1=6$, whence $\Pi^{+}=3$ and $\Pi^{-}=2$. Can it be inferred that there is no symmetry? A priori not, since the 5 inner ovals could have their complex orientations being reversed by the symmetry while one oval is kept invariant. Doing the same calculation for Hilbert’s curve we find $(\Pi^+ -\Pi^-)=1$ and $9=\Pi=\Pi^{+}+\Pi^{-}$, so $2\Pi^+=9+1=10$ and $\Pi^{+}=5$ while $\Pi^{-}=4$.

The symmetry could be not a reflection about a line but a rotation about a point. Yet from the projective viewpoint this seems to be equivalent. At any rate Gudkov’s curve in Viro’s realization is anyway not symmetric under a rotation.

Let us apply Rohlin’s formula to the $(M-2)$-schemes which are potentially asymmetric, e.g., $\frac{6}{1}2$. Then $2(\Pi^+
-\Pi^-)=9-k^2=0$, hence $\Pi^+ -\Pi^-=0$, but $\Pi^+ +\Pi^-=6$ so that $2\Pi^+=6$, and $\Pi^+=3=\Pi^{-}$. Hence the symmetry cannot exchange the $6$ inner ovals in pairs without fixing any of them.

This requires some explanation. Recall we are looking for holomorphic involutions of some plane dividing curve $C_m$ induced by an element of $PGL(3, \RR)$ exchanging both halves. Call such an involution a [*mutation*]{}. If a curve has a mutation then its chamber $O$ has nontrivial polarized covering $O_2\to O$.

For sextics (except the deep nest and the unnested curves $1,2,\dots,10$) there is a unique nonempty oval. Distinguished as a such, this must be preserved by the mutation $\tau$, which must reverse its orientation. If not, orientation is preserved and $\tau$ acts as a rotation on this circle. Taking an invariant tube-neighborhood one deduces that both halves are preserved as $\tau$ respects orientation of the surface (hence of this tube), violating the mutating assumption.

Supposed fixed a complex orientation of the dividing curve. The mutation reverses orientation of the nonempty oval, and also the complex orientations of all other ovals because $\tau$ preserves orientation but exchanges both halves. Symbolically, we may see this by writing $\tau(\partial
C^+)=\partial (\tau C^+)=\partial (C^-)=-\partial (C^+)$.

At this stage we are ripe for picturing. Imagine the mutation given by a symmetry about a line (this is probably no loss of generality in projective geometry, as the other candidate namely a rotation about a point fixes the line at infinity). Consider the following schematic pictures (Fig.\[Sym2:fig\]). The first (Fig.a) is not mutating the orientation, hence precluded. Fig.b is mutating the complex orientation, but violates Rohlin’s formula. In fact the mutation condition in case where no inner ovals are invariant imposes an even number of positive injective pairs of ovals and Rohlin’s formula cannot be fulfilled. Hence Rohlin’s formula forbids a mutation without invariant inner oval. Fig.c shows a configuration where both mutation and Rohlin’s formula are satisfied. This is good schematically but bad theoretically, as it fails obstructing mutability of a curve of type $\frac{6}{1}2$. We are not much advanced in our problem. The other figures of Fig.\[Sym2:fig\] show that for each of the other asymmetric types there is always a schematic symmetry compatible with both mutation and Rohlin’s formula. So Rohlin fails to detect any structural asymmetry. Paraphrasing, asymmetry may just be a defect of our models (Hilbert and Viro) yet not an intrinsic property of the chamber.

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In fact, the above Rohlin’s formula argument only shows for the two $(M-2)$-schemes of type I ($\frac{6}{1}2$ and its mirror), that if they have a mutation the latter must preserves 2 inner ovals.

\[12.02.13\] $\bullet$ A smooth dividing plane curve $C_m$ defined over $\RR$ is transmutable if there is a rigid-isotopy switching its half (called a transmutation).

$\bullet$ A mutation is a linear automorphism $\tau \in
G=PGL(3,\RR)$ of the curve $C_m$ permuting both halves of the curve, say in this case that the curve is mutable.

Since the group $PGL(3,\RR)$ is connected any mutation induces (non canonically) a transmutation. Indeed choose a path $c$ in $G$ joining $id$ to $\tau$ and its operation upon $C_m$ defines a loop in the corresponding chamber of the space of curves (past the discriminant) which is a transmutation. It is not essential that $\tau$ has order 2, but then speak of a $2$-mutation.

So any $2$-mutable (dividing) curve is mutable, and in turn transmutable. The converses looks a priori quite improbable. Are all dividing plane curves transmutable? or even mutable, or $2$-mutable after some rigid-isotopy? The question looks of interest because a non transmutable curve would have a preferred half (privileged so-to-speak) which looks a bit against the flavor of Galois-theory and French revolution “égalité, fraternité, etc.”.

A mutation (like any self map of $\RR P^2$) has a (real) fixed point (e.g. via Lefschetz fixed point theorem using homology over ${\Bbb Q}$), and any $2$-mutation is a mirror about a line fixing also a real isolated point, as inferred from linear algebra (existence of real eigenvalues for an endomorphism of a real vector space of odd dimension).

Perhaps the above questions can be handled via C. Segre’s classification of real structures on projective spaces, especially the fact that the plane ${\Bbb P}^2$ has a unique real structure, but looks unlikely as the curves are not taken into account.

One way to approach the problem in general would be to look at the action of $G=PGL(3, \RR)$ on the chamber past the discriminant containing a dividing curve. Existence of a mutation in each such chamber amounts this action being never free, i.e. with nontrivial isotropy subgroup $G_{C}$ at some suitable curve $C=C_m$. This is not enough for the automorphism in question needs not permute both halves. Omitting this difficulty, there would be a free Lie group operating, hence an induced foliation of the chamber by leaves of dimension 8 (=$\dim G$). Alas such a foliation also exists when the action is only locally-free, i.e. discrete isotropy are allowed. It would be nice to know if all chambers of the discriminant (not only the orthosymmetric chambers) contains a curve with (linear) automorphism in themselves. For orthosymmetric chambers we would further like to know if there is such an automorphism permuting the halves of the curve (i.e. a mutation).

In a remarkable article extending earlier work by Kharlamov, Itenberg 199X [@Itenberg_199X-monodromy-deg-6] is able to compute the monodromy groups of each chamber of sextics. Extracting from his tabulation, only the type I cases gives the:

[(Kharlamov, Itenberg)]{} The monodromy groups of smooth sextics of dividing type are given by the following list (where $\triv$ is the trivial group, $S_n$ the symmetric group on $n$ letters, and $D_n$ the dihedral group):

$\bullet$ $\frac{9}{1}1 \rightsquigarrow \ZZ_2 $, $\frac{5}{1}5 \rightsquigarrow \triv $, $\frac{1}{1}9 \rightsquigarrow S_3 $,

$\bullet$ $\frac{8}{1} \rightsquigarrow D_4 $, $\frac{6}{1}2
\rightsquigarrow \triv $, $\frac{4}{1}4 \rightsquigarrow \triv $, $\frac{2}{1}6 \rightsquigarrow \ZZ_2 $, $9 \rightsquigarrow S_9 $,

$\bullet$ $\frac{5}{1}1 \rightsquigarrow \ZZ_2 $, $\frac{3}{1}3 \rightsquigarrow \ZZ_2 $ $\frac{1}{1}5 \rightsquigarrow D_5 $,

$\bullet$ $\frac{4}{1} \rightsquigarrow S_3 $, $\frac{2}{1}2 \rightsquigarrow \ZZ_2 \times \ZZ_2 $, $\bullet$ $(1,1,1) \rightsquigarrow \triv$.

It is interesting to compare this result with our picture Fig.\[Symmetry:fig\], as sometimes the whole monodromy group can be realized by rigid projective motions.

Besides, it seems interesting to compare this monodromy of ovals to the monodromy upon the halves. Albeit the latter viewpoint is less rich in general (being only a representation on the group with 2 elements) it is sometimes of complementary nature in detecting non-triviality of the fundamental group of the fixed chamber. For the moment, our halves-monodromy is only more sensitive in the deep-nest case. (Question: does the $\pi_1$ of the deep chamber reduces to $\ZZ_2$?)

Finally, note that the oval-monodromy is also fairly small for the 4 exceptional schemes, asymmetric in Hilbert’s (or Viro’s) realization (again Fig.\[Symmetry:fig\]). Hence both methods oval-monodromy and half-monodromy (at least via rigid symmetries) fails to detect nontrivial elements in $\pi_1$ of the corresponding chamber for the schemes of Gudkov $\frac{5}{1}5$, of left-Rohlin $\frac{6}{1}2$ and $\frac{4}{1}4$. Can we extrapolate that those chambers are simply-connected? If yes then those 3 curves are not transmutable, hence not mutable and therefore structurally asymmetric (i.e. there is no model invariant under a mirror).

More factually, Itenberg’s calculation prevents a curve of type $\frac{6}{1}2$ to accept a mirror like Fig.\[Sym2:fig\]c. Indeed otherwise if $\tau$ is such a mirror, it suffices to take a path in $PGL(3, \RR)$ joining the identity to this $\tau$ to get a loop in the space of curves with non trivial monodromy. Likewise no curve of type $\frac{4}{1}4$ can accept a mirror like Figs.\[Sym2:fig\]d,e, and no curves of the Gudkov type $\frac{5}{1}5$ can accept a mirror like Figs.\[Sym2:fig\]i. Hence Itenberg’s calculation implies the following answer to one of our basic question:

None of the $3$ monodromically-trivial dividing curves—as listed by Itenberg, i.e. Gudkov’s, the left Rohlin curve $\frac{6}{1}2$, and that of type $\frac{4}{1}4$—can support a mirror.

A mirror is a linear involution of $PGL(3,\RR)$ which fixes a line (plus a point at $\infty$). By Bézout, the fixed line can intercept at most 3 ovals which are then invariant. Examining the following Fig.\[Sym3:fig\] shows that there is always some pair of ovals permuted by the mirror.

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Alas, an oval can be invariant under the mirror without having to intersect the fixed line, say by running to infinity. Then one can try to argue with the line at infinity and Bézout.

A better way to argue is that the nonempty oval of the sextic has to be invariant under the mirror, hence at most 2 inner ovals can be left invariant under the mirror (else Bézout corrupted). So we infer existence of inner ovals permuted under the mirror $\tau$. Taking a path from $id $ to $\tau $ yields a loop in the chamber whose monodromy is nontrivial. This contradicts Itenberg’s calculation of the monodromy.

In particular this shows existence of curves without a $2$-mutation, since a $2$-mutation is certainly a mirror. However it is not clear if our 3 curves lack a mutation. Given a mutation (i.e. a linear automorphism) permuting the halves it is not a priori of order 2. However it has finite order (by Klein-Poincaré-Hurwitz finiteness of the automorphism group of closed Riemann surfaces of genus $\ge 2$). Since $\tau$ permutes the halves its order must be even say $n=2 e$. Hence $\tau^e$ is an involution, and a mutation if $e$ is odd. Alas in the other case, i.e. when $n$ is divisible by $4$, we cannot say much.

Assume given a mutation $\tau$ on a Gudkov curve, then it has to preserve the unique nonempty oval, and the inner and outer ovals have to be respected. Further $\tau$ (being a mutation) it has to reverse the complex orientation. As we may assume the order $n$ of $\tau$ divisible by $4$, it should follow that $\tau$ permutes the ovals along a 4-cycle, etc. Via some group theory there is presumably an obstruction to mutate the Gudkov curve?

Alternatively assume there is a mutation on some Gudkov curve $C_6$ then it will permute the ovals while preserving the nonempty one and the inner/outer ovals subdivision. Further it must reverse the complex orientation. If any permutation of the ovals is detected (which is precluded by the Kharlamov-Itenberg’s calculation of the monodromy as being trivial), we are finished. Assume so that the mutation preserves all ovals. Since it has to reverse their orientations, the mutation has 2 fixed points on each of them (an orientation reversing transformation of the circle has 2 fixed points e.g. via Lefschetz or via covering theory). But globally our mutation is of finite order (since it induces an automorphism of the Riemann surface of genus $10\ge
2$), and any element of finite order in $PGL(3, \RR)$ preserves a line. So we get a line intersecting the $C_6$ in 22 points, overwhelming Bézout. As another variant, once our mutation is known to have 2 fixed points on each ovals we have $2(g+1)$ of them, which is the maximum permitted by Lefschetz trace formula. We would like to conclude that $\tau $ is the hyperelliptic involution, which cannot exist on a smooth $C_6$ (which is only $5$-gonal).

Let us clarify this argument with the:

[A curve is [*antidromic*]{} if it monodromy group is trivial. A [*symmetry*]{} of a plane real curve is a linear automorphism of the plane (i.e. an element of $PGL(3,\RR)$) preserving globally the curve.]{}

[(1)]{} Any symmetry of an antidromic curve must leave each oval invariant. Hence all the $3$ antidromic sextics listed by Itenberg can only admit a mutation preserving all the ovals.

[(2)]{} In particular all the $3$ antidromic dividing sextics of Itenberg (the deep nest being excluded) lack a mutation. So Gudkov’s curve, and the left-wing Rohlin curve $\frac{6}{1}2$ and the dividing curve $\frac{4}{1}4$ are asymmetric at least under a mutation (i.e. they cannot mutate).

\(1) If there is a symmetry $\tau$ permuting somehow the ovals, then the path in $PGL(3,\RR)$ connecting $id$ to $\tau$ induces a loop in the space of curves with nontrivial monodromy (namely the ovals-permutation induced by $\tau$). (2) As to the second assertion, our oval-preserving mutation must necessarily invert the orientation of all ovals since it exchanges both halves. (Recall that a mutation reverses the complex orientation in the sense of Rohlin, since it preserves the orientation induced by the complex structure while exchanging both halves of the dividing curve.) Since a sense-reversing transformation of the circle has 2 fixed points as follows from Lefschetz’s fixed-point formula. (Indeed the Lefschetz trace number is $(+1)-(-1)=+2$ so there is a fixed point and removing it one obtains an orientation reversing homeomorphism of the line which has another fixed point, e.g. by applying Bolzano to the graph of this continuous decreasing function.) (Is it true that any sense-reversing transformation of the circle is an involution? We do not need this anyway!)

Since Itenberg’s antidromic curves have $r=11$ or $r=9$ ovals, we get 22 or 18 fixed points created. Next we have:

A linear automorphism in $PGL(3, \RR)$ that fixes $4$ (or more) points of $\RR P^2$ fixes either a line plus an isolated point or is the identity.

This follows by looking at the 4 corresponding eigen-lines in $\RR^3$, two of which have to correspond to the same eigenvalue (pigeonhole principle), and so there is the required fixed projective line. The third eigenvalue left (necessarily real) gives a third eigen-line and the announced alternative follows depending on whether this 3rd eigenvalue differs or coincides with the former double eigenvalue.

It follows that among our 22 or 18 many fixed-points (at least so many less one) are aligned, but this corrupts Bézout.

A priori it is much harder to detect an obstruction to transmute the Gudkov curve (or its 2 antidromic cousins), and likewise hard to show that it may be transmuted.

Assume there is a transmutation. Then since by Kharlamov-Itenberg’s lemma the curve is antidromic the induced permutation of ovals is trivial, but the complex orientation is reversed. Perhaps some obstruction can be deduced from this...

\[15.02.13\] Two days ago, Kharlamov informed me that the Gudkov chamber has fundamental group $\ZZ_2$ compare his letter (dated \[13.02.13\]) in Sec.\[e-mail-Viro:sec\]. So I presume that the Gudkov curve can be transmuted, and that the isomorphism $\pi_1\approx \ZZ_2$ may be realized as the monodromy acting upon halves. Kharlamov’s messages also emphasize the issue that the $M$-curves case is somewhat easier than the other cases. Hence while the (oval)-monodromies are completely calculated by Itenberg, it may be the case that the determination of the fundamental group of each chamber is somewhat harder to obtain.

Some naive questions are as follows. We presume that the deep-nest chamber has $\pi_1=\ZZ_2$. Are the (fundamental) of chambers always finite? This would follow (theorem of Myers, Synge, Hopf, etc.) if there is a complete metric of positive curvature on the hyperspace of curves (which is the case) yet the natural elliptic metric is not complete when restricted to the chambers. Further since Gudkov’s chamber is not simply-connected, any Gudkov curve is presumably transmutable (this amounts to say that the Kharlamov isomorphism $\pi_1=\ZZ_2$ is realized by the monodromy of halves). If so is the case, perhaps that even all dividing plane curves are transmutable.

Isotopic rigidity of the empty scheme: connectedness of invisible curves {#rigidity-empty-scheme-via-dyna:sec}
------------------------------------------------------------------------

\[08.04.13\] We now come to a tortuous revelation of a basic truth, namely the fact that the empty chamber is connected, i.e. any 2 real smooth plane curves with empty real locus can be connected by a path of similar curves (avoiding the discriminant). This problematic covers no less than $6$ sections up to Sec.\[invisible-discriminant-codim-2:sec\], where it is elucidated that the portion of the discriminant inside the locus of empty(=invisible) real curves has codimension 2 hence cannot effect a disconnection. Pivotal in this search was a kind letter by Shustin explaining the linear homotopy argument between empty curves, which shows that the empty locus (including possibly singular curves) is connected, and actually much more like being contractible. So we warn the reader that those six sections are far from a geodesic toward the goal, but we had not the courage to censure any bit of our poorly organized material as it often ramifies toward considerations of independent interest.

\[23.01.13\] In fact to be honest I realize that even the foundation of our reasoning (in the previous section[^51]) is not completely sound, namely the following fact:

[(Folklore??? is it really true? If yes where is it proved?)]{} Any two empty curves are rigid-isotopic.

[*Insertion*]{} \[08.04.13\].—Folklore probably! True, certainly, compare Shustin’s letter, but also the codimension 2 lemma (poorly) established in our Lemma \[invisible-discriminant-codim-2:lem\]. Where it is proved? We still lack a detailed reference, but apparently the fact is so trivial that nobody took care writing down a complete proof. It would be of interest to make a deeper historical search of who knew first this simple result. Possible candidates: Schläfli 1863 [@Schlaefli_1863], Cayley, Klein 1873–1925, C. Segre, Hilbert ca. 1891, Berzolari 1906 [@Berzolari_1906], Severi e.g. 1921 [@Severi_1921-Vorlesungen-u-alg.-Geom-BUCH], Brusotti 1921 or earlier, Petrovskii 1933, etc.

(Sometimes instead of saying empty curves we shall say invisible curve, when the real locus $C_m({\Bbb R})$ is empty.)

I was sure this to be known (but completely forgot where I read this in case I am remembering well!!!) At first sight this looks trivial but is not. One could imagine the empty chamber to be starlike or even convex, but this is not even evident. Somehow invisible curves could be like the immersed half of an iceberg, hence connected, while the visible part of the iceberg may consist of several islands (peaks) corresponding to the menagerie of chambers past the discriminant, well-known in Hilbert’s 16th. Alas even if this metaphor ought to contain some truth, it is easy to construct an iceberg with disconnected immersed locus. Take a letter “E”, rotate it by $-\pi/2$ to get the symbol ($\Pi\!\!\Pi$) considered as a tripod with 3 legs immersed in the water. To go in 3D, just take the body of revolution of this symbol to get an iceberg with 2 immersed components[^52]. After having being puzzled for while one might suspect this to be a result à la Hilbert, that a form not representing zero is something like a sum of squares...(not clear).

\[24.01.13\] More geometrically, one can look at the distance between the complex locus of a real curve $C_m({\Bbb C})$ and the real plane ${\Bbb R}P^2$. It is natural to work with the Fubini-Study metric on ${\Bbb C}P^2$. Then look at “the” point of the real plane closest to $C_m({\Bbb C})$, which must be generically unique. If we let $E$ be the empty chamber (a priori not connected), we obtain so a random-map $\pi\colon E \to {\Bbb
R}P^2$ taking an invisible curve to its closest “projection” in ${\Bbb R}P^2$. Up to removing some subset of $E$, one could arrange $\pi$ to be single-valued, and one would check that $\pi$ is akin to a fibration with connected fibres. The connectedness of $E$ could follow. It is also tempting to imagine a flow driving invisible curves to solitary nodes. This would be just the gradient flow of the functional distance to the real locus ${\Bbb
R}P^2$. The corresponding trajectories of steepest descent could converge to a curve with a unique solitary node (generic case). At the level of the Riemann surface this isotopy (given by the path of trajectory) really amounts to the contraction of an anti-oval toward a solitary node. So this is just a special case of Klein’s Ansatz (\[Klein-1876:conj-noch-entwicklungsfahig\]), that a nondividing curve can acquire a novel solitary node (by a large deformation). (Of course recall this to be erroneous by Shustin 1985 [@Shustin_1985], yet it is perhaps true for empty curves). In fact:

Klein’s Ansatz is trivially true for empty curves (just by general position and surgery of the real locus).

Indeed, given any invisible curve $C_m$, take any pencil through it passing through a visible curve $D_m$ (with nonempty real locus), then making this line transverse to the discriminant we get Morse surgeries the first of which must necessarily be a solitary node formation.

[*Insertion*]{} \[08.04.13\] Maybe one can object again this proof, by arguing that the first contact with the discriminant could be through a pair of imaginary nodes. Paraphrasing a bit we could imagine that travelling along the pencil spanned by $C_m,D_m$ we hit the discriminant but then fall again in the empty chamber. Presumably both scenarios can be avoided if we know that the invisible discriminant has real codimension 2 (as we shall see in the sequel, cf. Lemma \[invisible-discriminant-codim-2:lem\].)

Our flow would precisely do this contraction yet in some more organic(ized) fashion (i.e. no choices). Yet notice that we could make the above pencil argument by choosing once for all some visible curve $D_m$, while driving all the invisible curves $C_m$ along the line spanned by $C_m$ and $D_m$. If this does not work look at the flow (discussed above).

Optimistically, this method may suffice to establish connectedness of the empty locus $E \subset \vert m H\vert$. One may wonder if a variant of the argument could not also establish Viro’s open problem on the connectedness of the pseudoline locus $P\subset
\vert m H \vert$ when $m$ is odd. (Added \[08.04.13\].—It seems that this conjecture really goes back to Rohlin, if we interpreted correctly a letter of Viro in Sec.\[e-mail-Viro:sec\], dated \[26.01.13\].)

Return yet to the case of the empty locus $E$ (non void only for curves of odd degree). By what could it be disconnected? A curve in the discriminant $\frak D$ may well have two imaginary conjugate singularities. But this locus has codimension 2, so it cannot disconnect $E$. Does this suffices to prove connectedness of $E$? Probably not as a priori it may have several components lying “far apart”.

Another idea is to fix an invisible conic, e.g. the “canonical” one $E_0\colon x_0^2+x_1^2+x_2^2=0$ (on which conj acts like an antipodal map). To each point of the plane one can attach the apparent contour of this ellipse (polar lines) as seen from the given point to get a group of two points on this ellipse which is a Riemann sphere. (This is the most synthetic way to establish the well-known isomorphism between ${\Bbb C}P^2$ and the second symmetric power of ${\Bbb C}P^1$.) Points of $E_0$ correspond to groups of superposed two points, while real points in ${\Bbb
R}P^2$ maps to antipodal pair (invariant under conj when seen as a pair). Define a function $\rho$ on ${\Bbb C}P^2$ which given a pair measures the distance on the round sphere $S^2$ between the corresponding 2 points. This is equal to $\pi=3.14\dots$ on ${\Bbb
R}P^2$ and vanishes on $E_0\colon x_0^2+x_1^2+x_2^2=0$.

Now given any compact sublocus of ${\Bbb C}P^2$ (in particular an invisible curve $C_m$) one can look at the maximum of $\rho$ on $C_m({\Bbb C})$, which is $<\pi$. This gives the functional $$\theta:=\max \rho\colon E \to [0,\pi[,$$ whose ascending gradient lines should converge to solitary nodal curves (perhaps with several such nodes). Note that the ground (invisible) ellipse $k \cdot E_0$ (counted $k=m/2$ times) is the unique absolute minimum of this functional. This $\theta$ is quite likely to be a Morse function (or a slight generalization thereof with Monkey saddles, etc.), yet the critical points (causing annoying stagnation of the dynamics) ought to be isolated (codimension 2 suffices), hence not affecting the connectivity of $E$. By construction our flow tends to make an invisible curve more “visible” by pushing it progressively closer to the real plane. In the limit we expect something visible having solitary nodes and generically just one should emerge. So upon excising from $E$ a small set (of codimension 2, since at least two nodes is bad) we find a (dense) subregion $E^{\ast}$ which maps ${\Bbb
R}P^2$ by assigning the unique solitary node of the limit of $C_m
\in E^{\ast}$ under the flow at time $\theta=\pi$ (using $\theta$ as time parameter as usual for gradient flow). Now the fibre of this map $E^{\ast}\to{\Bbb R}P^2$ is the same as the [*bassin d’attraction*]{} of the flow which is cone-like formed by several trajectories abutting to the same solitary node. So this cone is connected by the end point, and connectivity of $E^{\ast}$ (hence $E$) should follow.

Still, the main difficulty is (as usual) to show that the $\theta$-functional lacks a local maximum preventing convergence to a visible curve. So given any invisible curve one should produce a small perturbation with larger $\theta$. This is probably not too hopeless. Naively one could perturb $C_m$ inside the pencil spanned by $k E_0$ and $C_m$. Since $k E_0$ is the most invisible curve a deformation along it should decrease $\theta$, while one \[deformation\] away \[of\] it should increment $\theta$. Of course there is some objection to this, since in a projective (real) line (a circle) it is never clear what means “along” and “away”.

All this is somewhat confuse and unconvincing. Perhaps also there is a much more elementary argument without gradient lines. As we said this could involve deforming all empty curves along some fixed visible curve. But which direction of retraction should we choose in the corresponding pencil? Since $m=2k$ is even, $\vert m
H \vert$ is of dimension $\binom{m+2}{2}-1=(k+1)(2k+1)-1=2k^2+3k=k(2k+3)=:N$. So when $k$ is odd there must be another singular point in the foliation induced by the [*faisceau*]{} (sheaf, bundle) of all lines through $D_m$ (by Poincaré-Hopf). Can we orient this foliation? No because when $N=2$, we have a Möbius strip after puncturing the basepoint of the pencil. The situation would be somewhat simpler if we could find in the hyperspace of curves $\vert m H
\vert \approx {\Bbb R}P^N$ a hyperplane $H\ni C_m$ avoiding the empty locus $E$. I do not know whether this is possible? Then there would be a nice way to retract the whole complement of $H$ toward the point $C_m$ in some canonical way. In particular all points of $E$ (invisible curve) would mark a first impact on the real locus ${\Bbb R}P^2$. Alas it is not even obvious that connexity of $E$ follows.

\[23.01.13\] Recall that the related question for odd degrees is still an open problem.

[(Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical p.199])]{}.—[Are all non-singular real projective curves of a given [*odd degree*]{} with connected set of real points [*rigid-isotopic*]{} to each other?]{}

The emphasis is Viro, and may suggest that without odd degree the assertion is known to be false!? If so then our (OOPS) conjecture (\[OOPS:one-oval-rigid-isotopic:conj\]) would be oops!

[*Insertion*]{} \[08.04.13\].—The answer were given in Viro’s letter dated \[26.01.13\] in Sec.\[e-mail-Viro:sec\], and may be summarized as follows. First in the even degree case, the problem of rigidity of the curve with a unique oval is still open (but probably reducible to the contraction conjecture (\[Itenberg-Viro-contraction:conj\])). Second, as we already said, the question ascribed above to Viro (2008) truly goes back to Rohlin (unpublished as far as we know).

Rigidity of the empty scheme (Shustin’s letter)
-----------------------------------------------

\[27.01.13\] This section treats the following desideratum: given two empty (plane) curves (hence of even degree), it is always possible to find a path of curves linking them while avoiding the discriminant.

In fact, I read about this fact a long time ago (ca. 2000) but could not remember from which source. Recently (24.01.13) failing to recover the source, we started to doubt about the truth of this assertion. Very kindly Shustin communicated us the simple (forgotten) argument giving a positive answer. Alas we have little idea of who proved this first. Shustin’s proof looks at first sight extremely trivial, but on more mature thinking the story looks a bit more tricky than expected.

\[empty-chamber-connected-Shustin:lem\] Any two empty curves are rigid-isotopic.

(Courtesy of Eugenii Shustin \[26.01.13\]) The chamber of empty curves of a given (even) degree is connected. Two such curves are defined by homogeneous polynomials $P,Q$, supposed (w.l.o.g.) positive for all real variables not all simultaneously zero. The linear homotopy $(1-t)P+tQ$, $0\le t\le
1$ gives then a path in the chamber of empty curves. Indeed the linear path between two positive numbers consists of positive numbers, and so all intermediate curves of this homotopy are empty curves.

A priori one can imagine that some intermediate curve of this homotopy (while staying empty) crosses the discriminant by acquiring a conjugate pair of nodes. (This eventuality was not mentioned in Shustin’s letter, but we think that is is a slight obstacle to the argument. However it seems to be not fatal as we shall discuss at length.)

Maybe the above argument should be supplemented by an examination of the partial derivatives $\frac{\partial P}{\partial x_i}$. Smoothness of a curve amounts the 3 partial derivatives (of the defining equation) lacking a common zero. Recall Euler’s relation $m F=\sum_i \frac{\partial F}{\partial x_i} x_i$ valid for any homogenous polynomial (form) of degree $m$. (This requires only to be checked on monomials $x_0^i x_1^j x_2^k$ such that $i+j+k=m$.) Combining this we infer that if there is some time $t\in [0,1]$ such that $(1-t)P+tQ$ is singular, then all its 3 partials vanishes simultaneously at some point $(x_0,x_1,x_2)$, which by Euler’s relation would be also a zero of $(1-t)P+tQ$. This violates however the emptiness of this curve. However for this argument to hold good it is essential for the point $(x_0,x_1,x_2)$ to be real, which is however not the case a priori.

Another more qualitative argument would be to first perturb slightly $P$ and $Q$ so that the linear pencil spanned by them is transverse to the discriminant. In that case if some member of the pencil acquires a singularity it will be a simple node, which consequently must be real. This violates emptiness of the intermediate curves $(1-t)P+tQ$, $0\le t\le 1$.

In summary the lemma looks true, yet not in the strong sense that any two empty smooth curves are linked by a linear homotopy consisting only of smooth curves. This strong form amounts to convexity of the empty chamber, and not just connectedness. Convexity is perhaps wrong, as the curve may traverse a pair of conjugate nodes during the linear deformation.

Can we corrupt convexity of the empty chamber? We think yes as follows. For the empty chamber to be nonempty, assume the degree $m$ even. Suppose given an empty curve $C_m$ with a pair of conjugate nodes. We shall later explain how to construct this, but one can already imagine in 3-space (or in the ether) a diasymmetric Riemann surface acted upon antipodically without fixed point, on which two symmetric vanishing cycles are contracted. Now through the given curve $C_m$ (considered as a point in the hyperspace of $m$-tics), trace a little rectilinear segment transverse to the discriminant. Both extremities of the segment will be smooth empty curves, yet the linear homotopy connecting them hits the discriminant (hence fails to be entirely in the empty chamber). This argument works fine provided the linear homotopy coincides with our little segment, instead of being actually the “long” residual pieces of it in the pencil (which is a circle).

When $m=2$, a conic can only have one node, when degenerating to a pair of lines. As we require $2$ conjugate nodes, let us look at quartics. Start with a pair of empty conics with transverse complexifications intersecting in 4 points $p,p^\sigma, q, q^\sigma$ (cf. Fig.\[ShustinEmpty:fig\]a for a schematic view). By Brusotti 1921 [@Brusotti_1921] (or earlier workers like Plücker, Klein, etc.) we can smooth $q,q^\sigma$ away to create an empty quartic $\Gamma_4$ with 2 nodes nearby $p,p^\sigma$. The corresponding (singular) Riemann surface (complex locus $\Gamma_4({\Bbb C})$) is visualized as a genus 3 surface (acted upon by antipody) with 2 handles shrunk to points $p, p^\sigma$ exchanged by conjugation (Fig.\[ShustinEmpty:fig\]b). We can also imagine the structure of the discriminant near $\Gamma_4$ as being a single nappe (Fig.\[ShustinEmpty:fig\]a). Take a little rectilinear segment transverse to this nappe. Both extremities of this segment are smooth curves $C_4$, $D_4$ which are empty. This gives our counterexample provided the linear homotopy $(1-t) C_4 + t D_4$, $t\in [0,1]$ visits $\Gamma_4$.

-5pt0

-5pt0

It is worth trying to clarify the above proof. Given two forms $P,Q$ of degree $m$, define the [*linear homotopy*]{} as the path of forms $(1-t) P+t Q$ where $t\in [0,1]$. Denote it symbolically $P\to Q$. If we look at curves (i.e. homothety classes of forms) then between any two curves $C, D$ of degree $m$ there is a pencil of curves $\lambda C +\mu D$ which is the line $\overline{CD}$ through both points seen in the hyperspace of curves. A little drawing (Fig.\[ShustinEmpty:fig\]c) shows that if $P, Q$ represents $C,D$ resp., then the linear homotopy $P\to Q$ projects to one piece of the line $\overline{PQ}$, while the linear homotopy $P\to -Q$ describes the other road of access in the circle $\overline{CD}$.

Shustin’s argument shows that if two forms $P, Q$ not representing zero have the same sign then the linear homotopy $P\to Q$ consists of forms not representing zero. (Recall that the sign of even degree forms is well-defined, because $F(\lambda x_0, \dots,
\lambda x_n)=\lambda^m F( x_0, \dots,  x_n)$.) However it does not say that if $P,Q$ represents nonsingular curves, then so are all members of the linear homotopy $P\to Q$.

In our example with $C_4$ and $D_4$ we could argue that the corresponding polynomials $P,Q$ are very near (by Brusotti’s construction) so of the same sign, and further that the linear homotopy $P\to Q$ really passes through $R$ the defining equation of $\Gamma_4$. In that case Shustin’s argument would be in slight jeopardy.

Does our counter-argument work? We think yes we start from $R$ a form defining $\Gamma_4$, and perturb slightly the coefficients of $R$ by Brusotti to get the polynomials $P$ and $Q$ so that the linear homotopy $P\to Q$ passes through $R$. Note that $R$ has some well-defined sign on ${\Bbb R}P^2$, and by smallness of the perturbation $P$ and $Q$ have the same sign as $R$. Thus even when $P$ and $Q$ do have the same sign we are not ensured that the linear homotopy $P\to Q$ transits only through nonsingular curves.

In conclusion given two smooth[^53] empty curves $C,D$ and choose representing forms $P,Q$ (resp.) of the same sign, then the projection of the linear homotopy $P\to Q$ in the space of curves $\vert m H\vert$ correspond to empty curves yet not necessarily smooth.

[*Related literature for Shustin’s argument.*]{} Maybe Wilson, Shustin ICM, etc...

Of course even if the above Brusotti-type construction is correct, it does not prove that the locus of empty curves is disconnected, but merely that the proof via linear homotopies is insufficient.

One possible critique to our argument is that because $\Gamma_4$ has $2$ nodes it is not on a principal stratum[^54] (wall) of the discriminant of codimension 1. Yet in reality this is a pair of conjugate points so really one point in the sense of Grothendieck’s schemes (to which we are from adhering). Perhaps our segment not transverse but rather tangent to the nappe of $\frak D$. However this looks not so realist by construction. The key issue is to decide whether our binodal curve $\Gamma_4$ is a smooth point of the discriminant, which looks likely if we regard only real curves. Of course in the complexified discriminant $\frak D({\Bbb C})$ there is two nappes of passing through the binodal curves $\Gamma$.

If our reasoning is correct, we see that the problem of the connectedness of the empty chamber is not settled by the linear homotopy argument. Perhaps the empty locus is even disconnected? How to approach the problem?

Let $m$ be some fixed even degree. Consider $\vert m H \vert$ the space of all real curves, and $\frak D$ be the discriminant parametrizing real singular curves. Denote by $I$ the [*invisible locus*]{}, consisting of all empty curves, and let $E$ be the [*empty locus*]{} consisting of all smooth empty curves. Obviously $E\subset I$. In fact $E=I - \frak D$. The linear homotopy argument shows that $I$ is connected (even convex in some sense), but a priori the hypersurface $\frak D$ could split $I$ in several pieces.

In general a hypersurface does not need splitting a manifold (consider e.g. a (pseudo)line in ${\Bbb R}P^2$ or a meridian/parallel in a torus). In our case $\partial I$ the boundary (or frontier) of $I$ consists primarily of solitary nodal curves (principal strata) and further subsequent lower strata. Hence clearly, $\partial I \subset \frak D$. But what about $\frak D \cap \overline{I}$? A priori this does not reduce to $\partial I$. One can imagine additional nappes of $\frak D$ moving inside $I$, or even that $\frak D$ contains spheroids (or other closed manifolds) not directly connected to the boundary $\partial I$ (cf. Fig.\[ShustinEmpty:fig\]d).

Extending our Brusotti argument to $k=m/2$ pairs of empty conics having transverse complexifications shows that for any even integer $m\ge 4$, there is an empty curve $\Gamma_m$ of degree $m$ with 2 conjugate nodes $p,p^\sigma$. Such a curve belongs to $\frak D \cap {I}$, i.e. is both singular and invisible. Can such a curve be connected to $\partial I$ by a path in $\frak D$, and hence to the “visible world” $V:=\vert mH \vert- I$. The answer would be yes if $\frak D$ is connected. The latter is a real algebraic hypersurface, a priori with several components.

Looking at the (singular) Riemann surface (Fig.\[ShustinEmpty:fig\]b), we can try to contract algebraically the anti-oval winding around the middle hole toward a solitary node. This would give a path as required. This sort of problem was already discussed at length in another strangulation section, yet we lack a serious procedure.

Assume now the opposite, i.e., $\frak D$ disconnected in the sense of having a component inside $I$. The linear homotopy argument shows that $I$ is convex in the sense that between any two of its points the projective line joining them has one half contained in $I$. It follows that $I$ is a contractible manifold! (Warning since Whitehead 1936 do not draw hastily that $I$ is homeomorphic to ${\Bbb R}^N$). Such manifolds (more generally those which are simply-connected, or even under weaker homological condition) are subsumed to Jordan-Brouwer separation. Under our supposition that $\frak D$ has some component inside $I$, it would result a separation of $I$ by $\frak D$. In fact quite independently of this supposition even if $\frak D$ is connected there is still a separation.

Of course we need some lemma extending Jordan separation caused by a manifold, to a separation caused by a stratified variety (not necessarily smooth). This looks true either by Anschauung (cf. Fig.\[Shustin2:fig\]d) or by a reduction to the manifold case by selecting adequately strata as to manufacture first a topological (but piecewise smooth) manifold out of the strata (Fig.\[Shustin2:fig\]e). Of course this should rest upon Brusotti’s description of the discriminant.

Whatever the method used we get a morcellation of $I$ by the discriminant. Quite ironically the linear homotopy argument (reminded by Shustin) seems to do exactly the opposite job than its primary intention. More precisely it implies that $I$ is contractible, hence subsumed to Jordan-Brouwer separation. On the other hand our Brusotti-type construction shows that $\frak D$ appears inside $I$ (provided $m\ge 4$), hence must divide the invisible locus. Modulo details, we believe to have proved the following:

[(Revolutionary if true, but false!)]{}.—For any even integer $m\ge 4$ the empty smooth locus (past the discriminant) is disconnected.

If true this would wash up several misconceptions in the literature, e.g. that the rigid-isotopy type of quartics is unambiguously determined by the real scheme (this would be false for the empty scheme). This rigidity is due to Klein 1876 [@Klein_1876_Verlauf] and well-known to Russian geometers (e.g. Rohlin 1978, Viro 1984, 1989, 2008, etc.) Likewise it would corrupt the same assertion in degree 6, which is included in Nikulin’s theorem (1979 [@Nikulin_1979/80]).

Let us again examine our argument to find our probable mistake. It decomposes in 3 distinct steps. Remember that $I$ denotes the invisible locus consisting of all curves having empty real locus.

\(1) Linear homotopy implies that the invisible locus $I$ is a contractible manifold.

\(2) The discriminant $\frak D$ is visible inside the invisible locus $I$ for $m\ge 4$. (This follows via simple application of Brusotti.)

\(3) Jordan separation holds true in a contractible manifold (or more generally a simply connected one). This can be proved in several ways, either by homology or directly by building a certain double cover out of the hypersurface by a polarization trick going back to Riemann (cf. e.g. Gabard 2011 [@Gabard_2011-Ebullition], arXiv, “Ebullition in foliated surfaces vs. gravitational clumping”).

[*Insertion*]{} \[06.04.13\].—At the risk of killing some dialectic suspense, there is a 4th issue namely the codimension of the discriminant inside the invisible locus, as not being 1 but 2 instead!

The step which looks most fallacious is Step (1). The reason could be the following. While there is between any two $m$-forms $P,Q$ a path $(1-t)P+tQ$ of $m$-forms (called the linear homotopy $P\to
Q$), and which sweeps out forms not representing zero if $P$ and $Q$ have the same never changing sign, it is not clear that given (invisible) curves $C,D\in I$ there is always a consistent choice of sign for representing forms ensuring a global retraction of $I$ to a point. Claiming this amounts finding a section of the evident (tautological) bundle.

Let ${\cal F}_m$ be the set of all forms (=homogeneous polynomials) of degree $m$. If we include the zero polynomial this becomes a vector space, with the space of $m$-tics $\vert m H
\vert$ being its projectivization. Denote by $$\pi\colon {{\cal F}}_m \to \vert mH \vert$$ the corresponding projection.

Choose a basepoint $D$ in $I$ (e.g. the class of the form $Q=
x_0^m+x_1^m+x_2^m$). (This is akin to Fermat’s equation $x^n+y^n=z^n$ except for lacking real points.) $Q$ has positive sign on ${\Bbb R}P^2$. Given any point $C\in I$ choose a representing form $P$ which has positive sign. Then the linear homotopy $h_P\colon P\to Q$ defined by $h_P(t)=(1-t)P+t Q$ stays in $I$, which projected down to $\vert m H \vert$ gives a path joining $C$ to $D$. This path is actually independent of the chosen representative $P$ of $C$ (by a variant of Thalès, alias linearity). Define now $$H\colon I \times [0,1] \to I, \quad H(C,t)=\pi (h_P(t) ).$$ This would be the required contraction (retraction to a point) showing that $I$ is contractible. However the subtlety is whether we can choose $P=s(C)$ continuously as a function of $C$. This amounts asking if $\pi $ (the tautological projection) admits a continuous section above $I$. Of course $\pi$ lacks a (global) section by looking at the fundamental group $\pi_1$ while using functoriality. Indeed the base of the fibration has $\pi_1={\Bbb Z}_2$, while the total space has trivial $\pi_1$.

Over the smaller subregion $I$ the situation is less obvious. Can one compute $\pi_1(I)$? If it is non trivial then we cannot find a section, and we are annoyed.

Can we construct a section geometrically? We can look at the counter-image $\pi^{-1}(I)$ interpreted as the cone of forms not representing zero (so-called [*anisotropic*]{} forms, if we remember well some highbrow arithmetical jargon[^55]). While on ${\cal F}_m$ the sign of a form is well-defined at a point, on $\pi^{-1}(I)$ it is well-defined globally. So our cone $C:=\pi^{-1}(I)$ splits in two components $C^+, C^-$, each being connected by the linear homotopy argument.

Now choose the hyperplane $\Pi$ through $Q$ which is orthogonal to $Q$ seen as a vector. This could give a section.

In fact both $C^+$ and $C^-$ are contractile, being actually starlike and even convex by the linear homotopy argument. Each of them is fibred by rays (semi-lines=orbits under scaling by the positive reals ${\Bbb R}_{>0}$) and the quotient of each of these cones by the multiplicative group of positive reals ${\Bbb
R}_{>0}$ is naturally identified with $I$.

Abusing geometric intuition we could nearly conclude that $I$ is contractible. Yet, this is not so evident as we lack a global cross-section, e.g. by cutting by a hyperplane selecting globally a point in each fibres. This looks hazardous, so let us concede some little algebraic détour or rather homotopy theory (presumably the quintessence of topology since Jordan 1865, Klein 1882 [@Klein_1882], Poincaré 1895, Dehn-Heegard 1907 who coined the term, and then Brouwer, H. Hopf ca. 1926–30, Hurewicz, Borsuk ca. 1935, J.H.C. Whitehead, who else?[^56]).

The exact homotopy sequence of a fibration ${\Bbb R_{>0}} \approx
fibre \to C^+\to I$ gives $$0=\pi_1(fibre)\to \pi_1(C^+) \to \pi_1(I) \to \pi_0(fibre)=0,$$ and implies that $\pi_1(I)=0$ is trivial. So there is no algebraic obstruction to find a section over $I$ (but algebra is never enough to ensure geometric existence!). Pursuing in that way with the exact homotopy sequence of a fibration of the early 1940’s (Hurewicz, Hopf[^57], Stiefel, Eckmann, Steenrod, G.W. Whitehead, Pontrjagin, etc.) we get $$0=\pi_i(fibre)\to \pi_i(C^+) \to \pi_i(I) \to \pi_{i-1}(fibre)=0,$$ and so $\pi_i(I)=0$ for all $i=0,1,2,\dots, \infty$ (modulo “nihil est infinito”!). Note that $I$ is connected being the image of the connected set $C^+$. Now our space $I$ is not a bad one (remember Viro’s talk “Compliments to bad spaces”). More precisely, $I$ is a manifold (being an open set in the manifold $\vert m H \vert\approx {\Bbb R}P^N$). This manifold $I$ is metric moreover, hence it has the homotopy type of a CW-complex in the sense of J.H.C. Whitehead[^58] (compare Hanner, Borsuk, Milnor 1959 [@Milnor_1959], Palais 1962, Gabard 2006/08 [@Gabard-2006/08]). By a theorem of J.H.C Whitehead 1949, it follows that $I$ is contractile.

—From the 1940’s (Ehresmann-Feldbau=Laboureur[^59]-Hopf-Stiefel-Pontrjagin), etc., any locally trivial fibration over a contractile base (which is paracompact[^60]) is globally trivial, hence admits a continuous section. Applying this to $C^+ \to I$ gives the required section permitting to contract $I$ via $H$. However all this optional remark is not really logically required.

This proves the following:

The space $I$ of invisible curves is contractible, and so it is separated by the discriminant $\frak D$ in several components. In particular the “chamber” of empty curves $E_m$ is never connected as soon as $m\ge 4$. So call it rather the empty locus. (Insertion \[06.04.13\].—This last clause is probably erroneous.)

The determination of the number of components $\iota={\rm card}
(E_m)$ is probably another pleasant game. (Let us guess that $\iota_4=2$, and $\iota_6=3$?)

This theorem (especially its second clause) contradicts nearly everything what has been said about the empty locus. It shows (despite being a pure existence proof using primarily the exact homotopy sequence of a fibration and Whitehead) that there are obstructions to rigid-isotopy lying beyond the pure optical level. It is of course a marginal contribution to Hilbert’s 16th problem, who primarily asked the right opposite extreme (isotopy classification especially of $M$-curves). Here we live in the opposite invisible part of the mushroom (Arnold’s prose) of what could be called (by analogy with Petrovskii 1933/38 [@Petrowsky_1933], [@Petrowsky_1938]) $m$-curves, where $m$ stands for Harnack “minimal” or minimalist artwork (empty locus like Mark Rothko’s monochromes[^61]). In some sense our result of disconnectedness is reminiscent (albeit different in method) to Marin’s disproof (1979 [@Marin_1979]) of the rigidity of $M$-schemes in degree 7. In both cases the real scheme fails determining unambiguously a chamber of the discriminant, and this in situations where there is no duplication by Klein’s types I/II (what Rohlin 1978 [@Rohlin_1978] calls schemes of indefinite type).

Simplifying the previous section: disconnection of the empty locus via Jordan separation and the exact homotopy sequence {#Disconnection-of-the-empty-locus:sec}
------------------------------------------------------------------------------------------------------------------------

\[27.01.13\] As the former section reflects our discovery process (as “meandering” as it may be) we prefer to keep its shape unchanged. Since our conclusion contradicts all what was asserted about the empty locus $E$ (especially Klein 1876, and Nikulin 1979), we shall here try to be more formal and direct, leaving aside historical considerations, and actually simplifying much the proof (in particular Whitehead’s criterion of contractibility via the vanishing of homotopy groups $\pi_i$ is not needed). \[28.01.13\] Our intention is to prove the following disconnection of the empty locus of plane curves of even order $m\ge 4$.

\[Gabard-anti-Klein-Rohlin:thm\] [(Gabard, 27.01.13, but certainly false)]{} The “empty locus” $E$ of all real smooth plane curves having empty real parts of some fixed degree $m\ge 4$ is disconnected. In other words there exists, for any even integer $m\ge 4$, two empty smooth curves of degree $m$ which are not rigid-isotopic.

We shall emphasize that this conclusion is quite unexpected. It seems to contradict much that has been said about rigid-isotopy of empty (smooth) curves. In particular, it is incompatible with the assertion going back to Klein 1876 [@Klein_1876_Verlauf] (see also Rohlin 1978 [@Rohlin_1978 p.96], or Viro’s surveys 1986, 1989, 2008) that the real scheme of a quartic curve determines uniquely its rigid-isotopy class. (More on this at the end of this section.) It conflicts also with Nikulin’s result (1979 [@Nikulin_1979/80]) that for sextic real curves the real scheme enhanced by the type data (I/II) of Klein (1876 [@Klein_1876]) suffices to determine the rigid-isotopy class.

Hence it is very likely that our theorem contains a serious misconception, either at the conceptual level of definitions, and if not so, there must be a bug in the proof below. In fact it is well known that the empty chamber is connected. Eugenii Shustin was kind enough to recall us the simple argument of linear homotopy between two forms of the same sign. This is supposed to show connectedness, yet exploiting it systematically we arrived ironically at the opposite conclusion. In part, this discrepancy is merely a matter of deciding what we like to call the empty locus. The linear homotopy argument shows connectedness of what we call the [*invisible locus*]{} $I$ (consisting of all empty curves), whereas by the [*empty locus*]{} $E$ we really mean the sublocus of $I$ consisting of smooth curves. The latter space is the more relevant one when it comes to problems of rigid-isotopy, where the game is to travel as much as we can while avoiding the discriminant (i.e., never strangulate the underlying Riemann surface).

To avoid any misunderstanding, let us fix our jargon more precisely. A [*ternary form*]{} is a homogeneous polynomial in three variables of some degree $m$. We shall only consider those with real coefficients, and call them [*real forms*]{}. A [*real plane curve*]{} is a homothety class of real forms under scaling of the coefficients. This is nothing else that what A. Weil would call a plane curve defined over ${\Bbb R}$. Denote by ${\cal F}_m$ the set of all real forms of degree $m$, and by $\vert mH \vert$ the space of all real curves of degree $m$ (the latter being merely the projectivization of the former). Denote by $\pi\colon {\cal F}_m \to \vert mH \vert$ the tautological projection which is an ${\Bbb R}^{\ast}$-bundle over $\vert mH
\vert\approx {\Bbb R}P^N$, where $N=\binom{m+2}{2}-1$.

A plane curve is [*smooth*]{} (or [*nonsingular*]{}) if the three partial derivatives of any defining form do not vanish simultaneously on ${\Bbb C}^3-\{0 \}$; else it is said to be [*singular*]{}. The set of all real singular curves forms the [*discriminant*]{} (hypersurface) denoted $\frak D$. Elimination theory (or better some counting argument) shows the latter set to be an algebraic hypersurface in the hyperspace $\vert mH \vert$ of all $m$-tics. Note that a singular real curve may well have a smooth real locus (in the sense of differential topology), yet it will then have conjugate pairs of singularities exchanged by ${\rm
conj}\colon {\Bbb C} P^2 \to {\Bbb C} P^2$, $(x_0,x_1,x_2)\mapsto
(\overline{x_0},\overline{x_1},\overline{x_2})$.

A real form is [*anisotropic*]{} if it does not represents zero (non-trivially), i.e. the sole real solution of the equation $P(x_0,x_1,x_2)=0$ is $(x_0,x_1,x_2)=(0,0,0)$. This is tantamount to emptiness of the real locus $C({\Bbb R})$ of the corresponding curve. Say in this case that the real curve is [*empty*]{} or [*invisible*]{}. Intersecting with any line defined over ${\Bbb
R}$, one sees that any odd degree curve has non-void real locus. Let $I$ be the set of empty (invisible) curves. This is nonempty iff $m$ is even, and $\pi^{-1}(I)=C$ is the cone of anisotropic forms. Such a form has a well-defined sign $\pm$, and accordingly the cone $C$ splits in two halves $C^+,
C^-$ invariant under ${\Bbb R}_{>0}$-scalings. (We overuse the letter $C$, for being the cone, or the curve but no confusion should arise.)

The proof of our (dubious) theorem (\[Gabard-anti-Klein-Rohlin:thm\]) decomposes in 3 short steps.

$\bullet$ [*Step 1*]{} (Simple-connectivity of the invisible locus $I$).—We consider the fibration $\pi\colon C^+ \to I$, whose base is the set of invisible curves, while the total space is the space of positive-definite anisotropic form. The fibre is the space ${\Bbb R}_{>0}$ of positive reals.

The space $C^+$ is convex. Whenever we choose 2 points in it, say $P, Q \in C^+$, the barycentric combination $(1-t)P+tQ$ for $t\in
[0,1]$ belongs to $C^+$. Accordingly $C^+$ is certainly contractile, and in particular [*simply-connected*]{}. (Perhaps $C^{+}$, being starlike, is even diffeomorphic to a genuine cell, but we do not need that presently. This follows perhaps from J.W. Alexander’s lemma on isotopy, ask L. Siebenmann or A. Marin?)

The first stage of the exact homotopy sequence of the fibering ${\Bbb R}_{>0}\approx F\to C^+\to I$ reads $$0=\pi_1(fibre)\to \pi_1(C^+) \to \pi_1(I) \to \pi_0(fibre)=0,$$ and it follows that the space $I$ is also simply-connected.

$\bullet$ [*Step 2*]{} (Construction of invisible curves with singularities).—By a simple application of Brusotti 1921 [@Brusotti_1921], it is easy to construct invisible real curves $C_m$ of degree $m\ge 4$ having a pair of conjugate nodes, i.e. ordinary double points (cf. lemma below for details). (Abstractly, from the Riemann complexification viewpoint, imagine a pretzel acted upon by antipody with two handles strangulated to a pair of points $p, p^{\sigma}$ exchanged by conj.)

$\bullet$ [*Step 3*]{} (Jordan-Brouwer separation of the invisible locus $I$ by the discriminant ${\frak D}$).—Paraphrasing Step 2 in our notation, this means that $\frak D \cap I$ is nonempty. The space, we are really interested in, is the empty locus $E$ consisting of all smooth empty curves. By definition we have $E=I-{\frak D}$. So the empty locus $E$ arises from the invisible locus $I$ by removing a certain real algebraic hypersurface (or at least the portion ${\frak D} \cap I $ visible in $I$). Since the manifold $I$ is simply-connected (Step 1), it follows from the Jordan-Brouwer separation theorem that $\frak D \cap I$ disconnects $I$. We conclude that the residual set $E=I-{\frak D}$ has at least 2 components whenever $m=2k\ge 4$.

We now make more explicit the lemma required in Step 2 (for a schematic picture in the case $m=4$, cf. Fig.\[ShustinEmpty:fig\]a.):

\[Brusotti-binodal-invisible-curves:lem\] Given any even integer $m\ge 4$, there exists a degree $m$ real plane curve which is invisible (empty real locus) with $2$ ordinary nodes exchanged by complex conjugation.

Take a collection of $k=m/2$ real conics $E_1, \dots, E_k$ (degree 2) each having empty real locus such that the union of their complexifications has only normal crossings (ordinary nodes). By Brusotti’s theorem (1921 [@Brusotti_1921]), we may smooth away from $E_1\cup \dots\cup E_k$ all pairs of conjugate points safe one $p, p^{\sigma}$, where $\sigma={\rm conj}$ is complex conjugation. (Since $k\ge 2$, there is at least 4 nodes on our configuration of ellipses by Bézout.) The resulting binodal curves is real, invisible (being manufactured by small perturbation of an invisible curve). The proof of the lemma is complete.

In Step 3 of the proof of Theorem \[Gabard-anti-Klein-Rohlin:thm\], we use a slightly extended form of Jordan separation imposed by a variety (possibly singular) and not just by a manifold. In fact we may imagine that the structure of the discriminant permits one to deduce a sublocus of ${\frak D} \cap I$ which is a genuine topological manifold, yet piecewise smooth (nothing so crazy as Bing-Casson-Freedman). This would involve aggregating suitably some principal strata of the discriminant exploiting perhaps Brusotti’s description of the latter, or just general properties of algebraic sets.

It also conceivable that there is a direct proof by applying directly the homological apparatus involved in Jordan separation to the case of an algebraic hypersurface. Probably this is already implemented somewhere (maybe by H. Kneser, Bieberbach, Whitney, Thom, Milnor, Tognolli, Marin, Bochnak-Coste-Roy, etc.). Alas we do not know a more precise reference. Evidently the proof of separation within the simply-connected locus $I$ should just use some very basic properties of real algebraic hypersurfaces. Crudely speaking a real algebraic hypersurface cannot “stop” like a manifold with boundary (via the implicit function theorem), and so really effects a separation in the large (at least within the simply-connected subregion $I$). This elementary property of real algebraic variety was known for long (e.g. when Zeuthen 1874 [@Zeuthen_1874] speaks of a “branche complète”, etc.)

Rigid isotopy of quartics: classical sources (Schläfli, Zeuthen 1874, Klein 1873–76, Rohlin 1978) {#Klein-rigidity-of-quartics:sec}
-------------------------------------------------------------------------------------------------

\[28.01.13\] This section discusses in some more details some masterpieces of classical literature conflicting strongly with our conclusion (Theorem \[Gabard-anti-Klein-Rohlin:thm\]).

A first place is Rohlin 1978 [@Rohlin_1978 p.96], who ascribes the rigid-isotopy classification for $m=4$ to Klein, while writing the following:

“§4. Isotopy.—4.1. The classical problem. By virtue of the definition of a real plane projective algebraic curve of degree $m$, such curves form a real projective space of dimension $m(m+3)/2$. Singular curves, that is, curves with real or imaginary singularities, fill out in this space a hypersurface of degree $3(m-1)^2$, and non-singular curves fill out the complementary open set, which splits into a finite number of components[^62]. It is clear that curves that belong to one component have the same real scheme, that is, the class of all non-singular curves with a given real scheme consists of whole components. The investigation of these components is a very old problem, like the investigation of the classes themselves. It was known more than hundred years ago that for $m\le 4$ the components coincide with the classes[^63] (the least trivial case $m=4$ was considered by Klein; see \[4\](Klein 1922=Klein 1876 [@Klein_1876_Verlauf]), p.112). From the results of the previous section it follows that for $m\ge 5$ this is not so; for the complex scheme is constant on each component, but is capable of changing within one class for $m\ge 5$. \[…\]”

As usual Rohlin quotes Klein’s GMA=Ges.Math.Abhandl. (1922), yet the original source is the 1876 paper “Über den Verlauf der Abelschen Integrale bei den Kurven vierten Grades” [@Klein_1876_Verlauf]. Klein’s prose is as usual quite magical (like that of Rohlin), and reads as follows:

“ Eine wesentliche Eigenschaft dieser Einteilung der Kurven vierter Ordnung in sechs Arten ist in dem folgenden Satze ausgeschprochen, der weiterhin eine fundamentale Bedeutung für die Tragweite unserer Untersuchungen gewinnt:

[*Von jeder allgemeinen[^64] Kurve vierter Ordnung kann man zu jeder anderen, die derselben Art angehört, durch allmähliche reelle Änderung der Konstanten übergehen, ohne da[ß]{} bei dem Übergangsprozesse Kurven mit Doppelpunkt oder gar allgemeine Kurven, die einer anderen Art angehören, überschritten zu werden brauchten.*]{}

Ein direkter Beweis dieses Satzes hat keine Schwierigkeit[^65], aber er ist weitläufig. Es soll hier um so mehr Abstand genommen werden, als die bei ihm nötig werdenden Betrachtungen mit diejenigen, die im gegenwärtigen Aufsatze zu entwicklen sind, wenig Beziehungspunkte haben. \[So roughly Klein says that there is little connections between rigid-isotopy and Abelian integrals!\] Dagegen sei angedeutet, da[ß]{} man ihn vermöge kurzer Zwischenbetrachtungen führen kann, wenn man auf frühere Untersuchungen von Zeuthen und mir zurückkgreift. Ich habe \[in Abh. XXXV, S.24, 25\] gezeigt, da[ß]{} ein ähnlicher Satz gilt für die fünf Arten, welche man nach Schläfli bei den allgemeinen Flächen dritter Ordnung zu unterscheiden hat. Es hat dann Zeuthen bewiesen (Math. Ann., Bd.7 (1874), S.428), da[ß]{} die Arten der Kurven vierter Ordnung den fünf Flächen Arten in sehr einfacher Weise entschprechen. Projiziert man die $F_3$ von einem ihrer Punkte aus stereographisch auf eine Ebene, so tritt als scheinbare Umhüllung bei den Arten I, II, III, IV von Schläfli eine vierteilige, drei-, zwei-, einteilige Kurve vierter Ordnung auf. Die Art V ergibt, bei analoger Konstruktion, je nachdem man den Projektionspunkt auf ihrem unpaaren oder paaren Teile annimmt, die Gürtelkurve oder die imaginäre Kurve. Umgekehrt kann auch jede Kurve vierter Ordnung aus der entschpechenden Flächenart in der angegebenen Weise gewonnen werden. Hierin liegt der vor uns gewünschte Beweis. Um ihn völlig zu führen, hat man nur noch die Modifikationen zu untersuchen, welche die scheinbare Umhüllungskurve erfährt, wenn der Projektionspunkt auf der fest gedachten Fläche beliebig verschoben wird. Aber auch dieses hat Zeuthen ausgeführt \[Études des propriétés de situation des surfaces cubiques; Math Annalen, Bd.8, (1874/75).\]”

Let us summarize Klein’s proof of the rigid-isotopy of $C_4$’s: first Schläfli in 1863 [@Schlaefli_1863] found five isotopy class of real cubic surfaces $F_3$, and Klein showed them to be rigid-isotopic (in Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn]). Then Klein exploits the yoga of Zeuthen (which goes back to Geiser, compare Zeuthen 1874 [@Zeuthen_1874 p.428]) of looking at the apparent contour of the $F_3$ (projected from a point on the surface) to get a $C_4$ (all of them arising so). In fact the $F_3$ with 2 components produces both the Gürtelkurve or the empty $C_4$ depending on whether the center of vision is located on the pseudo-plane or on the spherical component. This is easily visualized if one imagine $F_3$, a small perturbation of a sphere union an equatorial plane (Fig.\[Zeuthen-Klein:fig\]). Note that the empty apparent contour arises from a phenomenon of total reality of the bundle of lines through the spherical component (this being again reminiscent of “Ahlfors”).

-5pt0

-5pt0

So a smooth $F_3$ with a marked point gives rise to a $C_4$, and all quartics arise so. One must still study the rôle of the marked point, and there is not just the five classes of Schläfli-Klein but one more due to the marking (being either on the pseudo-plane or the spheroid). Hence Klein’s argument looks quite convincing but we should understand the details more precisely. Maybe there is little gap in this proof when setting up explicitly the $F_3\leftrightarrows C_4$ correspondence. Is the contour apparent of a smooth $F_3$ always a smooth $C_4$ (and viceversa a singular cubic to a singular quartic), so that the Zeuthen-Klein correspondence really sets up a dictionary between the corresponding discriminants (hence rigid-isotopy classes). In particular to what sort of $F_3$ corresponds the empty quartic with a pair of conjugate nodes. Those are the essential guys which in our Theorem \[Gabard-anti-Klein-Rohlin:thm\] causes the disconnection of the invisible locus $I$.

Further as the center of projection is moving in 3-space the ZK-correspondence is somewhat non-canonical, i.e. there is not a fixed ${\Bbb P}^2$ on which to project. So the reduction proposed by Klein is perhaps foiled somewhere, at least requires to be modernized (and detailed) seriously. Naively our Theorem \[Gabard-anti-Klein-Rohlin:thm\] (if correct) could be an obstruction to completing Klein’s proof. If the ZK-correspondence is sound, it could be that Klein’s 1873 rigid-isotopy classification of $F_3$’s (cubic surfaces) is foiled.

Fixing the paradox
------------------

\[29.01.13\] Can it be that the discriminant $\frak D$ while penetrating inside the invisible locus $I$ appears there with (real) codimension 2 hence without separating $I$?

Recall that real loci of algebraic hypersurfaces (=primals in old British jargon, e.g. Semple-Roth) may look anomalously small. For instance a solitary node on a plane cubic curve is merely an isolated point of real dimension $0$ (hence real codimension $2$).

This phenomenon could foil Step 3 of the proof of Theorem \[Gabard-anti-Klein-Rohlin:thm\]. However via Brusotti description of the discriminant one could still hope that ${\frak
D} \cap I$ has real-codimension $1$. Naively the principal stratum corresponding to a curve with a conjugate pair of nodes looks at first of codimension 2 because there is two nodes. However by the reality condition one of them is forced and so it is really one closed point in the sense of Grothendieck’s schemes. Without Grothendieck such arguments of reality counting also abound in Klein, e.g. when it comes to coverings of the Riemann sphere with complex conjugate ramification. Here a complex ramification point count for 2 real dimensions, while a real branch point affords one freedom parameter. Yet under the symmetry condition both cases actually contribute to the same. It is with this sort of argument that Klein managed to compute the dimension of the moduli space of real curves by aping what did Riemann over the complexes.

Alas one can argue that the stratum of the complexified discriminant $\disc(\CC)$ with two nodes $x,y$ nearby $p,p^\sigma$ has geometric codimension 2 over the complexes, and it would follow that the real locus $\frak D$ has at most real codimension 2 when $x,y$ lye symmetric under $\sigma=$conj. Another way is to start the dimension count of the discriminant from the scratch. So we look at plane curves with a node or a higher singularity marked on it. This gives an incidence variety $(C,p)$ consisting of curves $C$ singular at $p$. Saying that $p$ is singular of $C$ amounts (via the Euler relation) to say that the 3 partials of a defining equation vanish at $p$, yielding 3 linear conditions on the coefficients. So looking at the projections $p\leftarrowtail
(C,p) \mapsto C$ we see fibres of codimension $3$, while moving the point $p$ gives codimension $1$ (hypersurface) in the space of curves.

Adapting this dimension count near our binodal empty curve $C$ shows that we have a pair of projections $p\leftarrowtail (C,p,
p^{\sigma}) \mapsto C$. Imposing a singularity at $p$ gives 3 linear conditions, but the point $p$ being any imaginary point of the plane its location depend upon 2 complex parameters (4 real parameters). When $C$ is defined over ${\Bbb R}$ the 3 equations for the partials are again linear (in the coefficients), but involve complex constants. Thus splitting into real and imaginary parts yields twice so many linear equations, hence 6 of them. Those being satisfied then $p^\sigma$ is also a node by symmetry, and so the variety ${\frak B}$ of (binodal) curves having a conjugate pair of nodes has real (co)dimension $4-6=-2$ in the space of curves. In that case it seems that the principal stratum ${\frak D}\cap I$ consisting of empty curves with a pair of conjugate node has also real codimension 2 in $\vert mH \vert$, and this would foil our theorem (\[Gabard-anti-Klein-Rohlin:thm\]).

For short let us call ${\frak D}\cap I$ the [*invisible discriminant*]{}, abridged $I$-discriminant. The above argument has to be polished by checking that the 6 linear equations are really independent conditions. The main issue is therefore to calculate the dimension of the invisible discriminant. If it has codimension 2 then $E=I-\frak D$ is connected, and so Klein-Rohlin were right. (Otherwise, if of codimension $1$, then it separates and Klein-Rohlin were wrong.)

If the $I$-discriminant has codimension 2, it is like a knot, and it seems interesting to compute its fundamental group (of its complement), and to look at the Picard-Lefschetz monodromic transformation arising when winding once around a meridian of this $I$-discriminant. Before adventuring we should first solve (more rigorously) the dimension problem of the $I$-discriminant. This must surely be done in Brusotti 1921 [@Brusotti_1921] (and known to Gudkov, maybe in the 1974 survey [@Gudkov_1974/74]). Perhaps this was already known in the era of Zeuthen-Klein-Harnack-Hilbert.

The invisible discriminant has codimension $2$ {#invisible-discriminant-codim-2:sec}
----------------------------------------------

\[29.01.13\] The goal of this section is to resolve our paradox, that the locus of empty smooth curves is disconnected (violating thereby well assessed knowledge of Klein, Rohlin-Nikulin, etc.). It seems that our sole mistake was based on the linguistical misconception of thinking that the discriminant-hypersurface is a hypersurface (throughout)! Names and terminologies are often misleading in mathematics. In fact we shall try to convince that inside the invisible locus (of curves with empty real loci) the discriminant has only codimension 2, hence too small to effect any Jordan-Brouwer separation. Once this is observed this raises some little questions about knowing which chambers residual to the principal strata of the discriminant (where it has really of codimension 1) contains such smaller strata of codimension 2. In more geometric terms, this amounts essentially deciding which smooth curves can acquire a pair of conjugate nodes.

Fix some even integer $m=2k$, and consider only real curves of fixed degree $m$. Let $\vert mH \vert\approx {\Bbb R}P^N$ be the corresponding parameter space of real $m$-tics. In this space we pay special attention to the space $I$ of [*invisible curves*]{} (those with empty real locus). This is clearly an open set in $\vert mH \vert$.

Let $\frak B$ be the variety of invisible real plane curves with at least one singular point (hence necessarily at least a pair thereof). We have $\frak B = {\frak D}\cap I$, the so-called [*invisible discriminant*]{} (abridged $I$-discriminant).

\[invisible-discriminant-codim-2:lem\] The $I$-discriminant has (real) codimension $2$ in the hyperspace of all curves.

Consider the incidence relation $B=\{ (C,p) \colon C \in I, p \in
Sing C \}$. We have natural projections $${\frak B}\buildrel{\pi_1}\over\longleftarrow B
\buildrel{\pi_2}\over{\longrightarrow} {\Bbb P}^2({\Bbb C}).$$ First study the fibre of the second projection $\pi_2\colon
(C,p)\mapsto p$ . This amounts to look at all curves having a prescribed singularity at $p$ an imaginary point. This imposes 3 linear equations (vanishing of the 3 partials, which suffices by Euler equation for the point to be on the curve). Splitting in real and imaginary parts gives 6 linear conditions, which looks linearly independent. So the fibre $\pi_2^{-1}(p)\approx (I \cap
{\Bbb R}P^{N-6})\times\{p\}$. Since $\pi_2$ is surjective onto ${\Bbb P}^2({\Bbb C})-{\Bbb P}^2({\Bbb R})$ it follows that the real dimension of $B$ is $\dim_\RR B= 4+(N-6)=N-2$. As the first projection $\pi_1$ is generically 2-to-1 (or finite-to-one except over special curves with multiple irreducible components), it follows that $\dim_\RR {\frak B}=N-2$, hence of codimension 2 in $I$ (or in $\vert m H \vert$).

More generally, imagine a visible curve (i.e. $C_m(\RR)\neq
\varnothing$) acquiring a conjugate pair of nodes. This will not affect the real scheme (=soft isotopy class of the embedding $C_m(\RR)\subset \RR P^2$). A priori inside each “class” can penetrate a portion of the discriminant of codimension 2.

Let us be more formal. Split the discriminant $\disc=\disc^+\sqcup \disc^{-}$ in two parts depending on whether $Sing C$ contains a real point or not. Precisely define $\disc^{-}$ as the set of singular curves lacking real singularities.

The argument of the above lemma shows that $\disc^{-}$ has real codimension two. We call it hence the [*hypo-discriminant*]{}. The set $\disc^+$ is defined as its complement, i.e. $\disc^+=\disc- \disc^{-}$. The latter has codimension 1 by a variant of the above argument. (Indeed a real singularity imposes 3 linear conditions, but moving the point create 2 dimensions, whence the defect of $-1$.) We call $\disc^+$ therefore the [*hyper-discriminant*]{}.

The assignment $\vert mH \vert -\disc \to \frak S$ of the real scheme to a non-singular equation is more generally defined on the larger space $\vert m H\vert-\disc^+$ residual to the hyper-discriminant, while being locally constant there. Crudely put, in problems of rigid-isotopies the hypo-discriminant can be neglected (being only of codimension 2, hence effecting no additional separations). However when we would like not only to study the connectivity of the chambers but also their topology then the hypo-discriminant ought to be considered again.

A first question is whether any chamber residual to the hyper-discriminant (abridged [*hyper-chamber*]{}) intersects the hypo-discriminant. (This is true for the empty chamber residual to $\disc^+$, by our Brusotti-style lemma \[Brusotti-binodal-invisible-curves:lem\].)

The general problem looks again to involve a contraction principle of Riemann surfaces, now under a symmetric pair of vanishing cycles. For $M$-curves, there seems to be a topological obstruction, since strangulating two imaginary cycles $\beta,
\beta^{\sigma}$ causes a disconnection of the Riemann surface in 2 algebraic pieces $C_m\to C_k\cup C_l$ of degree $k, l$ (hence cutting themselves in $k\cdot l$ points by Bézout). But $C_k,
C_l$ cuts transversally in $2$ points only, hence $k=2$, $l=1$ (up to renumbering). Hence, this eventuality can only occur for $m=3$ (cubics), where it does occur when an $M$-cubic degenerates to $E_2\cup L$ a conic union a disjoint line. This curve $E_2\cup L$ belongs to the hypo-discriminant since it lacks real singularities. (\[06.04.13\] Further the dimension of such split cubics is $5+2=7$, which is indeed of codimension 2 in the hyperspace of cubics of dimension $\binom{3+2}{2}-1=\frac{5\cdot
4}{2}-1=9$.)

Given a curve in the hypo-disc $\disc^-$, it seems likely that by genericity we may assume the latter to be a binodal curve with a conjugate pair of nodes. By Brusotti 1921 [@Brusotti_1921], smooth them away. Interpreting the process backward in time we see a pair of imaginary conjugate vanishing cycle $\beta,
\beta^{\sigma}$ strangulating toward the nodes $p,p^{\sigma}$. If this argument holds true we see that each hypo-discriminantal component gives rises to a bistrangulation along imaginary cycles. In particular:

Safe for $m= 3$, the hyper-chamber of an $M$-curve of degree $m$ never contains the hypo-discriminant.

Perhaps this is the sole obstruction, in the sense that any other hyper-chamber (than those of $M$-curves) intersects the hypo-discriminant. In fact there is perhaps still such an obstruction for $(M-1)$-curves. Naively the latter look like an $M$-curve safe that one “oval” is masked. Still if we imagine the corresponding symmetric Riemann surface it seems that a pair of imaginary cycles must divide.

\[30.01.13\] So we are led to the following general topological question:

\[bicycle-existence:ques\] (Existence of bicycles).—[Given a symmetric surface in the sense of Klein 1876 (i.e. an oriented closed surface $X$ with an orientation reversing involution $\sigma$), when is it possible to find a pair $\beta,\beta^{\sigma}$ of imaginary cycles(=Jordan curves) such that $\beta \cup \beta^{\sigma}$ does not divide $X$?]{} Here imaginariness means that $\beta$ has no point fixed under $\sigma$. It is also required that $\beta$ is disjoint from its conjugate $\beta^{\sigma}$. We call such a pair an (imaginary) [*bicycle*]{}.

As well-known since Klein 1876 [@Klein_1876], symmetric surfaces are either ortho- or diasymmetric (equivalently dividing or not, or of type I resp. II). The type together with the number $r$ of “ovals” (pointwise fixed circuit) and the genus $g$ fixes the equivariant topology of a symmetric surface.

As to our question it is plain that in the type I (dividing) case then there is a bicycle provided the surface is not maximal $r=g+1$ (“$M$-surface”). In the maximal case such a bicycle is not available, because the quotient $X/\sigma$ is planar hence schlichtartig (i.e. divided by any Jordan curve). Literally “schlichtartig” means planar like.

Incidentally recall the implications:

\[Gabard-five-lemma:lem\] “simply-connected$\Rightarrow$schlichtartig$\Rightarrow$ orientable” for a topological surface.

Exaggerating a bit the only rigorous proof, we are aware of, uses the five lemma and homology, cf. e.g. Gabard-Gauld 2011 [@Gabard-Gauld_2011-Dynamics Lemma 4.17], Dynamics of non-metric manifolds. It works universally without having even to assume the surface metrizable. Hausdorffness is however crucial (consider a branched plane and in it a Jordan curve on the upper sheet, then you can evade via the lower sheet so as to reach the outside of the upper sheet.)

The five lemma implies that when we have a Jordan curve $J\subset
U \subset M$ included in two nested spaces, then if it divides the large space $M$, it must divide the small one $U$, and viceversa provided $H_1(U)\to H_1(M)$ is onto. This latter fact prompts the first implication “$\Rightarrow$” by taking $U$ the tubular neighborhood of $J$ (which must be trivial since otherwise there would be an indicatrix-reversing loop violating the assumption $\pi_1(M)=0$). The second implication “$\Rightarrow$” follows from the first fact, namely a division in the large $M$ implies a division in the small $U$, hence is particular of the tubular neighborhood which must therefore be trivial.

When applied to the quotient of a symmetric surface, Lemma \[Gabard-five-lemma:lem\] gives for the latter: “ortho-sphere$\Rightarrow M$-surface$\Rightarrow$ dividing”.

Note also that if there is a bicycle on $(X,\sigma)$, then its projection in the quotient $\bar X=X/\sigma$ is a cycle $\bar
\beta$ interiorly traced which does not divide and preserves the indicatrix (=local orientation). (The non-division just follows from the fact that the image of the connected set $X-(\beta \cup
\beta^{\sigma})$ has to be connected.)

It remains to answer our question (\[bicycle-existence:ques\]) in the diasymmetric case. The lowest diasymmetric case $r=0$ is easy since then $(X,\sigma)$ may be visualized in 3-space as a pretzel invariant under central symmetry (antipody). Hence a bicycle exists provided the genus $g\ge 2$. (For $g=1$ there is a pair of cycle exchanged, but collectively they do divide. It cannot be otherwise by Riemann’s definition of the genus.)

For the other cases there are several models. One way due to Klein-Weichold-Kervaire (private communication of the latter in 1999) amounts to look at a Möbius band embedded in 3-space (make holes in it) and look at a thickening of the normal bundle of thickness vanishing along the boundary (Fig.\[Kervaire:fig\]a).

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Alternatively, we may start from the Harnack-maximal case visualized as a planar membrane with $r=g+1$ contours (Fig.\[Kervaire:fig\]b), and kill successively contours by cross-capping them (à la von Dyck 1888 [@von-Dyck_1888 p.479], another of Klein’s student). This operation does not alter the Euler characteristic $\chi$, and so keep the genus of the double unchanged as $\chi (X)=2 \chi
(X/\sigma)$. The symmetric surface is constructed abstractly via the usual process of the double orientation cover (without duplication of the boundary points by local orientations).

So imagine a disc with $g$ holes while cross-capping them successively. If no cross-cap we have an $M$-curve[^66], if one cross-cap an $(M-1)$-curve, if two cross-caps an $(M-2)$-curve of type II, etc. As soon as there is $2$ cross-caps, connect them by a path and closing it back (Fig.\[Kervaire:fig\]d) gives a cycle $\bar \beta$ which preserves the indicatrix (as it traverses twice the cross-caps) and which does not divide $\bar X$ (because its apparent inside is connected with the outside via the cross identifications). Lifting $\bar \beta$ to $X$ gives the desired bicycle. Another, but slightly weaker argument, is that as soon as there is 3 cross-caps available, they can be traded against one cross-cap and one handle (as both contribute identically to the Euler characteristic). Then it is enough to take the meridian (or parallel) of that handle (Fig.\[Kervaire:fig\]e).

The case of where there is only one cross-cap is a bit more tricky, and it seems that we cannot find a nondividing cycle $\bar\beta$ which is indicatrix-preserving. (Trace a picture which can be either like a figure “8” crossing the cross-cap (Fig.\[Kervaire:fig\]f) or like a figure “$\omega$” with extremity linked together (Fig.\[Kervaire:fig\]g). This is abstractly just the figure 8, except that the one loop envelopes the other one. In both cases it is seen that a division is produced.)

Here is the obstruction:

An $(M-1)$-surface (with $r=g$) cannot have a bicycle.

Since the bicycle $\beta\cup \beta^{\sigma}$ does not divide the surface $X$, its image $\bar \beta$ in the quotient $\bar X$ does not divide it. The covering $X-{\rm Fix}(\sigma)\to \bar X-\partial \bar X$ restricted to the complement of $\bar \beta$ shows that $X-{\rm Fix}(\sigma)-(\beta \cup \beta^{\sigma})$ has at most $2$ components (exchanged by $\sigma$). Yet it suffices to add one oval (of ${\rm Fix}(\sigma)$) to make it connected. Hence we have $(g-1)+2=g+1$ retrosections not disconnecting the surface, violating Riemann’s definition of the genus.

If such a $\bar beta$ existed then lifting it to the symmetric surface would give 2 cycles $\beta,\beta^{\sigma}$ on $X$ of genus $g$. Since $\bar\beta$ does not divide $\bar X$, the covering $X-{\rm Fix}(\sigma)\to \bar X-\partial \bar X$ shows that $X-{\rm Fix}(\sigma)-(\beta \cup \beta^{\sigma})$ is still connected since it is the orienting cover of the connected surface $(\bar X-\partial \bar X)-\bar \beta$ which is still non-orientable. If orientable then the double orientation cover would be trivial on both pieces hence on their union.

In conclusion we exhibit $(g-1)+2=g+1$ cycles on $X$ the surface of genus $g$ not disconnecting it. This overwhelms Riemann’s definition of the genus.

In summary we have proven:

A symmetric surface admits a bicycle iff $g\ge 2$ and $r\le g-1$. In other words iff it is not an $M$-surface nor an $(M-1)$-surface.

Via Brusotti this seems to afford obstructions to the presence of the hypo-discriminant in certain hyper-chambers. More precisely:

Inside a pre-maximal hyper-chamber (i.e. $r\ge g$) the hypo-discriminant is vacuous. In particular a premaximal curve (i.e. an $M$- or an $(M-1)$-curve) cannot acquire a conjugate pair of nodes by continuous variations of its coefficients (among smooth curves safe for the extremity). \[Warning: the case $m=3$ is the sole exception.\]

More risky is the (converse) assertion that via a suitable (but very hypothetical) contraction principle the topological presence of a bicycle suffices to create an algebraic deformation toward a curve $C_m$ with an imaginary pair of nodes. If optimistic about the freedom of the joy-stick this supports the:

\[hypo-discriminant:conj\] Any (smooth real plane) curve $C_m$ which is not premaximal can acquire a conjugate pair of nodes (bi-node) via continuous deformation among smooth curves safe for its extremity. In particular the hypo-discriminant appears in all the corresponding (“ante-maximal”) hyper-chambers.

This is another large-deformation principle a bit akin to the Itenberg-Viro contraction conjecture (\[Itenberg-Viro-contraction:conj\]) for empty ovals. To get serious prohibitions one would perhaps even require a strengthened collective form of it. For instance if $r=2$ and if we are dividing, we could contract several bicycles which collectively split the Riemann surface. Browsing through increasing degrees $m=3,4,5,6,7, \dots$ gives the genus $g=1,3,6,10,15,\dots$. By virtue of Klein’s congruence $r\equiv_2 g+1$ we look especially at $m=4$ or $m=7$ (or $m=3k+1$). When $m=4$, we find no obstruction to the splitting (since $g=3$ and we have 4 vanishing cycles contracting to $4=2\cdot 2$ points in agreement with Bézout, see Fig.\[Pretzel:fig\]a). If $m=7$, then $g=15$ and so there is $8$ bicycles (Fig.b) which strangulated toward nodes gives 16 (simple) intersections between both pieces of the degeneration $C_7\to C_k\cup C_l$, where $k+l=7$. Testing all values $(k,l)=(1,6),(2,5),(3,4)$ gives always the wrong number of intersections $k\cdot l=6,10,12$ never equal to $16$. This contradiction with Bézout reproves that a dividing septic cannot have $r=2$. (Of course this is best proved as a consequence of Rohlin-Mishachev’s formula).

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Extending this to all degrees requires another form of contraction. In the case of $C_5$’s: then assuming $r=1$, there is 3 bicycles and 1 ortho-cycle (Fig.c). We have a splitting $C_5\to C_k\cup C_l$, and as $g=6$ we have 7 intersections in $C_k\cap C_l$ after strangulation of the Riemann surface. But this is never equal to $k\cdot l$ for $(k,l)=(1,4), (2,3)$. So this would prove again that a quintic with one circuit ($r=1$) cannot be dividing. Compare (\[Klein-Marin-quintic:lem\]) for another proof.

[*Insertion*]{} \[06.04.13\].—Applying the same method to a quintic with $r=3$, while imagining the underlying Riemann surface orthosymmetric and spliced by 2 bicycles and 1 orthocycle (Fig.d), the strangulation process leads to 2 algebraic pieces $C_k\cup C_l$ intersecting in 5 points. This is again not of the form $k\cdot \ell$ (equal as above to $4$ or $6$). This reasoning proves that a quintic with $r=3$ cannot be dividing, which is [*nonsense*]{} (remind the deep nest and its total reality). So the methodology (of such imaginary contractions) appears jeopardized. It seems at first that by “rotating” the pretzel of genus $6$ as to make the five cycles into reals circuit prompts a corruption of Itenberg-Viro (\[Itenberg-Viro-contraction:conj\]) by the same device. This is not so because one of the circuit is a pseudoline (since we are in degree $m=7$ odd). But of course we could imagine an example in even degree. In fact what protects a direct corruption of the Itenberg-Viro conjecture (\[Itenberg-Viro-contraction:conj\]) is Rohlin’s formula which in case of no-nesting forces $r=k^2$, and intersecting the half of the strangulated Riemann surface (now of degree $k$ since exchanged by Galois) is concomitant with Bézout.

More generally, one can perhaps by this method get another derivation of the Rohlin-Marin inequality $r\ge m/2$ for a plane dividing curve of degree $m$.

One could also imagine more radical degeneration by a bicycle $\beta, \beta^{\sigma}$ such that already $\beta$ divides $C_m({\Bbb C})$ (fig.f). A such is easy to visualize in the dividing case and would separate all imaginary handles from the real contours. However in the case of a $C_4$ of type I, such a pair $\beta,\beta^\sigma$ would strangulate the surface of genus $3$ in three pieces of genus 1, so the degree must be at least $3+3+3=9$, which is much greater than $4$. Hence such contractions are unlikely to exist algebraically.

\[31.03.13\] When $m=4$, the above conjecture (\[hypo-discriminant:conj\]) looks trivial, e.g. because all ante-maximal schemes $r\le M-2=2$ admits realization as pair of conics (either nested or disjoint).

Rigidity index
--------------

\[31.03.13\] Another naive remark concerns the rigidity of the “one-oval scheme” $1$. Once the empty scheme is known to be rigid, then via the contraction conjecture CC (\[Itenberg-Viro-contraction:conj\]), the one-oval scheme ought to be rigid as well. Naively via CC one could pursue inductively and all schemes would be rigid (which is not true as best and first shown by Rohlin via Klein’s type for $m\ge 5$). So there are subtle obstructions coming from separation between chambers at the next level. Despite such difficulties we call this method the [*rigidification procedure by reduction to the empty chamber*]{}.

So we start from the empty chamber $0$, and then there is the chamber $1$ (which should be still connected). Then “attached” to this there is the “chamber” $\frac{1}{1}$ and $2$, etc. Of course here “chamber” should rather be “isotopy class” and it should be proved that such schemes are rigid, i.e. that their respective isotopy classes correspond to a unique chamber of the discriminant.

For a fixed integer $m$, the smallest integer $r=r(m)$ such that all real schemes of degree with less than $r$ ($\le r$) real branches are rigid, is called the rigidity index in degree $m$.

As usual in mathematics, when we are unable to prove something we just introduce a:

[The [*bifurcation index*]{} $r(m)$ in degree $m$ is the smallest integer $r=\rig(m)$ such that there is a real scheme of degree $m$ with $r$ real branches which is non-rigid, i.e. represented by 2 real curves of degree $m$ which are not rigid-isotopic. It is set equal to $+\infty$ if all schemes are rigid. If finite, and diminished by one unit it could be called the [*rigidity index*]{}, since below it all schemes would be rigid. (Our terminology is a bit awkward, because a high rigidity index truly means that the video game is flexible.)]{}

Basically $\rig(m)$ measure the critical level at which the above rigidification algorithm fails surely. Very little is known on it as exemplified by the (still open) conjecture on the rigidity of the one-oval scheme (\[OOPS:one-oval-rigid-isotopic:conj\]), which traduces into the assertion $\rig(m)\ge 2$ for all even $m$. Even this modest estimate is pure speculation to present knowledge.

It is trivial that $\rig(1)=+\infty$, $\rig(2)=+\infty$ (because $PGL(3,\RR)$ acts transitively on lines or conics provided the latter are defined by quadratic forms with the same signature). $\rig(3)=+\infty$, i.e. all schemes of degree 3 are rigid is already somewhat more sophisticated since there are moduli. Yet this must follow either form Newton-Plücker or from the theory of elliptic functions (Euler-Legendre-Abel-Weierstrass, etc.) It is probably not completely trivial to write down the details, but looks evident if we keep in mind a reduction to the Weierstrass normal form $y^2=(x-a_1)(x-a_2)(x-a_3)$, where the distinct $a_i$ can either be all real or two of them imaginary conjugate.

The result of Schläfli-Zeuthen-Klein (cf. Klein 1876 [@Klein_1876_Verlauf]) implies that $\rig(4)=+\infty$. This is already less evident, and was first proved by Klein via cubic surfaces as discussed in Sec.\[Klein-rigidity-of-quartics:sec\].

For $\rig(5)$ we have a scheme of indefinite type (namely $4\sqcup
J$) which is elementary to find (see Fig.\[Gudkov-Table-quintic:fig\]) and first described in Rohlin’s era (cf. e.g. 1974 [@Rohlin_1974/75] and 1978 [@Rohlin_1978]). It suffices to smooth a pair of conics plus a line in two different ways as indicated on the left of Fig.\[Indefiniteodd:fig\]. It follows that $\rig(5)\le 5$. By Rohlin’s inequality ($r\ge m/2$ if type I) and Klein’s congruence holding right above, the above example is clearly minimal to detect an obstruction to rigid-isotopy via Klein’s types (see also the argument in Sec.\[quintic-table-Klein-Gudkov:sec\]). Thus it is fairly clear that $\rig(5)=5$, but we know no proof without appealing to Kharlamov’s version (1981/81 [@Kharlamov_1981/81]) of Nikulin’s classification in degree 6 (cf. next item).

Looking at the Gudkov-Rohlin table (Fig.\[Gudkov-Table3:fig\]) of sextics, it is clear that $\rig(6)=5$. The proof of this rests the deep result of Nikulin 1979 [@Nikulin_1979/80] that the type enhanced real scheme suffices to encode the rigid-isotopy class. It would be interesting to know if the CC conjecture is able to reprove the rigidity of all sextic schemes lying below $r<
5$ the bifurcation index. Of course Nikulin tells much more.

In view of the Rohlin-Marin inequality $r\ge m/2$ for a dividing curve, Klein’s types afford no obstruction to rigid-isotopy for curves with few branches. A naive optimist can expect no bifurcation below this value. By a bifurcation of a scheme we simply mean it being stretched apart in two chambers of the discriminant. So $m/2$ is a sort of ebullition temperature, below which everything is frozen, i.e. only type II schemes are represented apart from the deep nest scheme with $r=m/2$. The latter scheme is (pure) of type I by the simplest form of total reality, hence does not cause a type bifurcation, while being actually rigid by Nuij’s theorem (1968 [@Nuij_1968]).

The critical temperature $r=m/2$ can be augmented by unit, because the next $r$ is forced belonging type II by Klein’s congruence. (All this extends to the case where $m$ is odd by taking as critical temperature $r=(m+1)/2$ the number of branches of the deep nest).

Hence all schemes with $r\le [(m+1)/2]+1=[(m+3)/2]$ are necessarily of type II, safe for the deep nest. In particular there is no indefinite schemes below this level, and the first such indefinite scheme is expected to be found at height $[(m+1)/2]+2=[(m+5)/2]$. (The height of a scheme is merely its number of components, a jargon suggested by the diagrammatic of the Gudkov table Fig.\[Gudkov-Table3:fig\].)

[The [*type bifurcation index*]{} (or just [*indefiniteness*]{}) $\indef=\indef(m)$ is the minimal height of an indefinite scheme. It is set equal to $+\infty$ if all schemes of degree $m$ are definite (i.e. either of type I or II in the sense of Rohlin 1978).]{}

For $m\ge 5$ it seems evident that indefinite schemes always exist, but this requires some proof. (This and more will follow from Figs. \[Indefinite:fig\] and \[Indefiniteodd:fig\] below.)

The following is all what can be said at first sight:

We have $1 \le \rig(m)\le \indef(m)$, and $\indef(m)\ge [(m+5)/2]$. (In fact it is a simple matter to show that the latter is sharp for $m\ge 5$, cf. Figs.\[Indefinite:fig\] and \[Indefiniteodd:fig\].)

\(1) The first estimate is trivial when $m$ odd, and when $m$ is even it follows from the rigidity of the empty scheme, which is a consequence of the fact that the invisible discriminant has real codimension 2 (cf. Lemma \[invisible-discriminant-codim-2:lem\]).

\(2) The second estimate is a trivial consequence of the fact that a rigid-isotopy induces an equivariant isotopy between the allied symmetric Riemann surfaces. Formally the proof may require the Ehresmann-Feldbau-Pontrjagin, etc. trivialization of a fiber bundle over a contractible base (which is paracompact, else false tangent bundle to the (simply-connected) Prüfer surface). Actually it requires an equivariant version thereof, but by passing to the quotient and reconstructing the symmetric surface as the double orientation cover we can reduce to the classical setting.

\(3) The third estimate follows from Rohlin’s inequality, conjointly with Klein’s congruence, interpreted as obstructing type I right above the height of the deep nest (of type I being totally real under a pencil of lines).

The second estimate could be sharp. This dream is still much out of reach, and so is the nightmare of refuting this via the Fiedler-Marin method. At first the problem may look tractable, yet it certainly does not follow from Gabard 2000 [@Gabard_2000] where a sole classification of symmetric surfaces realizable as plane curves is given.

So below the height $[(m+3)/2]$ things are nearly pure and frozen (no indefinite types) and naively we could expect that all schemes are rigid below this altitude, i.e. when $r\le [(m+3)/2]$. Hence:

All schemes of degree $m$ with $r\le [(m+3)/2]$ are rigid.

(Check if this was not disproved by Fiedler, but we do not think so.)

We know (e.g. by the conceptual argument of Morse surgeries as exposed in Viro 1989 [@Viro_1989/90-Construction]) that all values of $r$ (number of real circuits) below Harnack’s bound are realized, by taking a generic pencil between an $M$-curve (e.g. Harnack’s) and an empty or Fermat curve with $r=1$. Likewise either by the pedestrian argument in Gabard 2000 [@Gabard_2000] or perhaps a variant of Viro’s conceptual argument we know that for all intermediate values there is a representative of type II (safe for $M$-curves). Of course the conceptual argument involves the Klein-Marin theorem (\[Klein-Marin:lem\]), since when lowering its number of component through a Morse surgery a curve of type II cannot become of type I. (Note yet that this is not enough to reprove the little theorem of Kharlamov-Viro-Gabard exposed in Gabard 2000 [@Gabard_2000] in a conceptual Morse-theoretic fashion.) At any rate it shows that the type II is ubiquitous at all levels of the “pyramid” as measured by the basic invariant $r$ (number of real circuit), safe at the maximal $M$-level ($M=g+1$).

The behavior of the function $\rig(m)$ is highly mysterious. It gives a measure of the flexibility of the video game allied to Hilbert’s sixteenth problem. You see on the screen $\RR P^2$ (essentially our retina) two curves of some fixed degree $m$ presenting the same isotopic topology (i.e. distribution of ovals), can you pass continuously from one to the other by moving the joystick in the hyperspace of all curves while avoiding the discriminant $\disc$? If you can [*always*]{} achieve this goal $\rig(m)=+\infty$ (you win always the game). If not $\rig(m)$ measure the smallest number of “ovals” (better real branches) where you can loose the game. A priori $\rig(m)$ could be as low as $\rig(m)=1$ when $m$ is large say $m>10^3=1000$, but some topologist expect the Hilbert-video-game to be a more flexible one, e.g. Rohlin-Viro-Itenberg positing rather that $\rig(m)\le 2$ for all $m$.

What is (inside each chamber of the discriminant) the curve with largest systolic ratio, i.e. the most healthy against infarctus (=hearth attack). Of course all this would be computed w.r.t. the Fubini-Study metric, or maybe the uniformizing metric. It seems likely that flows allied to such functionals ought to give some information on the above problem, essentially because when the systole shortens we approach the discriminant.

We learned the following from an e-mail of O.Viro (dated \[26.01.13\] in Sec.\[e-mail-Viro:sec\]):

If the one-oval scheme (unifolium) is rigid then the contraction conjecture holds true for curves with one oval.

Let $C_{2k}$ be an even order smooth curve with one oval of degree $2k$. It is enough to construct a contraction for a specific curve $F_{2k}$ of degree $2k$. One can consider the Fermat contraction (in affine equation) $x^{2k}+y^{2k}=\rho^{2k}$ with $\rho\to 0$. The latter shrink to a point but alas the tangent cone is not really that of an ordinary solitary node but rather possess $2k$ branches which form $k$ conjugate pairs corresponding to the $2k$-th roots of $-1$. So we need a slightly different contraction toward a solitary node.

This can probably be done explicitly or more loosely by taking a curve of the form a circle $E_2$ union a $D_{2k-2}$ which is invisible, while shrinking the radius of $E_2^{\rho}\colon
x^2+y^2=\rho^2$ and smoothing by Brusotti the union $E_2^{\rho}\cup D_{2k-2}$.

Now let us estimate the indefiniteness $\indef(m)$, i.e. the lowest altitude at which there is a scheme of indefinite type. This amounts to construct an indefinite scheme of minimum height. This is fairly easy as shown by the following series of type I curves of even degree (upper row of Fig.\[Indefinite:fig\]). The bottom row of this figure exhibits type II curves with the same schemes. This is obtained by starting with a type II sextic while adding conics. Since type II is a genetically dominant character, all successors are also of type II. Hence all those schemes are indefinite and have the minimal height permissible by Rohlin’s inequality (\[Rohlin’s-inequality:cor\]), namely two units above the corresponding deep nest.

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A similar series is easy to find in odd degrees (Fig.\[Indefiniteodd:fig\]) and hardly requires any further comments. This proves the:

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For $m\ge 5$, $\indef(m)=[(m+1)/2]+2=[(m+5)/2]+2$.

For $m\ge 5$, we have the estimate $\rig(m)\le [(m+5)/2]$.

This implies a certain rigidity in the video game, and is merely a consequence of Rohlin’s work. Of course it would be miraculous if this estimate is sharp, prompting a maximal flexibility of the video game. For $m=5,6$ it is certainly sharp by the work of Kharlamov 1981 [@Kharlamov_1981/81] and Nikulin 1979 [@Nikulin_1979/80], respectively. So we must concentrate on degree 7, or 8, and by Rohlin’s inequality the required obstruction must necessarily be of a somewhat deeper nature than via Klein’s types.

Searching obstructions to rigid-isotopy below height $DEEP+2$
-------------------------------------------------------------

\[31.01.13\] Here the technology is due to Marin-Fiedler and involves the lock allied to the subscheme $S$ of degree 7 of symbol $\frac{3}{1}\sqcup J$ (3 ovals enveloped in one oval and a pseudoline $J$ outside). If we trace the triangle of 3 lines through the 3 empty ovals (Fig.\[Locks:fig\]a), each line has 7 real intersections (saturating Bézout). It follows that 2 schemes of degree 7 enlarging $S$ cannot be rigid-isotopic as soon as the distribution of the remaining ovals past the 3 lines is different. Alas $\indef(7)=4+2=6$, hence to beat this we must find a pair of isotopic but non rigid-isotopic curves with $r\le 5$ circuits, which is already the height of the Marin-Fiedler lock $S$. Hence this method seems not suited to our goal.

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We could change the lock into $\frac{2}{1} 1\sqcup J$ (Fig.\[Locks:fig\]b), but this has still height $5$. Another choice is Fig.\[Locks:fig\]d but this has also height 5. Another lock could involve a conic through 5 ovals (Fig.\[Locks:fig\]e) but this is not locked as $10< 2\cdot
7=14$.

Another idea is to use [*pseudo-locks*]{} like on the second row. Alas a line is not dividing $\RR P^2$. Perhaps one can construct a lock by aggregating the pseudoline $J$ to the lock (cf. Fig.\[Locks:fig\]f), whence our name [*pseudo-lock*]{}. Then the red line union the pseudoline $J$ divides $\RR P^2$, and so the location of a fifth oval could be an obstruction to rigid-isotopy. Of course Fig.\[Locks:fig\]g is not interesting being saturated (maximal scheme). Fig.\[Locks:fig\]h could be employed as the former Fig.f. For this to work one should have an isotopic-invariant way to distinguish both residues to the [*augmented-lock*]{} consisting of the red line plus the pseudoline. Alas in view of the symmetry of the lock it seems that there is little chance to distinguish invariantly both halves (of the augmented lock). One could imagine to move from the empty oval to the deep oval (on Fig.\[Locks:fig\]f) along the line while choosing the route not intersecting the pseudoline $J$. W.r.t. this oriented segment there would be a left and right hand side residual to the lock. This concept is perhaps invariant under isotopy, and there is some little chance to detect 2 septics with $r=5$ which are isotopic but not rigid-isotopic.

Such a pair of septics is constructed on Figs.\[Locks:fig\]i,j, where the remaining oval lies either of the left (Fig.i) or on the right (Fig.j) of the oriented red segment from [*the*]{} empty unnested oval to the empty nested oval. Does this prove both curves being not rigid-isotopic? Maybe not since both are mirror images under a symmetry in $G=PGL(3,\RR)$, which is a connected group ($\RR P^2$ being non-orientable there is no way to reverse orientation) and so there is a path in $G$ from the identity to the mirror transformation. Applying this path to the first curve yields a rigid-isotopy to the second curve. So where is our former argument faulty? Culpability seems to be the italicized “the” some few line above. Indeed there is on Fig.i no canonical choice for the origin of the arrow, and if instead we had chosen it in the other (outer) oval then the free (unlocked) oval would of course sit on the right (instead of left) of the red arrow.

We can try to remedy this defect by allowing only one outer oval, but then there is another inner oval and there is no canonical way to choose it (cf. Fig.\[Locks:fig\]k). One could hope that one of both inner ovals is distinguished, say by complex orientations (but no chance as we are in the “post deep-nest case” $r=5=4+1$ hence nondividing).

\[01.02.13\] The situation becomes more favorable if we look at locks in degree 9, especially the one depicted on Fig.\[Locksdeg9:fig\]l. Then there is a canonical way to trace an arrow between the deep ovals (say from the less profound to the more profound one as on Fig.\[Locksdeg9:fig\]l). This is invariantly defined in case the remaining oval (dashed) lies outside the largest nonempty oval (as on Fig.\[Locksdeg9:fig\]l). Then the choice of this arrow is canonical and it is hoped that the position of the dashed oval on the left versus right of the arrow (augmented by the pseudoline) affords an obstruction to rigid-isotopy. It is easy to manufacture an algebraic curve realizing this schematic lock, cf. Fig.\[Locksdeg9:fig\]m where the free oval is righthanded. It causes no trouble to find a similar picture with the free oval lefthanded. This would give a nontrivial obstruction to rigid-isotopy below Rohlin’s temperature $\indef(m=9)=5+2=7$, namely at $r=6$. In particular Fig.m would not be rigid-isotopic to its mirror image. This violates however the above argument using connectedness of the group $PGL(3,\RR)$. Of course our mistake is that in the nonorientable $\RR P^2$ there is no consistent way to distinguish the left from the right. More precisely while it is possible to orient the red line from the less massive to the deepest oval, when the latter intercept the pseudoline there is no way to choose a left or right sense to bifurcate as the pseudoline itself lacks a preferred orientation.

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The method becomes more effective if we permit one more ovals. Then the two “free” (dashed) ovals can either be separated by the augmented lock $L\cup J$ or not (Fig.\[Locksdeg9:fig\]n). Both cases do occur as shown by Figs.\[Locksdeg9:fig\]o,p. Both depicted curves are of type II (inspect the little 3 arrows and the negative smoothing right above it on Fig.\[Locksdeg9:fig\]o). The same local pattern appears on Fig.\[Locksdeg9:fig\]p, which is thus also of type II. However both curves are not rigid-isotopic, because during the rigid-isotopy the two free “dashed” ovals of Fig.\[Locksdeg9:fig\]n cannot traverse the red line which is Bézout-saturated nor can they traverse the pseudoline. This is a little success of the Fiedler-Marin method, alas occurring at the same height as the indefiniteness $\indef(m=9)=7$.

A similar example can be found already in degree 7, since we do not actually require to orient the line, compare Fig.\[Locks2:fig\]a,b which should be self-explanatory. Note again that both septics on Fig.\[Locks2:fig\]c,d are of type II, yet not rigid isotopic. This would be worth stating as a lemma since it is a little variant of the Fiedler-Marin method (with now separation caused by the added pseudoline). However this does not answer our puzzle of detecting obstruction to rigid isotopy below the critical temperature $DEEP+2$.

It is then tempting to lower to degree 5, while considering the lock Fig.\[Locks2:fig\]e,f, but then alas we lack a canonical choice for the red line. One can try other locks in degree 7, like Fig.\[Locks2:fig\]g, but then we lack again canonicalness. Still one could make some choice and propagate it consistently during the isotopy. So we get Figs.\[Locks2:fig\]h,i and arrive at the fallacious conclusion that the curve is not rigid-isotopic to itself. This nonsense helps emphasizing the importance of the lock being somehow God-given by the curve, and we (human beings) making minimalist intervention upon the creation.

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It seems that detecting obstructions to rigid-isotopy beyond Klein’s type and below the critical temperature (=indefiniteness $\indef(m)$) is a hard business requiring completely new ideas, or at least some better acquaintance with the Marin-Fiedler obstruction.

\[01.02.13\] Paraphrasing, the method of the lock does not seem to obstruct rigid-isotopies below the indefiniteness $\indef(m)$, i.e. the lowest height of an indefinite scheme. So perhaps the first obstruction to rigid-isotopy is given by Klein’s type and occurs at height $\indef(m)=[(m+1)/2]+2=[(m+5)/2].$ In that case the rigidity index $\rig(m)$ would be highest possible equal to the indefiniteness $\indef(m)$.

\[02.02.13\] Let us summarize the discussion. For any degree $m$, there is a deep nest with $r=[(m+1)/2]=:DEEP$ real branches. Two units above the latter’s height it is easy to construct curves having the same real scheme yet different types (I vs. II) hence not rigid-isotopic. Using the method of the lock it is even possible to exhibit at this height curves of degree 7 or 9 having the same real scheme and the same type II, yet not rigid-isotopic. Probably the method described extend to all other odd degrees. However, it seems much more tricky and actually the locking method seems incapable detecting obstruction below this height, starting thus at height one unit above the height of the deep nest. Could it be that all schemes at or below this height are rigid, i.e. any two curves representing it are rigid-isotopic.

[*Minor question (skip)*]{}.—As a minor problem we suspect that for all odd degrees $m\ge 7$ there is a non-rigid scheme at height $DEEP+2$ containing a pair of type II curves which are not rigid-isotopic. This is probably easy and merely involves extending into series the examples of Figs.\[Locks2:fig\]c,d and \[Locksdeg9:fig\]o,p.

\[03.02.13\] [*Main problem*]{}.—So we first focus on the case $m=7$. Let us denote by $\Delta=DEEP=[(m+1)/2]$ the height of the deep nest. Our goal is to find obstruction to rigid-isotopy (strictly) below height $\Delta(m)+2$. For $m=7$, we have $\Delta=4$, and so we look at schemes with height $r=5$. Several cases occur and are primarily the schemes $$\frac{3}{1}J,\quad \frac{2}{1}1J,\quad \frac{1}{1}2J,\quad 4J,$$ where we use Gudkov’s notation and $J$ denotes the pseudoline (unique up to isotopy). The corresponding schemes are depicted on Fig.\[Locksdeg7:fig\], where the locks are depicted as red thick-lines which are Bézout saturated, while dashed-lines are not. The philosophy of the locking method is that a free oval cannot traverse during a rigid-isotopy the lock (without violating Bézout) and therefore the distribution of additional ovals [*among the residual components of the lock*]{} ([*past the lock*]{} for short) has to be respected. If is [*not*]{}, then we have an obstruction to rigid-isotopy. The dramaturgy in our case, where the height is as low as $r=\Delta+1$, is that we do not have any such additional ovals available (all having been consumed by the lock so-to-speak). On Fig.a we could kill the nonempty oval, but then we loose Bézout-saturation of the red-lines.

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So we need some much deeper idea. One idea is to look how the locked ovals themselves are separated by the lock. This seems however to lead nowhere. Indeed examine the case of the scheme $\frac{1}{1}2J$ (i.e. Fig.\[Locksdeg7:fig\]c), where there is a menagerie of possible disposition of the pseudoline $J$ (Fig.\[Locksdeg7:fig\]e). To effect a nice separation we include the pseudoline into the lock. The pseudoline plus the 2 thick Bézout-saturated lines effects a separation in 4 zones, yet whatever the situation of $J$ the disposition of ovals in those zones is still the same. At least so are the number of residual components in each of these zones, weighted on Fig.\[Locksdeg7:fig\]e by the corresponding number of components $2,3,3b,4$. One can even play more sophisticated games by choosing one of this zone in some invariant manner. For instance given the 2 points of intersections of the thick lines with $J$, we may link them to the deep nest along the thick lines while choosing the way avoiding the dashed line, and close this by the piece of $J$ cutting the dashed line an even number of times (counted by multiplicity). Since this canonical curve $J_0$ cuts the dashed lines an even number of times, it is null-homotopic and bounds a unique disc, which is our canonical region. Alas one checks (experimentally) on Fig.\[Locksdeg7:fig\]e that it always contains 2 components of the scheme. There is a dual curve constructed by taking the segments linking the points of $J \cap
L_i$ to the deep nest via the path cutting once the dashed line, and aggregating the same portion of $J$ as above. This Jordan curve still cuts $L_3$ (dashed line) an even number of times, and so bounds a unique disc. The latter (alas) always contains 4 components of the curve.

Another little idea we had, is to mark for each point in $L_i\cap
J$ the vertices of the locking triangle which looks closest to the intersection point while travelling on the given $L_i$. However as shown by Figs.\[Locksdeg7:fig\]f,g this is completely insensitive to a variation of the position of the pseudoline.

Repeating ourselves, it seems that the [*method of the lock*]{} fails to detect any obstruction to rigid-isotopy at height $\Delta+1$ (or below). Accordingly one may suspect that there is no such obstruction.

Here is an idea. Given a smooth $C_7$, there is a unique pseudoline $J$. Let us speculate about a large deformation $C_7 \to C_6 \cup L_1$ toward a sextic plus a line. This is supposed to be a path in the space of curves avoiding the discriminant sole for its extremity. In particular the split curve is isotopic to the original $C_7$. We call this the rectification conjecture:

[(Rectification conjecture=RC)]{} \[rectif-conj:conj\] Given any (smooth, real, plane) curve of odd order $C_{2k+1}$ there is a deformation in the large toward a curve $C_{2k}\cup
L_1$ where $L_1$ is a line.

[*Objection*]{} \[07.04.13\] Already for quintics, this formulation is sloppy: take an $M$-quintic (hence with symbol $6J$), while quartics can have at most 4 ovals.

If this large deformation is implementable (more about this soon), then we deduce that $r(C_7)=r(C_6)+1$. If the given septic scheme has height $r\le \Delta(7)+1=4+1=5$, then the sextic has $r\le
4=\Delta(6)+1$. But in this low range sextic schemes are rigid by Nikulin’s rigid classification enhancing the Gudkov-Rohlin table (cf. Fig.\[Gudkov-Table3:fig\]). Hence we are inclined to think that septic schemes are rigid below height $\Delta+1$.

Indeed given 2 septics which are (soft) isotopic, i.e. belong to the same real scheme, we apply the rectification conjecture (\[rectif-conj:conj\]) twice to deduce sextics with the same real scheme and of low height $r\le 4$, hence rigid-isotopic. Now using a path between the split curves of degree $6+1$ and using a version of Brusotti’s theorem with parameters (yet to be formulated) one could argue that the 2 given septics are rigid-isotopic. The proof would be completed.

A brief word in favor of the conjecture (\[rectif-conj:conj\]). Given an odd order curve there is a unique pseudoline, and one may measure its length (w.r.t. the round elliptical geometry on the real projective plane $\RR P^2$). Obviously the (genuine) line is the pseudoline of minimum length, namely $\pi=3.14\dots$ if we work on the unit sphere as preferred double cover of $\RR P^2$. Hence for this functional (length of the pseudoline) the gradient lines ought to converge toward curves splitting off a line. (Maybe one can also look at the total geodesic curvature of the pseudoline as another competing functional doing the same job.)

Having this we may dream of a grand inductive process reducing the whole problem of rigid-isotopy (at least below the range $\Delta+1$) to Nikulin’s seminal theorem on sextics (itself relying on deformation theory of K3 surfaces). This would lead to a sharp estimation of the rigidity index $\rig(m)$ of the previous section as being equal to $\Delta(m)+2$.

However even for degree 8, this looks hazardous. One could imagine two modes of deformation of a $C_8$ to either a septic plus a line $C_7\cup L_1$ or a $C_6\cup E_2$. The latter looks dubious for the (8)-scheme consisting of 3 nests of depth 2 (of height $r=6=\Delta+2$), since removing one oval one has still the line through the two remaining nests creating 8 intersections (too much for a $C_6$). Yet the latter is precluded as we restrict to schemes of height $\le \Delta+1$. Listing all of them we find in Gudkov’s notation the following list of schemes (cf. Fig.\[Locksdeg8:fig\]): $$\frac{4}{1},\rau\frac{3}{1}1,\rau\frac{2}{1}2,
\rau\frac{1}{1}3,\rau
5,\rau
(1,\frac{1}{1}2), \rau
(1,\frac{3}{1}), \rau (1,\frac{2}{1}1), \rau (1,\frac{2}{1})1,
\rau (1,\frac{1}{1}1)1, \rau (1,\frac{1}{1})2,\rau
\frac{1}{1}\frac{1}{1}1, \rau \frac{2}{1}\frac{1}{1}.$$

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Albeit messy, our picture (Fig.\[Locksdeg8:fig\]) is supposed to take the census of all possible degenerations $C_8\to C_6\cup E_2$ which are Bézout permissible. Of course we do not claim that all these moves are algebraically realized, but at least Bézout gives no obstruction. Alas it is far from obvious (unlike in the odd degree case) which functional is capable effecting the large structural deformation (LSD) of “conification” $C_{2k}\to
C_{2k-2}\cup E_2$ splitting off a conic $E_2$. Naively we may expect that it is always some of the empty oval which shrinks to a solitary node, but that soon before getting extinct he splits off an infinitesimal circle (or ellipse). This could involve a sort of isoperimetric functional measuring rotundity of ovals, and the allied lines of steepest descent (or ascension). Note however that for the $8$-scheme $(1,\frac{3}{1}),
(1,\frac{2}{1}1),(1,\frac{2}{1})1$ (the 3 firsts of the third row on Fig.\[Locksdeg8:fig\]) we cannot “conify” the empty ovals without violating Bézout. Indeed removing one of the 3 possible empty ovals leads to scheme containing the deep nest of depth 3 as a (strict) subscheme. Of course those (8)-schemes really exist, as depicted on Fig.\[Locksdeg8:fig\]b. A priori nothing precludes a degeneration like Fig.\[Locksdeg8:fig\]c, where a nonempty oval would be “conified”.

[(Conical/ellipticity conjecture=EC) \[inserted 05.02.13\]]{} Given a (smooth, real, algebraic, plane) curve $C_{2k}$ of even degree $m=2k$ with few ovals (i.e. $r\le \Delta(m)+1$ where $\Delta(r)=k$ is the number of ovals of the deep nest of degree $m$) there is a deformation (=rigid-isotopy safe its extremity) toward a curve $C_{2k-2}\cup
E_2$ where $E_2$ is an ellipse, or equivalently a circle up to projectivity. Alas this cannot always occur by extinction of an empty oval, but sometimes by inflation of a large oval (perhaps via an isoperimetric gradient-flow).

The difficulty with this conjecture is that unlike for its odd degree avatar (\[rectif-conj:conj\]) we lack a canonical functional to be minimized like the length of the pseudoline. (The line is the shortest pseudoline, and being non-null-homotopic it is like a systole.) In the even degree case all ovals are null-homotopic and there is no systole in $\RR P^2$. Of course there could be a systole on the Riemann surface of the complexification. Alternatively one may replace the systolic problem by an isoperimetric one taking also area into account. Let us introduce the isoperimetric ratio ($\isop$) of an oval as its length squared divided by the area of its bounding disc, all in reference to the round elliptical geometry on $\RR P^2$. In Euclidean geometry this is minimum for a circle $(2\pi
\rho)^2/(\pi \rho^2)=4 \pi=12.566\dots$. For a large circle near the equator this can be as close as we please to $(2\pi)^2/ (2
\pi)=2\pi=6.28\dots$, which is smaller. This is probably the absolute infimum if we demand the oval on the unit sphere to be disjoint from its antipode. Now we could hope that the minimum isoperimetric ratio of all ovals leads to a functional whose gradient lines tend to inflate the most rotund oval toward an ellipse (rotundity being measured by the isoperimetric ratio). This could give the required degeneration. Perhaps in the limit the most rotund oval degenerate to a pair of lines (double line) and suppressing one of those leads to a odd degree curve of degree one less. This would give the other mode of degeneration:

Given any $C_{2k}$ of even degree of height $\le \Delta+1=k+1$, there is a rigid-isotopy safe extremity toward a curve $C_{2k-2}\cup (L_1)^2$ splitting off a double line $L_1$. More precisely the orthogonal trajectories of the rotundity functional (measured by the minimum isoperimetric ratio) drives any such curve toward such a curve in a canonical fashion.

If this conjecture holds true then we would have a sharp estimate of the rigidity index $\rig(m)$ for all degrees \[end insert 05.02.13\].

All what we are saying sounds very optimistical, and we are still very far from having a decent understanding of this problem of rigid-isotopy (strictly) below the height $\Delta+2$.

We can hope that the method of the lock is more efficient in degree 8 than it was in degree 7 (still confining our attention to heights $\le \Delta+1$). Fig.\[Locksdegree8:fig\] depicts some of them. The method of the lock is a jewel discovered in the late 1970’s by Marin and Fiedler independently (all being inspired by V.A. Rohlin’s work). It involves basically the idea of attaching in the most canonical way to a given curve a certain red configuration acting as a separator. More precisely special attention is paid to red thick lines which are Bézout-saturated, so that the remaining ovals of the curve cannot traverse this line during the isotopy. So basically we choose a triad of points inside some “deep” ovals and link them by a triangle of lines. Of course the choice of the points is not perfectly canonical, but we choose them inside the disc bounding an empty oval. The Marin-Fiedler trick is quite reminiscent of what Grothendieck calls “le principe des choix anodins” (in Esquisse d’un programme 1984 [@Grothendieck_1984/1997-esquisse-d'un-programme]) that whenever we make some choices within a contractible space the construction is nearly canonical, hence robust and fruitful. It is also reminiscent of the moving-frame method of Darboux-Cartan (repère mobile), since during the rigid isotopy will really move the whole triangle.

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On the 3 first pictures of the 2nd row of Fig.\[Locksdegree8:fig\] we have a perfect lock by a triangle consisting of 3 lines which are Bézout-saturated. Alas we have no more ovals left to separate and the method looks inoperative. So let us look at the next degree $9$, and list all the schemes at height $\Delta+1=5+1=6$. It seems plain that this merely amounts to add a pseudoline to the former configurations listed in degree 8 (cf. Fig.\[Locksdegree9:fig\]).

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Again the big deception is that no elementary obstruction given by locking appears in view. Of course one could interpret the figure as a rectification (\[rectif-conj:conj\]) toward octics which are (hypothetically) rigid at height $\Delta+1$ (say via a reduction to sextics), and so would be our curves of degree $9$.

The next real jump in complexity involves degree 10. Let us tabulate all schemes at the critical height $\Delta+1=5+1=6$ while avoiding any Gudkov symbolism (cf. Fig.\[Locksdegree10:fig\]). This is elaborated as follows. Start from any configuration, especially the maximum elements sembling highly concentric and protected medieval settlements like $(1,1,1,1,1,1)$, or $(1,1,1,1)(1,1)$, $(1,1,1)(1,1,1)$, $(1,1)(1,1)(1,1)$, and then apply basically two moves freeing an oval. Vertical moves correspond to liberating a deep oval, while horizontal moves freed a superficial oval (of small depth). Sometimes there are ovals at 3 different depths so that we have also a 3rd oblique move. The red framed schemes are prohibited by Bézout, yet are useful as generator (under the described moves) of other schemes that otherwise are easily overlooked.

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\[04.02.13\] It seems evident at this stage that there is a combinatorial law (which overwhelms my intelligence) impeding the the locking method to act as an obstruction to rigid-isotopy. Looking at all possible locks on Fig.\[Locksdegree10:fig\], no obvious obstruction to rigid-isotopy strikes the vision. In contrast on the basis of the same picture, one may argue that erasing a suitable oval all our (10)-schemes reduce to one of degree 8, and if this cancellation is geometrized via the conification (elliptization) conjecture we could deduce rigidity of all the (10)-schemes at height $\Delta(10)+1=6$, from that of the corresponding (8)-schemes, which in turn was reduced to (6)-schemes where low-height rigidity holds true by virtue of Nikulin’s theorem (1979 [@Nikulin_1979/80]).

As a little experiment imagine the curve of degree $2k$ to have 3 nests of depth $d_1,d_2,d_3$. Since we are at height $\Delta(2k)+1=k+1$, we have $k+1=r=d_1+d_2+d_3$. Let $L_1,L_2,L_3$ be 3 lines passing through the deep nests and suppose them Bézout-saturated, then $d_1+d_2,d_2+d_3,d_1+d_3$ are all equal to $2k$, and thus summing and dividing by two we infer that $d_1+d_2+d_3=3k$, which is much greater than $r=k+1$.

If instead of 3 deep nests we have one deep nest containing 3 little ovals, then the 3 lines through them supposed Bézout-saturated cut the curves in $4+2d=2k$ real points where $d$ is the depth of the nest. Hence $r=d+3=(d+2)+1=k+1$, so that all ovals are exhausted by the lock (and nothing remains left to be separated).

Such arguments seem to extend to all other schemes of Fig.\[Locksdegree10:fig\]. There sometimes we lock with only two totally real lines like e.g. on the scheme $(1,1,1,1)2$ lying near the center of Fig.\[Locksdegree10:fig\] (right above the 2 anti-Bézoutian schemes). Two lines suffice to separate the plane $\RR P^2$, but here again the construction of the lock consumes all the ovals at disposal. In summary it seems hopeless to find an obstruction to rigid-isotopy at or below the height $\Delta(m)+1$ (at least via the lock-method of Marin-Fiedler).

As a last chance, consider the (10)-scheme of Fig.\[Locksdegree10:fig\] right before the “mild” arrow, that is $(1,(1,\frac{1}{1}1)1)$. This is distinguished by having 3 empty ovals at different depths. So we can link them by a $2$-simplex with boundary oriented as going from the deepest to the middle deep and then to the less profound oval closed back to the deepest one. This would induce a certain orientation on the inside of the largest oval (as usual ovals being ordered by inclusion of their insides). The problem however is that while the 2 lines through the deepest oval are saturated (hence there is preferred pathes joining them in the inside of the maximal oval), the third is not and so there is no preferred way to join the middle empty oval to the less deep one (compare Fig.\[Lock10:fig\]). However we could argue that whatsoever the way chosen we get the same orientation (compare Figs.\[Lock10:fig\]b and c). On the latter figures we follow the line until reaching the maximal oval $O_m$ and then follow the latter. The problem is which direction to choose when we meet $O_m$. A priori there is no preferred sense to bifurcate, but we may choose the path such that the circuit $1\to 2\to 3 \to 1$ does [*not*]{} enclose the deep oval of depth 3 (i.e. the one containing the point $1$). This has no intrinsic meaning unless we take the precaution of first rounding the corner at the vertices 1 as shown on Figs.\[Lock10:fig\]e,f. Note that there is a unique way to put near $1$ an arrow circulating on the deepest oval in such a way that we do not intercept the lines $1,2$ and $1,3$ too frequently (i.e. only twice instead of 4 times). This as an intrinsic meaning since those lines are saturated.

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As a result any ten-ics $C_{10}$ belonging to the discussed scheme would have a canonical orientation of the inside disc of its maximum oval (hence of the latter as well). Of course this (semi-)local orientation of the maximal oval propagates continuously through a rigid-isotopy, but it seems that as $\RR
P^2$ is nonorientable no obstruction to rigid-isotopy can be derived from this complicated trick. So even if two such curves $C_{10}$ would have opposed canonical orientation over some region of overlap of there maximal discs (bounding the maximal oval) this would not impede them being rigid-isotopic.

Notwithstanding since the maximal disc (the inside of the maximal oval) is oriented canonically, we may look at the deep line $1,2$ through the deepest empty ovals. This line does not separate $\RR
P^2$, but certainly separates the maximal disc. Further the deep line is oriented by going from $1$ to $2$ while staying inside the maximum oval. Using the canonical orientation there is a left and right hand side of this deep line inside the maximal disc. The location of the superficial empty oval as being right- or left-sided could give an obstruction (since the 3rd superficial oval $O_3$ is not permitted to traverse the deep line during the rigid-isotopy). So if like on Fig.\[Lock10:fig\]a the superficial (empty) oval $O_3$ is left-sided with respect to the oriented line $1,2$ and the canonical orientation it will stay so during for all curves explored by the isotopy, in particular for the end curve. So naively it would suffices to apply a horizontal axis $1,2$ symmetry to Fig.\[Lock10:fig\]a (and realize the scheme geometrically which causes no difficulty via Brusotti) as to find a curve with $O_3$ sitting on the other (right) side. However we must really work with the canonical orientation of the maximal disc, looking at Fig.\[Lock10:fig\]abis shows that the oval $O_3$ really sits on the left albeit sembling on the right (where of course left has to be interpreted as the half pointed by the canonical orientation). With all these confusing remarks, it should be clear that there is no hope to detect an obstruction to rigid-isotopy.

\[05.02.13\] We can also study the embryology of the scheme as shown on Fig.\[Lock10bis:fig\] depicting a nearly exhaustive list of collision which an oval can acquire with the non-saturated line $2,3$ through the 2 most superficial ovals. This represents the possible cytoplasmic expansions of the ovals, but does not [*per se*]{} afford obstructions to rigid-isotopy since all configurations are linked to the initial one in some starlike fashion.

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At the opposite extreme of such Bézout permissible moves, we have the following 3 motions forbidden by Bézout where one of the empty oval cannot traverse the saturated thick red line (Fig.\[Lock10tris:fig\]). So if we transgress the Bézout obstruction by letting the oval traverse the dead-line then we get the configuration of the second row of Fig.\[Lock10tris:fig\] which are priori could be non-rigid-isotopic to the initial one. Alas there is still this argument of symmetry using the connectedness of the group $PGL(3, \RR)$ which prevents one to conclude that the configuration pre- and post-transgression are not rigid-isotopic.

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Trying in vain to corrupt Nikulin (via Marin-Fiedler) {#Nikulin-corruption:sec}
-----------------------------------------------------

\[05.02.13\] It is quite tempting (for dummies) to see if the method of the lock (Marin-Fiedler 1979–1980) can parasite Nikulin’s rigid-isotopy classification of sextic (1979 [@Nikulin_1979/80]). Of course this is not to palish the glory of Nikulin’s theorem which is perhaps the deepest jewel ever obtained along the lines of Hilbert’s 16th problem, but rather an experimental game emphasizing the profundity of Nikulin’s result. Usually, the more a theorem looks unbelievable, the deeper it stands.

For instance we may start with the basic scheme $\frac{3}{1}$ of degree 6 (locked by the triad of lines through the 3 pairs of deep ovals), and enhance it by adding 2 outer ovals to get the scheme $\frac{3}{1}2$. We look at the distribution of outer ovals past the locking triangle, which a priori can be as on Fig.\[Lock6:fig\] either monopartite or bipartite. If one is capable to exhibit two curves $C_6$ with distinct distributions then both curves are not rigid-isotopic, for during a rigid-isotopy the unlocked ovals cannot traverse the (moving) triangle which is already Bézout-saturated. (Of course the locking triangle works as well for $\frac{2}{1}1$, but then there is nothing to separate, and if we add ovals then canonicalness of the triangle is spoiled.) On tracing explicit sextic curves $C_6$ via the small perturbation method applied to configurations of 3 conics we always find the same mono-partite arrangement where both ovals lies in the same component residual to the triangle (Fig.\[Lock6:fig\]).

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It seems impossible to corrupt Nikulin’s result. As the scheme $\frac{3}{1}2$ has height $r=6=\Delta+3$, three units above the deep nest it is necessarily of type II (by Klein’s congruence) and therefore Nikulin’s theorem actually implies the:

Any sextic $C_6$ belonging to the scheme $\frac{3}{1}2$ is such that the triangle through the $3$ deep ovals does not separate the outer ovals.

(We do not know whether this can be proved in an elementary fashion without using the technological arsenal behind Nikulin’s theorem.) \[07.04.13\] Update: yes we can, cf. Le Touzé in Sec.\[LeTouze:sec\].

\[06.02.13\] Of course we may also add to $\frac{3}{1}$ more ovals and examine the resulting distributions past the deep triangle. That is we consider the schemes $\frac{3}{1}\ell$, where $2\le
\ell \le 5$ according to Gudkov’s table (Fig.\[Gudkov-Table3:fig\]).

Consider first the scheme $\frac{3}{1}3$. Smoothing 3 ellipses we can realize this scheme in two fashions either of type I or II (Fig.\[Lock6bis:fig\]). However in both cases the locking triangle through the deep (odd) ovals does not separate the 3 outer ovals. This is quite surprising as both curves are not rigid-isotopic, one could have expected that the lock-method to detect the obstruction. We may also realize this scheme via a variant of Hilbert’s oscillation method, but again the distribution of the 3 outer ovals is the same mono-partite one (at least on the Walt-Disney depiction of Hilbert “à la Gudkov”).

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Since both types I, II have representatives with the same distribution, Nikulin’s theorem implies the:

Any sextic $C_6$ with scheme $\frac{3}{1}3$ (be it dividing or not) is such that the triangle through the $3$ deep ovals does not separate the $3$ outer ovals.

Next examine the scheme $\frac{3}{1}4$. Here we start with a schematic picture à la Hilbert-Gudkov producing the curve $\frac{4}{1}4$ (cf. Fig.\[Lock3-14:fig\]a) which has too much inner ovals (4 instead of the 3 desired). Such a Hilbert-vibration is realized by Fig.b. A suitable smoothing gives Fig.c. The latter has actually a companion generated by smoothing differently the 3 inner nodes. In both cases however the deep triangle does not separate the 4 outer ovals. Fig.d depicts a Hilbert vibration perturbing the union of both ellipses to the Zeuthen-Klein Gürtelkurve, but the quartic $C_4$ would then intersect too frequently (at least 10 times) the conic. Such a vibration is therefore precluded. The dual vibration however (Fig.e) is Bézout compatible (as the $C_4$ intersect $8$ times the two conics). It is questionable if such a vibration exists as the dual does not. Anyway let us (somewhat liberally) smooth Fig.e to get Fig.f, a somewhat funny curve belonging to the scheme $\frac{3}{1}3$. Tracing the triangle through the deepest ovals is somewhat challenging, but does not seem to effect a division of the 3 outer ovals. A priori the depiction could be like on the surrealist détail (i.e., the median oval lying on the “left” of the line through the other 2 inner ovals), but this does not even seem to affect our issue about distribution of outer ovals past the lock. Fig.g depicts another mode of vibration which still overwhelms Bézout. Fig.h depicts yet another mode of vibration essentially dual of Fig.b, but which also overwhelms Bézout. It is a bit puzzling that not any admissible vibration seems to admit a dual vibration.

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At any rate if we believe in Nikulin’s theorem (as we should since it is Soviet mathematics of the best stock) our sole Fig.\[Lock3-14:fig\]c suffices to imply (since by Klein’s congruence our scheme $\frac{3}{1}4$ is of type II) the following:

Any sextic $C_6$ belonging to the scheme $\frac{3}{1}4$ is such that the triangle through the $3$ deep ovals does not separate the $4$ outer ovals.

Next we consider the scheme $\frac{3}{1}5$. For this we can either look at Gudkov’s construction (Fig.\[GudkovCampo-5-15:fig\]) or at the easier construction via a variant of Harnack’s method. In Gudkov’s setting, we must presumably consider the pull-back of the triangle under the Cremona transformation and this a bit tricky to depict. This should be manageable if one is in good form but perhaps there is a more elementary direct construction via the Harnack method.

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So it seems fundamental to construct the over-scheme $\frac{4}{1}5$ via the variant of Harnack’s method mentioned in Gudkov 1974 [@Gudkov_1974/74 p.42], where alas no details are supplied. The smaller scheme $\frac{3}{1}5$ we are interested in should then easily be deduced by taming the smoothing. Since this has some independent interest we devote the next section to the topic.

Gudkov’s variant of Harnack: construction of the $(M-1)$-scheme $\frac{4}{1}5$
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\[06.02.13\] We now try to fix Gudkov’s claim (in 1974 [@Gudkov_1974/74 p.42]) that a suitable variant of Harnack’s method produces the $(M-1)$-scheme $\frac{4}{1}5$. [*Per se*]{} this is not extremely original for we already managed (on the shoulder of Gudkov’s “original” construction of $\frac{5}{1}5$, cf. Fig.\[GudkovCampo-5-15:fig\]) to exhibit this scheme, yet now a more elementary method is demanded. Despite elementariness, if one is not so clever (like the writer) this game can be pretty time consuming as demonstrated by the following section. This consisted in a sequence of failing trials, and alas TeX forced us to censure most of these instructive trials as otherwise our text was not anymore synchronized with the images.

For convenience the first picture (Fig.\[Harnack-Gudkovvariant:fig\]a) reminds the classical implementation of Harnack’s method of 1876 [@Harnack_1876] (little warning: in the original paper the depiction is much left to the imagination of the writer, and our picture though standard is really inspired by nice drawings available in Viro’s papers). A first idea is to put the oscillation inside the ground circle, but this looks too naive and we recover exactly Harnack’s scheme $\frac{1}{1}9$ (cf. Fig.\[Harnack-Gudkovvariant:fig\]b).

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Next we may consider the following variant (Fig.\[Harnack-Gudkovvariant3:fig\]a) but after much effort it only leads to the scheme $9$.

Now we try another variant, and after much efforts manufacture the following picture (Fig.\[Harnack-Gudkovvariant3:fig\]b), upon which one arrives at a meta-mathematical contradiction. Namely we have constructed a sextic (of even degree) with a pseudoline! What we do is really real time mathematics with mistakes, irritation, dirty fingers, etc. At this stage the moral is double: first Harnack’s original method is a quite boring inductive process, and further it is not so easy to implement the variant advocated by Gudkov. Probably it is not so surprising that himself found not the place in his survey to make a decent picture. Geometry without pictures is not geometry it becomes alchemical arithmetico-symbolism impossible to assimilate. So we need to make more pictures and more careful ones. In fact the mistake we did is that we omitted to neutralize the curve $C_4$ of the previous step. What a grotesque mistake!

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So let us make a better picture, and correcting the previous one gives Fig.\[Harnack-Gudkovvariant5:fig\]a. Alas the scheme so obtained is $\frac{1}{1}8$.

Our next idea was to invert the order of some oscillation as on Fig.\[Harnack-Gudkovvariant5:fig\]b, but alas this does not help much, and we obtain the same $(M-1)$-scheme $\frac{1}{1}8$ close to Harnack’s.

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So it seems we are in need of a more drastic form of intertwining the oscillation. This lead us to the following Fig.\[Harnack-Gudkovvariant7:fig\]a which alas still represent the same scheme. (At this stage we really feel unclever or at least unlucky.)

The next part b. of the figure still remains to be done by slight permutation of the oscillation.

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\[07.02.13\] In fact one may argue that if by our choice of starting from an $(M-1)$-cubic we would arrive at the scheme $\frac{4}{1}{5}$ which is pre-Gudkovian (i.e. $\frac{5}{1}{5}$) then we would also be likely to realize the latter by a Harnack construction. This seems unlikely, at least experimentally and there is perhaps a known theoretical obstruction. Accordingly it is perhaps better to start from the maximal vibration and loose one oval latter through the disposition of vibrations.

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If one is somewhat tired (or hung over) one is even able to produce via Fig.\[Harnack-Gudkovvariant7:fig\]b the scheme $\frac{2}{1}8$ corrupting Gudkov’s knowledge or theoretical avatars like Arnold 1971’s congruence mod 4.

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And in reality we only get the scheme $8$:

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After much efforts and trials we ultimately found (the next day \[07.02.13\]) the solution as Fig.\[HarnaGudkov4-15:fig\]. The trick is to leave much room in between the vertical lines effecting the oscillation of Harnack’s method, so as to place the subsequent vibration in between. One of the difficulty we encountered before finding the solution is that since the desired configuration $\frac{4}{1}5$ is an $(M-1)$-curve one is tempted to start with an $(M-1)$-cubic. Then one can apparently loose much energy in the desert.

Instead we start form a Harnack-maximal cubic obtained by slight perturbation of an ellipse $E_2\cup L$ union the horizontal line $L$ and perturb this by a triplet of vertical lines. It results the black depicted $C_3$ on the first row of Fig.\[HarnaGudkov4-15:fig\] intersecting thrice the horizontal line.

The quartic curve $C_3\cup L$ is then perturbed by a quadruplets of lines. Those could be a priori be located everywhere, but we choose them in between as depicted on the figure. Here it seems quite crucial that as the number of lines is even we may concentrate the vibration on a single oval. After this vibration the large central oval looks like a pair of Ray-Ban eyeglasses (viewed in perspective). We have now a $C_4$ oscillating 4 times across the (horizontal) line $L$ and we perturb again the union. How to do this? Always by the same method but we are free to choose the location of the vibrator. A priori since an oval can vibrate an even number of times across a line we may want to choose only 4 vibrations in the “nearby glass” of the Ray-Ban and one outside. This leads to something interesting namely the scheme $\frac{4}{1}4$ (compare Fig.\[indef414:fig\], which we transported earlier in this text). Here instead we keep the 5 vibratory lines close together (this usually maximizes their vibratory impact), and all inside the big glass of the Ray-Ban, cf. second row of the figure. It remains to depict the resulting smoothing of $C_4\cup L$. The result is the red curve $C_5$ depicted but it is essential to choose this oscillation (and not the opposite one) in which case you destroy many ovals (this will be depicted concretely on the next Fig.\[HarnaGudkov3-14:fig\]). So there is something like a snake visiting the nearby glass of the Ray-Ban. This gives a $C_5$ traversing 5 times the line $L$.

Smoothing the union $C_5\cup L$ produces the sextic of the 3rd row (of Fig.\[HarnaGudkov4-15:fig\]). Note that the two (red-colored) branches nearby the horizontal line are linked together at $\infty$ to form a single circuit, which we call the median circuit. More generally all branches going to infinity are connected with the diametrically opposite branch. The median circuit of the $C_6$ is clearly the unique nonempty oval. What appears naively in its interior is in reality a Möbius band (due to the diametral identification), hence its interior really contains 4 ovals. This shows that the constructed curve realizes the desired scheme $\frac{4}{1}5$.

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Somewhat against our expectation this curve cannot be simplified toward the scheme $\frac{3}{1}5$ we were interested in (in the previous section), as the 4 inner ovals are not coming from a vibration. Nonetheless during our exploration up to finding this premaximal scheme $\frac{4}{1}5$ nearly Gudkovian, we found a variant of the exposed construction yielding the scheme $\frac{3}{1}5$ (cf. Fig.\[HarnaGudkov3-15XXL:fig\] below).

First let us choose the opposite mode of vibration as the lucky one we first depicted. This gives Fig.\[HarnaGudkov3-14:fig\]. Now the snake oscillates around the nearby glass, wind around the nose of the investigator, to loop around the second (distant) glass of the Ray-Ban, etc. (As we must optically smooth the union of both black curves $C_4\cup L$, we could a priori hope to close up an oval with the bottom half of the first close glass, but this forces a 6th intersection in $C_5\cap L$ violating Bézout.) On smoothing $C_5\cup L$ we find a curve realizing the scheme $\frac{3}{1}4$. Although not so exciting as $\frac{3}{1}5$, this is already interesting for the purpose of the previous section (namely trying to corrupt Nikulin).

So the game is to trace the triangle through the 3 inner ovals and look upon the separation it effects upon the 4 outer ovals. Of course our picture has poor metrical qualities as we blew it up topologically as to see what happens within the viscera of Harnack’s method. Notwithstanding the naive green triangle depicted seems to leave unseparated the 4 outer ovals (which to me remembered appears “inside”). However upon dragging the upper vertex below while staying in the outer oval residual to the upper semi-circle, we can easily (at least on our topological picture) effect a separation. Remind (from the reasoning of the previous section) that if such a division occurs, then the rigid-isotopy classification of Slava Nikulin 1979 [@Nikulin_1979/80] is violated. Can we infer anything serious from such a topological picture of Harnack method? Maybe we can via a mental contraction of some ovals restore some metrical faithfulness in the depiction as to be sufficiently accurate to answer the (non)separation question by the fundamental triangle through the 3 (deep) inner ovals.

Let us start with the observation that the initial cubic $C_3$ looks on our distorted picture (Fig.\[HarnaGudkov3-14:fig\]) more like a quintic (consider a line “parallel” to the horizontal one passing through the unique oval of the $C_3$). So in the real picture the central oscillating bump of the cubic is much less pronounced. Imagine the oval of the cubic as a sun radiating light, then there cannot be shadow lying behind the hill formed by this bump (otherwise 4 intersections with a line too much for Bézout).

So the real picture is heuristically like the 4th row of Fig.\[HarnaGudkov3-14:fig\]. In particular the Ray-Ban glasses (=vibrating oval of the $C_4$) is much stretched vertically. One may argue that the Ray-Ban glass traps the oscillation, and also the resulting 4 outer ovals created in the last step of Harnack’s iteration. Accordingly it seems sufficient to use the $C_4$ as a sort of envelope. Since the 3 ovals of the quartic $C_4$ distinct from the Ray-Ban are actually (modulo infinitesimal perturbations) the 3 inner ovals of the final sextic $C_6$, and noting also that the line through two of them cannot intersect the Ray-Ban oval (Bézout), we may conclude that any triad of lines through the inner ovals of our $C_6$ does not separate the 4 outer ovals. This no-separation scenario is in accordance with our previous depiction of such a curve via the more user friendly Hilbert’s method (cf. Fig.\[Lock3-14:fig\]). Alas or fortunately our reasoning does not foil Nikulin’s theorem. We summarize this trapping argument by the:

For the sextic curve $C_6$ of scheme $\frac{3}{1}4$ realized via Harnack’s method the fundamental triangle through the deep ovals does not separate the $4$ outer ovals.

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At this stage, it is with a mixture of happiness and disappointment that Nikulin still seems to resist our naive aggression via the Fiedler-Marin method. It remains however to look at the scheme $\frac{3}{1}5$.

\[08.02.13\] How to realize it? Again several tests are required and usually we (at least the writer) lack an understanding of the predestination governing Harnack’s method. Using the technique of the microcosmic vibration “in between” we realized the schemes $\frac{4}{1}5$, $\frac{4}{1}4$ and $\frac{3}{1}4$ all hitting quite central positions of Gudkov’s pyramid (Fig.\[Gudkov-Table3:fig\]). But how to get $\frac{3}{1}5$ lying more “eccentric” on this table? Incidentally one could dream that this Harnack method we are using leads to the eclectic Gudkov scheme $\frac{5}{1}5$. Of course this would corrupt experimental evidence assembled along centennial working tradition (Harnack 1876, Hilbert 1891, Rohn 1888–1913, Brusotti 1910–1945, Gudkov 1954–1973, etc.). However we do not know (personally) a theoretical obstruction impeding Harnack’s method to produce Gudkov’s scheme. Arguably if well assimilated Harnack’s method reduces to a finite collection (for a fixed degree say $m=6$) of combinatorially distinct locations for the vibratory lines. So it suffices to explore all choices and notice that Gudkov’s scheme never appears. We do not claim to be clever enough to complete this boring exercise, but our microfilm picture perhaps contributes to this (cf. Fig.\[HarnaGudkov3-15MICRO:fig\]). \[08.04.13\] It is not clear at this stage if this picture will be publishable in the arXiv due to size limitations.

But let us return to our main duty of exhibiting $\frac{3}{1}5$. Here a series of tests given in micro-film format (Fig.\[HarnaGudkov3-15MICRO:fig\] only consultable on a PC where one can zoom and alas unreadable on the paper). Alas we cannot give the pictures in decent format for otherwise the flow of pictures overrun dramatically what we have to say on the topic. We are in the realm of pure geometry were only pictures have some weight, but alas this does not seem to please my TeX-compilator, who accept at most two pictures per page. Here the second column picture of this microfilm shows an interesting variant of the scheme $\frac{3}{1}4$ where the 4 outer ovals are not directly enveloped by the “Ray-Ban” oval, and so our former argument does not readily apply here. It seems however dubious to expect a corruption of Nikulin. Without getting sidetracked by this issue, keep in mind our goal of realizing $\frac{3}{1}5$.

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After several trials (cf. again the microfilm Fig.\[HarnaGudkov3-15MICRO:fig\]) we arrived at the idea of using the same vibratory configuration of lines as for $\frac{4}{1}5$ safe that instead of starting from an $M$-cubic we start from an $(M-1)$-cubic. This seems to require locating one of the vibratory line inside the circle. Our final picture is Fig.\[HarnaGudkov3-15XXL:fig\]. It hardly deserves to be commented upon once it is found, except for saying that the initial cubic is to be thought of as a small perturbation of the circle $E_2$ union the line, despite sembling a large deformation thereof. The trick in tracing Harnack’s curves is always to exaggerate small perturbations as to create some free room to depict the next stage of the inductive process (vibratory pudding). This is of course possible due to the malleability of the continuum $\RR$ of real numbers.

On this figure (Fig.\[HarnaGudkov3-15XXL:fig\]) we recognize again our Ray-Ban oval except that it has now acquired a “branch” (compare 2nd row of Fig.\[HarnaGudkov3-15XXL:fig\]). Again our interest is to apply the lock method of Fiedler-Marin. So we trace the triplet of lines through 3 points in the deep (inner) ovals $1,2,3$, and examine whether and how this triangle splits apart the outer ovals [*1,2,3,4,5*]{}. (Notice the importance of italicization in our notation: italics are outer ovals while roman-arabic numbers are the inner ovals.) In contrast to the Harnack curve realizing $\frac{3}{1}4$ where all the 4 outer ovals were encapsulated in the Ray-Ban oval, we notice now that the oval [*5*]{} lies outside this (Ray-Ban) oval. So our former argument does not readily apply. Notice also that the inner oval $2$ lies inside the Ray-Ban. All this is a bit puzzling but should not discourage us attempting to study the division of the outer ovals by the locking triangle for our Harnack-modified curve $C_6$. Note incidentally that the latter is not perfectly well defined as a curve unless we specify exactly the deformation constants involved in Harnack’s small perturbation method. Yet it seems natural to expect that the combinatorial data involved in our Harnack-Gudkov style description is enough to determine unambiguously a rigid-isotopy class. Hence by the Fiedler-Marin locking argument (involving merely Bézout saturation) we infer that the distribution of outer ovals within the 4 components past the lock is well-defined. It remains only to determine it.

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In the sequel we shall often speak of “the line passing through two disjoint ovals”. This is a slight abuse of language for such a line is not uniquely defined, and is really intended to mean choose 2 points in the insides of the 2 disjoint ovals and trace the line joining them. Of course the phrasing “the line” becomes somewhat sloppy, but when the two ovals are inner ovals then any such line is Bézout saturated, and so from the viewpoint of analysis situs there is some canonicalness.

Since “the” (or a) line through the ovals $1,3$ regarded on the quartic $C_4$ cannot cut more times the $C_4$, we infer that it does not cut the Ray-Ban oval of the $C_4$ (the one oscillating 4 times across the horizontal line $L$). Next the line through the ovals $2,3$ interpreted on the quintic $C_5$ cannot cut more this curve safe for a point on its pseudoline. A similar remark holds for the line through $1,2$. All this looks a bit sterile and we really need the geometry of the picture to understand the distributional question. For this purpose, look at the 3 green-colored lines on Fig.\[HarnaGudkov3-15XXL:fig\], while enlarging slightly oval 3 as to adjust the picture. It seems then that the triangle separates oval [*5*]{} from the ovals [*1,2,3,4*]{}. Of course upon stretching further oval $3$, we could arrange that the line $2,3$ passes above oval [*5*]{} in which case the locking triangle effects no subdivision of the outer ovals. Which of both scenarios corresponds the reality? A priori the first scenario looks more likely (at least in line with our picture). Remind however the slogan (anonymous, Poincaré, etc.) “La géométrie c’est l’art de bien raisonner sur des figures mal dessinées”.

Let us attempt a more realist depiction on the following figure (Fig.\[HarnaGudkov3-15XXLTEST:fig\]). Even the first right-side picture (fig.d) is not Bézout permissible (the green-line cut the cubic $C_3$ five times). Further it may be observed that the line through $2,3$ may pass “below” the series of ovals [*1,2,3,4*]{}. This is a third possible scenario in which there is no subdivision.

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Admittedly this question looks quite tricky to decide and requires some good idea or high optical acuity. Harnack’s method seems not ideally suited to clinch the matter. Hilbert’s method would be more convenient, yet does not seem capable producing the scheme $\frac{3}{1}5$ which is slightly more on the Harnack right-hand side of Gudkov’s pyramid (Fig.\[Gudkov-Table3:fig\]). This is surely no intrinsic reason since Harnack’s scheme itself is accessible to Hilbert’s method. Inspecting carefully our former Fig.\[GudHilb8:fig\] cataloging several variants of Hilbert’s method it is pretty clear why Hilbert’s method fails producing the scheme $\frac{3}{1}5$. Indeed what comes closest to $\frac{3}{1}5$ is the scheme $\frac{4}{1}4$ depicted near the center of Fig.\[GudHilb8:fig\], and one may argue that the vibrating oval has always an even number of (cytoplasmic) expansions coming across the fundamental ellipse $E_2$ of Hilbert’s construction thereby creating an odd number of ovals. So to have 3 inner ovals requires 2 inner expansions like on the scheme $\frac{3}{1}3$ on the bottom of Fig.\[GudHilb8:fig\] but this dissipates too much of the oscillating energy and not enough outer ovals are created.

As an attempt to corrupt Nikulin it would be interesting to detect a second realization of the scheme $\frac{3}{1}5$ besides Harnack’s presented above, and study on it the distribution of outer ovals past the fundamental triangle through the deep ovals. Even without any scepticism about Nikulin the net effect is that if the latter is correct the determination of this distribution for a single curve $C_6$ belonging to the scheme $\frac{3}{1}5$ which is of type II would by the Fiedler-Marin argument determines this distribution for all curves belonging to the scheme. (That such a curve is of type II necessarily, follows from Arnold’s congruence (\[Rohlin-implies-Arnold:lem\]) $(3=1-3+5=)\chi=p-n\equiv k^2
\pmod 4(=3^2=9\equiv 1)$ valid for all type I curves.) We get so some nice geometric theorem as a consequence of the truth of Nikulin. (Insertion \[08.04.13\]: a much more elementary argument is given by Le Touzé in Sec.\[LeTouze:sec\].)

\[09.02.13\] We now present some tricky argument in favor of non-separation of the outer ovals on the Harnack model constructed above.

Referring to Fig.\[HarnaGudkov3-15XXLTEST:fig\]a, let us look at the quartic $C_4$ with $r=3$ ovals occurring as an intermediate step of Harnack’s construction. The unique oval of the $C_4$ which oscillates 4 times across the horizontal line $L$ is referred to as the [*Ray-Ban oval*]{}. Label the intersection points in $C_4\cap L$ as $p_1,p_2,p_3,p_4$. Consider the 3 green-lines through the 3 inner ovals of the sextic $C_6$ (Fig.\[HarnaGudkov3-15XXLTEST:fig\]c) but imagined traced on this $C_4$ (i.e. on plate Fig.a). Since the oval $2$ of the $C_6$ is enveloped by the Ray-Ban oval (of the $C_4$) it may be inferred that the line $2,3$ does not intercept the Ray-Ban oval outside of the arc $p_3,p_4$ of the $C_4$ on Fig.a). Hence the line $2,3$ is actually much more horizontal than on our Fig.c. More precisely we infer the following.

Since the ovals [*1,2,3,4*]{} are encapsulated in the Ray-Ban oval (of the $C_4$), the line $2,3$ passes below them. (Of course “below” as no absolute sense in projective geometry, but here it has since we have another line $L$ as reference.)

Even better than that consider also the lines $1,2$ and $1,3$ as depicted on Fig.c. Here we see clearly (and if in doubt argue as above with Bézout saturation w.r.t. the Ray-Ban oval) that all 5 outer ovals [*1,2,3,4,5*]{} are not separated by the union $1,2 \cup 1,3$ of those lines. Once those both lines are removed passing below really means not separating them. Another way to formalize this is to look at the intersection $2,3 \cap L$ which should be either very negative or positive when $L$ is identified to the real line. This again not so appealing. In fact instead of trying to define formally what means passing below, we should merely say avoid the Ray-Ban oval of $C_4$ and hence cannot separate $1,2,3,4$. In fact the inside of the Ray-Ban smashed along $p_3,p_4$ linearly is a topological disc (say $D$) away the 3 green-lines, and containing the 4 ovals [*1,2,3,4*]{}.

Further it also passes below oval [*5*]{}, for otherwise it passes above but having to avoid the Ray-Ban it would then have to lounge the nasal portion of the Ray-Ban while passing between oval [*5*]{} and the curvilinear arc $p_2,p_3$ of $C_4$. (Recall that our line $2,3$ is de facto Bézout-saturated (w.r.t. to $C_6$ or even w.r.t. the $C_5$), hence cannot intercept any outer oval, in particular [*5*]{}.) But then the line $2,3$ would intersect twice the horizontal line $L$, violating the simplest case of Bézout.

In conclusion the line $2,3$ passes below all outer ovals [*1,2,3,4,5*]{}, and the real picture could be more like Fig.\[HarnaGudkov3-15XXLTEST:fig\]e. Alas this depiction does not seem possible because the line $2,3$ already crosses 6 times the sextic so the oval $1$ cannot cross this line. Since it moves above it on the right side of the picture it must resurface on the right side below the line $2,3$, which is not the case on our picture (Fig.e). Accordingly Fig.\[HarnaGudkov3-15XXLTEST:fig\]f might be more realistic, and the conclusion would be that the fundamental triangle does not separate the outer ovals.

Have we proved anything? Let us say “yes” and state the following lemma of which we shall supply a more formal proof right below.

The fundamental triangle consisting of the $3$ lines passing through the $3$ inner ovals of Harnack’s curve (depicted above as Fig.\[HarnaGudkov3-15XXL:fig\] or Fig.\[HarnaGudkov3-15XXLTEST:fig\]) realizing the scheme $\frac{3}{1}5$ does not separate the $5$ outer ovals.

The trick toward a more formal proof is to consider the topological disc $D$ obtained from the inside of the Ray-Ban oval by expanding it at $p_2,p_3$ linearly while smashing it inside at $p_3,p_4$ (compare the shaded region on Fig.\[HarnaGudkov3-15XXLTEST:fig\]a). Since this region contains the 5 outer ovals [*1,2,3,4,5*]{} it suffices to check that this disk avoids the 3 green lines through the 3 inner ovals.

This is clear for the line $1,3$ which is Bézout-saturated on the $C_4$, hence can only attack our modified disc $D$ through the arc $p_2,p_3$, but as the inside of the Ray-Ban oval is avoided it results a second intersection with $L$ violating Bézout.

The same argument works for the remaining two lines $1,2$ and $2,3$ after noticing that since those lines intercept twice the oval $2$ of the sextic, it may be assumed that they intercept twice the curvilinear arc $p_3,p_4$ of $C_4$. This follows merely from the nature of the method of small perturbation. [*Warning*]{}.—In fact a priori we could imagine that the line $2,3$ penetrates in the oval $2$ much more vertically than on Fig.c meaning really that it intercepts the segment $p_3,p_4$ of $L$, but in that case too, it is clear that the fundamental triangle does not separate the outer ovals. In fact in this case our line $2,3$ cuts the Ray-Ban oval only twice, and we may excise from $D$ the half of the trace of our line on $D$ containing $p_4$, plus a little tubular neighborhood thereof. During this excision it is clear that we do not loose the covering of the outer ovals, since the line $2,3$ is Bézout-saturated on $C_6$.

Now each of our lines through oval $2$ is Bézout-saturated with the $C_4$ hence must avoid the inside of the Ray-Ban. However our line cannot penetrate the arc $p_2,p_3$, for otherwise a 2nd intersection with $L$ is created, hence has void intersection with $D$.

Viro’s construction specialized to the scheme $\frac{3}{1}5$
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\[08.04.13\] This section explores other realizations than Harnack’s (especially of the scheme $\frac{3}{1}5$) which is somewhat cumbersome. It is primarily a matter of exploring Viro’s method, but the latter turns out to be not much more suited than Harnack’s model to fix the distribution question. As we already said the royal road is Le Touzé’s argument in Sec.\[LeTouze:sec\]. Of course Viro’s method has supernatural appeal too, but our exposition is far from explaining the true core of the dissipation method which is a secret to us. Hence this section can be omitted with loss of continuity.

\[08.02.13\] A first idea is to use Marin’s variant of Hilbert’s method but this seems only able to produce the scheme $\frac{3}{1}4$ (cf. Fig.\[Viro3-15:fig\]a). Another idea is to use Viro’s dissipation of 3 ellipses tangent at 2 points. This being again a small perturbation method like Harnack’s it is not a priori clear that we will be in a better position to tackle the distribution question past the deep triangle. The charming feature of Viro’s method is its ability to create nearly all sextics as perturbation of this configuration of 3 coaxial ellipses. To implement this, look at Fig.29 in Viro 1989/90 [@Viro_1989/90-Construction p.1103] showing all the possible dissipations of a germ of curve singularity of type $J_{10}^-$ consisting of 3 real branches having a second order tangency like on our global model of the 3 coaxial ellipses. This Viro figure is reproduced as Fig.\[Viro3-15:fig\]b below, which includes 5 modes of dissipation denoted by us V1,V2, …, V5 (V standing for Viro of course). Each of them admits an array of permissible values for spontaneous “champagne bubbling” of ovals created out of the blue. Then we can patchwork such smoothing [*independently*]{} at both singular points of the configuration of 3 ellipses to create a global curve with controlled topology. (This is of course highly reminiscent of Brusotti-Gudkov’s independence of smoothing, based on Severi and in turn upon Riemann(-Roch) via possible détours through the Plato cavern of Brill-Noether. Compare Brusotti 1921 [@Brusotti_1921] and Gudkov 1974 [@Gudkov_1974/74].)

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For instance the dissipation V1-V2 with $(\alpha,\beta)=(4,0)$ and $(\alpha,\beta)=(4,0)$ resp. yields Hilbert’s scheme $\frac{9}{1}1$. Choosing instead for V1, $(\alpha,\beta)=(0,4)$ and for V2 $(\alpha,\beta)=(0,4)$ yields Harnack’s scheme $\frac{1}{1}9$. If we choose for V1, $(\alpha,\beta)=(4,0)$ and for V2 $(\alpha,\beta)=(0,4)$ yields Gudkov’s scheme $\frac{5}{1}5$. Pause a little moment at this stage, to be puzzled by the fact that Gudkov’s rare bird—which escaped the attention of all experts during 8 decades (from Hilbert to Gudkov)—appears in the fingers of Viro as a species not much more tropical than common birds like Harnack and Hilbert. Perhaps this banalization of Gudkov by Viro is against the philosophy expressed in Sec.\[Diophantine-and-proba:sec\] that Gudkov’s curve(s) ought to have some statistical and Diophantine rarity.

How to realize $\frac{3}{1}5$? It suffices to choose at V1 $(\alpha,\beta)=(2,0)$ and at V2 $(\alpha,\beta)=(0,4)$ (cf. Fig.c, right). Does Viro’s method help to solve our distributional problem. A priori not as shown by our naive depiction Fig.\[Viro3-15:fig\]d exhibiting both distribution (separating or not) as logically possible a priori. Recall yet that on behalf of Nikulin’s theorem both options cannot occur simultaneously. Of course naive geometric intuition tells us that the upper part of Fig.d is more likely with small ovals spread horizontally as a vestige of the horizontal tangent line at the singular point of the initial configuration of 3 coaxial ellipses. But of course even in this situation where the 2 microcosmic ovals (generated by the dissipation of the bottom singularity “V1”) are nearly horizontal (and so is the line through them) this does not prevent the “vertical” green-lines to separate the upper series of 4 ovals. But again on ground of some microscopical geometric intuition it seems realist to argue that even if the top vertex of the green-triangle is chosen very near to the top of the banana oval (i.e. the large inner oval of the curve $C_6$ resulting from the fusion of the two branches $2,3$ of ellipses when labelled from left to right), the little 4 top ovals will condense themselves as to be non-separated by the deep triangle. Actually if a separation would occur, then by sweeping the nearly vertical green-line inside the pencil of lines rooted at a basepoint in the bottom oval gives by continuity an intermediate line with 8 intersections with the sextic $C_6$ overwhelming Bézout (or the smoothness of the $C_6$). Indeed if the line through one of the bottom micro-oval has slope of ca. 135 degree it cuts twice the micro-oval, twice the banana and twice the (largest) nonempty oval, hence 6 times the curve. If we let this angle diminishes to 90 degree (plus $\varepsilon$) by dragging the upper (banana) vertex up to the top of the banana while supposing that the pair of lines effects a division of the 4 top micro-ovals for a suitably small value of $\varepsilon $ then both lines have again 6 intersections, but in between $135$ degrees and $90+\varepsilon$ degrees there must be a line cutting 8 times the curve $C_6$ namely the one line sweeping the separated oval. Sorry for this messy argument. Of course the key is just to observe that the green-lines are Bézout-saturated, hence the distribution of outer ovals cannot change.

By the same sort of argument, precisely by tracing the line through two points near the bottom of the inner banana and the outer banana we get a line which is already Bézout-saturated. Pushing this line to its ultimate confinement we get the bitangent through the most “meridional” points of both bananas. Following this motion by continuity implies that the 2 bottom ovals are pressed down below this bitangent, and so looks nearly horizontal. Alas this does not prevent the situation of the bottom half of Fig.\[Viro3-15:fig\]d where both microscopic ovals are sitting nearly one above the other but of course at much lesser height than depicted, that is below the bitangent to the most meridional portion of both bananas.

All this is quite exciting for the imagination, but does not seem to answer our puzzle on the distribution of outer ovals past the fundamental triangle. To analyze better the situation we should introspect in more detail the quantitative geometric aspects of Viro’s construction which certainly includes answers to our basic question.

Ignoring that issue for the moment, we can introduce the concept of the bundle spanned by two disjoint ovals. This is the collection of all lines traced through a pair of points chosen inside the respective ovals (“boundaries” included). This bundle is often supported by (or spanning) a bordered surface homeomorphic to a Möbius band. If we think of both ovals as celestial bodies (like Earth and Moon) then this bundle (or rather its support, i.e. what is swept out by this collection of lines) is essentially the region where one oval masks the other (at least partially) like during an eclipse. We call thus this region the [*eclipsus*]{} of both given ovals, or just their mutual [*shadow*]{}.

The shadow of the 2 bottom micro-ovals in case their mutual disposition is nearly vertical (like on the bottom half of Fig.\[Viro3-15:fig\]d) cannot intercept any further oval than the 3 obvious one (each of the 2 protagonists plus the nonempty oval enclosing them). In particular in that case of nearly vertical alinement of both “meridional” micro-ovals their shadow must find its way out through the little room left vacant between the top part of the outer banana and the top 4 micro-ovals. This is actually possible as suggested by Fig.\[Viro3-15shadows:fig\]c, provided both ovals really live at the microscopic scale. Fig.\[Viro3-15shadows:fig\]a depicts the shadow of both bananas. Each line in this shaded region is Bézout-saturated, hence no ovals can survive in this region, hence the situation is forced to be like on Fig.\[Viro3-15shadows:fig\]b. Once we have Fig.c then we must still analyze further shadows, but some thinking at the nanoscale should convince the reader that the verticality scenario posited by Fig.c is not further obstructed by Monsieur Étienne Bézout. Philosophically algebraic plane curves are like celestial configurations not liking to have their horizon too much saturated by galactic nebulosity. They express a principle of economy and purity.

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Our conclusion is that the topological aspect of Viro’s method alone does not seem sufficient to settle the distributional question of a $C_6$ of type $\frac{3}{1}5$. However it is more likely that for some Viro curve $C_6$ the distribution of the bottom ovals is nearly horizontal, and therefore that the fundamental triangle through the 3 inner ovals does not separate the 5 outer ovals of this $C_6$. If this is true (and Nikulin’s theorem also) then we deduce first the following lemma and next the following theorem by uniting the forces of all lemmas of the previous Sec.\[Nikulin-corruption:sec\]:

Any sextic $C_6$ with scheme $\frac{3}{1}5$ is such that the fundamental triangle through the $3$ inner ovals does not separate the five outer ovals.

\[Nikulin-Fiedler-Marin-Gabard-no-separa:thm\] Any sextic $C_6$ belonging to a scheme of the form $\frac{3}{1}\ell$ (with $0\le \ell \le 5$ according to Gudkov’s table) is such that the fundamental triangle through the $3$ inner ovals does not separate the outer ovals.

If this theorem is true, one may of course wonder if there is an elementary proof circumventing the highbrow intervention of K3 surfaces, Torelli, etc. i.e. all the technology involved in Nikulin’s proof. \[13.02.13\] Such an elementary proof is given in the next section (\[LeTouze:sec\]) and was communicated by Le Touzé.

Fiedler-Le Touzé’s answer (conical chromatic law) {#LeTouze:sec}
-------------------------------------------------

\[13.02.13\] Three days ago, I received the following answer from Séverine Fiedler-Le Touzé (née Le Touzé, and often abridged as a such too avoid confusion with her husband Fiedler, who also worked in the field “a long time ago”).

$\bullet$$\bullet$$\bullet$ \[10.02.13\] Bonjour Alexandre, Thomas m’a transmis ta question. La réponse est toute simple: soient $A$, $B$, $C$ trois ovales intérieurs et $D$, $E$ deux ovales exterieurs de ta sextique. Le triangle fondamental $ABC$ est entièrement contenu dans l’ovale non-vide. Si $D$ et $E$ sont dans deux triangles $ABC$ (non-fondamentaux) différents, alors la conique passant par $A$, $B$, $C$, $D$, $E$ coupe la sextique en $14$ points, contradiction. Avec des coniques, on montre plus généralement que: Les ovales vides de la sextique sont distribués dans deux chaines (int, ext), l’ordre cyclique est donné par les pinceaux de droites basés dans les ovales interieurs. Les ovales interieurs sont disposés en position convexe dans l’ovale non-vide. Bon dimanche, Séverine

Translated in my poor English this gives the:

\[LeTouze:lem\] Let $C_6$ be a sextic with $3$ inner ovals, and at least $2$ outer ovals, then the latter are in the same component past the fundamental triangle consisting of the $3$ lines through the deep ovals. In particular any sextic of type $\frac{3}{1}\ell$ has all its ovals distributed in the same component past the fundamental triangle.

[*Insertion*]{} \[08.04.13\].—As a loose idea it could be interesting to see if the method can be boosted as to prohibit the scheme $\frac{3}{1}6$. Of course this follows also via total reality of the scheme $\frac{2}{1}6$ also due to Le Touzé.

Assume on the contrary that the 2 outer ovals are in different subregions past the fundamental triangle. The idea is to look at the conic passing through the $3+2=5$ ovals (3 being deep and 2 being outer ovals). This conic certainly cuts the $C_6$ in at least $(5+1)\cdot 2=12$ points, like say on Fig.\[LeTouze:fig\]a albeit of course no depiction is required since those intersections are so-to-speak topologically forced. However from our supposition that the 2 outer ovals are separated by the fundamental triangle the real picture is rather like Fig.\[LeTouze:fig\]b or Fig.\[LeTouze:fig\]c yielding $2\cdot 5+4=14$ intersections. Bézout is overwhelmed.

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It remains to find the intrinsic reason of why this holds true. The key is to look at the order of the 5 assigned points on the conic. On Fig.a the 3 inner points (in black) are not separated by the outer ones (white-colored), while on Fig.b the 3 inner points are separated by the 2 outer points. This explains the formation of extra intersections whenever we have to salesman-travel from the inside to the outside of the nonempty oval. More precisely when the 3 black inner points are separated by the 2 white outer points then we see 4 elliptical arcs with dichromatic boundary, each of which contributing for an intersection (with the nonempty oval separating the outer from the inner ovals). So the whole story is reduced to the following Hilfssatz somewhat hard to state elegantly:

[(Chromatic law for conics)]{}.—\[LeTouze-Gabard-Hilfssatz:lem\] Let us be given $3$ black-colored points in the plane $\RR P^2$ which are not aligned, and let $T$ be the triangle through them. Take $2$ additional (white colored) points outside the triangle. Then the unique conic $E$ through the $5$ points considered is smooth. Further if both white points are in the same component past the triangle $T$, then the $5$ points are monochromatically distributed on the conic as white-white-black-black-black (with only $2$ chromatic transitions when reading cyclically). If instead both white points are separated by the triangle, then the distribution is dichromatic as white-black-white-black-black, which read cyclically gives $4$ chromatic transitions (from black to white or viceversa).

Applying this Hilfssatz concludes the proof of the lemma (\[LeTouze:lem\]).

The philosophical outcome of this brilliant argument (communicated by Séverine Le Touzé) is that we cannot hope any corruption to Nikulin by the Fiedler-Marin locking technique. In fact the distribution of outer ovals past the fundamental triangle is always monopartite. In particular we get a more conceptual and lucid proof of several lemmas that we tried hard to establish on the cumbersome models of Harnack, Viro. In particular, we get an elementary proof of Theorem \[Nikulin-Fiedler-Marin-Gabard-no-separa:thm\] without the whole transcendental apparatus behind Nikulin’s theorem (K3 surfaces, Torelli, etc.).

Trying to extend Nuij-Dubrovin rigid-isotopy of the deep nest via total reality {#Nuij-Dubrovin-extended:sec}
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\[23.01.13\] Apart from the beautiful result of Nikulin on sextics, stating that the real scheme enhanced by Klein’s types suffices to determine unambiguously the rigid-isotopy class (\[Nikulin:thm\]) and the rigidity of the empty scheme, the only positive general result available is Nuij-Dubrovin’s theorem stating that the deep nest constitutes a unique rigid-isotopy class.

\[rigid-scheme:defn\] [Let us say that an ($m$)-scheme (i.e. a scheme of order $m$) is [*rigid*]{} if any two $m$-tics curves representing the scheme are rigid-isotopic.]{}

Since the deep nest is totally real (à la Ahlfors) under a pencil of lines one might wonder if other totally real schemes also enjoy rigidness. Examples of such total schemes include all the $M$-schemes by Bieberbach-Grunsky, the $(M-2)$-schemes $\frac{6}{1}2$ and its mirror $\frac{2}{1}6$ by an unpublished argument of Rohlin (nobody is able to reconstruct). More easily it includes the scheme of degree 8 $\frac{1}{1}\frac{1}{1}\frac{1}{1}\frac{1}{1}$ (4 nests of depth 2) which is total under a pencil of conics.

Conjecturally for a scheme, type I implies maximal (Rohlin 1978).

$\bullet$ A scheme of degree $m$ is [*total of order $k$*]{} if any curve $C_m$ representing the scheme admits a total pencil of $k$-tics. (For instance the $2k$-scheme $(1,1,\dots,1)$ consisting of $k$ nested ovals is total of order 1.)

$\bullet$ Say that a scheme of degree $m$ is total if any curve representing the scheme admits a total pencil.

It seems natural to expect the:

If a scheme of degree $m$ is total then it is total of order $k$ for some universal $k$ depending only on $m$.

At first glance this could follow from Ahlfors 1950 [@Ahlfors_1950]. At any rate we have the implications:

Total of some order $k$ $\Rightarrow$ total $\Rightarrow$ type I (perhaps implying maximal).

Marin convinced me that the transition from the abstract to embedded viewpoints might be not so easy, hence the converse of the second arrow might not be a trivial corollary of Ahlfors 1950, but we naively still hope so.

One may speculate on an extension of Nuij-Dubrovin’s theorem as follows:

For a scheme, totality of order $k$ (for some fixed $k$) implies rigidness.

[*Insertion*]{} \[08.04.13\].—This conjecture (like the subsequent one (\[M-schemes-rigid\])) is probably also disrupted by Marin’s obstruction (discussed in the next Sec.\[Marin:sec\]). Indeed by our extrinsic variant of Riemann-Bieberbach-Grunsky (\[total-reality-of-plane-M-curves:thm\]) the total reality of plane $M$-curves of degree $m$ is exhibited by a pencil of curves of degree $(m-2)$, hence we have totality for some universal degree, namely $k=m-2$ (depending only on the degree $m$ of the scheme and not on the geometry of the curve), yet no rigidity can be observed by Marin’s obstruction.

Perhaps it suffices to assume type I, or maybe even maximal implies rigidity (in increasing order of hazardousness). Here maximality is interpreted in the sense of Rohlin (as opposed to Harnack’s more specialized sense). Of course a priori there is very little evidence for a direct correlation between those concepts. (Again try to look if Fiedler, Marin give some counterexample, more on Marin soon.)

In particular we may have something like:

\[M-schemes-rigid\] [(Too Naive!!!, completely false as shown by Marin, Fiedler)]{}.—Any $M$-scheme is rigid.

A priori if the devil of algebra does well his job this ought to be completely false in high degrees $m\ge 8$, or say perhaps $m\ge
10^3=1000$.

Marin’s lock 1979: obstruction to rigid-isotopies {#Marin:sec}
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\[23.01.13\] Alas it seems that the above conjecture (\[M-schemes-rigid\]) is completely wrong for $m=7$ already, compare Marin 1979 [@Marin_1979 p.60–61]. Alas I was on a bad day and could not completely understand his argument, which looks however fairly simple involving the prose: “La distinction des deux courbes se fait en étudiant la position des ovales extérieurs par rapport aux droites joignant les ovales impairs.” (cf. p.60, of ). At first this looks sloppy justification as we are not just playing with projectivities but with rigid-isotopies which (in marked contrast to their names) are completely soft pathes in the residue of the discriminant. Yet I am sure that Marin is right (as usual) but his argumentation is for highbrow readers?

Ah yes the argument must be that when dragging the curve in the parameter space (with the joystick) while choosing a triangle through the deep ovals as on Marin’s picture (reproduced as Fig.\[Marin:fig\]), then the forced intersection with the pseudo-line gives already total reality (or Bézout-saturation) of these lines with the septic $C_7$ preventing the remaining (outer) ovals to traverse the triangle during the motion (=rigid-isotopy). Hence the distribution of ovals past the deep triangle is an invariant of the rigid-isotopy class, i.e. any 2 curves exhibiting distinct distributions of ovals past the fundamental triangle cannot be deformed into the other.

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So this is a bit like the moving frame of E. Cartan, and seems indeed to corroborate Marin’s clever observation!!! (Compare also Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000] who call this trick a “lock”, while ascribing it as well to Fiedler.) Note that even the complex orientations agree on Marin’s example. This method of the lock (or moving frame/traingle) affords therefore an obstruction (à la Bézout) to rigid-isotopy. It uses the fact that a triangle in the projective plane subdivides it in 4 pieces. One can wonder if other (more complicated) locks are also useful. This method surely deserves to be better explored and assimilated (as remarked in Degtyarev-Kharlamov ). For instance one can wonder if it is enough to a lock with a pair of lines which suffices to separate the outer ovals of the top figure. During the isotopy we can keep track of them (at least the ovals where they are passing through). Of course Marin’s choice has the advantage of canonicalness. What is crucial is that the lock do not degenerate during the isotopy, which is ensured by the fact that the 3 inner ovals cannot become aligned without violating Bézout. We have proved Marin’s result:

[(Marin 1979, or Fiedler)]{} There is two isotopic $M$-septics, i.e. having the same real scheme (and in fact the same complex orientations, but that requires adding the arrows on Fig.\[Marin:fig\]), yet which are not rigid-isotopic (i.e., belong to distinct chambers past the discriminant).

[*Insertion*]{} \[08.04.13\].—This raises of course the question of counting the number of septics chambers corresponding to this scheme (of degree 7). Perhaps variants of Marin’s figure (Fig.\[Marin:fig\]) produce more than 2 chambers, but we are not sure.

Still some link between totality and rigidity? Highbrow Nuij’s principle
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\[23.01.13\] Is there still some link between total reality and rigidity? A priori imagine the simplest situation of total reality under a pencil of line (ensured whenever we have a deep nest). Then very naively one could imagine to contract progressively the curve to some normal form like concentric circles and then drag it as a such toward the other center of perspective and blow it up again along the radial foliation toward the other curve. Of course doing so we meet reducible curves hence the discriminant yet perturbing the path there is some hope to avoid it completely, proving thereby the Nuij theorem [@Nuij_1968]. Though surrealist this argument is the best we can give in favor of a connection between totality and rigidity.

How does Nuij or Dubrovin prove their fantastic results? Can we “do-it-yourself” by making precise the above idea? One trick would be to take a total pencil (vision from the innermost oval) and perturb the nest toward concentric circles around the center of perspective. Since both curves are (softly) isotopic there is some chance that one path along the pencil (this being a circle abstractly there is 2 such pathes) does not cross the discriminant while affording a rectilinear rigid-isotopy. Then one finishes as above.

$\bigstar$ [*Long Insertion*]{} \[08.04.13\].—In quintessence, the deep nest for which rigidity holds true by Nuij 1968, is the satellite of the conic (cf. Sec.\[satellite-total-reality:sec\]) whose rigidity can be nearly ascribed to ancient Greeks (or Descartes, Newton, Sylvester’s law of inertia for quadratics forms, etc.). By analogy the quadrifolium schemes of degree $m=4k$ consisting of 4 nests of depth $k$ are total under a pencil of conics (as is fairly trivial, and explicitly remarked in Rohlin 1978 at least for $m=8$). This in turn is the satellite of the quartic quadrifolium (degree $m=4$ with $r=4$ ovals) whose rigidity is known since Klein 1876 (Sec.\[Klein-rigidity-of-quartics:sec\]). Hence this gives some evidence that totality in degree 2 implies rigidity. Extrapolating further along a stability of rigidity under satellites it could follow from the Rohlin-Le Touzé $(M-2)$-schemes of degree 6 (whose rigidity is ensured by Nikulin’s theorem deeper than Klein but sharing with it the rôle of surfaces, viz. K3 quartics vs. cubics for Klein) that:

\[satellite-Rohlin-(6)-schemes-rigid:conj\] All satellites of Rohlin’s sextic schemes $\frac{6}{1}2$ (and its mirror $\frac{2}{1}6$) are rigid.

Of course if rigidity is stable under satellites this would not only applies to the Rohlin (totally real) schemes but to all schemes of degree 6 which are of definite type as tabulated on the Gudkov-Rohlin table (Fig.\[Gudkov-Table3:fig\]). There are precisely $64-2\cdot 8=48$ many such schemes. $\bigstar$ [*End insertion*]{}.

Another method of proof (of Nuij’s theorem) could be dynamical like the one (we attempted) for CCC (\[CCC:conj\]), yet involving another functional a priori. Very loosely the functional ought to have a unique attractor consisting of a series of concentric circles (or perhaps ellipses). If the flow can be shown to have this unique attractor (itself a certain manifold) then every nested curve converges there and going forth and then back we link rigidly our both curves.

Can we adapt the above synthetic method (or find an even more synthetic method) to construct the (hypothetical) rigid-isotopy between two (8)-schemes consisting of 4 nests of depth 2? The naive canonical form would be the same with circle or ellipses yet its degree is twice too big (namely 16). ([*Added*]{} \[08.04.13\].—Perhaps as suggested above, the canonical form is the satellite of a quartic which has the correct degree.)

We can first observe that such an octic curve has, like the deep nest, all the nested ovals forming negative pairs (this can be seen either à la Fiedler on a model or à la Ahlfors via the total pencil which forces the orientation to gyrate in the same sense as swept by the pencil). As a such there is no topological obstruction to shrink them at the microscopic scale (or apply alternatively a variant of CCC). Once contracted at the microscopic scale our configuration moves without resistance in the free vacuum and then may be re-expanded at the next curve. This is very sloppy heuristic of course, sembling much like inter-sidereal travelling, but there may be some truth in this. At least one sees a connection with CCC. If this works we get a proof of the:

\[rigidity-sat-quadrifolium:conj\] The $8$-scheme $4\times\frac{1}{1}$ of $4$ nests of depth $2$ is rigid (i.e. any two of its representatives are rigid-isotopic). More generally the $4k$-scheme consisting of 4 nests of depth $k$ is rigid.

This could be the “degree 2” avatar of Nuij’s theorem, and ought to be proved by an iterated variant of CCC (like C++). As yet the total reality of the scheme was only involved to ensure that all the ovals belonging to some nest gyrate in the same sense according to complex orientation (what Rohlin 1978 calls negative pairs). Those negativity may be interpreted as some depressiveness permitting precisely the collapse to the microscopic scale where then we can travel without friction within the “ether”.

Given any collection of deep nests with negative pairs of ovals (gyrating in the same sense), we may hope to contract them at the nanoscale (via CCC or C++) and then travel and re-expand to reach any other curve with the same complex orientation. This looks topologically plausible yet the drawback of ignoring the total pencil (‘à la Ahlfors-Rohlin) is that our assumption does not only pertain on the real scheme but also upon its complex characteristics. So the assertion gain in generality but loose some elegance.

As yet we have merely considered schemes which are towers (i.e. without branching in their nesting graph). However Rohlin (1978) claims that the $(M-2)$-schemes of degree 6 $\frac{6}{1}2$ and its mirror $\frac{2}{1}6$ are total of order 3. Can we deduce that those schemes are rigid by a method independent of Nikulin 1979 [@Nikulin_1979/80], and analog to the one sketched above? One should be in position to visualize the cubic pencil so as to draw the complex orientation. Bypassing this difficult task we may appeal to Rohlin’s formula to deduce the complex orientation. Assume the scheme to be $\frac{2}{1}6$, hence $2(\Pi^+
-\Pi^-)=r-k^2=9-9=0$ tell us that one pair is positive and the other negative. So both inner ovals gyrate “differently” (in accordance with Fig.\[GudHilb8:fig\]).

At this stage one is quite puzzled, i.e. one does not see how to bring the curve at the nanoscale via contractions. Of course granting CCC we can shrink the empty ovals, but a priori cannot shrink the nonempty one. So our heuristic method looks here quite impuissant! Another quasi-paradox of our heuristic method arises for a curve of type I belonging to the scheme $9$, which under a total pencil of cubics (easy to visualize, cf. Fig.\[Fcubic:fig\]) could be isotoped to the other curve(s) of this scheme of type II. Yet of course the second curve lacks a total pencil to re-expand.

All this is just supposed to illustrate that we see no direct relation between total reality and rigidity, at least via the naive contraction approach. However this does not preclude a deeper relationship. That would maybe involve exploiting more the total pencil as a tool to construct a first reduction to some normal shape, which ought to be then easily tele-transported and then re-expanded via the second total pencil. So the total pencil should act as some sort of (contracting) wormhole or as a railway guiding the curve to some canonical shape easier to tele-transport (at the speed of light). Alas this is much too vague to convince us about any implication like “total $\Rightarrow$ rigid”.

\[26.01.13\] After Shustin’s e-mail, who remembered me the reason why the empty scheme is rigid, one can suspect two basic scenarios ensuring rigidity. Taking the deep nest as prototype, where rigidity holds true by Nuij 1968 [@Nuij_1968] one could suspect that rigidity is causal either of total reality à la Ahlfors or by the proximity to the empty chamber (which is connected, and actually baricentrically “convex”). Both phenomena could explain Nuij’s rigidity of the deep nest while affording basic intuition about guessing further rigidity results. For instance total reality could explain the rigidity of schemes swept out by pencil of conics (e.g. $4\times \frac{1}{1}$ in degree 8), while the proximity to the empty locus could via CCC prompt rigidity of the scheme with one oval (in even degree at least).

Another naive idea I had (but which is now quite outdated) is that while total reality could imply rigidity via Ahlfors, the avatar of the latter for empty curves (namely Witt 1934 [@Witt_1934]) could be involved in the rigidity of the empty scheme.

More total reality
==================

Another attempt to prove Rohlin’s total reality claim
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[*Editorial note*]{} \[08.04.14\].—The prose of this section starts a bit abruptly, due to a permutation of section. Prior this material came right after Le Touzé’s section (Sec.\[LeTouze:sec\]), explaining why the chromatic law for conics impedes a direct corruption of Nikulin by Fiedler-Marin’s trick. This miracle of extra intersection created by dichromatism gave me some hope to attack Rohlin’s highbrow claim, but the difficult turned out to be immense to fill.

Further the impact of this method of extra intersections gained by dichromatism must probably also be the key behind Rohlin’s proof of the universal orthosymmetry of the sextic schemes $\frac{6}{1}2$ and $\frac{2}{1}6$. We call any curve having one of these schemes a [*Rohlin curve*]{} as the latter Academician in his 1978 article [@Rohlin_1978] was the first (and actually the unique creature in the universe except for possible extraterrestrial intelligences) to state the universal orthosymmetry of such curves.

Through the 8 deep ovals (equivalently the empty ones) of such a Rohlin curve we let pass a pencil of cubics. As above (\[LeTouze-Gabard-Hilfssatz:lem\]) we imagine the inner basepoints black colored while the outer basepoints are white colored.

Let $C_3$ be any cubic of the pencil, which we assume smooth for simplicity. If $C_3(\RR)$ is connected, then $C_3$ intersects $8\cdot 2=16$ plus twice the nonempty oval of $C_6$, and so the intersection is totally real. If $C_3$ is not connected then its splits an oval and a pseudoline. A priori the oval could visit the 6 inner points while the pseudoline the 2 outer ovals. In this case there is no forced extra intersections.

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By Le Touzé’s lemma \[LeTouze:lem\] we may infer that the triangle through any 3 of the 6 inner ovals (assuming $\frac{6}{1}2$) does not separate the 2 outer ovals (Fig.\[LeTouzeRohlin:fig\]c). There are $\binom{6}{3}=20$ such triangles. But this does not seem very useful.

Let us start again. Consider a $C_6$ with scheme $\frac{6}{1}2$, a so-called [*left-wing Rohlin curve*]{} (in view of its position on the Gudkov table, Fig.\[Gudkov-Table3:fig\]). Consider the pencil of cubics through 8 deep basepoints selected inside the $6+2=8$ empty ovals of the $C_6$. We claim (with Rohlin 1978 [@Rohlin_1978]) that this pencil is totally real (and consequently the curve is of type I).

Total reality of $C_3\cap C_6$ is clear when the cubic is connected for then there are $2\cdot 6+2=18$ intersections, the last two being created while crossing the nonempty oval.

So let us assume the cubic disconnected. Two cases are possible either it is smooth or not. If singular then the cubic may either have a solitary point or even be a conic union a disjoint line. However since we are free to perturb the 8 basepoints the pencil can probably be assumed to be transverse to the discriminant so that its singular members are uninodal curves. This rules out the second case. In the first case of a solitary cubic then the solitary node cannot be one of the 8 basepoints, and so the connected pseudoline of this singular cubic visits all 8 points, creating thereby 2 additional intersections with the nonempty oval.

So may assume the cubic smooth and as soon as its pseudoline visits both inner and outer points we are finished (2 bonus intersections are created). On the other hand if the pseudoline visits only inner points then it must evade out of the nonempty oval (otherwise it would be null-homotopic) and so we score again 2 extra points, and gain total reality. Hence we may assume:

We have to show that this is contradictory, but are still far from the goal. In fact at the time of writing these lines the writer does not know if he will ever be able to complete this argument.

A first remark is that our pencil of cubic has another (non-assigned basepoint). Where is it? We think that it must be on the pseudoline of $C_3$ for simple vibratory reasons. Indeed look at Fig.\[LeTouzeRohlin:fig\]d and imagine a nearby cubics $C_\epsilon$ in the real locus of the pencil. The corresponding oval will have to oscillate about that of $C_3$ and since 6 basepoints are on the oval the oscillation closes up perfectly. (In savant terms the oval has a trivial tubular bundle.)

Another thing natural to do is to cut out the inside of the oval $O$ of the cubic $C_3$ of Fig.d in order to apply the Poincaré index formula to the foliation ${\cal F}$ induced by the pencil. Then the situation is a bit messy but as follows. The pencil hits the discriminant of cubics 12 times over the complexes as $\deg \disc_3=3(m-1)^2=12$ for $m=3$. A priori not all intersections are real, but can occur in conjugate pairs. Those singular curves which are real are either solitary cubics or have two real branches crossing transversally (“real bitangent” and we call them [*nodal cubics*]{}). Denote their respective number $\sigma$ and $\beta$ (where $\sigma$ stands for “solitary” and $\beta$ for “bitangent”). We have $\sigma+\beta=12-2k$.

Using the Poincaré local index formula $j=1+\frac{I-E}{2}$ where $I$, $E$ are the number of internal resp. external tangencies of the foliation with a small circle surrounding the singularity, it is a simple matter to compute indices. Of course a basepoint gives a foyer of index $+1$, a solitary cubic gives a “centre” of index +1 (since $I=E=2$), while a nodal cubic gives a hyperbolic saddle of index $-1$ (as $I=0, E=4$). Applying Poincaré’s index formula (cf. Poincaré 1885 or Gabard 2011 [@Gabard_2011-Euler-Poincare-obst-pretzel-long-tentacles], arXiv, “long tentacles”) it follows $$9+\sigma - \beta=\chi (\RR P^2)=1.$$ It may be deduced that $\beta \ge 8$, that is the following:

\[nodal-cubics-8-many-in-a-pencil:lem\] Any generic pencil of cubics contains at least $8$ nodal cubics.

Further we easily tabulate the possible value as $(\beta,
\sigma)=(8,0),(9,1), (10,2)$.

It would be however probably more interesting to apply the Poincaré formula in the inside of the oval $O$ (of the cubic $C_3$) doubled to get a sphere. (This doubling is merely a trick yet useful to eliminate the boundary.) The difficulty in doing so is that we do not really know a priori how the singularities of the foliation $\cal F$ induced by the pencil are distributed inside the oval $O$. So let us denote with subscript “naught=0” the corresponding quantity of singularities inside the oval $O$. Then we have $$6+2 \sigma_0- 2 \beta_0=\chi(S^2)=2,$$ where we used implicitly the fact that the 9th basepoint is not inside our oval (nor on its periphery since it is rather located on the pseudoline).

Alas at this stage the situation looks confuse. One idea is to imagine a solitary node inside the oval $O$. Then there is a unique time direction so that this oval inflates while moving inside the pencil. Since the oval $O$ has a tube neighborhood like Fig.\[LeTouzeRohlin:fig\]e this oval cannot hit the oval $O$, and must rather collide with the pseudoline component to form a nodal singularity of type $\beta$. So to each solitary node is canonically assigned a non-solitary node (all this occurring inside $O$). It seems evident that the corresponding map is injective, and so $\sigma_0\le \beta_0$. Alas this gives no contradiction when injected in Poincaré’s relation displayed above.

\[scholie:Rohlin-does-not-boils-to-Poincare\] [\[14.02.13\]]{}—In fact it seems unlikely that there is a proof of Rohlin’s total reality assertion (for $\frac{6}{1}2$ and its mirror) using Poincaré’s formula only.

We arrived at this conclusion after tracing a rather complicated foliation of the plane $\RR P^2$ containing the bad cubic $C_3$ as a leaf. (More about the discussion of the relevant pictures soon.) The bad cubic is one which is not totally real, hence whose oval is necessarily enclosed in the nonempty oval of the $C_6$.

In reality such a foliation is fairly easy to construct just by starting with the bad cubic and then merging its components together to a nodal cubic and pursuing the depiction in a more or less canonical fashion. At each step Bézout for $C_3\cap C_6$ is respected and Poincaré index formula is of course verified.

What should we deduce? Could it be that Rohlin’s total reality assertion is false, while its theorem on the type I of his schemes is right as follows from some highbrow topological congruence (due to himself, Kharlamov and Marin, cf. (\[Kharlamov-Marin-cong:thm\]))? If the latter super-classical congruence is correct then via Ahlfors theorem the total reality assertion is likely to hold true yet perhaps not for [*all*]{} pencil of cubics through the 8 empty ovals (or a priori for pencils involving curves of higher order). Perhaps the proof should involve Abel’s theorem applied on the cubics, yet they vary so seems unlikely. Perhaps Abel has to be used on the $C_6$?

All this is puzzling and we frankly confess our poor understanding which calls for a synthesis between the abstract viewpoint of Riemann-Klein-Ahlfors and the embedded viewpoint of Harnack-Hilbert-Gudkov-Rohlin. This subdivision of our science is still vivid today, compare e.g. in Russia the tradition along Natanzon vs. Kharlamov-Viro.

As we noticed earlier in this text (very optionally see Sec.\[sec:Total-reality-Harnack-max-case\]), it is also tantalizing to trace a total pencil on the $M$-sextics, already on those of Harnack and Hilbert. It would be of interest to know what is the degree of curves forming a total pencil in the $M$-case, whose existence follows in principle (modulo Marin’s private communication objection) from Ahlfors even in the simple schlichtartig variant of Bieberbach-Grunsky. ([*Added*]{} \[08.04.13\].—This question should now be settled via (\[total-reality-of-plane-M-curves:thm\]).)

Since an $M$-sextic has 10 empty ovals it looks quite improbable that total reality is exhibited by a pencil of cubics which has only 9 foyer-type singularities. This must, we believe, easily follow from Poincaré’s index formula. Indeed each empty oval must contain a singularity of positive index while thorn singularities of index $+1/2$ are precluded for an algebraic pencil.

Now let us discuss our picture leading to the announced Scholium \[scholie:Rohlin-does-not-boils-to-Poincare\]. The game is to foliate the plane by “flexible” cubics in the sense that we depict only the singular nodal curves abstractly like a figure “8”. A cubic cannot have a node plus an oval. Using this we sometimes have the impression that there is an obstruction to complete the foliated structure like on Fig.\[LeTouzeRohlin:fig\]. (Could it be the case that Rohlin made such a mistake, in the sense of a too hasty inference?)

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However deleting curves and starting again one finds Fig.\[LeTouzeRohlin2:fig\] which is topologically admissible. On it each cubic looks like a cubic, Bézout is respected as well as Poincaré’s index formula (as it should). In fact our solution shows no solitary cubics. Of course we do not claim that this free-hand drawing foliation is algebraic (in which case Rohlin’s claim would be erroneous), but we are also not able to exclude this eventuality. What this picture really shows is that we cannot expect to prove Rohlin’s assertion via the sole apparatus of the combinatorial topology of foliations (à la Poincaré). So if true Rohlin’s statement has some deeper geometric significance, and it is quite tantalizing to imagine its complexity. Further by Ahlfors theorem it is likely that Rohlin’s claim is just the top of the iceberg of a plethora of another phenomena of total reality in higher degrees which must all be very delightful to visualize if not quickly overburdening any human intelligence. Again our prophecy (cf. Introd. of this text) is that this is linked to the stability of matter at the nano-scale, or at least that such totally real pencils describe the dynamics of electrons about an atomic nucleus, as ellipses described the trajectory of Mars about the Sun in Kepler’s days (ca. 1605).

Total reality from the elementary viewpoint
-------------------------------------------

\[15.03.13\] Broadly speaking our main Leitmotiv is the question of examining if there is any relation between total reality à la Ahlfors (or rather Riemann-Schottky-Klein-Bieberbach-Grunsky-Ahlfors, etc.) and Hilbert’s 16th problem. Rohlin’s claim about his sextic curves posits such a deep relation. In this section we try to explore more systematically this relation.

Before entering into the details let us pose some of the guiding questions. Given an (abstract) dividing curve $C$ there is according to Ahlfors 1950 [@Ahlfors_1950], a totally real map $f\colon C\to \PP^1$ of the curve to the projective line $\PP^1$ (i.e. $f^{-1}(\PP^1(\RR))=C(\RR)$).

When this curve is plane does this map extends to the projective plane $\PP^2$ as to be induced by a pencil of curves? The question is actually pure geometry primarily meaningful over the complexes. Given a plane curve $C_m$ defined over $\CC$, and a holomorphic map $C_m(\CC) \to \PP^1(\CC)$ is it true that there is a pencil of curves so that the map induced by the pencil is the given holomorphic map.

If this is true then it is certainly true equivariantly and Ahlfors theorem implies that [*any plane dividing curve admits a total pencil of curves.*]{}[^67]

The next question is then to trace such pencils, and try to control its order (i.e. the degree of the curve constituting it). Tracing them in case of $M$-curves looks an especially hard exercise. (\[08.04.13\] Okay but see (\[total-reality-of-plane-M-curves:thm\]).)

Without appealing to Ahlfors theorem one can also study totally real pencils [*per se*]{}, as a tool to detect the dividing character of curves. Actually it is this trivial criterion (as applied to the Gürtelkurve, or also hyperelliptic curves) which lead the writer to discover Ahlfors’ theorem in ca. 2000–01 independently and prior of knowing about Ahlfors’ work. So whenever a curve $C_m$ is swept out by a total pencil of curves it is dividing.

So one can examine which sort of curves are exposed to such a total pencil to derive in principle an infinite series of orthosymmetry criterions. The prototype is the case of the deep nest totally swept out by a pencil of lines through the deepest oval. Then one would like to study pencil of conics, cubics, quartics, etc.

Some extra difficulty arises from the distinction between universal total reality where it is forced by the sole knowledge of the real scheme as in the case of deep nests or Rohlin’s sextics $\frac{6}{1}2$ and it mirror $\frac{2}{1}6$, and “versal” total reality where the detailed geometry of the curve is required to exhibit total reality. This is of course much allied to what Rohlin calls schemes of indefinite type.

For instance the octic scheme $4\times \frac{1}{1}$ consisting of 4 nest of depth 2 is universally totally real, under the pencil of conics through 4 basepoints inside the empty ovals. This example easily extends to schemes of degree $4k$ having 4 nests of depth $k$. Existence of such curves for each $k$ is demonstrated by Fig.\[Total:fig\]. (\[14.03.13\] A somewhat more conceptual reason is given by taking the algebraic satellites, i.e. nearby levels of the quadrifolium quartics, so $P_4\cup
P_4+\epsilon_2\cup\dots \cup P_4+\epsilon_k$ and smoothing this union of reducible curve.)

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[*Insertion*]{} \[08.04.13\].—It may be observed (Fiedler’s smoothing law) that those curves are of type I for surgical reasons, providing another proof independent of total reality. Further the total pencil induces the complex orientations due to the holomorphic character of the underlying total map. It suffices then to imagine the pencil of conics to see that the intersection series will move along Fiedler’s arrows (this we shall vaguely refer to as dextrogyration). Understanding this properly in general could be the source of some progresses in the field, maybe?

Any curve $C_{4k}$ of degree $4k$ whose real scheme consists of $4$ nests of depth $k$ (for short) is total under the pencil of conics through the $4$ empty ovals.

Let $C_2$ be any conic of the pencil. First, $C_2(\RR)$ is connected, e.g. because it is a rational curve, aka as [*unicursal*]{} in Cayley’s jargon, cf. optionally the Introd. of Harnack 1876 [@Harnack_1876], which is of course not really required on the case at hand since it suffices like in Antiquity to sweep out the conic by lines from one of its point. It follows that $C_2(\RR)$ has to cut our curve $C_{4k}$ in $4\cdot 2k$ real points for topological reasons. But this is the maximum permissible according to Bézout, hence the pencil is totally real.

It seems likely that the converse statement holds true, namely any curve $C_{4k}$ totally real under a pencil of conics with 4 real basepoints has this scheme of 4 nests of depth $k$. Note at least that the depth of the ovals cannot be distributed otherwise without violating Bézout for lines. For instance if a $C_{12}$ instead of having 4 nests of depth 3, had nests of depths say $2,4,3,3$ then the line through the nests of depth $4$ and $3$ would have too much intersection (namely $8+6=14>12$).

It is worth noticing that the degree of such total maps are in accordance with the bound $r+p$ announced in Gabard 2006 [@Gabard_2006]. Indeed if $C_{4k}$ is a quadrifolium, then $r=4k$. Hence by the obvious Klein relation $g=(r-1)+2p$ and the genus formula $g=\frac{(m-1)(m-2)}{2}$, where $m$ is the degree, we find $$g=\textstyle\frac{(4k-1)(4k-2)}{2}=(4k-1)(2k-1), \quad \textrm{
and }$$ $$p=\textstyle\frac{g-(r-1)}{2}=\textstyle\frac{(4k-1)(2k-1)-(4k-1)}{2}=
\textstyle\frac{(4k-1)(2k-2)}{2}=(4k-1)(k-1).$$ On the other hand by letting degenerate the 4 basepoints against the deep oval we find a total morphism of degree $2\cdot
4k-4=8k-4=4(2k-1)$. This has to be compared with the $r+p$ bound $$r+p=4k+(4k-1)(k-1),$$ which is indeed much greater as shown e.g. by evaluating for $k=1,2,\dots$. We find for $k=1$, $4(2k-1)=4\le r+p=4$. For $k=2$, $4(2k-1)=12\le r+p=8+7=15$, and so on due to quadratic growth of $r+p$. In fact the gonality of such curves is probably $\gamma=4(2k-1)$ (at least majored by this quantity) and so significantly lower than the universal upper bound $r+p$ stated in Gabard 2006 [@Gabard_2006].

In fact it is worth testing the truth of this $r+p$ bound on sextics already. Then we shall basically pass into review all the dividing curves of the Gudkov-Rohlin table Fig.\[Gudkov-Table3:fig\]. It would be natural to start from the top of this table but as $M$-curves are paradoxically tricky to understand from the viewpoint of total reality, we start from the bottom. The paradox is that the total reality phenomenon for abstract $M$-curves is basically the schlichtartig ($p=0$) case of Ahlfors theorem which is pretty much easier than the positive genus case. ([*Added*]{} \[08.04.13\].—This paradox is settled via (\[total-reality-of-plane-M-curves:thm\]).)

So starting from the bottom we have first the deep nest $(1,1,1)$. Then $r=3$ and $p=[g-(r-1)]/2=[10-2]/2 = 4$. The gonality $\gamma=5 \le r+p=7$ is exhibited by the pencil of lines through a point on the deepest oval.

For the scheme $\frac{4}{1}$ (when of type I) total reality comes from the conics pencil through the deep nest. It leads to a total series of degree $\gamma \le 2\cdot 6-4=8 \le r+p=5+3=8$ and Gabard’s bound is sharply realized. (Trick: while it is sometimes boring to compute $r+p$ it may be remembered that this is also $\frac{r+(g+1)}{2}$, i.e. the mean between $r$ and Harnack’s bound $g+1$.)

In fact checking total reality involves here the exact geometry and not merely knowledge of the real scheme. More specifically the 2 prototypes of curves of type $\frac{4}{1}$ are depicted on Fig.\[R4-1:fig\]. One would like to have a geometric criterion for deciding a priori the type.

In both cases we have 4 deep (=empty) ovals. Now we may choose 4 points in them. Given such a tetrad there are 2 cases to be distinguished (cf. Fig.\[Total2:fig\]a). Crudely put, either one of the 4 points can be inside the triangle spanned by the 3 other or not. However projectively this is a misconception as shown by Fig.b. Yet as we are given a curve of type $\frac{4}{1}$ it may look either like one of the 2 versions of Fig.c. Here the deep triangles (those traced through 3 deep ovals) have always a distinguished 2-simplex traced inside the nonempty oval. We call any such a (fundamental) simplex and there are 4 of them. Now two cases are possible: either one fundamental simplex is the union of the 3 others or not. Alternatively one oval is contained inside a fundamental simplex or not. (Check that this is well-defined requires keeping Bézout in the background memory.) Hence Fig.a recovers some intrinsic significance (and amounts essentially to the 2 possible visions we may have of a 3D-tetrahedron when projected on our 2D-retina). We call the second option a (as a short cut for stable tetrad like a prism stably posed on the sheet of paper, in contrast to the unstable tetrad posed on its edge hence in unstable equilibrium).

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This being said we have the following recognition lemma of Klein-Rohlin’s type by pure geometry:

If the $4$ empty ovals of a sextic curve of type $\frac{4}{1}$ form a then the pencil of conics through the deep (empty) ovals is total, and in particular the curve is dividing (type I).

Let us choose any conic $C_2$ of the pencil (through the 4 deep points inside the empty ovals). Like on Fig.e this curve will appear inside the largest fundamental simplex. Then the idea is to surger the conic into 2 pseudolines $C_2=J_1+J_2$ as shown on Fig.f or g. There is several way to do this but choose one. This surgery amounts to aggregate a certain edge of the tetrahedron which is exempt of intersection with the nonempty oval $N$ (because it is already Bézout-saturated). Therefore $C_2\cap N=(J_1\cup J_2)\cap N=(J_1\cap N)\cup ( J_2\cap
N)$, but as each $J_i$ is a pseudoline each must intersect twice the nonempty oval $N$, and we gain 4 extra intersections. On the other hand $C_2$ intersect twice each empty oval and so we totalize $2\cdot 4+ 4=12$ intersections the maximum permitted by Bézout. Total reality follows, and the proof is complete.

Albeit not perfectly hygienical our proof shows how to gain extra intersections by this splitting method. (Ideally we could hope that this method is also the key to Rohlin’s total reality claim for the curve $\frac{6}{1}2$, but this is not clear a priori. Imagine the bad cubic whose oval is contained inside the nonempty oval of the $C_6$, and what to do next!!!?)

Next we would like a similar optical recognition criterion of the type for the sextic scheme $\frac{2}{1}2$. Here looking at Fig.\[R2-12:fig\] reproduced below as Fig.\[Total2-12:fig\]a,b below (which I borrowed from Degtyarev-Kharlamov’s survey 2000 [@Degtyarev-Kharlamov_2000]) suggests that what distinguishes both types is whether the line through the 2 outer ovals separates or not the two inner ovals within the inside of the nonempty oval.

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Somewhat more formally, let us extract from both prototypical curves (Fig.a, b resp.) some combinatorial datum. We mark in black inner ovals by choosing a point inside, and choose also 2 white points in the outer ovals. Unlike in the previously studied case, there is no preferred fundamental simplex inside the nonempty oval $N$, but we have a $1$-simplex entirely traced inside the nonempty oval $N$, which we mark by a double stroke. So we extract Fig.c resp.d and what distinguishes both is the issue that the line through the white vertices intercepts the line through the black vertices along its double marking or not. Let us call the first case (like Fig.c) a [*crucifix*]{} and then we have the:

If a sextic curve $C_6$ of real scheme $\frac{2}{1}2$ has a crucifix, then the pencil of conics through the empty ovals is totally real (and the curve is of type I).

We have defacto $4\times 2=8$ real intersections coming from the empty ovals. Take any conic of the pencil. Two cases may appear. Either the conic is [*dichromatic*]{}, that is when we follow it along some orientation we visit the 4 basepoints in the sequence black-white-black-white (BWBW) in this alternating way, or it can be monochromatic if this sequence reads BBWW (compare Fig.e). In the dichromatic case 4 intersections are created, and total reality is ensured. In the monochromatic case, we apply the splitting method which decomposes the conic $C_2$ as an union of two pseudolines $J_1,J_2$ traced on Fig.f obtained by cutting the conic at the two black points and adding the fundamental $1$-simplex linking both black vertices in the inside of $N$ (the nonempty oval). Since this $1$-simplex does not cut $N$, the intersection with $N$ remains the same after this surgery, but each pseudoline forces 2 intersections with $N$, and we gain the 4 required extra intersections. Total reality of the whole pencil is proved.

[*Insertion*]{} \[09.04.13\].—On a second reading of this proof, it is not clear what prevents to apply the same argument to the other configuration. Try to clarify this at the occasion.

To summarize the proof is the same as the previous one safe for an intervention of chromatism (black and white reflecting the inner and outer ovals). Note also that as $r$ is the same as in the previous case, Gabard’s bound $r+p$ is likewise verified (at least not quashed=invalidated)! $\bullet$ Can we continue this game? According to Gudkov-Rohlin’s table (Fig.\[Gudkov-Table3:fig\]) the next specimen to study is $\frac{5}{1}1$. As we have now $6$ empty ovals it is evident that a pencil of conics will not exhibit total reality. We probably have to move to cubics with 8 basepoints assignable.

Let us use the same naive device of combinatorial extraction from two prototypes (cf. Fig.\[Total5-11:fig\]). We get some beautiful bi-pyramid (or octahedron) and one should imagine each face (or interface) shaded whenever the corresponding triangle is fundamental (i.e. included in the outer oval of the sextic $C_6$). Alas both configurations so obtained look combinatorially equivalent, and we feel puzzled. The next idea that comes to mind is to look at the conic through the deep black points rooted in the inner ovals. It seems that what distinguishes both types (I vs. II) is the location of the outer points as being resp. inside or outside this conic. Another feature distinguishing both models is the absence resp. presence of a line through the outer oval missing the nonempty oval $N$.

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In fact even on the model it is quite difficult to guess which pencil of cubics will exhibit total reality of the type I configuration (as predicted by Ahlfors’ theorem). One could take the horizontal line through the white point which cut the sextic 6 times, and take two extra basepoints on this line (perhaps in the 2 lower wings of the butterfly). Then at least the split cubic consisting of the conic through the 5 black and 3 white points would be totally real. Another puzzling point is that such a pencil will have mapping-degree $3\cdot 6- 8=10$ when the 8 basepoints degenerate on the curve $C_6$, whereas Gabard predicts one of degree $r+p=(r+g+1)/2$ the mean value of $r$ and Harnack’s bound that is $(7+11)/2=9$. So perhaps this constitutes a (potential) counterexample to Gabard 2006 [@Gabard_2006], at least if all abstract pencils are concrete and realized by cubics pencils. If we look at quartics pencil with $\binom{4+2}{2}-1-1=13$ free basepoints then the degree would be $4\cdot 6- 13=11$, still higher than Gabard’s bound. For quintics there are $\binom{5+2}{2}-1-1=19$ free basepoints and so the degree is $5\cdot 6-19=11$, for sextics $\binom{6+2}{2}-1-1=26$, so the degree is $6\cdot 6-26=10$, for septics $\binom{7+2}{2}-1-1=34$, so the degree is $7\cdot 6-34=8$. Gabard seems rescued, yet it looks quite tantalizing to understand the geometry of such a total pencil if it exists. (If Gabard’s bound is true and the Riemann-Hilbert transition from the abstract to the concrete viewpoints equally holds true then such a total pencil should exist of order at least seven!)

[*Insertion*]{} \[09.04.13\] Let us summarize this as follows:

\[(M-4)-sextics-corrupt-Gabard:scholium\] $(M-4)$-sextics of type I are perhaps a good place where to corrupt Gabard’s bound $r+p$. And if not it is at least a pièce de résistance against the principle that any abstract pencil is concrete, and therefore Ahlfors abstract theorem is unlikely to apply without friction in Hilbert’s 16th problem. In other words Riemann’s canary feels claustrophobic in the Plato cavern of Brill-Noether-Hilbert. Perhaps the above example merely corrupts the conception that the mapping-degree of a total pencil is minimized when the order of its constituting curves is. However it could still be true that any dividing plane curve of degree $m$ has its total reality exhibited by a pencil of order $(m-2)$ (or less), compare e.g. [(\[total-reality-of-plane-M-curves:thm\])]{} for the case of $M$-curve.

\[16.02.13\] Let us leave aside this problematic concerning the truth of Gabard’s bound $r+p$ to concentrate on the existence on a cubics pencil which is total on our sextic of symbol $\frac{5}{1}1$. Of course the existence of the latter is more an act of faith than a truth a priori, as it is not obviously implied by Ahlfors’ theorem. The latter probably gives the existence of a total pencil and one may wonder what is the least possible order of the curves in the pencil.

$\bullet$ Then we can look at the next curve $\frac{3}{1}3$ of the Gudkov tabulation (Fig.\[Gudkov-Table3:fig\]). Two models are depicted on Fig.\[Total3-13:fig\]c, and one may hope to distinguish them by some combinatorial recipe (perhaps by looking at the inner fundamental simplex and some outer simplex).

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Another idea is that since our curve has 6 empty ovals one should look at the corresponding hexagon and at pencils of cubics spanned by 2 triangles. Specifically, we may choose a hexagon which visits the 3 black inner points and the 3 white outer points in dichromatic alternation (BWBWBW), cf. e.g. Fig.d. Then we may expect that the pencil of cubics spanned by the red and green cubics is total. Alas Fig.e refutes this expectation. Of course there are other dichromatic hexagons but this is unlikely to be the right method. For instance Fig.f is another dichromatic hexagon, yet the corresponding pencil is still not total as shown by Fig.g.

In conclusion those $(M-4)$-schemes are a bit puzzling from the viewpoint of total reality as there is no (obvious) canonically defined pencils since we have 6 empty ovals, which is not the number of basepoints of a pencil of plane curves, namely $4$ for conics and 8 for cubics. The two extra virtual basepoints for cubics could be chosen as high-order contacts imposed to the pencil and this done properly could exhibit total reality. It remains however to understand the natural geometric condition that are so-to-speak imposed by the geometrical vision of the curve. Remind indeed that total reality always amounts to place the ocular system “inside” of the glass so that the latter has no apparent contour (compare the baby case of the Gürtelkurve, Fig.\[Guertel-saturated:fig\].)

[*$(M-2)$-curves*]{}.—We may hope that the situation is improved when moving to $(M-2)$-curves. The first case to study is the scheme $\frac{8}{1}$. Fig.\[Total8-1:fig\] shows models of both types I vs. II, but it is again quite puzzling to decide which intrinsic criterion distinguishes both configurations. Of course a loose answer could be that the type I configuration is characterized by the fact that the pencil of cubics through the 8 empty ovals is total, however one could desire a more optical recognition algorithm. Perhaps what distinguishes the type I is the possibility of tracing a convex octagon through the empty ovals. Note that convexity has some meaning since given two points in the nonempty oval $N$ we shall always select the half-projective line (segment) which is inside this oval $N$. ([*Added*]{} \[09.04.13\].—But the oval $N$ can be non-convex, and so this is meaningful only when the 2 points are inside the deep ovals of course.)

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For this scheme $\frac{8}{1}$, one could nearly argue that the pencil of cubics through the deep 8 points is always total, for we have $8\cdot 2$ automatic intersections, plus the 2 coming from the fact that the cubic is not null-homotopic hence must cut twice the nonempty oval. However this would contradict Rohlin’s remark that this scheme is indefinite as shown by the above constructions (pictures). However the sole obstruction to total reality of a cubic in this deep pencil is that the cubic has a small oval entirely inside some of the empty ovals. Indeed if the cubic is connected total reality is clear as $2\cdot 8+2=18=3\cdot 6$, and if not yet the oval of the $C_3$ visits at least two ovals of the sextic then each oval visited contribute for 2 intersections and total reality is evident.

[*Insertion*]{} \[09.04.13\].—This “soleness” looks inexact: another obstruction occurs when the cubic splits off an oval visiting all 8 basepoints and the pseudoline stays confined outside the nonempty oval of the $C_6$.

One would like to show that under a suitable geometric hypothesis (capturing the essence of the type I scheme) this sole obstacle cannot occur.

\[17.02.13\] All these questions are fairly delicate and have to be extended to all type I schemes listed by Rohlin (see again the Gudkov-Rohlin Table=Fig.\[Gudkov-Table3:fig\]). Precisely what is demanded is an optical recognition procedure of the type in the sense of Klein (orthosymmetry vs. diasymmetry) via a synthetical device ensuring total reality of a certain class of pencils naturally attached to the curve (or its schemes). This would extend somehow Rohlin’s claim of the absolute orthosymmetry of the sextic schemes $\frac{6}{1}2$ and $\frac{2}{1}6$, which is the purest manifestation of the phenomenon. Meanwhile (yesterday), Séverine Fiedler-Le Touzé informed us (and several other colleagues, cf. letter in Sec.\[e-mail-Viro:sec\] dated \[16.02.13\]) that she was able to prove Rohlin’s claim for the scheme $\frac{2}{1}6$. Probably her argument contains crucial ideas that solve as well our slightly generalized problematic.

[*Insertion*]{} \[09.04.13\].—This is nearly true, safe that it turned out that Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics] proves a slightly weaker assertion than the full Rohlin claim, namely she relies on the RKM-congruence ensuring type I a priori.

Concretely we have then the scheme $\frac{4}{1}4$ for which one need to formulate an optical recognition, and the scheme $9$. For the latter it seems that the type I configuration is characterized by the fact that the pencil of cubics through 8 deep points inside some of $8$ ovals is such that the 9th basepoint lands in the 9th oval. A lucky stroke!

[*Insertion*]{} \[09.04.13\].—It is nearly implicit from the above that we posit:

Any dividing $(M-2)$-sextic has a total pencil of cubics. In a stronger shape: any cubics-pencil assigned to pass in the $8$ deep ovals is total, and it is permissible to let degenerate the basepoints on the ovals themselves.

Then it is also plain to see that this is much in line with Gabard’s bound $r+p=10$, since $3\cdot 6-8=10$ is the mapping degree when the 8 basepoints are degenerated upon the ovals.

Total reality of plane $M$-curves
---------------------------------

\[17.02.13\] Next we have the case of $M$-sextics. This puzzled me for a while, yet it seems clear that now cubics pencils will not exhibit total reality. The reason is that we have 10 empty ovals but only 9 basepoints available. On the other hand it seems evident that each empty oval must contain a singularity of foyer-type corresponding to a basepoint of the pencil. So this follows from Poincaré’s index formula applied to the foliation induced by the pencil (cf. Lemma \[Poincare-lower-bound\] much earlier in this text, but restituted below in perhaps clearer fashion).

This being said we shall move to pencil of quartics. The crucial idea is to remind the synthetic proof in the abstract context of the schlichtartig avatar of Ahlfors’s theorem, i.e. the theorem due to Riemann 1857-Schottky 1875–77-Bieberbach 1925-Grunsky 1937. More precisely we have in mind the simple argument via Riemann-Roch rediscovered by Huisman and Gabard, yet first clearly enunciated in Enriques-Chisini 1915. Bypassing all these historical details, the logical argument is simply given in our Lemma \[Enriques-Chisini:lemma\] (prior in this text). The idea is merely that if one has an abstract $M$-curve (not necessarily plane), then choosing one point on each oval (=real circuit which is linguistically better in this abstract context) one has a group of $g+1$ points which therefore move in its linear equivalence class by Riemann(-Roch), or just by Abel’s theorem since there are $g$ Abelian differentials (holomorphic one-forms) imposing magneto-hydrodynamical constraints upon the motion of a divisor in its linear equivalence class.

So our effective divisor of degree $g+1$ moves on the curve of genus $g$. Since there is only one point on each “oval”, it is like a miniature railroad, in which there is only one train one each track, and so there cannot be collisions and total reality is automatic.

Now when the $M$-curve is plane, I was frustrated to know nothing on the degree of such total maps. However the answer is “[*toute simple*]{}”(=very simple). Indeed inspired by the abstract proof, choose one point on each oval (and also one on the pseudoline if there is one). Then there is a standard recipe to construct the linear series spanned by a given group of points on a plane curve $C_m$ (due to Brill-Noether 1873/74?, Enriques-Chisini’s book 1915, Severi’s book 1921 [@Severi_1921-Vorlesungen-u-alg.-Geom-BUCH], van der Waerden’s book 1939/73 [@van-der-Waerden_1939/73], Walker’s book 1950 [@Walker_1950/62], who else?): just choose an integer $k$ large enough so as to have enough free parameters to pass a $k$-tics through the given group of points. Choose such a curve $C_k$ and look at the residual intersection with the curve $C_m$. Consider then all $C_k$’s passing through this residual intersection and the latter cut on the curve groups whose mobile part are divisors equivalent to the given one. This method clearly belongs to the genre of a sweeping method (balayage).

Applying this to an $M$-curve leads to the following very modest theorem (stated as a such just because it escaped my attention for several months, if not years):

\[total-reality-of-plane-M-curves:thm\] Given any plane $M$-curve of degree $m$ there is a total pencil of $k$-tics of degree $k=m-2$ (two units less than the given degree $m$). In fact exactly like in the abstract Bieberbach-Grunsky theorem, any equidistribution of points (i.e., one point on each real circuit) moves in a linear system of dimension $\ge 1$ and induces a totally real pencil by the sweeping method. In particular each $M$-sextic is total under a pencil of quartics.

As the proof involves some arithmetical nonsense it is didactic to first handle the case of sextics. Then Harnack’s bound (in Petrovskii’s notation) is $M=g+1=\frac{(m-1)(m-2)}{2}+1=11$. The space of $k$-tics has dimension $\dim \vert k H \vert=\binom{k+2}{2}-1$, that is $5$ for conics, $9$ for cubics, 14 for quartics, etc. Choose an equidistribution of $11$ points one on each oval of the $C_6$. Then quartics have enough freedom to visit them. Choose a $C_4$ passing through the 11 points, and the residual group has $4\cdot 6-11=24-11=13$ points. But this is exactly one less than the dimension of all quartics, and so the residual series—consisting of all curves passing through the residual group—gives the required pencil. The total reality of the latter follows by the non-collision principle involving the continuity argument implicit in Enriques-Chisini’s anticipation of the Bieberbach-Grunsky theorem. Of course the impossible-to-beat anticipation is Riemann’s 1857 Nachlass [@Riemann_1857_Nachlass]!

The general case is merely the same numerological coincidence worked out in general. Given any $M$-curve $C_m$ of degree $m$, Harnack’s bound is $M=g+1=\frac{(m-1)(m-2)}{2}+1$. Choose $M$ points on the real locus $C_m({\RR})$, one on each oval. Locate the least integer $k$ such that $\dim \vert k H \vert \ge M$. Since both the genus and the dimension of this complete linear system are given by binomial coefficients, this traduces into $\binom{k+2}{2}-1 \ge \binom{m-1}{2}+1$ which is first satisfied for $k=m-2$ (but not at $k=m-3$). Indeed this amounts to $\binom{m}{2}-1 \ge \binom{m-1}{2}+1$ which is plain as $\binom{m}{2}=1+2+3+\dots+(m-1)$. Now the residual intersection of a $C_k$ through the $M$ points with $C_m$ gives so many points as the following expression, which turns out to be the dimension of the system $\vert k H \vert $ less one unit, as shown by the following boring calculation: $$\begin{aligned}
k\cdot m-M&=(m-2)m-\textstyle\frac{(m-1)(m-2)}{2}-1 \cr
&\textstyle =(m-\frac{m-1}{2})(m-2)-1
=(\frac{m+1}{2})(m-2)-1=\dots=\dim \vert k H \vert -1.\end{aligned}$$ Somewhat more elegantly, $$\begin{aligned}
k\cdot m-M &=(m-2) m-[1+2+\dots+(m-2)]-1 \cr
&=[(m-1)+(m-2)+\dots+2]-1=\textstyle\binom{m}{2}-2=\dim\vert kH
\vert-1,\end{aligned}$$ for $k=m-2$.

Several questions arise as usual after discovering a trivial truth. (Derrière les montagnes encore des montagnes: Proverbe des iles créoles, if I remember well). In the case of sextics one can probably say therefore (modulo a more careful analysis of the case of $(M-4)$-sextics that all dividing sextics have their total reality exhibited by a pencil of degree $\le 4$). Perhaps it is true in general that:

Any dividing $m$-tic has its total reality exhibited by a pencil of curves of order less than $(m-2)$.

A more serious game would be to see if the above theorem essentially due to Riemann-Enriques-Chisini-Bieberbach-Grunsky-Wirtinger-Huisman-Gabard does not imply when suitably complemented by foliation theory à la Poincaré the well-known obstruction of Hilbert-Rohn-Petrovskii-Gudkov for $M$-sextics (e.g. the highbrow Gudkov-Rohlin congruence mod 8 or at least the weak version thereof mod 4 due to Arnold $\chi=p-n\equiv k^2 \pmod 4$). For higher degrees one may even dream of new results along this method, but all this requires more serious work. One could even dream that the method extends outside the realm of $M$-curves, as to recover e.g. Rohlin’s claim (meanwhile Le Touzé’s theorem) but this is unlikely because the continuity principle of no collision meets then serious difficulties, which are precisely those making Ahlfors theorem harder than Bieberbach-Grunsky’s theorem. Gabard 2006 gives an abstract topological algorithm overcoming this difficulty of collisions, but it looks hard to transplant this to the context of Hilbert’s 16th problem. [*Added*]{} \[09.04.13\].—The key is of course that if one has an overpopulation of trains circulating on a track one must ensure dextrogyration so as to avoid collisions. Then total reality is granted.

\[02.03.13\] As a lovely special instance of the above theorem (\[total-reality-of-plane-M-curves:thm\]), Le Touzé (1 March 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]) observes the:

[(Le Touzé 2013)]{} \[LeTouze-quintic:scholie\] “For an $M$-quintic $\langle J
\sqcup 6 \rangle$, one finds a suitable pencil of cubics with six basepoints distributed on the six ovals, and two further chosen on the odd component $J$. As this component must cut any cubic an odd number of times, the required $15$ real intersections are granted.”

—[Gabard’s addendum \[09.04.13\]:]{} As it will be observed in the sequel, but can already be noted here, one can also avoid this topological argument with the pseudoline, by noting that since $2\cdot 6+2=14$ intersections are granted, the remaining one is forced to reality by algebra (Galois-Tartaglia[^68] involution).

[*Long Insertion (ca. $2\frac{1}{2}$ pages)*]{} \[09.04.13\].—How to generalize this Le Touzé’s Scholium to an $M$-septic? Pencil of quintics have 19 basepoints assignable. Assign 17 of them on ovals and 2 on the pseudoline, then there is $34+3=37$ real intersections granted, overwhelming the $5\cdot 7=35$ of Bézout. But by Harnack $M_7=g_7+1=\frac{6\cdot 5}{2}+1=16$, so we do not have as many ovals as 17. Actually our sloppy argument reproves Harnack’s bound with one unit less. This is, by the way, not so surprising as Enriques-Chisini 1915’a purpose was precisely to re-derive a proof of Harnack via Riemann-Roch. Okay, but this sounds too modest and we can surely expect more clever generalizations of Le Touzé’s Scholium. Indeed, by Harnack we have $15$ ovals on the $C_7$ (recall from Möbius-von Staudt that odd order curves have exactly one pseudoline). Distribute the 19 basepoints on the $15$ many ovals, plus 4 on the pseudoline. Then $30+4=34$ real intersections are granted by topology, and the last one is forced by algebra (Galois-Tartaglia symmetry of complex conjugation). Hence total reality is granted. We have proven the:

Given any $M$-septic $C_7$, the pencil of quintic assigned to pass through the inside of all $15$ ovals (warning: it is more prudent to assign them directly on the ovals themselves) of $C_7$ and $4$ points marked on the pseudoline is totally real.

Two questions arise. A deep one is whether this can be used to infer something about Hilbert’s 16th (distribution of ovals solved for $m=7$ essentially by a single hero, Viro ca. 1979). Another question is whether Le Touzé’s Scholium extends to all (odd) degrees. More philosophically, it seems that our abstract argument of Theorem \[total-reality-of-plane-M-curves:thm\] works in all degrees and gives the required total pencil of degree $(m-2)$, yet Le Touzé’s scholium (\[LeTouze-quintic:scholie\]) looks sharper as it tells precisely where to assign basepoints, without using the sweeping method and the Restsatz à la Brill-Noether.

In degree 6, Le Touzé’s method suggests looking at a Harnack-maximal $C_6$ swept by $C_4$’s. Those (quartics) have 13 basepoints assignable. Distribute $11$ of them on the ovals, but where to place the remaining 2? A priori only 22 real intersections are granted. Of course we still have 2 more basepoints, but they do not force new intersections. How to ensure total reality in this case?

Let us look at degree $m=9$. Then septics have $\binom{7+2}{2}-2=34$ basepoints assignable (for a pencil). For $m=9$, Harnack’s bound is $M_9=\frac{8\cdot7}{2}+1=29$. So distribute the 34 bases on the 28 ovals plus 6 on the pseudoline, granting so $2\cdot 28+6 = 56+6=62$ intersections (just one less than Bézout’s $7\cdot 9=63$), but algebra forces reality of the last man. At this stage it is evident that Le Touzé’s Scholium extends to all odd degrees as follows:

\[Le-Touzé-extended-in-odd-degree:scholium\] (Extended Le Touzé’s Scholium).—Given any $M$-curve of odd degree $m$, the pencil of $(m-2)$-tics assigned to visit once all ovals and with residual collection of basepoints assigned on the pseudoline is totally real. Further, the mobile part of the pencil has exactly one point moving on each real circuit, and the allied circle map has lowest possible mapping-degree namely the number $r=M$ of real circuits (exactly like in the Riemann-Bieberbach-Grunsky theorem).

What is first chocking is that when $m=5$ (Le Touzé’s Scholium) there is really an extra intersection gained by topology (of the pseudoline), while for $m=7,9$ our argument merely uses algebra. So a priori the argument could split in two cases depending on some (sordid) periodicity modulo 4. The geometer dislikes intrusion of capitalism and arithmetics in his garden. However it should be observed that even in Le Touzé’s argument one can use algebra, which is alas more capitalistic than her geometric argument. So it is still reasonable to expect an unified proof (without mod 4 stories) along pure arithmetical nonsense.

This is as follows: let $C_m$ be our $M$-curve of odd degree $m=2k+1$. We look at curves $C_{m-2}$. Those can be assigned $\binom{m}{2}-2=:B$ many basepoints. But by “Möbius-von Staudt” $C_m$ has $M-1$ ovals (just omit the pseudoline of course), and by Harnack $M-1=g=\binom{m-1}{2}$. We distribute the $B$ basepoints on the $g$ ovals and the $B-g$ remaining ones on the pseudoline. Let us calculate $B-g=[(1+2+\dots+(m-1))-2]-[1+2+\dots+(m-2)]=(m-1)-2=m-3$. So we have $2g+(B-g)$ real intersections granted by topology (of the ovals), and this is equal to $(m-1)(m-2)+(m-3)=m(m-2)-1$. But this is one unity less than Bézout’s number $m(m-2)$ of complex intersections in $C_m\cap C_{m-2}$, so that the last intersection is forced to reality too! Did we used the assumption that $m$ is odd in any dramatic fashion? I would say no, but we did! Probably the argument adapts to even degrees as well if one is a bit more clever than we were for $m=6$.

As to the last clause, it is evident by construction. Indeed since one basepoint is assigned on each oval, some extra (mobile) intersection is created on the oval. Further the last intersection granted by the Galois-Tartaglia symmetry conj, is forced to live on the pseudoline, since each real circuit contains at least one mobile point (by an evident sweeping principle or just the fact that any point has a well defined image).

Hence a problem of interest is to understand the even degree case, and we hope that someone will easily tackle this question. It is quite beautiful at this stage to feel some big harmony between Riemann, Harnack, Brill-Noether as well as Le Touzé, or Rohlin, at least a sort of unity between conformal and algebraic geometry. Poincaré wrote something like the following: [*“La pensée n’est qu’un éclair dans la nuit, mais c’est ce qui éclaire tout”*]{}.

In fact, when $m=6$ we have 13 basepoints (for quartics) and 11 ovals on the $C_6$. Could it be useful to impose the 2 additional basepoints as imaginary conjugate on the curve $C_6$. Those points will not be real, yet since they are statical they do not spoil total reality which merely involves dynamical points. (Note: this is not a new idea, cf. e.g. p.7 of Gabard’s Thesis 2004 [@Gabard_2004].) Note also that in the above proof (\[total-reality-of-plane-M-curves:thm\]) adapting Bieberbach-Grunsky to plane $M$-curves nothing grants that basepoints are real. In fact we start from any group of $g+1$ points equidistributed on the $M=g+1$ circuits, pass a curve of sufficiently large degree (i.e. $(m-2)$) through them and look at the residual intersection, which is a priori not totally real (but just real, stable under conj). This gives perhaps some evidence that we should for $m=6$ permit a pair of conjugate basepoints. Doing so we really have 11 real points moving on each oval in accordance with the train-track principle (i.e. Bieberbach-Grunsky, or Enriques-Chisini, etc.)

So we state:

\[Le-Touzé-scholium-deg-6:lem\] Given any $M$-sextic, the pencil of quartics assigned to visit any $11$ points marked on the $11$ ovals and a pair of conjugate points of $C_6$ is totally real and induces a (circle) map of degree $11$.

Quartics depend upon $\binom{4+2}{2}-1=14$ parameters and so $13$ basepoints may be assigned. Distribute them on the 11 ovals available and fix the 2 remaining ones as a conjugate pair of points of the $C_6$. For topological reason each real curve of the pencil cuts once more each oval (usual closing lemma for ovals), and so $2\cdot 11=22$ real intersections are granted. By Bézout there is a total of $4\cdot 6=24$ intersections. Hence, [*all*]{} intersections are under control, i.e. either the 22 real ones or 2 imaginary ones which are statical. The latter do [*not*]{} perturb total reality, since they are not moving “electrons”.

Probably the statement extends to all other even degrees (by working properly the arithmetics eventually by using what we already calculated in the odd degree case).

More geometrically (and returning to $m=6$), one may wonder if we could not by continuity push the 2 imaginary basepoints on the real locus so as to impose a tangential contact in the limit (zusammenrücken). Then the modest advantage is that all basepoints would again be visible on the reals, and total reality should be conserved by continuity. So we arrive at the:

For any $M$-sextic, the pencil of quartics assigned to visit any $11$ points marked on the $11$ ovals and tangent at a $12$th point of the $C_6$ is totally real and induces a (circle) map of degree $11$.

Hence some contact can be imposed at any point, and the resulting foliation will look like a dipole at this point of tangency. It would be interesting to see if we can infer any of the deep classical obstructions on the distributions of ovals due to Hilbert, Rohn, Gudkov, from this method.

Another idea (to be explored better than what follows) is to assign such imaginary basepoints on the puzzling case of $(M-4)$-sextics (of type I), cf. Scholium \[(M-4)-sextics-corrupt-Gabard:scholium\]. Then we had a pencil of cubics, of which we distribute the 8 basepoints on the 6 empty ovals and 2 points remain left. A priori we could imagine that an imaginary pair counts just for one linear condition (since after all the passage through one of them forces passing through the conjugate). So we could impose 2 imaginary pairs of additional basepoints, and the mapping-degree of the pencil would be $3\cdot 6- 6 -4=8$, again in accordance with Gabard’s bound $r+p=9$ (best interpreted as the mean of $r$ and Harnack’s bound). This looks however dubious since the pencil of cubics would then have $6+4=10$ basepoints overwhelming Bézout. This can be repaired if we impose only 5 real points and 2 imaginary pairs, and then Gabard’s bound is (exactly) verified, since $3\cdot 6-5-4=9$. However it another piece of work to check that total reality can be ensured. So let us be happy with a conjectural (and admittedly vague) statement:

Any $(M-4)$-sextic of type I admits a total cubics-pencil of degree $9$ (like Gabard) with $5$ basepoints assigned on the oval and $2$ pairs of imaginary basepoints assigned on the curve.

Here only $10$ real intersections are granted (among the $18-4=14$ which are moving) and so much work remains to be done to ensure totality under suitable assumptions. If we impose 6 real basepoints then 12 are granted, etc. [*End Long insertion.*]{}

\[18.02.13\] Let us try the following strategy. \[We now come back to Rohlin’s total reality problem.\] Suppose the sextic to have the scheme $\frac{6}{1}2$. Let us choose 8 points $p_i$ on the $8$ empty ovals, one on each oval, and consider the corresponding pencil of cubics. We would like to show total reality of this pencil. It is clear that any cubic cuts at least $2\cdot 8=16$ times the $C_6$. This is because the $C_3$ can be either tangent at $p_i$ to $C_6$ or transverse. In the first case, intersection multiplicity is two, while in the second, one side of an infinitesimal analytic arc of the curve is inside the oval while the outer is out, hence an extra intersection is gained by closing the real circuit.

Take any cubic in the pencil which is [*connected*]{} (and smooth). The latter is clearly totally real as 2 bonus intersections are created on the nonempty oval of the $C_6$ (which acts as a separator between the inner and outer basepoints). Note that we use the lemma that any pencil of cubics contains a connected cubics (which we nearly proved via Lemma \[nodal-cubics-8-many-in-a-pencil:lem\] showing that a pencil contains in general 8 nodal cubics).

Now a simple idea ensuring total reality would be to look at a nearby cubic $C_3'$ and look if the 2 extra intersections [*gyrate in the same sense*]{} (dextrogyrate) on the nonempty oval $N$. In case of dextrogyration, total reality follows because there would be no collision between both points on $N$. Indeed when we move the cubic curve in the pencil then the intersection points move continuously and none of them can suddenly change its sense of motion, for otherwise there would be 2 curves of the pencil (nearby hence distinct) passing through the same point.

So it suffices checking that the 2 intersections $p',q'$ of $C_3'\cap C_6$ located on $N$ move in same sense (dextrogyrate) on $N$, equivalently that $p',q'$ are separated by the corresponding 2 intersections $p,q$ for $C_3\cap N$.

Since the curve $C_3'$ is a small perturbation of $C_3$ it oscillates about it (in a slaloming fashion). Now a simple picture shows that the gyration is good (occurs in the same sense, or dextrogyre) iff the number of basepoints inside $N$ is odd. So the whole question reduces to knowing if the 9th (non-assigned) basepoint of the pencil is located inside $N$ (or not). If it is inside then we are finished and total reality follows. Alas I know about of no argument ensuring the inside-ness of the 9th basepoint.

The above argument (or rather strategy) relies on the existence of a connected cubic in the pencil which must be a simple matter. This can be bypassed if we argue differently. It is clear that the sole obstruction to total reality is a disconnected cubic whose oval lies inside $N$. Such a cubic is smooth except if it has a solitary node, yet in that case total reality is evident for the pseudoline of the solitary cubics has to connect an inner and outer point so contribute for an extra 17th intersections. Then either by algebra or topology the 18th intersection is real too. Given such a bad cubic $C_3$ which is smooth and whose ovals lies inside $N$, we can again look at a small perturbation $C_3'$ which will oscillate about $C_3$, and so do the corresponding ovals. Now it is clear by a slaloming argument that the oscillation is possible iff the number of inner basepoints (inside $N$) is even. Hence again we would have a contradiction, if we knew that the 9th basepoint of the cubics-pencil lies inside $N$.

Whatever the strategy adopted, Rohlin’s claim seems to require innerness of the 9th basepoint. So this gives a 2nd reduction of Rohlin’s claim.

As a metaphor it seems that such total reality proofs à la Rohlin-Le Touzé are akin to an Eiger-Nordwand ascension. There are several base-camps where to rest, but as the climbing goes on they become rarer and rarer and one is forced to follow a nearly canonical route, \[more and more vertiginous and perilous, by the way.\] It should be noted yet that our approach is slightly weaker than the Rohlin-Le Touzé claim for we do not check total reality of all pencils with 8 deep basepoints [*inside*]{} the empty ovals, but merely the case when the latter 8 points are located [*on*]{} the ovals. Our weaker variant suffices yet to detect total reality and so the type I of such Rohlin’s schemes (e.g. $\frac{6}{1}2$). Our tactic looks simpler, since when basepoints are assigned in the interior of ovals, they do not create defacto real intersections, because the cubic’s oval could be microscopically nested inside one oval of the sextic. [*A dubious strategy with the hexagon (skip the next 2 paragraphs)*]{}.—Another idea was suggested by a naive look at the Hilbert-style construction of $\frac{6}{1}2$. Here it seems that the hexagon through the inner points is not convex. $\bigstar$[*Inserted Objection*]{} \[09.04.13\].—This is so on the naive Walt-Disney picture Fig.\[GudHilb8:fig\], but less clear on the more realist picture Fig.\[GudHilb6-12:fig\].$\bigstar$ On the other hand if there is a bad cubic (one whose oval is inside the nonempty oval of the $C_6$) then we know (since at least Zeuthen 1874 [@Zeuthen_1874]) that the oval of the cubic is convex. This is to mean that whenever we join two points inside the oval by the rectilinear segment inside the oval it stays entirely inside the oval. This is not well phrased and should be formulated by saying that whenever we take 2 points inside the cubic-oval the line through it dissected in 2 pieces by the 2 intersection points with the oval is such that the half not meeting the pseudoline is entirely within the inside of the oval.

So Rohlin’s claim would follow if it can be shown that the fundamental hexagon of our $C_6$ of type $\frac{6}{1}2$ is non-convex. Here the fundamental hexagon is defined as the union of all fundamental 2-simplices (triangles). Recall that given 3 points on the inner ovals (inside $N$) the lines joining them are Bézout-saturated and there is a unique full-triangle traced inside $N$, which we call fundamental. The fundamental hexagon of our $C_6$ (with $8$ marked points $p_i$ on the empty ovals) is the union of all these fundamental triangles rooted at the 6 inner points.

Another strategy is as follows. Choose any 8 points on the sextic $C_6$, one on each empty oval. To show: the pencil of cubics through them is totally real.

[*Step 0*]{}.—The sole obstruction to total reality is the presence of a bad cubic, i.e. one with an oval entirely traced inside the nonempty oval of $C_6$.

[*Step 1*]{}.—Assume that there is a bad cubic then the 6 inner points are hexagonally distributed on the oval.

[*Step 2*]{}.—Imagine the 6 inner points black and the 2 outer points white-colored. Then try to infer existence of a conic $C_2$ through 3 inner points and the 2 outer points which is dichromatic, i.e. such that the 2 white points split the 3 black points in 2 groups.

[*Step 3*]{}.—Such a conic has 4 transitions from black to white, hence cuts the $C_6$ in $10+4=14>12=2\cdot 6$ violating Bézout.

The difficult step is Step 2. To exhibit a dichromatic conic it suffices by Le Touzé’s lemma (\[LeTouze:lem\]) that some fundamental triangle through 3 black points separates the 2 white points.

So by contradiction assume that all black triangles does not separate the 2 white points. But alas I do not know why this circumstance (which is actually forced by Le Touzé’s lemma) implies a contradiction with the bad cubic assumption.

Yet another strategy via long run evolution of the bad cubic
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\[19.02.13\] In this section we explore another strategy toward a proof of the Rohlin-Le Touzé claim of total reality for Rohlin’s curve $\frac{6}{1}2$. The argument perhaps adapts to its mirror $\frac{2}{1}6$, yet we concentrate on $\frac{6}{1}2$ for simplicity. As above, we rather attack the somewhat weaker total reality assertion for a pencil with basepoints assigned [*on*]{} the ovals of the curve. By letting basepoints degenerate on the ovals, this is probably logically implied by the Rohlin-Le Touzé’s theorem (of which at the time of writing we have not seen a proof). It seems also that Le Touzé proves rather the case of the mirror $\frac{2}{1}6$ but probably her argument adapts to $\frac{6}{1}2$. Our argument is just a strategy far from a complete proof, trying to study the dynamical evolution of a pencil lacking total reality while hoping to detect a contradiction. So it is a dynamical approach, but perhaps the real proof (of Rohlin and Le Touzé) is more clear-cut or based perhaps on the same idea.

Start by recalling certain trivialities, which we repeat for convenience. In all this section, $C_6$ denotes a “Rohlin curve” of type $\frac{6}{1}2$, i.e. 6 ovals enveloped in a larger one with 2 ovals outside. The 6 ovals are said to be inner ovals and the 2 ovals outside called outer ovals.

A pencil of cubics passing through $8$ basepoints injectively distributed on the $8$ empty ovals is said to be deep.

The following statement (for me still hypothetical) is a variant of the Rohlin 1978–Le Touzé 2013 theorem, and probably weaker than it, yet which seems to us easier to prove as there is not the possibility of microscopic ovals passing through the assigned basepoints yet without creating real intersections. Albeit weaker it is sufficient for detecting the type I of the given scheme, and in some sense stronger as it yields circle maps (or totally real maps) of smaller mapping-degree.

Any deep pencil on a sextic $C_6$ of real scheme $\frac{6}{1}2$ is totally real, i.e. any (real) curve of the pencil cuts only real points on the $C_6$.

The sequel is an (unsuccessful) attempt of proof of this (hypothetical) statement.

It is divided in several steps, each justified in the subsequent paragraph.

$\bullet$ [*Step 1*]{}.—Each cubic $C_3$ of such a deep pencil has at least 16 (real) intersections with the $C_6$.

Indeed $C_3$ is assigned to pass through 8 points $p_i$ distributed on the 8 empty ovals of $C_6$. Two cases can occur. Either the cubic is tangent to the sextic at $p_i$ in which case we have intersection multiplicity $2$, or the $C_3$ is transverse to $C_6$ in which case there is through $p_i$ a small analytic arc of the $C_3$ with extremities both inside and outside the corresponding oval of $C_6$. By basic properties of algebraic projective curves, this arc of curve has to close up itself and so a 2nd (real) intersection with $C_6$ is created. A priori $C_3$ could visit $p_i$ via just a solitary node (isolated real ordinary double point). In that case the intersection multiplicity is still $2$, and by the way I suspect that this case cannot occur by elementary properties of pencils which have a foyer-type singularity at the basepoints preventing an isolated singularity to appear there. (All this is clumsy due to a lack of profound algebro-geometric knowledge of the writer.)

Step 1 shows that we are quite close to total reality, where [*each*]{} cubic curve is required to have 18 real intersections (counted by multiplicity) with the sextic $C_6$.

$\bullet$ [*Step 2*]{}.—The sole obstruction to total reality is the presence of a [*bad cubic*]{}, i.e. a smooth cubic with 2 components whose oval is contained in the nonempty oval $N$ of the $C_6$.

If the cubic $C_3$ is connected (i.e. $C_3(\RR)$ is connected), then as it must visit both inner and outer points a 17th intersection is created and the 18th follows either by algebra or topology. If $C_3$ is not connected then it is either smooth with 2 components, or a solitary cubic with a solitary node. In the latter case the solitary node passes at most through one of the eight $p_i$ (though this is improbable), yet even in that case the pseudoline of the solitary cubics visits both inner and outer points so has to be total. Hence the sole curve possibly failing total reality is a smooth cubic with 2 real branches. It has further to be monochromatic in the sense that the outer and inner points $p_i$ have to be “purely” distributed on both real circuits of the $C_3$. Else if both an inner and an outer point among the $p_i$ land on a same circuit of $C_3$ then a 17th intersection is created by topology, and so an 18th one by algebra. Further if the inner points are on the pseudoline of $C_3$, then topology forces a 17th intersection (else the pseudoline would be contractible inside the bounding disc of the nonempty oval $N$). So the inner points are on the oval of $C_3$, and Step 2 is completed.

So from now on we shall assume that our deep pencil contains a bad cubic $C_3$, and try to infer a contradiction. Several basic remarks are perhaps useful.

1\. The unique oval of a cubic with 2 components is convex in some obvious sense. (Perhaps this already implies a contradiction, but need to be detailed.)

2\. The oval of our bad cubic $C_3$ will vibrate during an infinitesimal motion along the (deep) pencil $\Pi$. As $6$ basepoints are assigned on the oval $O$ of $C_3$, a vibratory (slaloming) principle implies that the oval oscillates an even number of times across itself. (Of course this may also be reduced to homological intersection mod 2.) It follows that the 9th basepoint of the pencil $\Pi$ is located on the pseudoline of the bad cubic $C_3$.

Now our strategy is the naive one of studying the long-run evolution of the bad cubic as time evolves, i.e. as the cubic is dragged along the pencil. Probably the real argument of Rohlin-Le Touzé is more clear-cut Bézout-style obstruction without dynamical process.

So what may happen to our bad cubic as time evolves? The discriminant of plane cubics has alas even degree $3(m-1)^2=12$ for $m=3$ (or more generally when $m$ is odd) so that we cannot infer presence of a singular curve in the pencil for basic degree reasons. Yet there is surely a deeper argument either like Klein-Marin (1876–1988 [@Marin_1988]) or via Poincaré’s index formula (1885) prompting the existence of a connected curve in any pencil of cubics. Cf. e.g. (\[nodal-cubics-8-many-in-a-pencil:lem\]).

Accordingly two scenarios may occur when the bad cubic is dragged along one of the two possible sense along the real locus of the pencil:

SC1.—The bad cubic has its oval coalescing with its pseudoline.

SC2.—The bad cubic sees its oval shrinking to a solitary node.

Of course SC2 seems unlikely since the oval of the bad $C_3$ passes through the 6 inner points so a shrinking looks impossible at least in the near future of $C_3$. So SC1 is the first thing to occur when the bad cubic is propagated along the deep pencil.

A qualitative picture (without high precision tracing instrument) may give something like Fig.\[Total-qualitative:fig\]a showing the coalescence of the oval of the bad (black) cubic with its pseudoline via transition through a nodal cubic (in red). On tracing naively the next lilac curve one seems to get a corruption with Bézout as the lilac curve seems intersecting 4 times the horizontal line. This is fairly naive and there must be ways to avoid such a trivial accident.

Another optical illusion is the following. On looking Fig.\[Total-qualitative:fig\] one may get the impression that in the transition from the red curve to the lilac one along the segment $A,B$ the cubic must necessarily split off the line $A,B$ (and accordingly a so-called residual conic $C_2$). If so, then the 6 remaining (assigned) basepoints have to lie on the residual conic $C_2$ which intersects $12+2=14$ times the $C_6$, since 2 bonus intersections are forced with $N$ (by dichromatism). (Note also to complete the argument that none of the 3 (assigned) basepoints can be aligned as then we get $6+2=8>6$ intersection of $C_6$ with a line.)

However on zooming (violently) the segment $A,B$ one arrives at Fig.\[Total-qualitative:fig\]b showing a transition from red to lilac by an undulating family of (qualitative) cubics respecting Bézout (at least as far as the intersection with line $A,B$ is concerned). During this undulation no splitting off of a line is forced.

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A similar depiction could settle the pseudo contradiction with Bézout of Fig.a (involving the line $C,D$ and the lilac curve). This is suggested on our loose picture Fig.c. In reality nobody tell us that the picture is like this, being possibly rather like Fig.d or even different. It is clear at this stage that the argument becomes much involved if possible to complete at all. Philosophically the drawback of our strategy is that it is indirect (by contradiction). One could dream of a direct argument, but this surely requires different ideas. Our indirect argument requires solid consolidations perhaps by enumerating carefully the several Morse surgeries implied by the evolution. By genericity those could be assumed of elementary type (uninodal curves only). Also during the time the oval of the bad cubic stays an oval, its expansion seems, by convexity, confined within the fundamental triangle of Fig.a. Finally, in the limit when we encounter the first nodal curve of the pencil, the inside of this loop (which is also the geometric limit of the insides of the ovals past the bad cubic) has also to be convex and therefore contained in the fundamental triangle of Fig.a, plus its companion (forming a “David star”).

Another strategy to Rohlin-Le Touzé’s phenomenon
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\[21.02.13\] (based on hand-notes of the past 3 days).

We consider again a sextic $C_6$ of type $\frac{6}{1}2$. We distribute $8$ points on the empty ovals of the $C_6$. The phenomenon in question claims that the pencil of cubics through those 8 points is totally real, i.e., each real curve of the pencil cuts only real points on $C_6$.

First one notices that each curve of the pencil (denoted $\Pi$) cuts at least 16 points on the empty ovals. (Here and in the sequel, intersections are always counted by multiplicity.) Denote by $N$ the nonempty oval of $C_6$.

If the pencil $\Pi$ is not totally real, then it contains a bad cubic $C_3$, i.e. such that $C_3(\RR)\cap
N=\varnothing$.

If all cubics of $\Pi$ cut $N$ then all have 2 extra intersections located on $N$, and so the pencil is totally real.

Such a bad cubic is necessarily smooth, because singular cubics are either connected or have a solitary node, but in the latter case the real pseudoline connects inner and outer points so an interception of $N$ is forced by continuity.

Assume (by contradiction) that there is a bad cubic in $\Pi$. One idea is to look at the future of this bad conic along the pencil $\Pi$. One can introduce the projection induced by the pencil as the map $$\pi\colon C_6 \to \Pi$$ taking a point of the curve to the unique curve of the pencil passing through it. $\pi(N)=:G$ is the set of good conics, whose complement is $B$ the set of bad conics. Under our assumption that $B\neq \varnothing$, it is clear that $G$ is a (compact) interval in the circle $\Pi$. In fact as the pencil is defacto nearly total with 16 real intersections (over the 18 maximum permissible), the map $\pi\colon N\to G$ is two-to-one. Hence given $C_3$ our (initial) bad cubic we may let it degenerate toward one of the 2 extremities of the interval $G$ along 2 pathes consisting only of bad cubics safe for their extremities. As only 2 extra intersections are possible, this may occur in 3 fashions only (by a simple continuity argument):

\(I) Inner touch: the oval of $C_3$ inflates inside $N$ and ultimately touch it from the interior.

\(D) Double touch: the oval of $C_3$ inflates from inside and collides with the pseudoline of $C_3$ on a point of $N$.

\(O) Outer touch: the pseudoline of $C_3$ touches $N$ (necessarily from outside) while the oval stays inside $N$ disjoint from it.

Further when dragging the curve along $\Pi$, at some stage (first touch or contact) two real points eventually appear on $N$, and subsequently move apart along $N$ (without possible return by the property of linear systems or holomorphic maps) to merge again on the opposite first contact of $C_3$ with good cubics. This looks attractive but is probably only a first step toward a contradiction. In fact a simple picture (Fig.\[Tot1:fig\]a) shows that such a scenario is perfectly permissible, topologically at least.

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This figure suggested another idea as follows. While the above picture (Fig.\[Tot1:fig\]a) is topologically legal, the thick traced blue curve seems to violate Bézout upon tracing a line through its node intercepting it 4 times. A tactic would be to argue by Poincaré’s index formula (applied to the inside of the egg $E$, i.e. the unique oval of our bad cubic $C_3$) that there is necessarily such a nodal curve in the pencil (with node located inside $E$), and by some messy combinatorial argument such a curve would necessarily corrupt Bézout, heuristically because it has to visit too many points forcing high-contortion like the thick blue curve above.

As to the Poincaré argument, look at the inside $E^{\ast}$ of the egg $E$ with the (mildly singular) foliation induced by $\Pi$ and double it to a sphere, $2E^{\ast}\approx S^2$. We see on the boundary (doubled!) 6 foyers of index $+1$ (locally like the pencil of lines through a point). A priori there could be centers (locally like concentric circles) with index $+1$ and arising from a solitary cubic. Finally nodal cubics (with non isolated ordinary singular point) contributes for (hyperbolic) saddles (locally like the levels of $x^2-y^2$) which are of index $-1$. Poincaré’s index formula tells the sum of indices being equal to the Euler characteristic of the manifold. Hence $6-2 S\ge \chi(S^2)=2$, where $S$ is the number of saddles, and we deduce that there is at least two of them inside $E$. (As $E$ is smooth they cannot be located on the boundary of $E^{\ast}$.)

The above programme sounds good (albeit requiring alienating combinatorics!) until the moment, one realizes that a nodal cubic is able to salesman-travel through the 6 basepoints on $E$ without being contorted. Remember at this stage that the 8 (assigned) basepoints of $\Pi$ determine (by Bézout) a 9th one, which for vibratory reasons has to be outside $N$ (otherwise a slight perturbation of $E$ would intercept an odd number of times $E$, violating the depiction or if you prefer homological intersection modulo 2.) Of course if the 9th unassigned basepoint lands on $N$ then total reality is evident.

Let us now depict such a nodal cubic able to visit the 6 inner points without being contorted (Fig.\[Tot1:fig\]b). \[22.02.13\] On the latter all the (rational) nodal cubics occurring in the pencil have relatively decent looks. To formalize the lack of contortion of such a cubic one can uses the pencil of lines through the node which cuts a group of 3 points with 2 of them statically monopolized by the node while the 3rd moving along the curve. So when one looks from a nodal cubic from its node one always see at most (an in fact exactly one) point forced to be real. Our idea was that at least one of the nodal cubics (ensured via Poincaré’s index formula) would be contorted, i.e. violating this tightness of nodal cubics, yet our Fig.\[Tot1:fig\]b gives little hope to complete this.

Another strategy also jeopardized by the above pictures (Fig.\[Tot1:fig\]) is that there ought to be always a cubic of the pencil which is dichromatic in the sense that the 6 inner points (black colored) and 2 outer points (white colored) are lying mixed on some suitable cubic of the pencil $\Pi$ with the 2 white points separating the collection of all 6 black points. If so is the case, 4 extra intersections are gained on the nonempty oval $N$, and Bézout is violated. Perhaps this strategy is the right one but requires more geometrical argument à la Le Touzé.

Yet another idea is that by using the nodal cubics of the system we may infer that the outer basepoints are strongly stretched apart, while by contrast Le Touzé’s lemma (chromatic law for conics, cf. \[LeTouze-Gabard-Hilfssatz:lem\]) forces them to be much condensed, in the sense of not being separated by any triangle through any triplet among the 6 inner points. Remember (from Le Touzé’s Sec.\[LeTouze:sec\]) that if a separation occurs then the conics through the 3 corresponding inner points and the 2 outer points is dichromatic (with the 2 white points separating the 3 black points) so that the corresponding conic has $10+4$ intersections with $C_6$ (violating Bézout).

Another idea is that since the 9th basepoint is outside $N$ (for the vibratory reasons already explained), all cubics of the pencil have to oscillate about those 3 points. This is perhaps incompatible with the tightness of (rational) nodal cubics.

Yet another idea was that the 9th basepoint of our pencil $\Pi$ (almost canonically assigned to the $C_6$) has to land inside $N$ and this would contradict the vibratory properties of a bad cubic. However this miraculous property looks logically much stronger (i.e. not logically equivalent) to the Rohlin-Le Touzé total reality claim, so that this is perhaps not a realistic strategy, at least we were not able to implement it.

Maybe what is required is an avatar of the chromatic law for cubics instead of the version for conics (Lemma \[LeTouze-Gabard-Hilfssatz:lem\]). The logics would be as follows. Trace the “diamond” of all $\binom{6}{2}=15$ lines through the 6 inner points. By the chromatic law for conics, this diamond does not separate the 2 outer points. So by a hypothetical chromatic law for cubics it could follow that the pencil of cubics through the 8 points is dichromatic, i.e. contains a dichromatic cubic. The latter would overwhelm Bézout. Of course all this if it works should use the assumption of a bad cubic which implies an hexagonal (convex) distribution of the 6 inner points on the egg-shaped oval $E$ of the bad cubic.

The bottom foliation (Fig.\[Tot1:fig\]c) extends the right-part (Fig.b) of that figure (while changing slightly the colorimetry), and shows again that there is no topological obstruction in the large. So it seems that the contradiction (if it exists, i.e. if Rohlin-Le Touzé are right) must really involve some deeper geometry (presumably at the level of Bézout, or maybe Cayley-Bacharach, Jacobi, etc.).

Of course our global picture shows some new nodal cubics which are highly contorted, for instance the thick-blue curves. Reminding tightness of nodal cubics, the inside of the loop of that cubic must be convex. By the [*loop*]{} of a nodal cubic we mean the unique arc joining the node to itself via the half which is null-homotopic in $\RR P^2$. This being said, we may from the node of the blue-thick curve $B_3$ trace a rectilinear segment joining the top-point of the loop of $B_3$, and lying entirely in the inside of the loop of $B_3$ (cf. dashed line on Fig.\[Tot1:fig\]c). This segment which is linear (despite the appearances!) cuts for topological reasons at least 4 times the lilac-colored cubic, hence Bézout is corrupted, and perhaps the Rohlin-Le Touzé theorem is nearly proved.

Let us formalize the argument. Consider the pencil $\Pi$ of cubics through 8 basepoints (injectively) distributed on the 8 empty ovals of the $C_6$. If $\Pi$ is not totally real, there is a bad cubic $C_3$ whose real part is disjoint from $N$, the nonempty oval of $C_6$. Denote by $E$ the unique oval of this bad cubic which is necessarily smooth. By applying Poincaré’s index formula to $E$ (or the double of its inside) we infer that there is at least 2 saddle points inside $E$. On applying it to $\RR
P^2$ we infer that there is at least 8 saddle points on the whole projective plane. Such saddle points correspond to nodal cubics (and perhaps it is convenient to assume some genericity of the pencil after dragging slightly the 8 assigned basepoints).

By Bézout recall that $N$ is at most intercepted twice by each $C_3$ of the pencil, and actually exactly twice for each cubic which is not bad (i.e. which intersects $N$). Accordingly we get an involution with 2 fixed points on $N$ (namely the first contact of the bad cubic with good conics). This permits to fold the boundary of $N$ to get a topological sphere. Now depending on whether the first contact with good cubics are inner touch, or outer touch, or double touch (as discussed earlier) we get by the folding different type of singularities. Specifically an inner touch induce no singularity, and so do a double touch, while a outer touch induces a center (do some simple local pictures to get convinced).

So we may apply Poincaré inside $N$ (folded) and deduce that there is at least 4 saddles inside $N$ (in accordance with the picture) and perhaps at most 6 saddles (compare picture or think hard). All this to ensure that there is at least one saddle outside $N$ and the corresponding nodal cubic ought to have always a loop enveloping 4 transverse arcs of another nodal cubic with inner node (as on the picture). Remind that the existence of a lilac-colored cubic seems to be forced by Bézout. If all this works then we are finished and the general case is so-to-speak always reducible to the one depicted.

Of course we need to be slightly formal (and clever) for instance by defining the concept of a barred-pair of nodal cubics (or barrage for short). This is a pair of nodal cubics such that one of them appears 4 times inside the loop of the other. (For an example cf. again the thickest curves of Fig.\[Tot1:fig\]c.) It remains then to show that such a barred pair always exists, which requires some abstract self-confidence in combinatorics or a long discourse. Note on the picture (at least) that if we consider instead of the thick lilac curve the red one then there is also a barrage consisting of 4 disjoint arcs inside the loop of the blue curve. Hence the proof could decompose in the following 2 steps:

Step 1.—Show that there is always a nodal cubic whose loop visits all the 8 assigned basepoints.

Step 2.—Show that there is always another nodal cubic forming a barrage w.r.t. the nodal cubic of step 1, i.e. which appears in the inside of its loop as 4 pairs of transverse arcs joining the 8 basepoints in pairs.

This is perhaps a universal property of pencil of cubics (or maybe valid only in our special situation, where the 8 basepoints are on a $C_6$, with six of them hexagonally distributed of the convex egg of the bad cubic $C_3$). Universality would be better as then the proof could be simpler, but this looks too optimistic for in that case pencil of cubics would just not exist. Further if Step 1 looks too hard, one could imagine other types of barrages like the one depicted on the 3rd row of Fig.\[Tot1:fig\]d.

We hope that \[all\] this \[mess\] can be made clearer and perhaps there is a simpler argument (maybe Le Touzé’s proof).

Albeit difficult to make formal the above proof (if it is one!) shows the special rôle played by nodal cubics in the pencil which have lowest complexity from the viewpoint of algebraic geometry. Those are perhaps the unique “brèche par laquelle on puisse entrer dans une place réputée jusqu’ici imprenable”.

Doubling, Satellites and total reality {#satellite-total-reality:sec}
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\[23.02.13\] As discussed at length, Rohlin-Le Touzé’s theorem is somewhat elusive to prove but let us assume it to be correct. Why is it so important? Why is it fairly difficult to prove? How does it generalize? As a last remark we note that any proof using the bad cubic tends to be indirect, and this makes any proof a bit frustrating. One could dream of a direct proof using maybe the fact that any cubic of the pencil is dichromatic in the sense of having both inner and outer points one the same component of the cubic. This would give a direct proof but of course still much remains to be justified.

Though quite unable to complete the proof, we may try to speculate of what comes next, and what is the true phenomenology governing such phenomena of total reality.

According to Ahlfors theorem (1950 [@Ahlfors_1950]) what is behind total reality is basically the orthosymmetric character of the curve. More concretely (or in the spirit of Rohlin 1978 [@Rohlin_1978]), total reality seems to be sometimes forced by the sole knowledge of the real scheme. For instance, we have the prototype of the deep nest of depth $k$ and degree $2k$ which is totally real under a pencil of lines. The point here is that topology forces so many intersections as algebra permits whence total reality. Idem for a quadrifolium nest consisting of 4 nests of depth $k$ and degree $4k$ which is total under a pencil of conics assigned to pass through any 4 points distributed in the deepest ovals.

Modulo technicalities, some higher intelligence should be able to perceive total reality of Rohlin’s sextics with the same ease as in the above two examples. Of course certain aspects changes radically, like the 9th unassigned basepoint, as well as the issue that cubics are possibly disconnected, concomitantly with their irrationality, or positive genus of the underlying Riemann surfaces. Is this a sufficient reason to mistrust the ubiquitousness of the phenomenon of total reality, say as (partially) evidenced by Ahlfors theorem at the abstract level?

Typical to the basic cases of total reality—sweeping of deep nests via pencil of lines through a deep center of perspective, or the vision of quadrifolia through conics—is some concentric paradigm, namely an infinite series of species totally real under the same pencil. So one can start from a conic and imagine its unique oval (unifolium) doubled, then tripled, etc., and so we get the series of deep nests. The same “satellitosis” occurs by starting from the quadrifolium quartic and doubling each of its ovals in a tube neighborhood, to get an octic totally real, a twelve-tic, etc.

[Given a real scheme $S$ (of degree $m$) with only ovals (=nullhomotopic curves) we may abstractly define its [*$k$th satellite*]{} by replicating each oval up to a certain multiplicity $k\ge 1$ and get so the scheme $k\times S$ of degree $km$. (In Rohlin’s sense, a [*real scheme*]{} is primarily an isotopy class of embedding of a disjoint union of circles plus some integer $m$ given in the background memory, the so-called [*degree*]{} of the scheme.)]{}

Note that this abstract operation can be aped algebraically just by taking an equation of even degree realizing the scheme $S$ (we assume this to be possible) and then taking $k$ nearby levels close to zero while perturbing the union to get a smooth algebraic curve realizing the $k$th satellite schemes. This makes sense because the sign of an even degree form is well-defined.

In particular we can take Rohlin’s sextic scheme $\frac{6}{1}2$ and double it (second satellite) to get the scheme $2 \times
\frac{6}{1}2$ of degree 12, or triple it, and so on. It seems clear that this satellite is totally real under the same pencil of cubics as in Rohlin’s (unproven) phenomenon. This is evident when the satellite is realized as a small algebraic perturbation of parallel levels, because in that case we have on the original sextic curve a foliation transverse to the real locus \[\[10.04.13\] this is a bit sloppy but nearly true\], and transversality is topologically stable. So we get an infinite series of curves of type I (as forced by total reality), and it is likely that not merely the algebraic satellites are of type I but all the curves belonging to the schemes. This would be the case if the schemes were known to be rigid, i.e. each forming a unique rigid-isotopy class (\[rigid-scheme:defn\]). More pragmatically, the fact that total reality is exhibited by a synthetic procedure (namely by assigning 8 basepoints on the 8 empty ovals of the $C_6$ of Rohlin’s type or over any satellite of this scheme) makes that we have some robust recipe ensuring total reality. So it is likely that Rohlin-Le Touzé’s theorem implies the following:

[(Hypothetical)]{}.—Any $k$th satellite of Rohlin’s schemes of degree $6$ (they are 2 of them namely $\frac{6}{1}2$ and $\frac{2}{1}6$) is again total under a pencil of cubics and so of type I. In particular the $2$nd satellites of Rohlin’s schemes are schemes of degree $12$ which are of type I.

With some good faith (or pessimism) one could fear that this implies a corruption of Rohlin’s maximality conjecture (type I implies maximal). The idea would be to take a fairly complicated configuration of 6 ellipses and smooth it à la Brusotti to get a curve whose scheme enlarges $2\times(
\frac{6}{1}2)$. Fig.\[Tot2:fig\] includes inconclusive attempts along this naive tactic.

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Adhering to the opposite attitude, the 2nd satellite of any one of both Rohlin’s $6$-schemes are $12$-schemes (denoted $2\times R$ or just $2R$, cf. Fig.\[Tot2:fig\]a) which are totally real in some geometric way (pencil of cubics through the 8 empty ovals). Hence it is likely that those schemes cannot be enlarged without corrupting Bézout. More precisely assume a real $12$-scheme $S$ enlarging $2R$, then select in $S$ a replica of $2R$ and construct the allied total pencil. Let pass a curve through one of the deleted oval of $S$, and get a corruption with Bézout.

More generally if a scheme is of type I, one may expect its representing curves to be totally real under a pencil of curves in some geometrically controlled way. This posits both a concretization of Ahlfors theorem as well as an extension of Rohlin-Le Touzé’s theorem. The byproduct would be a general proof of Rohlin’s maximality conjecture. At this stage, we confess to have first understood the full swing of Rohlin’s prophetical allusion when formulating his maximality conjecture: “there is much to say in its favor” (cf. Rohlin 1978 [@Rohlin_1978]). This idea will be developed in the next section.

We can also look at the 3 possible $M$-sextics permitted by Gudkov’s classification and take their satellites to get schemes of type I, actually total under a pencil of quartics via (\[total-reality-of-plane-M-curves:thm\]). The emerging philosophy is that the phenomenon of total reality should be stable under satellitoses and possesses a series of minimal (or primitive) models in each degree. Pencils of lines correspond to deep nests. Pencils of conics (with 4 real basepoints) correspond to the quadrifolium quartics and its satellites. Pencil of cubics have two minimal models with Rohlin’s sextics. Pencil of quartics have (at least) 3 minimal models given by the $M$-sextics (cf. Theorem \[total-reality-of-plane-M-curves:thm\]), etc. All this is quite vague and need perhaps strong correction, but our intention is to suggest the idea of a big tower of total pencils, of which the Rohlin-Le Touzé phenomenon should just be one of the very first cornerstones supporting a big cathedral. Admittedly the latter may reach such altitudes, that its higher structure is still completely dissimulated behind the clouds.

The motive behind total reality seems to be a topological predestination forcing reality of all intersections. So the phenomenon ought to be fairly robust. Now if we are given a scheme of type I, then any curve representing it is totally real by Ahlfors theorem. \[$\star$ Not even obvious!\] Yet to attack Rohlin’s maximality conjecture (RMC) we need more namely total reality forced by topological reasons. This amounts essentially to a synthetic knowledge a priori of the location of the basepoints. In this case let us say that the scheme is photovoltaic, more precisely:

A real scheme is photovoltaic (PV) if there is a canonical recipe($\approx$algorithm$\approx$Turing machine) exhibiting a total pencil of curves on it. When the recipe is as simple as saying “by assigning basepoints on the empty ovals” of any representing curve of the scheme, the scheme is said to be photographic.

We have “photovoltaic” implies “type I”, and even “photovoltaic” implies “maximal”. Of course the problem is that our “canonical recipe” is poorly defined, but one may just understand some algorithm. For instance Rohlin’s $6$-schemes are photographic by the Rohlin-Le Touzé’s theorem, while the $M$-schemes of degree 6 are photovoltaic since there is an algorithm to construct a total pencil via some residual series (cf. Theorem \[total-reality-of-plane-M-curves:thm\]). \[$\star$ But compare also (\[Le-Touzé-extended-in-odd-degree:scholium\]) showing that, in odd degree at least, there is a more concrete recipe for the total reality of $M$-schemes.\] One chance to go around the conceptual difficulty of the ill-posedness of our definition would be the following miracle:

All schemes of type I which are not $M$-schemes are actually photographic.

\[$\star$ Again in view of (\[Le-Touzé-extended-in-odd-degree:scholium\]) it is likely that $M$-schemes have not to be excluded.\]

This is true for sextics (granting the Rohlin-Le Touzé theorem), and deserves to be investigated in general.

If the conjecture is true in general, type I implies photovoltaic (by virtue of Theorem \[total-reality-of-plane-M-curves:thm\]), and hence maximal, and Rohlin’s conjecture would be settled.

Of course we are using the implication “PV” implies maximal. It looks hard to prove it because “PV” is ill-defined, but we really may avoid this concept since $M$-schemes are automatically maximal (Harnack 1876), while the other are photographic (by the conjecture) so that Rohlin’s maximality conjecture follow form the:

(Hypothetical!!!) If a scheme is photographic then it is maximal.

 \[$\star$ too vague!\] Suppose by contradiction that $S\subset E$ is an enlargement of the photographic scheme $S$. Choose $E_m$ a real curve representing the scheme $E$, and select a sublocus $\Sigma_m$ of $E_m$ realizing the scheme $S$. Alas we loose algebraicity doing so. However this sublocus $\Sigma_m$ is total under a pencil of curves with basepoints assigned (say) on the empty ovals of $\Sigma_m$ for “robust” topological reasons. This is to mean that any curve of the pencil of $k$-tics cuts $k\cdot m$ points on $\Sigma_m$ for topological reasons (e.g., like for the deep nests). Then we could conclude to a contradiction with Bézout by letting pass a curve of the pencil through the extra oval of $E_m$.

The above clumsy proof imposes a refinement of the definition making the above lemma true with “photogenic” instead of the “photographic” assumption.

An $m$-scheme is photogenic if any (differentiable, or real analytic) curve $\Sigma$ representing it admits a “total” pencil of $k$-tics such that each curve of the pencil cuts at least $k\cdot m$ points on $\Sigma$. It may even be assumed that the basepoints of such a pencil are assigned in the insides of the empty ovals.

This “photogeny” is a violent evasion outside the algebro-geometric realm, yet the deep nest as well as the quadrifolium of depth $k$ are photogenic schemes in this sense. It would be interesting to know if the Rohlin’s $6$-schemes are photogenic amounting to say that Rohlin-Le Touzé’s theorem accepts a purely topological proof. This seems already quite unlikely, and the right part of Fig.\[Tot3:fig\] supplies a simple counterexample. Here we consider a smooth cubic curve $C_3$ and triad of lines $D_3$ (both black colored). We trace (in the smooth category) the blue curve $C_6$ realizing the $6$-scheme $\frac{6}{1}2$ with 8 small ovals about 8 of the 9 intersections of the cubics, plus one large oval enveloping the oval of the smooth cubic. Now the pencil of cubics spanned by $C_3$ and $D_3$ may be interpreted as the pencil of cubics assigned to pass through the insides of the 8 empty ovals of the flexible curve $C_6$, yet it fails to be total as $C_3\cap C_6$ has only 16 points (and not $18$ the product of their degrees).

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So we cannot expect to be so naive as to be photogenic. This relates to the fact that pencil of cubics (or higher order curves) generally contains disconnected curves. One crude way to ensure total reality could be to use degenerate pencils lying entirely in the discriminant and more than that consisting only of rational curves (forcing via Harnack-Klein or less, like Lüroth-Clebsch, or Cayley) the curve to be connected. However this looks overspecialized and probably not even suited to detect the universal orthosymmetry of Rohlin’s $6$-schemes.

It remains to clarify several aspects. Is total reality stable under satellites? In particular is there an infinite series of examples above Rohlin-Le Touzé’s phenomenon of total reality. What are the higher order avatars of Rohlin-Le Touzé’s theorem, and how frequent is the phenomenon? More precisely which schemes are photographic? This looks of course extremely hard requiring a highbrow extension of the Rohlin-Le Touzé’s theorem. Are photographic schemes stable under satellites? If yes this is the trivial part of an iterative propagation of each total reality phenomenon. A priori one can speculate that photographic schemes are quite rare and essentially exhausted by pencil of lines, conics and cubics. In contrast one may expect the phenomenon to be ubiquitous and so frequent that all schemes of type I (safe perhaps some $M$-schemes) \[$\star$ this proviso looks not justified anymore, cf. (\[Le-Touzé-extended-in-odd-degree:scholium\])\] are photographic. In that case there is some little chance to tackle Rohlin’s maximality conjecture (the part thereof post-Shustin’s disproof). Alas even that looks difficult. One may also wonder how frequent are schemes of type I, again rarity versus abundance is quite puzzling.

Stability of type I under satellites
------------------------------------

\[24.02.13\] Are schemes of type I stable under satellites? The first case to test is $2\times R$ the 2nd satellite of Rohlin’s $6$-scheme $R:=\frac{6}{1}2$. Of course taking a perturbation of the double of Hilbert’s realization of $\frac{6}{1}2$ (Figs.\[GudHilb8:fig\] or \[GudHilb6-12:fig\]) it is likely that we find a dividing curve, and perhaps Rohlin-Le Touzé’s theorem is sufficiently robust as to imply universally the type I of this $12$-scheme. If not it may be that the $12$-scheme $2\times R$ is indefinite. A priori curves of degree 12 could be sufficiently messy as to allow a type II realization of the 12-scheme $2R$, or in contrast Rohlin-Le Touzé’s phenomenon could be sufficiently robust as to propagate to satellites.

The data of a curve plus a totally real pencil of “adjoint” curves is called a flash, and we say that the curve is flashed by the pencil. If a curve of even degree is flashed by a pencil then the doubled curve (and more generally its $k$th satellite) obtained by small perturbation of $k$ concentric levels is flashed by the same pencil. Note that for an algebraic satellite to be defined it is convenient to take an affine chart in which the whole curve is visible. This works certainly for Hilbert’s realization of Rohlin’s $6$-schemes. Hence it is clear that the 12-scheme $2R$ contains a representatives of type I (hence is not a scheme of type II). The question is to decide if this scheme is of type I or indefinite.

One idea could be to realize the 8 nests of depth 2 by an octic and then add two ellipses to get $2R$. However, passing a (connected) cubic through the 8 deep nests creates $4\cdot 8=32>3\cdot 8=24$ many intersections, and Bézout is much overwhelmed. Replacing the octic by a curve of degree 10 is still insufficient $(32>3\cdot 10=30)$.

A priori one could hope to find a type II realization of the $12$-scheme $2R$ by perturbing an arrangement of 12 lines. This is a bit messy to depict. The most approaching object we could trace is shown on Fig.\[Tot4:fig\]. This is rather akin to the double of the (other) Rohlin scheme $\frac{2}{1}6$, but alas there is not enough free room left to build the prescribed configuration.

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Another idea is to use Hilbert’s method, but the latter does not seem ideally suited for the generation of nest of depth 2 (Fig.\[Tot5:fig\] of very poor quality). Let us shamefully leave this delicate question, as we sincerely hope that total reality is ubiquitous (in particular stable under satellites).

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Satellites of curves of odd degrees {#Satellite-odd-degree:sec}
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[*Inserted*]{} \[16.03.13\].—It seems evident that the construction of satellites extends to curve of odd degrees. Of course there is a slight complication coming from the fact that the pseudo-line lacks a trivial tube-neighborhood, and so we cannot replicate so canonically as in the even degree case. As a simple example consider a cubic with 2 circuits (one oval and a pseudoline). Doubling its oval and “doubling” its pseudoline will lead to a curve of degree 6 which (for a suitable smoothing) will be a nest of depth 3, hence again totally real under a pencil of lines.

By analogy if we look at the next odd degree, namely 5, we have examples of total reality given by the $M$-quintics (cf. Le Touzé’s Scholie \[LeTouze-quintic:scholie\]). So when taking its satellite we are supposed to find a nice example of total reality in degree 10 for a scheme of the form $(1,6\times
1)$, i.e. 6 nests of depth 2 enveloped in a larger oval. So it is natural to conjecture that this scheme of degree 10 is of type I.

If this is possible to prove this is quite interesting because the scheme in question has $r=13$ ovals which is fairly low in comparison to Harnack’s bound $M=37$, when $m=10$ as $g=\frac{(m-1)(m-2)}{2}=\frac{9\cdot 8}{2}=9\cdot 4=36$. Of course, the type I of this scheme is not covered by the RKM-congruence for $(M-2)$-curves. So this gives a certain addendum to Rohlin’s (somehow denigrating) remark that the method of total reality is somehow subsumed to the RKM-congruence (cf. his remark in 1978 which reads “However, all the schemes that we have so far succeeded in coping with by means of these devices are covered by Theorem 3.4 and 3.5. \[i.e., the congruences\]”, compare (\[Rohlin1978-total-reality:quote\]) for the integral citation.

Of course total reality (hence type I) is also observed for satellites of the unifolium or quadrifolium having the same property of being at lesser altitude than $(M-2)$-schemes. Thus the phenomenon under examination is formally not new, but those examples being so trivial they were probably not taken seriously enough. So:

\[satellite-of-M-quintic-total:conj\] The scheme of degree $10$ of symbol $(1, 6\times \frac{1}{1})$ (cf. Fig.\[satellite-of-Harnack’s-quintic:fig\]a) arising as the 2nd satellite of Harnack’s $M$-quintic with symbol $6 \sqcup
J$ (a unique rigid-isotopy class by Kharlamov-Nikulin) is totally real under a pencil of cubics, hence in particular of type I.

To prove this we use the method of Le Touzé’s scholie, namely to assign the 8 basepoints of a cubics-pencil on the 6 ovals of the quintic plus 2 on the pseudoline. Then we have 14 intersections and the last one is forced to reality either by algebra (Galois-Tartaglia) or topology (Möbius-von Staudt).

[*Optional side remark*]{}.—It may be observed that the scheme in question is not prohibited by Rohlin’s formula. Hint: decomposes the Hilbert tree of the scheme in $x$ and $y$ many branches of length 2 (so $x+y=6$) which are resp. positively or negatively charged. By Rohlin’s formula we have $2(\pi-\eta)=r-k^2=13-25=-12$. By the signs-law (cf. Fig.\[Signs-law-dyad:fig\]) we find $\pi-\eta=x-3y$, and thus $x-3y=-6$, $x+y=6$. Eliminating $x$ gives $-4y=-12$, so $y=3$ and $x=3$. Rohlin’s equation with signs is thus soluble.

A priori Le Touzé’s total reality should adapt to the double. Formally we assign 6 basepoints on the deep ovals of the 6 nests and 2 on the maximal oval. Crudely speaking we await $6\cdot
4+3\cdot 2=24+6=30=3\cdot 10$ and total reality would be granted. However in reality we get less than that on basic topological grounds. However it is clear that for a small deformation of the doubled quintic we can expect total reality and the hope is that this propagates to the full scheme (chamber which a priori is not even known to be connected!) Maybe there is some deep reason ensuring total reality like in Le Touzé’s argument.

[*Insertion*]{} \[10.04.13\].—It also interesting to compute the mapping-degree of the allied circle map. The number of mobile points of Le Touzé’s series will be $6+12+4=22$, with 1 point circulating on each of the $6$ deep ovals, 2 on the 6 ovals immediately surrounding them and 4 moving on the doubled pseudoline (maximal oval). This degree of $22$ can be compared with Gabard’s bound $(r+M)/2=(13+37)/2=25$ and turns out to be compatible with it.

A validation of the conjecture (\[satellite-of-M-quintic-total:conj\]) could be of interest for the following reason related to Rohlin’s maximality conjecture(=RMC). If we think globally at the satellite operation and the arithmetics of small integers factorized into primes ($1,2,3,4=2\cdot 2,5,6=2\cdot
3,7,8=2^3,9=3^2,10=2\cdot 5,11,12=2^2 \cdot 3=2\cdot 6$, etc.), we remark that the first nontrivial satellite [*not*]{} totally real under a pencil of lines or conics truly arises in degree 10. Degree 9 involves the prime $3$, yet the 3rd satellite of the cubic with 2 components is merely the deep nest $4\sqcup J$ (totally real under lines). In degree 6 we have indeed the 2 total realities of Rohlin-Le Touzé yet they are primitive manifestations (not satellites), and of course in adequation with Gudkov’s classification and Rohlin’s maximality conjecture. Hence degree 10 is the first case where (granting our conjecture) we get a type I scheme which possibly is not maximal (in case we are skeptical about the truth of RMC). Of course this would be against our own philosophy that Ahlfors has much to say about Hilbert’s 16th. Yet we must keep in mind this eventuality.

Thus we can ask for a curve of degree 10 enlarging the doubled $M$-quintic (with 6 bifolia, plus one large oval, Fig.\[satellite-of-Harnack’s-quintic:fig\]a). The naive construction of Fig.\[satellite-of-Harnack’s-quintic:fig\]b does not even reach more than 4 bifolia nested in a larger oval. So other techniques of construction are demanded, perhaps Harnack, Hilbert or Viro.

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If the posited phenomenon of total reality holds true, and is sufficiently explicit to imply maximality (as nearly evident by Bézout) then our scheme of degree 10 (Fig.\[satellite-of-Harnack’s-quintic:fig\]a called say the closed sextibifolia) would be maximal in the hierarchy of all schemes of degree 10. Hence this scheme could not be enlarged and it results, in one stroke, a myriad of prohibitions upon Hilbert’s 16th problem in degree 10 (still wide open), basically by virtue of the sole idea of total reality which goes back virtually to Riemann’s thesis 1851/57, then Schottky, Klein, Bieberbach, Teichmüller, Ahlfors just to name the heros.

So the question looks nephralgic. As a philosophical detail, while in Hilbert’s 16th there is some traditional focus upon curves maximizing the number of ovals (so-called $M$-curves since Petrovskii 1933/38), we see here in contrast that the lower the number of ovals is (for a curve subsumed to total reality) the stronger will be its prohibitive impact upon the higher stages of the pyramid. Of course the prototype is the deep nest, but this is merely the trivial case.

In degree 5 there are also $(M-2)$-curves which are totally real, typically under a pencil of conics, yet the corresponding scheme is not of type I. Its double will be of degree 10 and have a bi-quadrifolium (double couche) nested in a larger oval.

Toward a census of all type I (totally real) schemes or at least extension of the Rohlin-Le Touzé’s phenomenon prompted by the Rohlin-Kharlamov-Marin congruence for $(M-2)$-schemes {#census-and-extension-of-Rohlin-Le-Touzé:sec}
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\[24.02.13\] Another problem is to list all schemes of type I, or at least those totally real under a pencil of curves. We restrict attention to even degrees schemes ($m=2k$) and call [*order*]{} the degree $d$ of curves involved in the total pencil.

In degree $m=2$ we have a single oval (unifolium, denoted $1$ in Gudkov’s notation) which is total under a pencil of lines $d=1$ with center of perspective chosen inside the oval.

In degree $m=4$, we have the nest of depth 2 (denoted $\frac{1}{1}=(1,1)$ in Gudkov’s symbolism) total under a pencil of lines, and the quadrifolium $4$ total under a pencil of conics.

For $m=6$, we have the nest of depth 3 (Gudkov symbol $(1,1,1)$) total for $d=1$, and for $d=3$ the 2 Rohlin’s schemes $\frac{6}{1}2$, $\frac{2}{1}6$ (Rohlin-Le Touzé’s theorem), as well as the three $M$-schemes $\frac{9}{1}1$, $\frac{5}{1}5$, $\frac{1}{1}9$ of Hilbert, Gudkov, Harnack respectively. The latter are total for $d=4$ (cf. Theorem \[total-reality-of-plane-M-curves:thm\]).

Before attacking the case $m=8$, some remarks are in order. What is expected is that sometimes total reality is ensured by Bézout. This is the case when $d=1$, or $d=2$ where we have the extra knowledge of the connectivity of all members of the pencil due to the rationality (unicursality) of all genus $0$ curves. This property is lost when passing to high-orders $d\ge 3$ pencils. Still we can hope a priori that some other reasons prompt total reality in some favorable cases, which would be more elementary than the Rohlin-Le Touzé’s theorem for $(m,d)=(6,3)$. It is with this naive hope that we feel encouraged to adventure in the jungle of $m=8$. Another vague motivation is that schemes of type I are conjecturally maximal, so we are really attacking a sort of simplified Hilbert’s 16th problem (or rather surfing on its upper envelope). A last remark is that we (sentimentally) expect via Ahlfors theorem that the dividing character of curves is always exhibited by a [*linear*]{} pencil inducing a map to the projective line (whose complexification is the Riemann sphere with its standard “equatorial” real structure like our planet Earth).

[*Optional (non-linear pencils).*]{}—However any totally real map to more complicated dividing curve suffices to exhibit the dividing character of the covering curve. So perhaps we should keep in mind to explore also such nonlinear pencil. How do they arise concretely is another question. A naive guess of mine was via a plane cubic $E_3$ with 2 circuits and its dual curve (variety of tangents). We could hope that any point of some curve $C_m$ determines a unique tangent to $E_3$ but alas there are (generally) 6 of them passing through a given point (intersect with the polar curve, a conic here). Perhaps a suitable adaptation of this idea leads somewhere. However it is fairly standard, and we briefly discussed this (in Part I devoted to the abstract theory of Riemann-Klein-Ahlfors) that generally speaking curves mapping to irrational curves of positive genus have specialized moduli. Hence it is quite unlikely that we shall gain a general methodology, though plane curves themselves are modularly confined.

$\bigstar$ For $m=8$, we have the deep nest of depth 4, denoted $(1,1,1,1)$ total for $d=1$, and the doubled quadrifolium $\frac{1}{1}\frac{1}{1}\frac{1}{1}\frac{1}{1}=2\times 4$ which is total for $d=2$.

$\bullet$ We examine next $d=3$ (cubics-pencils). We have then 8 basepoints available assumed all real, and so we look at curves with this number of empty ovals. Imposing the 8 basepoints on the empty ovals, we are ensured for twice so many intersections (i.e. $16$) but this is still much less than $3\cdot 8=24$ (Bézout’s upper bound). Total reality looks hard to ensure. Of course we may envelop our empty ovals by some nonempty ovals. Remember that there is at most 4 nonempty ovals (as the doubled quadrifolium $2\times 4$ is total under a pencil of conics hence maximal). So we may range our 8 empty ovals in 4 groups of ovals and consider the scheme $\frac{k}{1} \frac{\ell}{1} \frac{m}{1} n$, where $k+\ell+m+n=8$. If optimistic each of the 3 nonempty ovals contributes for 2 intersections, and we arrive at $16+6=22$ which is still less than $24$. Hence it seems nearly impossible to find a (naive) phenomenon of total reality for $(m,d)=(8,3)$, but this does not of course exclude the possibility of such a phenomenon prompted by deep geometric reasons à la Rohlin-Le Touzé. Note also that the case $m=6$ showed that there is no (absolute) total reality for $d=2$, and so we cannot expect a priori to observe the phenomenon for each preassigned order and therefore let us skip the present value $d=3$.

$\bullet$ Assume next $d=4$. We have then $13$ basepoints assignable. Remember indeed that the space $\vert 4 H \vert$ of all quartics has dimension $\binom{4+2}{2}-1=\frac{6\cdot
5}{2}-1=14$, so that 13 conditions leave the mobility of a pencil. So we are again directed toward curves with 13 empty ovals (over which we shall as usual distribute our 13 basepoints), ensuring so 26 intersections, which is less than the $d m=4\cdot 8=32$ required. Enveloping our ovals in the at most 3 possible nonempty ovals (if 4 of them we reduce to the doubled quadrifolium) we get the schemes $\frac{k}{1} \frac{\ell}{1} \frac{m}{1} n$, where $k+\ell+m+n=13$. So the number of ovals is $r=13+3=16$ and we have an $(M-6)$-scheme (as $M=g+1=22$ for $m=8$). Perhaps like in the case $(m,d)=(6,3)$ some of them are total for deep geometrical reasons. If lucky, the 3 nonempty ovals creates 6 additional intersections, so reaching $26+6=32$ Bézout’s bound, and total reality would be granted. Remember yet that for $(m,d)=(6,2)$ there is no phenomenon of total reality, at least of the purest form where only knowledge of the real scheme is required. Repeating ourselves, we cannot expect a priori that total reality prevails for each value of $d$ given a fixed $m$.

$\bullet$ Examine next $d=5$. Then $\dim \vert 5 H \vert =
\frac{7\cdot 6}{2}-1=20$ so that 19 basepoints are assignable. By the same token, we look at schemes $\frac{k}{1} \frac{\ell}{1}
\frac{m}{1} n$, where $k+\ell+m+n=19$, ensuring $2\cdot 19=38<40=d
m=5 \cdot 8$ intersections. So in fact outside from the 19 empty ovals it is enough to have one nonempty oval being intercepted to gain total reality. We list the following candidates:

$\bullet$ $\frac{k}{1}\ell$, with $k+\ell=19$ and $r=20=M-2$.

$\bullet$ $\frac{k}{1}\frac{\ell}{1}m$, with $k+\ell+m=19$ and $r=21=M-1$ so cannot be total by Klein’s congruence $r\equiv g+1
\pmod 2$.

$\bullet$ $\frac{k}{1} \frac{\ell}{1} \frac{m}{1} n$, with $k+\ell+m+n=19$ and $r=19+3=22=M$, which are $M$-curves.

In the first class of $(M-2)$-schemes we may appeal to the Rohlin-Kharlamov-Marin congruence (\[Kharlamov-Marin-cong:thm\]) to select the serious candidates. (It seems fairly plausible that this mode of reasoning was the true motivation behind Rohlin’s Ansatz of total reality for the $6$-scheme $\frac{6}{1}2$ and its mirror, and that he found his (unpublished) proof a posteriori of this deeper knowledge.) This RKM-congruence states that an $(M-2)$-curve of degree $m=2k$ and type II satisfies the congruence $\chi=p-n\equiv k^2$ or $k^2\pm 2 \pmod
8$. This looks a priori undigest, but as merely to be interpreted as a deviation from Gudkov’s hypothesis(=congruence proved by Rohlin), compare, e.g., the diagrammatic in degree 6 (Fig.\[Gudkov-Table3:fig\]). So when the congruence is violated a scheme of type I is granted. Since $\chi=1-k+\ell$ and $k^2=4^2=16\equiv_8 0$ (beware the overuse of the letter $k$ but no risk of confusion). Recall the Swiss cheese algorithm for the Euler characteristic $\chi$ (that whenever we make a hole in the sense of removing a disc, $\chi$ drops by one unit, cf. Listing-Klein-von Dyck 1888, etc.). Starting with the scheme $\frac{19}{1}$, we find $\chi=1-19=-18\equiv_8 -2$. Then we have $\frac{18}{1}1$ for which $\chi=1-18+1$ equal to the former plus 2 units, and there is always an increment of 2. Running through the full list of such schemes we find that the condition $\chi\equiv_8 4$ ensuring type I occurs with periodicity 4 for the following schemes: $$\label{octics-five-examples-RKM:eq}
\frac{16}{1}3,\quad \frac{12}{1}7,\quad \frac{8}{1}11,\quad
\frac{4}{1}15,\quad 20.$$ The latter case (of the scheme $20$) seems to disprove our collective contraction conjecture (\[CCC:conj\]).

\[CCC-conj-disproof:thm\] [(ERRONEOUS—cf. $\bigstar$ below)]{}.—The collective contraction conjecture (of Gabard positing a wild extension of the contraction conjecture of Itenberg-Viro) is false in degree $8$ already.

$\bigstar$ \[05.03.13\] [*Corrigendum*]{}.—It follows easily from the so-called Thom conjecture that the scheme $20$ is not realized by an algebraic curve of degree 8 (necessarily of type I if it existed by the RKM-congruence), compare Theorem \[Thom-Ragsdale:thm\]. So the given argument is not a disproof of CCC. \[07.03.13\] In fact a simpler obstruction of this scheme comes from Rohlin’s formula (\[Rohlin-formula:thm\]), as $2(\pi-\eta)=r-k^2$ but the left-side is zero, so $r=k^2=16$, which is no the case. \[10.04.13\] Further this scheme is also prohibited (and this was historically the first proof available) by Petrovskii’s inequality (\[Petrovskii’s-inequalities:thm\]), which reads $\chi\le \frac{3}{2}k(k-1)+1=18+1=19$. This in contrast to the proofs via Thom or Rohlin does not use the dividing character of the curve prompted by RKM. Further it should be noted that our Thom-style theorem cited above is erroneous in the generality stated, yet sufficient to imply the present application as in the case at hand the filled surface is orientable (since we only glue disc to the half, and so there is no risk to create a twisted handle like in Klein’s bottle). For definiteness, let us briefly work out the argument. Since $20$ is an $(M-2)$-curve (of type I by RKM), we may split the Riemann surface and fill one half by the $20$ discs bounding the ovals. It will result a surface of genus $1$, whose fundamental class has degree $8/2=4$. However Thom conjectured (and Kronheimer-Mrowka, and others, proved) that the genus is minimized by algebraic (smooth) curves, hence at least $3$ in degree 4. Since our surface beats this bound, the real curve $20$ is prohibited.

 \[Outdated, but keep in mind the 2nd part of the proof (strangulation argument), which under CCC would provide another obstruction of the scheme $20$, of a fairly intuitive character, though hard to implement with present technology.\] —It seems clear that this 8-scheme $20$ is realized by (a variant of) Hilbert’s method.[^69] (I should still work out this in some more detail.) The resulting curve is of type I by the just cited congruence of Rohlin-Kharlamov-Marin, which is essentially based either on the deep Rohlin’s signature theorem for spin 4-manifolds, or perhaps on Kähler geometry in the presentation of Kharlamov (unpublished?). Alternatively on the model at hand (via Hilbert’s method) the type I of this curve realizing $20$ may be checked more elementarily via Fiedler’s sense-preserving smoothing law (elementary surgeries). However the resulting curve cannot be contracted collectively by shrinking simultaneously all its ovals to points, for if it could, then $C_8\to C_4 \cup
C_4^\sigma$ would degenerate to a pair of conjugate quartics obtained by strangulating the Riemann surface $C_8(\CC)$ along all the separating ovals, and so $C_4\cap C_4^\sigma$ would consist of 20 solitary nodes. Bézout is overwhelmed.

Moreover the above 5 schemes are avatars of the total reality claim of Rohlin-Le Touzé for $(m,d)=(6,3)$, i.e. sextics flashed by cubics, while now octics are flashed by quintics. In both cases we note the rôle of curves of order 3 units less than the given degree $m$, and one seems being sidetracked to the theory of adjoint curves à la Brill-Noether, etc. Recall indeed that adjoints of order $(m-3)$ cut out the so-called [*canonical series*]{} on the given plane curve, and thus there is perhaps some conceptual reason ensuring total reality of all these linear systems. This is perhaps the royal road to attack the Rohlin-Le Touzé’s assertion/theorem (and extension thereof prompted by the RKM-congruence).

In both cases $m=6$ or $8$ we have $(M-2)$-curves which are of type I, and swept out by a pencil of order $d=m-3$. The latter cuts the canonical series of the curve $C_m$ of degree $2g-2$ and dimension $(2g-2)-g=g-2$ or rather $g-1$?? In view of Gabard 2006 [@Gabard_2006] we may expect to find a total morphism of degree the mean value of $r=M-2$ and Harnack’s bound $M$, hence of degree $M-1$. So we could choose so many points on the ovals of $C_m$ while putting two of them on the nonempty oval. It is then hoped that the 2 points situated on the same oval will [*dextrogyrate*]{} (i.e. move along one orientation of the oval without entering in collision) and then total reality is ensured.

While any collection of $M=g+1$ points on a curve of genus $g$ moves in its linear equivalence class, only special collections of $M-1=g$ points will move but perhaps this is enough to ensure total reality hence recover the type I of the above list of schemes predicted by the RKM-congruence (\[Kharlamov-Marin-cong:thm\]).

It is not entirely clear if the Rohlin-Le Touzé’s phenomenon is true in full generality or only for special groups of points (at least this is the naive intuition coming from the abstract Ahlfors and Gabard viewpoint). Note in this respect that Rohlin’s claim is a priori less strongly formulated than Le Touzé’s assertion, in claiming only that a pencil of cubic exhibit total reality and not that all of them with deeply assigned 8 basepoints are total (compare Rohlin 1978 [@Rohlin_1978 p.94] with Le Touzé’s 2013 announcement in Sec.\[e-mail-Viro:sec\]). ($\star$ \[10.04.13\]—Meanwhile see also the article [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics].) It is likely (say by analogy with the trivial case $m=4$) that Rohlin had in mind the strong assertion of Le Touzé, and that it is only the extreme compression of Rohlin’s exposition that forced him to his somewhat looser version of the statement.

For $m=8$ we may assign 19 basepoints on the empty ovals of any of the $(M-2)$-scheme listed above (the scheme $20$ is exceptional in a sense that remains to be clarified[^70]). Then we can pass a quintic $C_5$ through these points and one extra point. Choose the latter on the nonempty oval $N$ of the $C_8$. Another intersection is created by topology. So we see 21 real points. The residual intersection of $C_5\cap C_8$ consist of $40-21=19$ points, etc...

Of course the difficulty looks immense but let us postulate the following avatar of Le Touzé’s theorem (i.e. strong form of Rohlin’s total reality claim):

\[tot-real-for-octics:conj\] For any octic representing one of the $8$-schemes listed above (Eq. \[octics-five-examples-RKM:eq\]) the pencil of quintics assigned to pass through the $19$ empty ovals (or even their insides) is totally real. (Of course the scheme $20$ deserves a modified statement of which we do not know yet the exact shape.)[^71]

If so is the case we get a total morphism $C_8\to \PP^1$ of degree $40-19=21$, which is the mean value of $r=20$ and $M=g+1=22$, in accordance with Gabard 2006 [@Gabard_2006]. Of course the latter affords only weak evidence as its result is subsumed to high suspicion.

It could be expected (granting Gabard’s result as correct) that a suitable interpretation thereof (at the level of extrinsic algebraic geometry à la Brill-Noether) could supply a proof of the conjecture at least in the weak form of a special configuration of $19+2=21$ points two of them being distributed on the same oval (while dextrogyrating). However ideally we would like a purely synthetical proof say as elementary as Bézout without incursion of such transcendental philosophers like Abel-Riemann. This is perhaps possible as some highbrow variant of the Rohlin-Le Touzé’s theorem but remains to be explored and is quite likely to be extremely elusive. On the other hand it could be of vital interest that the Abel-Riemann abstract viewpoint may help to see clearer what happens in the Plato cavern of Hilbert’s 16th problem as twisted by Rohlin’s synthesis with Klein’s Riemannian viewpoint. In this optimistic scenario we may hope to get when enlarging further $m$ above $m=8$ an infinite series of schemes of type I totally flashed by pencil of $(m-3)$-curves. So the phenomenon of total reality à la Rohlin-Le Touzé would be fairly frequent (and in some sense an extrinsic reflection of Ahlfors abstract theorem).

Perhaps Le Touzé’s proof (2013) adapts to give the above the conjecture, but alas as I do not know yet the details. It can also be the case that additional difficulties occur while the combinatorics viz. geometry becomes more involved and the argument more tedious. The argument is likely to start as follows. As we have $2\cdot 19=38$ intersections granted, only two are missing to reach total reality at $40=5\cdot 8$. The sole obstruction is therefore a bad quintic $C_5$ in the pencil disjoint from the nonempty oval $N$ of the $C_8$. But this mean that there is an Abelian differential without zero on this oval, and try to derive a contradiction.

Remember that the differential has $2g-2$ zeros (Riemann-Poincaré index formula) and we may hope to infer something. When we look at the trajectories of a (generic) holomorphic $1$-form we see only hyperbolic saddles of index $-1$ explaining the degree of the canonical class as boiling down to Poincaré’s index formula (1881/85). In the case $m=6$ for the scheme $\frac{6}{1}2$ we have 8 zeros assigned each creating a companion on the same oval (so 16) and a total of $3\cdot 6=18$ zeros, in accordance with $2g-2$ for $g=10$ the genus of sextics. The above looks a numerical miracle alike, but is not for $g=\frac{(m-1)(m-2)}{2}$ so that $\deg K=2g-2=(m-1)(m-2)-2=m(m-3)$, showing that adjoint curves of degree $m-3$ are indeed involved in the canonical class. There is of course a more intrinsic reason allied to the adjunction formula.

The dream would be that there is some metaphysical/toplogical principle ensuring total reality on the basis of holomorphic 1-forms which can often be interpreted in terms of incompressible fluids. Note yet that the argument cannot be too abstract (else could imply that all $(M-2)$-curves are of type I regardless of the isotopy class, a nonsense already for $m=4$), yet ideally it would be as simple as in the case $m=4$ so $d=m-3=1$ where we really see total reality on the Gürtelkurve $C_4$ with 2 nested ovals. Here we see some potential place of action for old stuff à la Abel-Riemann-Klein (\[10.04.13\] or maybe also Thurston’s argument in Gross-Harris 1981), but alas this escaped much from my memory or my curriculum. And still there is always this objection that the argument should really use the assumption on the real scheme, and so ought to be more in the Arnold-Rohlin spirit.

Have we listed all schemes of degree 8 of type I? Probably not as our search was far from exhaustive. It remains of course to list $M$-schemes (and this is a classical still open problem for some few exceptional cases). Compare works by Viro, Korchagin, etc. It is however likely that our list is exhaustive for $(M-2)$-schemes. (\[08.03.13\] Not even true, as we shall soon see!)

$\bigstar$ What is next? Degree $m=10$ of course. Here we have the deep nest of depth 5, totally flashed by a pencil of lines. The quadrifolium $4$ does not give nothing by taking its satellite (as our $m=10$ is not a multiple of 4). Next we move directly to adjoint curves of order $d=m-3=7$ (septics). One has $\dim \vert
7H \vert= \binom{7+2}{2}-1=\frac{9\cdot 8}{2}-1=35$. So $34$ basepoints may be assigned freely, and we can force $34 \cdot
2=68<70=7\cdot 10$ nearly all points to be real by assigning the basepoints to be located on distinct ovals. Again it is a reasonably folly (by analogy with Rohlin-Le Touzé’s assertion) to expect that under adding an extra nonempty oval enveloping some of the ovals and if furthermore the RKM-congruence is satisfied that the resulting scheme (being of type I) is totally real under the described pencil. Precisely we look at the schemes $\frac{k}{1} \ell $, with $k+\ell =34 $. So we have $r=35$ ovals and $M=g+1=\frac{(m-1)(m-2)}{2}+1=\frac{9\cdot8}{2}+1=37$ is 3 units above the number of basepoints (no surprise as the dimension of the space of curves and the genus both involve the same binomial coefficient). So we are again in presence of $(M-2)$-curves. The RKM-congruence says that type II forces $\chi$ to be either $k^2=25\equiv_8 1, k^2\pm 2\equiv 3,-1\equiv 7 \pmod
8$ so that $\chi\equiv_8 5$ forces type I. Applying the Swiss cheese recipe to $\frac{34}{1}$, we find $\chi=
1-34=-33\equiv_8 -1 $ and then running through all subsequent schemes $\frac{33}{1}1$, etc., $\chi$ always increases by two units. So we first met $\chi=5$ for $\frac{31}{1}3$, and find using fourfold periodicity the following list of schemes (potentially totally real): $$\label{RKM-schemes-deg-10:planar:eq}
\frac{31}{1}3, \quad\frac{27}{1}7, \quad\frac{23}{1}11,
\quad\frac{19}{1}15, \quad\frac{15}{1}19,
\quad\frac{11}{1}23,\quad\frac{7}{1}27,\quad\frac{3}{1}31.$$ (Like for $m=6$ (but unlike the case $m=8$) the list is symmetrical under the evident mirror of partnership in the jargon of Kharlamov-Finashin.) Again we expect the following total reality:

All curves $C_{10}$ of degree $10$ representing any one of the schemes of the previous display are totally real under the pencil of septics assigned to visit $34$ basepoints injectively distributed among the $34$ empty ovals of the ten-ics $C_{10}$. Actually the last item $\frac{3}{1}31$ of the list is prohibited by either Thom [(\[Thom-Ragsdale:thm\])]{} or by Rohlin’s formula $2(\Pi^+-\Pi^-)=r-k^2=35-25=10$, since $\Pi^+-\Pi^-\le \Pi:=\Pi^+
+\Pi^-=3$.

[*Insertion*]{} \[10.04.13\].—The last scheme of the series is (alas) [*not*]{} prohibited by Petrovskii’s inequality (\[Petrovskii’s-inequalities:thm\]), but it is by the strong Petrovskii estimate of Arnold (1971), cf. (\[Strong-Petrovskii-Arnold-ineq:thm\]). This states $p-n^{-}\le
\frac{3}{2}k(k-1)+1$, where $n^{-}=0$ here (negative hyperbolic ovals), so $p\le 31$ while our scheme as $p=32$. (It may be useful—if you are better in geography than in arithmetics—to visualize all this on Fig.\[Degree10:fig\]. Crudely put, Arnold is as strong as the Ragsdale conjecture.) Further our assertion regarding the prohibition by Thom is certainly foiled as there is no reason ensuring orientability of the Arnold surface in the present case, as we are really attaching a 3-holed disc to the half of the complexification. (More explanations in Sec.\[Thom:sec\].)

It is evident (due to the little arithmetical coincidence between the genus and the dimension of the curve-hyperspace, plus the universal validity of the RKM-congruence) that this series of $(M-2)$-schemes propagates in each degree $m\ge 10$, and so we get an infinite (nearly tautological) repetition of the above conjectures for each even integer $m$. [*Can all these conjectures be proven in a single stroke?*]{} This would be a highbrow extension of the Rohlin-Le Touzé’s theorem. This would give an infinity and certain abundance to the phenomenon of total reality as one could have suspected from Ahlfors’ theorem. Of course we do not claim that this will supply an exhaustive list of the phenomenon, but perhaps it is modulo the operation of satellites. More precisely:

\[primitive-manifestation-of-tot-real:conj\] Any primitive manifestation of the phenomenon of total reality on a curve $C_m$ of (even) degree $m$ arises either as an $(M-2)$-scheme totally real under a pencil of adjoint curves of order $m-3$ assigned to pass through the empty ovals of $C_m$, or as a pencil of curves of order $m-2$ if $C_m$ is an $M$-curve (cf. [Theorem \[total-reality-of-plane-M-curves:thm\]]{}).

If not primitive then the scheme is a satellite of either:

$\bullet$ the unifolium scheme $1$ of degree $2$ total under a pencil of lines (this gives the series of deep nests which exist in all degrees $m$), or

$\bullet$ the quadrifolium $4$ of degree $4$ total under a pencil of conics, with satellites in all degrees multiples of $4$, or

$\bullet$ the other $(M-2)$-schemes of Rohlin $\frac{6}{1}2$ or $\frac{2}{1}6$ with satellites in all degrees multiple of $6$, and so on inductively as the satellites of $(M-2)$-schemes predicted by the Rohlin-Kharlamov-Marin congruence, or finally

$\bullet$ as satellites of $M$-schemes of lower degrees always dividing the given one $m$.

If this conjecture is true we would have a complete classification of the phenomenon of total reality for plane curves. This is surely somewhat premature and probably requires some slight adjustments to reach more respectableness.

[*Insertion*]{} \[10.04.13\].—In particular, one must probably also takes into account satellites of curves of odd orders, cf. Sec.\[Satellite-odd-degree:sec\]. For instance in degree $m=10$, there is probably the 2nd satellite of Harnack’s $M$-quintic playing a rôle.

At any rate note that our initial expectation that some phenomenon of total reality is purely prompted by Bézout in a very primitive way is apparently never borne out. It seems rather that apart from the satellites of the elementary schemes (unifolium and quadrifolium) flashed resp. by the trivial pencil of lines and conics the phenomenon of total reality is at least as hard as Rohlin-Le Touzé’s theorem, but perhaps not much harder. At least both ought to be connected by the geometry of the canonical series.

More $(M-2)$-schemes in degree 8 of type I {#RKM-schemes-deg-8-MORE:sec}
------------------------------------------

\[26.02.13\] In fact it is clear that even for $m=8$ we have not listed all $(M-2)$-schemes of type I for we have only considered those with one nonempty oval, but we must also consider those with 2, or 3 nonempty ovals. Tabulating a complete list is merely an exercise of combinatorics.

Geometrically, it may not be essential to assign basepoints on empty ovals but some can be located on nonempty ovals, and we may expect total reality provided the RKM-congruence is fulfilled. The sole problem is that we then lack some recipe to assign basepoints, and so the game becomes somewhat obscure \[but quite challenging\].

First the RKM-congruence (\[Kharlamov-Marin-cong:thm\]) can be more conveniently paraphrased as:

\[RKM-congruence-reformulated:thm\] [(Rohlin 1978-Kharlamov 197?-Marin 1979)]{} Any $(M-2)$-scheme of degree $m=2k$ such that $\chi\equiv k^2+4 \pmod 8$ is of type I.

\[08.03.13\] [*Little Warning*]{}.—There is a minor metaphysical trouble with this statement. Indeed when $m=8$ (or for larger $m$) we have the scheme $20$ which satisfies the RKM-congruence, but which is not realized algebraically as follows either from Thom (\[Thom-Ragsdale:thm\]) or from Rohlin’s formula $2(\Pi^+-\Pi^-)=r-k^2=20-16=4$, since $\Pi^+-\Pi^-\le \Pi:=\Pi^+
+\Pi^-=0$ due to the absence of nesting in $20$. There is two ways to go around this trouble, either add the assumption that the scheme is algebraic, or interpret Rohlin’s definition of the types of schemes by declaring (usual logical nonsense allied to the empty set) that a non-realized scheme is simultaneously of type I and type II (but not of indefinite type which needs algebraic representatives in both types). Of course when tracing pyramids, e.g. the Gudkov-Rohlin table (Fig.\[Gudkov-Table3:fig\]) we ascribe the type I label (red-circle) only to those schemes which are of type I in the concrete sense that the scheme is [*algebraic*]{} (and all its realizations are of type I).

The RKM-congruence (Theorem \[Kharlamov-Marin-cong:thm\]) says that an $(M-2)$-curve of degree $m=2k$ and type II satisfies the congruence $\chi\equiv k^2 $ or $k^2 \pm 2 \pmod 8$.

So the theorem follows after checking the following basic fact: [*an $(M-2)$-curve of order $m=2k$ verifies universally $\chi
\equiv k^2 \pmod 2$.*]{} (From the diagrammatic of pyramids, e.g. Fig.\[Gudkov-Table3:fig\], this is the fairly trivial matter that the rhombic equilateral lattice underlying the pyramid is adjusted at $k^2$, and one may infer that the claimed congruence holds more generally on all $(M-2i)$-levels.)

This is easy to prove using the relations $\chi=p-n$, $r=p+n=M-2$ $$\chi=p-n=(p+n)-2n\equiv_2 (p+n) = r=M-2,$$ and by Harnack’s bound and the genus formula $g=\frac{(m-1)(m-2)}{2}$ we have $$M=g+1=\textstyle\frac{(2k-1)(2k-2)}{2}+1=(2k-1)(k-1)+1=2k^2-3k+2,$$ whence $$\chi\equiv_2 M-2=2k^2-3k \equiv_2 k \equiv_2 k^2.$$

Is the converse statement true? Remember that if a scheme of degree $m=2k$ is of type I, then it satisfies Arnold’s congruence $\chi\equiv k^2 \pmod 4$. Hence $\chi\equiv k^2, k^2+4 \pmod 8$, and the second option leads to type I, but I do not know if the first option necessarily implies type II or indefinite type. We know only that this converse holds true for $m\le 6$ by the Gudkov-Rohlin table (=our Fig.\[Gudkov-Table3:fig\]) which involve explicit constructions. So the RKM-congruence detects many $(M-2)$-schemes of type I, but it is not clear (to me) if it detects all of them.

\[RKM-converse:conj\] An $(M-2)$-scheme of degree $m=2k$ which is of type I necessarily satisfies the RKM-congruence $\chi\equiv k^2+4 \pmod 8$.

[*Insertion*]{} \[11.04.13\].—Perhaps an answer can be found in Rohlin 1978 [@Rohlin_1978 p.93–94], esp. Art. 3.5 and the end of 3.6, where it seems that an extremal property of the strong Arnold’s inequalities observed by Zvonilov-Wilson prompts type I in situation apparently not covered by the congruence. Alas, I had not yet the time to assimilate this properly, but look forward with great excitement to do so in the future (after some long editorial duty).

\[08.03.13\] Again there is little worry about definitions. For instance when $m=6$ we can consider the (non-algebraic) scheme $(1,1,1)6$ which is thus of type I (in the logical sense but of course also of type II), yet with $\chi=(1-1+1)+6=7\neq
3^2+4\equiv_8 5$. So we tacitely assume the scheme of type I in the strong sense that it is algebraically realized.

Of course the conjecture \[RKM-converse:conj\] is true for $m=6$ (look at the Gudkov-Rohlin Table=Fig.\[Gudkov-Table3:fig\]), which depends upon explicit construction of curves of type II for all schemes which are not RKM. Already for $m=8$, the conjecture seems to demand a menagerie of construction. One could hope that there is some theoretical argument.

Let us leave this question aside, as we merely want to list schemes of type I potentially subsumed to the phenomenon of total reality.

Let us now tackle the combinatorial aspect of dressing the list of all $(M-2)$-schemes of degree $m=8$ satisfying the RKM-congruence (hence of type I).

We may start with schemes with zero or one empty ovals and list all of them using the fourfold periodicity as we already did. Yet to be more systematic we start with $\frac{16}{1}3$ expand its Gudkov’s symbol as $\frac{16}{1}\frac{0}{1}\frac{0}{1}1$ to be of the shape $\frac{x}{1}\frac{k}{1}\frac{\ell}{1}m$ and then we trace a cubical lattice (Fig.\[RKM-schemes-deg-8:fig\]) in 3-space encoding all variations of this symbol for varied values of $(k,\ell,m)$. To aid visualization it turned useful to ascribe colors to the different levels: the ground floor is orange, the 1st floor is lilac, the 2nd floor blue, the 3rd floor is cyan, the 4th floor is yellow-green. As we are interested in $(M-2)$-schemes we have the relation $x+k+\ell+m+3=M-2=20$. The crucial point is that when $k$ or $\ell$ increases by one unity, then one hole is traded against another hole, so that $\chi$ is left unchanged. In contrast an increment of $m$ reduces $x$ by one, and so a hole in “$x$” is traded against a disc outside, so that $\chi$ increases by two. Hence as the RKM-congruence is modulo 8, we have a 4-fold periodicity until reaching the same value for $\chi$. (This explains the vertical motion along the cubical lattice.) Those symbols surrounded by dashed lines are doubloons (non-normalized Gudkov’s symbol), yet useful to stop the combinatorial proliferation. Underbraced symbols are those whose Gudkov’s symbol admits a shorter expression given below the brace (when enough room is left available).

-5pt0

-5pt0

All those schemes are avatars of the 2 Rohlin’s $(M-2)$-schemes of degree 6 (subsumed to total reality). It is a simple matter to count them. First collect on the nearby face of Fig.\[RKM-schemes-deg-8:fig\] all schemes lying in perspective beyond the red/thick numbers. Adding them vertically gives the blue/big numbers on the bottom row, yielding a total of $5 \cdot
11+9= 55+9=64$ schemes. (That this is a power of 2, incidentally the same as Rohlin’s count of all schemes of degree 6 decorated by types, is probably a mere coincidence.)

This is somewhat amazing combinatorics, and the geometrical conjecture would be that all these schemes are total under a pencil of quintics with suitably assigned 19 basepoints. The case which looks most appealing is when there are exactly 19 empty ovals. Those corresponds to the 4 schemes forming the vertical left 1-simplex of Fig.\[RKM-schemes-deg-8:fig\], which we call the monolith. The monolith has some obvious structure of a 3-simplex, stratified as follows into sub-simplices:

$\bullet$ 0-simplex corresponding to the scheme $20$ with zero nonempty oval,

$\bullet$ $1$-simplex corresponding to the 4 schemes with 1 nonempty oval and so admitting a Gudkov’s symbol $\frac{k}{1}\ell$,

$\bullet$ $2$-simplex corresponding to the 20 schemes with 2 nonempty ovals admitting a Gudkov writing $\frac{k}{1}\frac{\ell}{1}m$,

$\bullet$ $3$-simplex corresponding to the 39 schemes with 3 nonempty ovals admitting a Gudkov writing $\frac{k}{1}\frac{\ell}{1}\frac{m}{1}n$.

Only the schemes forming the one simplex have exactly 19 empty ovals. For the other categories (with resp. 20, 19, 18, 17 empty ovals) one may assign the 19 basepoints among the nonempty ovals (by choosing say the most massive nonempty oval, i.e. containing the largest number of empty ovals). Of course this is pure speculation and maybe the exact opposite has to be done.

There is probably here work for several generations of computing machines, unless one is able to crack all total reality phenomenon in a single stroke. Somewhat brutally in comparison to our low understanding of where to assign basepoints, we posit that whenever the RKM-congruence is verified then there is a phenomenon of total reality:

Suppose given an $(M-2)$-curve of degree $m=2k$ verifying the RKM-congruence $\chi\equiv k^2+4 \pmod 8$. Then the pencil of adjoint curves of order $(m-3)$ ascribed to pass through the empty ovals plus some other points distributed on the nonempty ovals is totally real.

As to the crude arithmetics, remember that the number $B$ of basepoints assignable to adjoints of order $(m-3)$ is given by the binomial coefficient $$B=\textstyle\binom{(m-3)+2}{2}-1-1,$$ while the pre-Harnack bound $$M-2=(g+1)-2=\textstyle\binom{m-1}{2}+1-2,$$ so that $$B=M-3.$$ This means that we have one basepoint less than the number of ovals, and so we may canonically distribute them when there is one nonempty oval.

\[M-2-curve-degree-like-Gabard:rem\] [\[03.03.13\] Assuming we are capable to ensure total reality of the pencil, it may be observed that the degree of the induced total map to $\PP^1$ would be in accordance with Gabard’s bound $r+p$, which is also the mean value of $r$ and $M=g+1$. This follows from a simple calculation. First $$2B=2(M-3)=2(\textstyle\frac{(m-1)(m-2)}{2}+1-3)=(m-1)(m-2)-4=m(m-3)-2,$$ and so the degree of the map is $$m(m-3)-B=2B+2-B=B+2=M-1,$$ which is Gabard’s bound i.e. the mean of $r=M-2$ and $M$. ]{}

Of course if one as some self-confidence in Gabard 2006 [@Gabard_2006], then there is a total map of that degree on each dividing $(M-2)$-curve (in particular those satisfying the RKM-congruence which are universally of type I). By some concretization yoga this map would be induced by a total pencil, and by a dubious reverse engineering of the above arithmetics this would be a pencil of $(m-3)$-tics. This gives some very weak evidence for the:

Any dividing $(M-2)$-curve of degree $m$ is totally real under a pencil of curves of order $(m-3)$.

\[27.02.13\] In fact we are not even sure that the above cubical lattice (Fig.\[RKM-schemes-deg-8:fig\]) gives an exhaustive list of RKM-schemes in degree 8, where we use the jargon:

A scheme of degree $2k$ is an RKM-scheme if it is an $(M-2)$-scheme satisfying the Rohlin-Kharlamov-Marin congruence $\chi\equiv k^2+4 \pmod 8$ which forces type I (alias orthosymmetry) of the scheme, i.e. that the real locus disconnects the complex one.

While any RKM-scheme is of type I, we do not know whether the converse holds true (for $(M-2)$-schemes). It is true for $m=6$ as follows from the Gudkov-Rohlin classification (Fig.\[Gudkov-Table3:fig\]). It seems that degree 8 is a perfidious iceberg killing any naive conjecture arising from contemplation of low order curves (say $\deg \le 6$). Specific illustration of this vague principle are Shustin’s disproof of the one-half of Rohlin’s maximality conjecture, and concomitantly the disproof of Klein’s Ansatz that nondividing curves may always acquire a solitary node. Another remark along the same line is the disproof (using the RKM-congruence) of the naive CCC-conjecture, cf. Theorem \[CCC-conj-disproof:thm\]. (\[06.03.13\] Alas this disproof of CCC is disproved by Thom’s conjecture, as remarked there!)

Now back to our classification of RKM-schemes of degree 8 we may wonder if there is one containing $(1,1,1)$ the nest of depth 3. As we focus on $(M-2)$-schemes and since $M=22$ when $m=8$, we may start with this configuration plus 17 outer ovals. In Gudkov’s notation this is the scheme $(1,1,1)17$. Any scheme $S$ of even degree is bounded by the Ragsdale orientable membrane $S^{\ast}$ with $\chi(S^{\ast})=p-n$. In our case $\chi((1,1,1)17^{\ast})=1-1+1+17=18\equiv 2 \pmod 8$ and not 4 as posited by the RKM-congruence. If we trade outer ovals against inner ovals lying deepest then $\chi$ is left unchanged. However if the trading is made for ovals at intermediate depth then the outer discs of the Ragsdale membrane becomes holes and $\chi$ diminishes by two. So the RKM-congruence is first arranged for the scheme $(1,\frac{1}{1}3)14$ (with $\chi=12$), and then using 4-fold periodicity the list is augmented as: $$\label{RKM-scheme-deg-8-four-primitive-type:eq}
(1,\frac{1}{1}3)14, \quad (1,\frac{1}{1}7)10, \quad
(1,\frac{1}{1}11)6, \quad (1,\frac{1}{1}15)2,$$ which are all RKM-schemes (containing the nest of depth 3). Once the Euler characteristic is adjusted to satisfy the RKM-congruence, we may trade outer ovals with innermost oval at depth 3 without altering $\chi$. So each of these schemes produces a list of derived schemes also RKM. Namely the 15 schemes $$\begin{aligned}
(1,\frac{1}{1}3)14,& (1,\frac{2}{1}3)13, (1,\frac{3}{1}3)12,
(1,\frac{4}{1}3)11, (1,\frac{5}{1}3)10, (1,\frac{6}{1}3)9,
(1,\frac{7}{1}3)8, (1,\frac{8}{1}3)7, \cr (1,\frac{9}{1}3)6,&
(1,\frac{10}{1}3)5, (1,\frac{11}{1}3)4, (1,\frac{12}{1}3)3,
(1,\frac{13}{1}3)2, (1,\frac{14}{1}3)1, (1,\frac{15}{1}3),\end{aligned}$$ and then the 11 schemes $$\begin{aligned}
(1,\frac{1}{1}7)10,& (1,\frac{2}{1}7)9, (1,\frac{3}{1}7)8,
(1,\frac{4}{1}7)7,  (1,\frac{5}{1}7)6,\cr  (1,\frac{6}{1}7)5,&
(1,\frac{7}{1}7)4, (1,\frac{8}{1}7)3, (1,\frac{9}{1}7)2,
(1,\frac{10}{1}7)1, (1,\frac{11}{1}7),\end{aligned}$$ and likewise the 7 schemes $$\begin{aligned}
(1,\frac{1}{1}11)6,\quad (1,\frac{2}{1}11)5,\quad
(1,\frac{3}{1}11)4,\quad (1,\frac{4}{1}11)3,\quad
(1,\frac{5}{1}11)2,\quad (1,\frac{6}{1}11)1,\quad
(1,\frac{7}{1}11),
$$ and finally, the 3 schemes $$\begin{aligned}
(1,\frac{1}{1}15)2,\quad (1,\frac{2}{1}15)1,\quad
(1,\frac{3}{1}15).\end{aligned}$$

(All together this gives $15+11+7+3=36$ additional schemes to be added to the 64 tabulated on Fig.\[RKM-schemes-deg-8:fig\], hence a total of $64+36=100$.)

If we neglect the largest nonempty oval we have 19 ovals and we may expect total reality of the quintic pencil ascribed to pass trough any (injective) distribution of the 19 basepoints on those 19 ovals (which albeit not all empty are the deepest items of the combinatorial scheme).

At this stage we hope to have exhausted the RKM-schemes of degree 8:

\[RKM-schemes-deg-8-census:lem\] Any RKM-scheme of degree $8$ has either depth $\le 2$ in which case it is catalogued as one of the $64$ schemes of Fig.\[RKM-schemes-deg-8:fig\], or it has depth $3$ in which case it is one of the $4$ displayed schemes or one of the $36$ derived products where an outer oval is traded against an innermost oval (cf. the last 4 display formulae tabulating the corresponding $36$ Gudkov’s symbols). In particular there are exactly $64+36=100$ schemes of degree $8$ which are RKM, and hence of type I (and therefore potentially subsumed to the phenomenon of total reality). \[06.03.13\] Addendum: One of them (at least), namely $20$ is not realized as it violates the Thom conjecture (cf. [\[Thom-Ragsdale:thm\]]{}), or better the Rohlin formula. \[13.03.13\] [*Warning*]{}.—The list of 100 schemes is far from exhaustive, cf. remarks right after that Lemma \[RKM-scheme-ruled-out:lem\].

(pseudo-proof) Alas we are not even sure that this list is now exhaustive albeit it might be likely by using the concept of depth of a scheme (the longest chain of ovals totally ordered by inclusions of their insides, i.e. the unique bounding disc given by the Schoenflies theorem in its smooth variant implicit in Möbius 1863 [@Moebius_1863], Hilbert (tacit), Dehn ca. 1899 (unpublished), Osgood 1902, Schoenflies 1906, etc., cf. e.g. Siebenmann 2005 [@Siebenmann_2005] for some historical background and the literature cited therein).

Given any scheme of degree $8$, its depth is at most 4. If equal to 4 it contains the deep nest and so the scheme is saturated (i.e. it cannot be enlarged without corrupting Bézout). If the depth is 3 then its contains $(1,1,1)$ the nest of depth 3, and if we were not too bad in combinatorics our recipe of 36 schemes above was exhaustive. For the same vague reason, when the depth is $\le 2$ then the catalogue of 64 schemes is exhaustive.

So we have one RKM-scheme of depth 1, $63$ such schemes of depth $2$, and 36 RKM-schemes of depth 3, while the unique scheme of depth 4 (deep nest (1,1,1,1)) is not an $(M-2)$-scheme hence not an RKM-scheme.

Further, it could be that sophisticated Bézout-style obstructions à la Fiedler-Viro (\[Viro-Fiedler-prohibition:thm\]) prohibit the realizability of some of those schemes in the algebraic realm. So perhaps several items albeit schemes in the abstract sense of Rohlin are not algebraically realized. (Improvising terminology and to conflict even more with the Grothendieck-Rohlin collapse of jargon we could speak of a Hilbert-scheme (H-scheme) when the scheme is realized algebraically.) So I do not know if the 100 RKM-schemes listed above are H-schemes. ([*Update*]{} \[06.03.13\] At least one of them $20$ is not realized as follows from Thom’s conjecture, cf. Theorem \[Thom-Ragsdale:thm\], or better just apply Rohlin’s formula.) Taking another naive look at the Gudkov-Rohlin table for degree $m=6$ (Fig.\[Gudkov-Table3:fig\]) shows that $(M-2)$-schemes are subjected to no restriction and so we can speculate the same for $m=8$, in which case all our 100 schemes would be $H$-schemes. At any rate we note that enlarging $m=6$ by just 2 units, involve a de-multiplication by the factor $50$ of all RKM-schemes.

(Skip this paragraph.) Have we really listed every RKM-schemes? We could start from another elementary configuration like 2 nests of depth 2 (Gudkov symbol $\frac{1}{1}\frac{1}{1}=(1,1)(1,1)$) and then add 16 outer ovals to get $(1,1)(1,1) 16$. Then $\chi=
1-1+1-1+16\equiv 0 \pmod 8$, while the good RKM-value is 4. So we trade outer ovals for inner ovals at depth 1, and so $\chi$ diminishes by 2 units. Hence we find first $(1,3)(1,1)14=\frac{3}{1}\frac{1}{1}14$ or $(1,2)(1,2)14=\frac{2}{1}\frac{2}{1}14$. Those are already catalogued on Fig.\[RKM-schemes-deg-8:fig\]. Then as the Euler characteristic is adjusted we may apply the same trick of trading outer ovals for innermost ovals without changing $\chi$, yet doing so we create schemes with 2 nests, one of depth 3 and one of depth 2, so that Bézout is violated by tracing the line through their “centers”. So it seems that no new candidates for total reality occurs along this way.

How to assign basepoints? {#How-to-assign-basept:sec}
-------------------------

\[01.03.13\] As a matter of extending the Rohlin-Le Touzé theorem (still unpublished and abridged RLT) to degree 8, we would like to know where to assign basepoints on each item of our list of 100 RKM-schemes. A priori not all of them are totally real in some uniform way despite the presence of Ahlfors theorem. Recall that our census of 100 RKM-schemes may be interpreted as five families:

(1).—the scheme $20$, (not realized by Thom \[Thom-Ragsdale:thm\], or Rohlin’s formula).

(2).—4 schemes of the form $\frac{k}{1}\ell$,

(3).—20 schemes of the form $\frac{k}{1}\frac{\ell}1 m$,

(4).—39 schemes of the form $\frac{k}{1}\frac{\ell}1 \frac{m}{1}
n$,

(5).—36 schemes enlarging the nest of depth 3.

The class (2) consist precisely of those elements having 19 empty ovals. And those are the most direct candidates for an avatar of the RLT-theorem. However in the class (5) there is also 19 preferred deep ovals, namely all those which are either empty or if nonempty which are not maximal (for the usual order on ovals given by inclusion of their bounding discs). So there is a family of 40 schemes where a direct extension of the RLT-theorem is straightforward (at least to state, but maybe not to prove).

On the other hand it could be the case that there is an extended formulation including all those 100 schemes. At least one idea would be to consider the notion of dextrogyre oval (abridged dextro-oval).

For a dividing curve, we say that an oval $O$ is a dextro-oval if its porous-inside $O^{\star}$, that is the inside minus the insides of all ovals directly inside it, has complex orientation matching $\partial
O^{\star}$ that arising as boundary of the porous-inside.

Of course any empty oval is dextro. As an example consider the Gürtelkurve $C_4$ of degree 4 with 2 nested ovals. Then either by using the total pencil of lines through a center in the innermost of the nest or by Fiedler’s law of positive smoothings the complex orientation consist of 2 concentric circles with the “same” orientation. So the nonempty oval is not dextro, while of course the inner oval is (being empty). Here we see that the pencil of lines is total precisely when its basepoint is assigned on the dextro-oval.

We may therefore expect that the pencil of quintics on our curves of degree 8 is total whenever the $19$ basepoints are distributed on $19$ dextro-ovals supposed available.

Consider e.g. the scheme $\frac{3}{1}\frac{1}{1} 14$. Then by Rohlin’s formula $2(\Pi^{+}-\Pi^{-})=r-k^2=20-16=4$, the difference $\Pi^{+}-\Pi^{-}$ is 2, while $\Pi^{+}+\Pi^{-}=4$, so that $\Pi^{+}=3$ and $\Pi^{-}=1$. From this one infers (picture) that there is at least one dextro-oval which is not empty. And so we have here precisely 19 dextro-ovals.

Obviously one can extend this to some other schemes, and running through the catalogue one could dress an exhaustive list of all schemes with 19 dextro-ovals (of course 20 will not belong to it), and expect the phenomenon of total reality for the latter.

Of course this method is somewhat ad hoc as it uses the dividing character of the curve while in its purest form (say as a way of taking independence of the RKM-congruence) one would like to avoid this knowledge.

Back to degree 6: weak form of RLT {#RLT:sec}
----------------------------------

\[01.03.13\] Obviously we were moving too fast by looking at degree 8 and need to return to degree 6, to get rid off of those combinatorial difficulties while concentrating on the geometry of total reality.

Our point now is that we may be pessimistic about Rohlin-Le Touzé’s theorem as being false in the generality announced by Le Touzé 2013[^72] (or at least difficult to prove). Recall moreover that Rohlin’s cryptical statement is not as strong as Le Touzé (at least leaves some free room for interpretation). Even if Le Touzé’s claim of total reality is correct, it could be that total reality is easier to prove for special 8 assigned basepoints. This weaker statement would still be sufficient to detect the dividing character of curves having an RKM-scheme, i.e. $\frac{6}{1}2$ and $\frac{2}{1}6$.

\[9th-basepointI:lem\] Suppose given a sextic $C_6$ of type $\frac{6}{1}2$. Assign $8$ basepoints on the empty ovals, and look at the corresponding pencil $\Pi$ of cubics. Then there is a 9th basepoint $p_9$ of $\Pi$. To ensure total reality of $\Pi$ it is enough that $p_9$ is either on $N$ the nonempty oval of $C_6$ or more generally in its inside $N^{\ast}$.

As 16 intersections are forced by topology, it remains only to gain 2 extra intersections for totality. If $p_9\in N$ then this is clear, and if $p_9\in N^{\ast}$ then by taking a cubic $C_3$ of $\Pi$ through a point $p\in N$ which is connected then it is easy to see for vibratory reasons that the other point $q$ of $C_3\cap
N$ (whose existence is ensured either by topology or by algebra) will dextrogyrate on $N$. This is to mean that when $C_3$ is slightly perturbed both points $p,q$ will move on $N$ along the same orientation. This suffices to ensure total reality, as both points cannot then enter in collision to disappear in the imaginary locus.

This argument uses existence of a connected cubic in any pencil of cubics. Another slight variant is to argue by contradiction by appealing to a bad cubic $C_3$, i.e. disjoint from $N$. If $p_9$ is inside $N$ then the oval of $C_3$ cannot vibrate properly, and we reach a contradiction.

So the whole problem of Rohlin-Le Touzé in weakened form can be reduced to the:

Given any $C_6$ of type $\frac{6}{1}2$, there exists an (injective) distribution of $8$ points $p_1,\dots, p_8$ on the $8$ empty ovals such that the pencil of cubics $\Pi$ interpolating them has its $9$th basepoint located in the (sealed) inside $N^{\ast}$ (i.e. $N$ included) of the nonempty oval. (In particular such a calibrated pencil is totally real.)

One could hope to prove this weaker assertion by pure topology without having to enter in fine Bézout-style considerations (upon which Le Touzé’s proof is likely to rest).

How to prove this conjecture? One very naive way would be just to use that $N$ divides the plane and so it would suffice to find two octuplets $p_i$ such that $p_9$ is resp. inside and outside $N$. By continuity of the 9th basepoint as a function $\beta$ of the 8 assigned one, we would be ensured of another intermediate octuplet such that $p_9\in N$.

A less naive way would be to assume by contradiction that $p_9$ always misses $N^{\ast}$ (the sealed interior) and then retract on the core of the residual Möbius band. This seems to give an essential (non null-homotopic) map. On the other hand as $\beta$ extends to a (16-dimensional) cell (given by allowing the $p_i$ to explore the insides of the 8 empty ovals) it must be null-homotopic. Alas the first step of argument is not easy to complete. At any rate, designating by $E_i$ the 8 empty ovals of the $C_6$, we can define the Rohlin body of the $C_6$ as the image of the Rohlin map $$\beta \colon E_1\times\dots\times E_8 \to \RR P^2$$ taking the 8 assigned basepoints of $\Pi$ to the 9th unassigned basepoint $p_9$.

This map is well-defined by virtue of the following easy lemma.

\[independent-cond:lem\] Our $8$ basepoints impose independent conditions on the space of cubics, and this holds true more generally when the basepoints are allowed to vary in the insides $E_i^{\ast}$ of the empty ovals $E_i$.

Let $\Pi_i$ be the linear system of curves passing through the first $i$ points $p_1,\dots,p_i$, $i=1,\dots, 8$. We have a filtration $$\vert 3H \vert \supset \Pi_1 \supset \Pi_2 \supset \Pi_3 \supset
\Pi_4\supset \Pi_5\supset \Pi_6\supset \Pi_7\supset \Pi_8=\Pi,$$ and one checks that all inclusions are strict by exhibiting an appropriate curve. Strictness of the first inclusions is trivial by taking appropriate configuration of 3 lines (Fig.\[TotFiltra:fig\]). For strictness of $\Pi_6\supset
\Pi_7$, consider a conic $C_2$ passing through 5 inner points plus a line through the 6th inner point, yet missing the 2 outer points $7,8$. We have to check that this $C_3$ does not pass through $7$. If it does then $C_2\cap C_6$ would consist of $5\cdot 2+2+2=14>2\cdot 6$ points as two extra intersections are created with $N$, so that Bézout is violated. For the last strictness $\Pi_7\supset \Pi_8$ one takes the same $C_2$ and aggregate the line $L$ through the points $7,8$.

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Denoting by $E:=E_1\times \dots \times E_8$ the Cartesian product of the empty ovals and by $E^\ast:=E_1^{\ast}\times \dots \times
E_8^{\ast}$ that of their insides $E_i^{\ast}$, we have a factorization of $\beta$ as $$\beta\colon E \to E^{\ast}\to \RR P^2,$$ where the first map is the inclusion and the second $\beta^{\ast}$ is given by the same recipe as $\beta$. It follows that $\beta$ is null-homotopic (since $E^{\ast}$ is a 16-cell).

On the other hand we may hope to show that if $\beta$ avoids $N^{\ast}$ the inside of the empty oval that the induced co-restriction map $\bar\beta\colon E\to \RR P^2-N^\ast$ whose target is homotopically a circle is [*essential*]{} (i.e. not null-homotopic). For this it would be enough to show that the induced morphism $\pi_1(\bar\beta)$ hits a odd multiple of the generator of the $\pi_1$ of $\RR P^2-N^\ast$ which is a Möbius band. This contradiction would prove the conjecture and so total reality of a suitable pencil. However this strategy demands some geometric understanding that presently elude us.

Another more naive strategy (using less topology) would be that the map $\beta\colon E \to \RR P^2$ is open (say as a vestige of the holomorphic character of the underlying complexification). Then it would be plain that $\beta(E)$ is compact (hence closed) and open, hence equal to all $\RR P^2$ by connectedness of the latter space. (As to be soon discussed this surjectivity of $\beta$ would however conflict with the Le Touzé theorem)

At any rate we see that the problem of total reality à la Rohlin-Le Touzé in its weakened form due to us (or perhaps Rohlin depending on the interpretation of his cryptical statement) is fairly basic at first sight. We are given a curve $C_6$ of type $\frac{6}{1}2$. Given any 8 points on the empty ovals $E_i$ we have some predestination mapping $\beta$ assigning to them the 9th basepoint (by a somewhat elusive recipe) and total reality of the pencil $\Pi$ of curves passing through the 8 points is ensured if $p_9$ lands in the inside $N^{\ast}$ of the nonempty oval $N$ of $C_6$. So crudely speaking we have one chance over two that total reality holds true, for a given configuration of point. We would like to show that it is always possible to have total reality for a clever configuration of 8 points, while the stronger Le Touzé’s announcement claims it for all choices of 8 points.

Our hope is that independently of whether this stronger statement is right or false there ought to be a simpler proof of the weaker assertion by say essentially topological methods. By using the dextrogyration argument it is clear that we have the:

\[9th-basepoint-totalII:lem\] The pencil $\Pi$ is total iff $p_9$ its non-assigned basepoint belongs to $N^{\ast}$ (the sealed inside of the nonempty oval $N$).

If $p_9\in N$ total reality is clear. If $p_9$ is in the open inside (interior) of $N$, then we have an odd number of basepoints insides. Looking at the oscillation of a connected member $C_3$ of the pencil about its basepoints it results that both points of $C_3\cap N$ (there cannot be more than two by Bézout) will dextrogyre, i.e. move along the same orientation of $N$ (compare Fig.\[Dextrogyre:fig\]a). Hence no collision can occur in the long run, since by the holomorphic character of the map a point cannot reverse spontaneously its sense of motion as the curve $C_3$ is dragged along the real locus of the pencil.

[*Insertion*]{} \[11.04.13\].—We can dispense connectedness of the $C_3$ (though quite easy to arrange) as follows. Choose any point on $N$ and consider the cubic $C_3$ through this point. If $C_3\cap C_6$ is not totally real, the oval of $C_3$ is necessarily inside $N$. For vibratory reasons this oval must visit an even number of basepoints, and actually must visit all 6 assigned inner basepoints (otherwise total reality of $C_3\cap C_6$ is granted). The 9th basepoint cannot be located on the oval of the $C_3$ (else it cannot vibrate properly), hence it is situated on its pseudoline and we may again conclude dextrogyration by the slaloming argument across an odd number of basepoints (compare Fig.\[Dextrogyre:fig\]b).

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Conversely if the 9th basepoint $p_9$ lies outside of $N$ then there is an even number of basepoints inside $N$ and the 2 points of a connected $C_3$ located on $N$ will anti-dextrogyre. In that case there will be a collision in the long run and total reality is foiled.

Hence the option $\beta$ surjective would contradict the Le Touzé’s theorem. Of course this is not a serious objection against her theorem because usually holomorphic maps restricted to real loci fails blatantly to be surjective (a key prototype of this phenomenon opposite to total reality, is when one projects on a line an ellipse from an outer point).

[*Optional paragraph.*]{}—Another slight variant: we may assign 7 basepoints on all safe one empty ovals and the 8th one $p_8$ on $N$. Then a little advantage is that whenever $p_8$ is collinear with 2 points of the first seven $p_i$, we get a special split cubic in the pencil $\Pi$, namely the line plus a residual conic. It is not very clear if this little advantage is really useful, and leave open this discussion.

New meditation after reception of Le Touzé’s article
----------------------------------------------------

\[02.03.13\] Yesterday (01.03.13), we conjointly received (with several other Russian colleagues Kharlamov, Viro, Nikulin, etc.) a copy of Le Touzé’s article vindicating Rohlin’s cryptical assertion of total reality of the RKM-sextics of type $\frac{6}{1}2$ and $\frac{2}{1}6$ under a pencil of cubics.

This fascinating paper helped me to rectify several misconceptions of mine about the content of her earlier announcement. In particular, she writes the following illuminating remark (Le Touzé[^73] 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics p.3]):

“By a congruence due to Kharlamov \[8\](=Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91]), the real schemes $\la 2 \vc 1 \la
6\ra\ra$ or $\la 6 \vc 1 \la 2\ra\ra$ are both of type I. We use this fact in the proof, so we confirm that the sextics with this two schemes do not contradict Rokhlin’s conjecture. Rokhlin claimed that he could prove the very same statement [*without*]{} using the fact that the sextics are dividing. It’s a stronger result. Unfortunately, his proof was never published and is now lost[^74].”

The Rokhlin conjecture alluded to by Le Touzé is made explicit in (p.2–3 of ), which is again worth quoting:

“The $M$-curves are clearly dividing. The so-called hyperbolic curves are also dividing. A hyperbolic curve of degree $m=2k$ or $2k+1$, consists in $k$ nested ovals, plus one pseudo-line if $m$ is odd. A pencil of lines whose base point is chosen in the innermost oval sweeps out the curve in such a way that the $m$ intersections are always real. One says that this pencil of lines is totally real with respect to the hyperbolic curve. Starting from this observation, Rokhlin \[10\](=1978 [@Rohlin_1978]) presents a beautiful argument[^75] proving that if an algebraic curve is swept out by a totally real pencil of lines, then this curve is dividing. The argument generalizes to pencils of curves of higher degrees. Can conversely any dividing curve be endowed with some totally real pencil?[^76] A weaker conjecture suggested implicitly in \[10\](=Rohlin 1978 [@Rohlin_1978]) is that any curve whose real scheme is of type I may be endowed with a totally real pencil[^77]. It turns out that the $M$-curves may indeed be endowed with suitable pencils of degree $(m-2)$, see \[6\], page 348(=Theorem \[total-reality-of-plane-M-curves:thm\] in this work=Gabard 2012/13 [@Gabard_2012/13], pagination may have fluctuated meanwhile).”

Le Touzé’s article clarified several misinterpretation of mine about her announced result (dated 16 février 2013, cf. Sec.\[e-mail-Viro:sec\]) but which we reproduce now as we misunderstood it:

$\bullet\bullet\bullet$ samedi 16 février 2013 17:54:55

Dear Alexandre, dear other colleagues,

I have managed to prove that a pencil of cubics with eight base points distributed in the eight empty ovals of a sextic $2 \cup 1(6)$ is necessarily totally real. Details will follow soon in a paper. Yours, Séverine

In fact my misconception was to think that Le Touzé claims total reality for [*any*]{} such pencil. So shame on me for not having read her message more carefully as she expressly writes [*a pencil*]{}. For me it is still unclear if the stronger claim of total reality for all such pencils holds true. Such a strong form of total reality holds in the basic cases (pencil of rational curves of degree $\le 2$) but perhaps the case of cubics is completely different for such curves need not being connected and also there is a 9th predestined basepoint which cannot be freely assigned (magneto repulsion well-known at least since Euler 1748 as reported e.g. in Griffiths-Harris 1978 [@Griffiths-Harris_1978/94 p.673]).

Taking (deliberately) the risk of being too cavalier let us put forward the strongest form of total reality (à la Rohlin-Le Touzé but perhaps too coarsely interpreted than what those authors ever wrote):

[*Added in proof*]{} \[08.03.13\].—Meanwhile Le Touzé validated the following conjecture, for explanations, cf. her message (dated 5 March 2013 in Sec.\[e-mail-Viro:sec\]).

\[SRLT:conj\] [(SRLT=Strong Rohlin-Le Touzé total reality, as misinterpreted by Gabard)]{} Given any (smooth) sextic $C_6$ of type $\frac{6}{1}2$ or $\frac{2}{1}6$ and any (injective) distribution of $8$ points on the $8$ empty oval of $C_6$, the pencil of cubics through the $8$ assigned basepoints is totally real.

First, the basic Lemma \[independent-cond:lem\] (on independency of conditions) seems to imply that the given linear system is really a pencil. The next step would be to prove (or disprove) this strong conjecture of total reality. The (more cautious) statement of Le Touzé does not imply the conjecture, yet it would be interesting to know if Le Touzé is aware of an obstruction refuting the conjecture. Let us reproduce Le Touzé’s theorem (cf. Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]):

\[LeTouze-big-thm-01-March-13:thm\] [(Le Touzé 1st March 2013, announced 17 February 2013)]{} Any $(M-2)$-sextic with real scheme $\frac{2}{1}6$ or $\frac{6}{1}2$ may be endowed with a totally real pencil of cubics with $8$ basepoints distributed in the $8$ empty ovals.

Also very interesting (though somewhat disappointing) is the issue that Le Touzé’s proof uses the RKM-congruence (as brilliantly nuanced by herself), and so does not recover the [*synthetic a priori*]{} character of Rohlin’s claim (which so becomes even more cryptical than it ever was). This dependence of Le Touzé’s proof on the RKM-congruence (which she ascribes to Kharlamov alone, like in the original text Rohlin 1978) has some overlap with thoughts I had yesterday (especially Sec.\[How-to-assign-basept:sec\]).

Le Touzé’s work makes very acute the desire to find the so-called [*lost proof of Rohlin*]{}. We thought on the question from various angles yet our present understanding is very confused due to overwork and nervous collapse (i.e., plethora of strategies and panoply of statements of varying strength).

The naive strategy (toward conjecture SRLT=(\[SRLT:conj\]) or just a weakened form thereof) at which we arrived yesterday (Sec.\[RLT:sec\]), was to use the map $\beta$ assigning to the 8 assigned basepoints the 9th (unassigned) basepoint. With this the lost proof of Rohlin could be harpooned by an essentially topological argument, yet we failed to overcome the last difficulty.

For convenience let us repeat some of our ideas along this topological tactic. Given a $C_6$ say of type $\frac{6}{1}2 $ for simplicity. Denote by $E_i$ the 8 empty ovals. For any injective distribution of points on those 8 ovals we have a pencil $\Pi$ of cubics through them (Lemma \[independent-cond:lem\]), and the latter is totally real iff the 9th basepoint of $\Pi$ lands in the (sealed) inside $N^{\ast}$ of $N$ the nonempty oval of $C_6$ (cf. Lemmas \[9th-basepointI:lem\] and \[9th-basepoint-totalII:lem\]).

Formally we can so introduce the [*$9$th basepoint map*]{} $$\beta\colon E_1\times \dots \times E_8 \to \RR P^2$$ taking the octuplet $(p_1, \dots , p_8) $ to the 9th basepoint of the pencil $\Pi$ of cubics passing through $p_1, \dots, p_8$.

A priori we could hope this mapping to be onto for reasons of Brouwer’s topological degree of a map (in homology modulo 2). Precisely if the induced morphism $H_2(\beta, \ZZ_2)$ is non-trivial, the mapping $\beta$ would be surjective (for otherwise a point is missed and the map factorizes through a punctured (hence open) Hausdorff manifold whose top-dimensional homology vanishes). Of course all this general theory can be dispensed as we just have a punctured projective plane homotopically equivalent to a circle. In this surjectivity scenario for $\beta$, it would suffice to take an inner point $p_9\in N^{\ast}$ and any lift of it via $\beta$ yields an octuplet inducing a total pencil on the $C_6$. In contrast, taking a point outside $N$ would imply that the strong form of total reality (SRLT)=(\[SRLT:conj\]) for all octuplets fails.

However it is unlikely that this Brouwer-style surjectivity criterion works because the map $\beta$ extends (still by virtue of Lemma \[independent-cond:lem\]) to the (sealed) insides $E_i^{\ast}$ (bounding discs) of the $E_i$ as to give the map $$\beta^{\ast}\colon E_1^{\ast}\times \dots \times E_8^{\ast} \to
\RR P^2,$$ also defined by taking the 8 assigned basepoints to the 9th one. Since the source of the map $\beta^{\ast}$ is a contractible $16$-cell (hypercube) the induced map $H_2(\beta, \ZZ_2)=0$ is trivial.

Still there is some hope to do something good. If the range(=image) of the map $\beta\colon E:=\times_{i=1}^8 E_i \to
\RR P^2$ meets $N^{\ast}$ we have total reality at least in the weak form (RLT). (Asserting the strong form SRLT amounts knowing that $\beta(E)\subset N^{\ast}$.) Hence to prove RLT it suffices to show that the option $\beta(E)$ disjoint form $N^{\ast}$ leads to a contradiction. Our naive idea is then that the map $\beta$ would have its range confined to the Möbius band $M:=\RR
P^2-N^{\ast}$ residual to $N^\ast$. If one is able to show that $\beta\colon E \to M$ is not null-homotopic (e.g. by showing that it hits an odd multiple of the generator of $\pi_1(M)=\ZZ$), then it would follow that $\pi_1 (\beta)$ is nontrivial, violating the above factorization $\beta^{\ast}$ through a contractible 16-cell. This contradiction would prove the weak form of RLT, i.e. existence of an octuplet inducing a total pencil.

Of course the above tactic to be completed requires a proof of the hypothetical fact that there is a loop in the 8-torus $E$ taken by $\beta$ to the nontrivial element of $\pi_1(\RR P^2)$. This seems hard to prove and requires at any rate some geometric understanding of pencil of cubics, especially of the predestination process creating the 9th basepoint as a function of the 8 assigned ones.

More pragmatically we could define the [*Rohlin body*]{} of a given $C_6$ (of RKM-type) as the image $B:=\beta (E)$. This compactum and especially its location w.r.t. the nonempty oval $N$ will govern much of the total reality question of the $C_6$. Essentially we have a trichotomy of alternatives:

$\bullet$ Either $B \subset N^{\ast}$ in which case the $C_6$ is strongly totally real, in the sense that any octuplet (in $E$) induces a totally real pencil, or

$\bullet$ $B$ overlaps $N^{\ast}$ without being contained in it, in which case some octuplets induce a totally real pencil and some other do not, or finally

$\bullet$ $B$ is disjoint of $N^{\ast}$ in which case all octuplets fail inducing a total pencil. (This is nearly incompatible with the RLT total reality phenomenon).

In the first scenario we could say that the sextic $C_6$ is [*strongly totally real*]{}, in the second that is [*(weakly) totally real*]{}, and in the 3rd case that is “anti-real”.

We believe that a continuity/degeneration argument applied to Le Touzé’s theorem (2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]) prevents the 3rd option, by letting degenerate the 8 basepoints to the empty ovals. Alas it does not seem that Le Touzé’s theorem (\[LeTouze-big-thm-01-March-13:thm\]) gives sufficiently many inner permissible basepoints so as to ensure a degeneration to the ovals.

As to the first two scenarios, we do not know if both of them do occur, and if not which one is ubiquitous. Paraphrasing, we do not know if there is a single sextic of type $\frac{6}{1}2$ (or it mirror) such that any octuplet chosen on the empty ovals induces a totally real pencil of cubics, nor can we preclude the option that all sextics verify this property.

[*Insertion*]{} \[12.04.13\].—If we interpreted correctly the last news of Le Touzé, it seems that total reality holds in the strongest possible sense, i.e. for all octuplets and even when the latter are inside of the ovals and not on themselves directly. Compare her article (2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics], plus eventually her letter in Sec.\[e-mail-Viro:sec\].

Esquisse d’un programme (déjà esquissé): the Ahlfors-Rohlin Verschmelzung {#Esquisse-dun-prog-deja-esquiss:sec}
=========================================================================

\[07.03.13\] The programme in question was probably nearly implicit in Rohlin 1978 [@Rohlin_1978], safe that he seems to have missed the possible connection with Ahlfors theorem. It is only in this respect that our programme bears some originality, yet presently we are unable to substantiate it in any serious fashion.

Large scale structure of total reality as it pertains to Hilbert’s 16th problem
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\[04.03.13\] Let us brush a sloppy summary of the situation. Rohlin 1978 [@Rohlin_1978] in somewhat cryptical fashion asserted existence of a totally real pencil of cubics on all sextics curves of type $\frac{6}{1}2$ or its mirror, yielding therefore a geometrization of the RKM-congruence asserting that $\chi\equiv
k^2+4 \pmod 8$ forces the curve of degree $2k$ being of type I. After consulting several specialists (Viro, Marin, Kharlamov, Fiedler, Le Touzé), it seems that this proof is now lost forever (or dormant in some celestial Eden). It seems extremely challenging to rediscover it if it ever existed, i.e. if Rohlin’s argument was sound and complete, as opposed to a Fermat-style cryptical allusion destined to challenge geometers. On reading the survey Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000], it seems that this is not the sole prophetical allusion of Rohlin in his fantastic 1978 survey. Also the question of estimating the number of points through which one can pass a rational connected curve is also sloppily stated by Rohlin without proof, and Degtyarev-Kharlamov consider the problem as still open. Perhaps the status of this problem evolved meanwhile.

Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics] supplied the first (and unique) written proof of a weak form of Rohlin’s total reality phenomenon, yet assuming the RKM-congruence, and so the curve to be of type I. As suggested in Le Touzé’s article (2013 ), Rohlin seems to have cultivated a large expansion of the phenomenon of total reality, as follows:

[TOR=Total reality (Le Touzé 2013, Gabard 2004 [@Gabard_2004], who ascribed this as implicit in Ahlfors 1950, and Teichmüller 1941 [@Teichmueller_1941], who loosely claims this to be found in Klein’s works).]{}—Any dividing (plane) curve admits a totally real pencil of curves.

[ROTOR=Rohlin’s total reality (implicit in Rohlin 1978 according to Le Touzé 2013)]{}.—Any dividing curve representing a scheme of type I admits a totally real pencil.

[RMC=Radio Monte Carlo=Rohlin’s maximality conjecture]{}.—Any real scheme of type I is maximal in the hierarchy of all schemes (of some fixed degree).

As noted by Le Touzé (always ), TOR implies ROTOR, and it is always tempting to believe that the latter implies RMC.

At the very source of that string of implications, we should have the Ahlfors theorem (ATR=abstract total reality) (which Teichmüller 1941 [@Teichmueller_1941] ascribes to Klein)

[ATR=Abstract total reality (Ahlfors 1950, Gabard 2004–06).]{}—Any (abstract) dividing curve $C$ (or what Klein calls an orthosymmetric Riemann surface) admits a totally real map $C\to \PP^1$ to the projective line, i.e. such that $f^{-1}(\PP^1 (\RR))=C(\RR)$. Further the degree of such a map can be arranged $\le g+1$, where $g$ is the genus of $C$. According to Gabard 2006, but still deserves to be better analyzed, there should even be such a total map of degree $\le \frac{r+(g+1)}{2}$ where $r$ is the number of real circuits of the curve $C$.

ATR should imply TOR modulo some little difficulties. A first difficulty is merely due my incompetence, of knowing if an abstract morphism from a plane curve to the line is necessarily induced by a pencil of curves. Another little difficulty is that a priori some basepoints (of a total pencil given by Ahlfors) may be imaginary conjugate, so that total reality has perhaps to include a slightly broader definition (admittedly more cumbersome) than the one used in Le Touzé. So probably the correct definition of a totally real pencil has to involve total reality of the moving points of the linear series, while the (statical) basepoints themselves being permitted to be non real(=imaginary, as we say since Tartaglia-Cardano (1535/39), essentially).

At the very end of the string ATR$\Rightarrow$TOR$\Rightarrow$ROTOR$\Rightarrow$RMC, we seem to have (accepting the elusive RMC) a sort of subordination of all highbrow congruences modulo 8, that is the Gudkov-Rohlin congruence for $M$-curves, the Gudkov-Krakhnov-Kharlamov congruence for $(M-1)$-curves (\[Gudkov-Krakhnov-Kharlamov-cong:thm\]), to the (Rohlin)-Kharlamov-Marin congruence (\[Kharlamov-Marin-cong:thm\]). To appreciate this fact in degree 6, contemplate once more the Gudkov-Rohlin Table(=Fig.\[Gudkov-Table3:fig\]). Then we see that (virtually) all congruences are subordinated to that of Kharlamov-Marin modulo the truth of RMC, safe the prohibition of the schemes $11$ (Hilbert) and $\frac{10}1$ (Rohn). The latter may be prohibited either via Arnold’s congruence or Rohlin’s formula.

So in crude approximation (and modulo RMC) “all” the prohibitions of Hilbert’s 16th problem (say perhaps apart from refined Bézout-style obstructions à la Fiedler-Viro virulent in degree $8$ or higher) are subsumed to the Kharlamov-Marin congruence ensuring type I whenever $\chi\equiv k^2+4 \pmod 8$ (abridged RKM, where R stands for Rohlin, albeit I am not sure about his exact contribution, while most writers credit Kharlamov and Marin only, cf. e.g. Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91]). ([*Insertion*]{} \[12.04.13\].—In Rohlin’s 1978 [@Rohlin_1978 p.93] the result is credited to Kharlamov alone, but it is remarked that the proof of this theorem was still unpublished.)

Accepting the above string of implications, we arrive essentially at the conclusion that all of Hilbert’s 16th problem could be governed by the phenomenon of total reality. This would be especially true if there is a geometrization of the RKM-congruence via total reality. This requires a highbrow extension of the Rohlin-Le Touzé theorem to all other $(M-2)$-schemes verifying the RKM-congruence. This looks hard but according to the previous sections it is likely that the total pencil is always of order $(m-3)$ involving the so-called adjoint curves of Brill-Noether incarnating the canonical series. The resulting degree of the map would then also be in accordance with Gabard’s version of Ahlfors theorem (compare Remark \[M-2-curve-degree-like-Gabard:rem\]).

To caricature a bit, as to emphasize our philosophy (especially as it pertains to the title of the present text devoted to Ahlfors), we formulate the:

All (or most) of Hilbert’s 16th problem can be reduced to the phenomenon of total reality, due to various authors. In in its most primitive schlichtartig form, this involves primarily Riemann 1857 [@Riemann_1857_Nachlass], Schottky 1875–77 [@Schottky_1877], Bieberbach 1925 [@Bieberbach_1925], Grunsky, etc. and in general to Klein according to Teichmüller 1941 [@Teichmueller_1941], Ahlfors 1950 [@Ahlfors_1950]).

Basically the phenomenon of total reality involves a linearization in the large of the curve via the concept of branched coverings, where any dividing curve is reduced to its most primitive incarnation, namely the line $\PP^1$. Any extraterrestrial planet, possibly with handles and several equators ([*exoplanet*]{}), yet not so exotic as to share with our planet Earth the character of orthosymmetry(=type I=dividing) involving 2 distinguishable hemispheres, can be conformally shrunk to our equatorial sphere so that fibers above the equator $\PP^1(\RR)$ are totally real, i.e. on the exo-equators of the exoplanet[^78].

This is an abstract theorem yet it should imply a concrete result of total reality like the above TOR. This process could be termed [*descent*]{} from the Riemannian universe (Gromov’s prose) to the terrestrial Plato cavern (of Hilbert’s 16th problem). Especially important would be a quantitative control on the order of those total pencils obtained by descent of the abstract Riemann-Schottky-Klein-Teichmüller-Ahlfors circle maps (equivalently total maps to $\PP^1$). For a simple implementation in the case of $M$-curves, cf. our Theorem \[total-reality-of-plane-M-curves:thm\], which is a trivial adaptation of the no collision principle of Riemann-Enriques-Chisini-Bieberbach-Wirtinger, etc.

Next when we move down to $(M-2)$-curves the first concrete (and nontrivial) phenomenon of total reality is the Rohlin-Le Touzé theorem (abridged RLT after their author’s names or for (total) ReaLiTy). Historiographically, it is noteworthy that Rohlin does not seem to have ever been aware of Ahlfors theorem, and it seems that the gap between both traditions (Riemann vs. Hilbert) as not yet been fully bridged. It is also notorious (either from the viewpoint of geometric function theory à la Riemann, or the algebro-geometric perspective) that total reality is much easier for $M$-curves as it involves a schlichtartig semi-Riemann surface (planar orthosymmetric half). This is also evidenced by the fact that there is no collision between a group of $g+1$ distributed on the $M=g+1$ ovals. In the non-Harnack maximal case, any group of $g+1$ points moves (Riemann-Roch) but then several of them being distributed on the same oval a risky collision can occur in the long run, foiling total reality. A subtle condition of dextrogyration must be ensured to gain total reality.

So what remains to be done?

Project 1.—Try to clarify the above logical implications between ATR, TOR, ROTOR, RMT. This is basically a Riemann-ification of Hilbert’s 16th problem.

Project 2.—Try to understand better the lost proof of Rohlin, and how the statement extends to curves of higher order.

Two routes toward (Project 2) seems a priori possible making the exploration somehow elusive or at least time consuming. Either work from the scratch in the Plato cavern of plane curves grooving the plane $\PP^2$ where Hilbert’s problem is formulated, or attempt a descent of the abstract result à la Riemann-Schottky-Klein-Enriques-Chisini-Bieberbach-Grunsky-Wirtinger-Teichmüller-Ahlfors-Gabard. The method of descent looks a priori delicate but worked for (plane) $M$-curves, cf. again Theorem \[total-reality-of-plane-M-curves:thm\]. The drawback of this method of descent is that it would (like Le Touzé’s proof) depend upon a knowledge a priori of the dividing character of the curve. If optimistic it could be that Gabard’s theorem (2004-2006 [@Gabard_2006]) could imply total reality for dividing $(M-2)$-curves of degree $m$ via a pencil of (adjoint) curves of degree $(m-3)$ (cf. again Remark \[M-2-curve-degree-like-Gabard:rem\] for some weak evidence). As a special case it could be the case that Gabard’s theorem implies a weak-form of the RLT-theorem, yet much work along ATR$\Rightarrow$TOR is required to bridge the gap.

Rohlin’s intuition vindicable via Ahlfors? Algorithmic rôle of RMC for plotting machines
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\[05.03.13\] All what we have to say is now essentially well-known, but alas still in embryonal state. Let us try to make the philosophy behind the grand dessin imagined by Rohlin more palatable, than in the previous section.

What is Hilbert’s 16th problem at all about? Answer: Topology of real plane algebraic curves. But in reality we are geometers and topology is merely a weakening of what wanted to study earlier geometers (say Diophante, Euclid, Archimedes, the algebro-geometrization of Fermat, Descartes, Newton, etc.) What is called upon is an understanding of the big video game where given an equation one traces the corresponding curve in real time. One should imagine a powerful enough machine showing us in real time the curve evolving when dragging with a joystick the coefficients of the equation.

Any machine able to do this presumably request at an algorithm telling when to stop the tracing procedure as the real locus has been represented within sufficient accuracy as to infer the exact topology of the curve.

It is at this stage that Rohlin’s maximality conjecture may enter into the scene. Indeed for a given degree there exists certain distinguished schemes (in the sense of Rohlin) representing so-to-speak fully crystallized extremal shapes not susceptible of any further apparition of ovals (as small as they may be). As popularized by Rohlin 1978, this intuition of saturated schemes truly goes back to Klein 1876, yet in some primitive sense distinct from Rohlin’s interpretation.

Consider for instance a curve of degree 6 whose real locus contains a deep nest of depth 3, then it is already saturated and there cannot be any further oval. More generally we have certain schemes of type I, which according to Rohlin’s intuition ought to be maximal, hence incarnating the maximum topological complexity permissible for the given degree.

The problem of understanding this ontological truth splits in two parts: Why is it true, and why is it useful?

First we try answering the second question. As already pointed out, Rohlin’s maximality conjecture (RMC) incarnates a sort of stopping process for the plotting machine (realizing the dynamical video game or just its statical variant). From the viewpoint of Hilbert’s 16th problem, the fact that the deep nest of depth 3 is a maximal scheme of degree 6 forbids a menagerie of other schemes enlarging it which a priori could exist, but do not essentially by virtue of Bézout’s theorem. Hence Rohlin’s type I schemes (granting their maximality) are like advanced sentinels prohibiting schemes of higher topological complexity. Without such prohibitions Hilbert’s 16th problem would be even more intractable than it already is. Hence Rohlin’s maximality conjecture is a sort of upper bound for the complexity of Hilbert’s 16th problem. This should be sufficient reasons for answering the utilitarian aspect.

As to the first question, it is somehow ironical that Klein seems to have been much in touch with both aspects of our question. First, as we said he is regarded (by Rohlin himself) as a precursor of Rohlin’s maximality conjecture. Second (but this is more elusive to testify with high accuracy), Klein is credited by Teichmüller 1941 [@Teichmueller_1941] as the true forerunner of Ahlfors theorem. At some broader scale, all goes back to Riemann (especially if his life would not have been so short).

Rohlin’s maximality conjecture should according to our intuition (Gabard ca. 1st January 2013) reduces to Ahlfors theorem, via what Le Touzé calls the Rohlin total reality conjecture, cf. (ROTOR) of the previous section. So it is quite interesting to see that the extremal shapes (maximal schemes) of Rohlin are induced by schemes of type I, and what makes this possible is the phenomenon of total reality. Behind the latter there is of course Ahlfors circle maps, and so basically an extension of the Riemann mapping theorem. This in turn is governed by potential theory, itself concomitant of the calculus of variation of Euler-Lagrange as applied to Laplace’s equation. All this détour to make apparent that the algebro-geometric extremal principle posited by Rohlin’s maximality conjecture seems governed by another extremal principle, namely those ensuring solvability of Dirichlet’s principle. So be it via Abel (and what some like to call Hodge theory) or directly via Euler-Lagrange-Laplace-Dirichlet-Riemann-etc-Ahlfors we have the phenomenon of total reality for curves of type I, and when the scheme itself is of type I this phenomenon acquires an extra punch of universality, making the scheme maximal in the Hilbert-Gudkov-Rohlin hierarchy.

Coarsely, our thesis could be that what missed to Rohlin to complete his programme (amounting essentially to bound the complexity of Hilbert’s 16th problem) is merely a rather simple theorem of function theory (due basically to Riemann-Schottky-Klein-Teichmüller-Ahlfors). The latter in turn being not much more than a bordered avatar of Riemann existence theorem exhibiting any closed Riemann surface as a branched cover of the projective line. In other words any abstract Riemann surface (à la Riemann-Prym-Klein-Weyl-Radó) becomes a concrete one (à la Abel-Galois-Cauchy-Puiseux-Riemann-Weierstrass, etc.)

Next it comes to make this Ahlfors-Rohlin fusion a reality. Here starts alas some little difficulties which we hope to able to overcome in the future. The problem breaks in two steps:

$\bullet$ Step 1. Make the Ahlfors theorem concrete by specializing it to dividing plane curves, and conclude the existence of a total pencil of “adjoint” curves exhibiting total reality.

$\bullet$ Step 2. Prove that a totally real pencil (often abridged total pencil) implies maximality of the scheme.

Start with a scheme of type I of degree $m$. This means by definition that any curve $C_m$ representing it is of type I. By ROTOR (i.e. a concretized version of Ahlfors theorem), there is a total pencil of curves $\Pi$ of order say $k$.

This means that each curve $C_k$ of the pencil cuts $C_m$ only along real points \[as soon as they are mobile\]. \[[*Brackets Added*]{} \[12.04.13\].—Probably imaginary basepoint have to be allowed, yet the mobile part of the pencil should be totally real.\] Such a balayage seems actually to supply a fast algorithm to trace the real locus by reduction to a problem in one variable (consider e.g. the Gürtelkurve swept out by a pencil of lines), which in turn could involve the Newton root-finding algorithm via linearization. Of course in general, curves of the pencil are not rational and so this asks for a tricky extension of Newton.

Once existence of a total pencil is granted we seems nearly finished, because if there are more ovals then passing a curve of the pencil through the additional point would corrupt Bézout. The notorious bug of that argument is that we assume implicitly the over-scheme to be realized by an augmentation of the given algebraic curve, which is priori not the case. \[12.04.13\] Further a total pencil exist as well on some type I curves belonging to indefinite (yet non-maximal) schemes, compare the case of sextics. This is of course another obstruction to completing crudely the just sketched argument.

However what makes that the argument works for the deep nest or the satellites of the quadrifolium? Is it the fact that the pencil curves are rational? or is it a sort of geometric intuition of the pencil, or perhaps some canonicalness of it? Maybe to get the RMC we need not only existence of a total pencil but some sort of uniqueness (up to anodyne choices like center of perspectives chosen in the “deepest” ovals).

It seems that a scheme of type I incarnates a family of curves which are so-to-speak totally real in some canonical way, and the total pencil is virtually God-given. (Beware yet that the family in question is not necessarily connected in the hyperspace of curves, cf. Marin 1979 [@Marin_1979] or Fig.\[Marin:fig\].)

At this stage it seems important to remember a metaphor allied to total reality. Total reality means that all intersections are visible on the reals. Using a pencil means essentially that we choose a mode of vision of the curve. Basepoints are eyes of some insect having several eyes and curves of the pencil are optical rays enhancing how the animal perceives the curve (Gebilde). What is strange is that total reality amounts saying that the vision is purely transverse and so the object is in reality invisible (no apparent contour). To make this concrete consider the example of the Gürtelkurve $C_4$ (2 nested ovals) projected from an innermost point inside the deepest oval. Paraphrasing in a real life metaphor, looking at a glass of wine from outside you see its apparent contour, but when placed inside of it, it suddenly becomes invisible. This is total reality.

It is tantalizing that total reality (via Ahlfors theorem) seems so close to prove RMC(=Rohlin’s maximality conjecture) but apparently fails. As we (or better Rohlin) suspected the assumption that the scheme is of type I must impose the corresponding curves being strongly harpooned by total reality. But how to make this idea precise?

We can imagine the space of all curves representing the scheme, and think about this a universal curve of type I. There should then be a version of Ahlfors theorem for family of curves or (as Teichmüller, Ahlfors, Bers, liked to say) for a [*variable*]{} Riemann surface.

The net effect would be that total reality is genetically imbued in the curve(s) itself in such a strong fashion that the scheme is maximal. So any scheme of type I has a canonical vision making it totally real, amounting essentially to look at the world form inside the glass (or bottle) of wine. This is akin to the photoelectric effect. (Compare with the known examples of the unifolium and its satellites, alias deep nests in the jargon of Hilbert and the Russian school, or the quadrifolium, and its satellite total under a pencil of conics, plus the (elusive) Rohlin-Le Touzé phenomenon for sextics).

Once this photoelectric vision of the curve is given then nothing more can appear in the blue sky and so the scheme is maximal. This is the intuition of why RMC holds true, but how to convert this in a mathematical proof. What seems to be in demand is a mechanism which from the shape alone of the scheme identifies the total vision of the curve. This we call the photorealism or photogenism.

If a scheme is of type I then it is photogenic, and then it must be maximal.

This seems to request for a general mechanism of where to assign basepoints which would extend the total reality of unifolium, quadrifolium, and 9-folium of Rohlin-Le Touzé flashed resp. by by pencil of lines, conics and cubics. As we discussed in a earlier section the case of degree 8 schemes looks a bit puzzling, where by the RKM-congruence we have plenty of $(M-2)$-schemes of type I (ca. 100 if we were not too bad in counting). The center of vision (basepoints) are then quite hard to predict. In general there are $B=M-3$ of them (where $M$ is Harnack’s bound), so $B=19$ for $m=8$, and alas it is presently not very clear where to assign them in full generality. Making all this explicit could solve the question of giving a precise sense to our notion of photogeny, and as a by product crack the RMC.

Is this a realist strategy? Is there a more abstract argument? If not, we really need some highbrow extensions of the Rohlin-Le Touzé theorem dictating us for all schemes of type I where to assign basepoints. This seems to call first for a classification of schemes of type I.

[*Long (paragraph) Insertion*]{} \[12.04.13\].—To tell the truth it should be remarked that even in the case of Rohlin-Le Touzé (degree $m=6$) we lack presently a proof of the desideratum that the vision of total reality (via the pencil of cubics) is strong enough as to ensure maximality of the scheme. (This conclusion is of course true via the Gudkov census but we lack a direct proof along the philosophy of total reality.) Perhaps Rohlin knew a proof, but as far as we know it was not published too.) Note two things. First, the more naive principle of maximality of Klein-Marin 1876/1988, when combined with Itenberg’s contraction affords another approach to the problem of RMC, which is perhaps easier to implement (though in general only based on the conjectural principle of contraction). Second, it seems that in Rohlin’s approach we lack some flexible medium to carry the enlarged scheme to the original one. This could involve trying to approximated a diffeomorphism of $\RR P^2$ by something more algebraic (maybe a Cremona transformation), but then it looks hard to finish the job. So maybe $A+B+C=Rmc^2$, i.e. Ahlfors, plus Bézout, plus Cremona implies Rohlin’s maximality conjecture. One may also wonder if there is not a much more flexible proof of RMC say akin to Rohlin’s formula where merely soft topology is used (while avoiding any contraction principle). We have then basically 2 curves, one enlarging the other, and one of which universally of type I. So one could fill the half à la Rohlin, i.e. by all discs (like in the proof of Rohlin’s formula) and inspect the intersection of the homology class of degree $k$ with the homology class of the enlarging curve. The sequel is certainly hard to complete. (It is at this stage that we had the idea of using Mangler to isotope the enlarged curve back to the original, cf. Sec.\[RMC-via-Mangler:sec\] for more detail on this strategy to attack RMC via Ahlfors, plus Mangler.)

Here we know quite little, but as said earlier in this text, it could be the case that the RKM-congruence is a universal detector of $(M-2)$-schemes of type I, while all other type I schemes arise as satellites of schemes of type I of lower orders (dividing the given degree $m$). So when $m=2p$ is twice a prime number there should be no such satellite (except that of the unifolium) and all new type I schemes would be concentrated at the $(M-2)$-level. Of course we can always make abstraction of the $M$-schemes where RMC holds trivially true. So we see some sort of higher arithmetic structure emerging in Hilbert’s 16th problem as boosted by Rohlin’s conceptions, namely a sort of inductive process that could progressively step-by-step enumerate all schemes of type I, merely as $(M-2)$-schemes of type I satisfying the RKM-congruence mod 8, or as satellite of earlier such schemes, and for all of them expect a synthetical revelation of the type by a canonical pencil à la Rohlin-Le Touzé incarnating primitive forms of the phenomenon of total reality. This deserves nearly the name of Rohlin’s divination.

As the list of photogenic schemes increases at each step $m$, we may conclude RMC by having exhibited in some ad hoc fashion the total reality of all type I schemes, and the RMC would follow step-by-steps. Needless to say this requires an immense effort, and the induction required to validate RMC in all degrees looks a priori extremely tricky.

Furthermore one could imagine that all this ascension effected in autarchy from Ahlfors theorem by using rather ad hoc optical recognition procedure via total reality. This would be parallel to the evident total reality of the satellites of the unifolium (alias deep nests) and idem for the quadrifolium, or Rohlin’s schemes of degree 6 (modulo the lost proof of Rohlin). In that case the theory would be purely Rohlinian and this is probably essentially what Rohlin envisioned.

In contradistinction, when attacking RMC, we know a priori the scheme being of type I so there could be some inference of Ahlfors theorem permitting to shortcut the (pure) total reality vision of Rohlin. This inference could increase the (ascensional) speed conceding some abstractness in the verification of RMC. Yet, as observed, even this looks hard unless we get a better grip upon the abstract total reality of Ahlfors.

A first modest (but nontrivial) exercise is to write down a clear version of Rohlin-Le Touzé’s total reality claim, and using it deduce the maximality of those 2 schemes. Here again notice that exploiting the type I assumption as do Le Touzé is not a concession since we are interested in RMC. So here total reality seems sufficiently strong (canonical) to ensure maximality of the schemes and we rederive so from Le Touzé’s result the prohibition of Gudkov, etc. (compare Table \[Gudkov-Table3:fig\]). In particular RMC holds true in degree 6 for some intrinsic reason allied to total reality, as opposed to being a byproduct of the full classification of Gudkov.

A more highbrow project would be to inject the function theory à la Ahlfors in the problem. Assume given a scheme of type I, we can for each representing curve $C_m$ choose a total pencil $\Pi$ which is a line in the space $\vert kH\vert$ of $k$-tics curves. It seems plausible that the dependence can be made continuous. Then we have a universal family of photoelectric effects on $C_m$ and its deformations (possibly in different chambers of the discriminant) in which case the line $\Pi$ may jump, a priori even in different hyperspaces indexed by different $k$.

On applying the $k$-tuple Veronese embedding $v_k$—i.e. the holomorphic map $\PP^2\to \PP^N$ induced by the linear system of all k-tics—the total reality of $v_k(C_m)$ would appear under a pencil of hyperplane, hence the curve would be total real under a pencil and therefore located as several spires gyrating around the base locus (plane of codimension 2 in $\PP^N$). Now it may be expected that the phenomenon of total reality is as evident as it was for the deep nest (i.e. reduction to the case of a linear pencil) and that we may conclude maximality from Bézout (applied of course now in the Veronese hyperspace).

After this little psychoanalysis of Rohlin’s secret garden, we see that “la réalité totale nous colle à la peau.” In some sense the phenomenon ought to be so inherent to a curve belonging to a scheme of type I that maximality of the scheme should follow via the photoelectric effect. By the latter we really mean that the total pencil being saturated nothing more is allowed to appear in the blue sky without corrupting Bézout. (Prototype: a deep nest with a pencil of lines through the deepest oval, or satellites of the quadrifolium in degree $4k$.)

As a foundational detail, I always thought that possibly imaginary basepoints have to be permitted in the definition of a totally real pencil, so that merely moving points of the series are real (cf. Gabard 2004, p.7). Now I am not sure that this is really required.

In all basic examples of total reality (i.e. the deep nests interpretable as satellites of the unifolium or the quadrifolium and its satellites) the permissible basepoints ensuring total reality are always varying through a contractible union of cells as they are located inside the deepest ovals. This is probably also true for the sextics of Rohlin-Le Touzé. If this is a general phenomenon then this is a bit in line with our desideratum that the total pencil ought to be almost canonically associated to the dividing scheme of type I. If instead we are interested in total maps of lowest possible degree then we are inclined to let degenerate the basepoints on the ovals themselves and so the total pencils of this sort are parametrized rather by tori.

Another idea \[developed in Sec.\[Thom:sec\]\] is to fill the plane curve by the (orientable) membrane of $\RR P^2$, to get a certain smooth surface in $\CC P^2$ whose fundamental class is $kH\in H_2(\CC P^2, \ZZ)$. Smoothing its corner and applying Thom’s conjecture (=Kronheimer-Mrowka’s theorem) could lead to some interesting consequence. (More about this soon, cf. Theorem \[Thom-Ragsdale:thm\].) Of course this is basically related to the ideas of Arnold and Rohlin.

Following our main theme, the idea would be that there is always for a scheme of type I some preferred (up to the ambiguity of a contractible space of parameters) total pencil, which we call a photon. This would naively speaking be obtained by assigning basepoints among the deepest ovals. All this works good for degrees $\le 6$. In degree $8$, the RKM-scheme 20 already affords a little problem as quintics have 19 basepoints assignable and it is not clear which ovals have to be used as “anchor” basepoints. \[But this scheme is prohibited by Thom, cf. again Theorem \[Thom-Ragsdale:thm\], or argue via Rohlin’s formula.\]

Once we have a photon (i.e. a canonical total pencil) then we would like to argue that its satisfies the photoelectric effect, and RMC would follow.

In the case of $(M-2)$-curves of type I we have $B=M-3$ basepoints for a pencil of $(m-3)$-tics that are freely assignable. This is one unit less than the number of ovals and it is not clear which one can be dispensed of being marked by a basepoint. We could imagine that we could always dispense the oval whose porous inside has the most negative Euler characteristic. To make this serious compare Fig.\[RKM-schemes-deg-8:fig\], where we find however schemes where such a dispensed oval is not uniquely defined, e.g. $\frac{6}{1}\frac{6}{1}\frac{4}{1}1$. So our recipe is certainly dubious.

We hope to have made the nature of the question clear enough. It seems first that there is no direct reduction of RMC to Ahlfors theorem, except perhaps if one as some deeper grasp upon the geometry of a total pencil (photon) so as to ensure via the photoelectric effect the RMC. As we said at the beginning of the section, the net impact would be a sort of upper bound upon the complexity of Hilbert’s 16th problem. In fact it would really be the clef de voûte yielding some insights upon the architecture of the pyramid of all schemes of some fixed degree $m$, namely type I schemes ought to be maximal element (yet not the sole ones cf. Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]). This conjectural maximality explains nearly all prohibitions (at least in the case $m=6$).

Despite our obsession to harpoon RMC via Ahlfors’ total reality (what Viro was sceptical about) and we have only the vague suggestions of the present section to propose \[see also maybe the next Sec.\[RMC-via-Mangler:sec\]\], it may be argued that even if it worked by a clever trick, it would perhaps not be as satisfactory as the full and slow progression along the menagerie of all schemes that must be tabulated along the higher order cases of Hilbert’s 16th problem. In other words we would like to know the whole pyramid and not just its maximal element. From the viewpoint of the French revolution we would like to know the whole folk and not just the aristocrats. More seriously we want to “see” the exact geometry of the phenomenon of total reality, and not just its capitalistical/hierchical impact via the photoelectric effect upon a validation of the RMC. \[12.04.13\] Also interesting is the question of the density of schemes below the aristocrats. Of course it seems that the pyramids as dense below the maximal elements: philosophically because an aristocrat seems unable to provide for its wants without the force of all its servitors.

An isotopic attack on RMC via Mangler 1939 {#RMC-via-Mangler:sec}
------------------------------------------

\[12.04.13\] As we often experimented RMC would follow from Ahlfors if we knew that the enlarged scheme lies in a tube neighborhood of the given curve of type I. The difficulty is that a priori the enlarging curve is very distant and hard to compare to the original one. Crudely put one could expect to reduce always to the easy case by using an isotopy.

So one could try to isotope the diffeomorphism taking the small curve to its enlargement to the identity. Recall this to be possible since the mapping class group of $\RR P^2$ is trivial) \[Mangler 1939 [@Mangler_1939], often used by Teichmüller 1939\]. One could then perhaps try to extend this isotopy to $\CC
P^2$ so as to get reduced to the case where the enlargement is a small perturbation (actually the identity). In this case Ahlfors suffice to imply RMC, since Ahlfors’s pencil affords something like a transverse structure, and one gets an easy corruption with Bézout by letting pass a curve of the pencil through the new oval (not within the tube neighborhood).

The difficulty is of course to check that Bézout (which is some something algebraic rigid) is conserved during the very plastical deformation of isotopy. Yet perhaps we may reinterpret intersections homologically as to gain more flexibility. Further the isotopy could be compatible with reality (equivariant and respecting $\RR P^2$ and its complement of imaginary points). Finally due to the geometric interpretation of intersection numbers (in homology) their values will be clearly conserved by the isotopy. So we arrive at the:

There is perhaps a trivial proof of RMC via isotopy of $\RR P^2$ equivariantly extended to $\CC P^2$, so that RMC reduces trully to Ahlfors.

Suppose $C_m$ to be a curve of degree $m$ belonging to a scheme of type I. Let $D$ be a curve (of the same degree) whose scheme enlarges that of $C_m$. Fix a diffeomorphism $f$ of pair $(\RR
P^2, C_m)\to (\RR P^2, D_{\ast})$ where $D_{\ast}$ is $D$ less one oval (w.l.o.g. or more ovals in general).

By Mangler 1939, we can isotope $f$ to the identity of $\RR P^2$. Now it seems reasonable to expect that there is a natural way to extend an isotopy of $\RR P^2$ to one of $\CC P^2$. This does not need to be strongly unique but merely to exist in some sense that it preserves real parts and maybe can be chosen equivariant w.r.t. conj. I.e. carrying a point along the isotopy up to time $t\in[0,1]$ commutes with the symmetry conj. Knowing that the quotient $\CC P^2/ conj$ is $S^4$ could be of valuable assistance to construct the extended-isotopy.

So we have $f_t$ an isotopy of $\RR P^2$ say with $f_1=f$ and $f_0=id$, and $F_t$ and extension thereof to $\CC P^2$. The map $f$ pushes injectively the ovals of the first good curve $C_m$ into those of the hypothetical enlargement $D$. So operating backward in time along the isotopy $F_t$ we may retract the complexified curve $D(\CC)$ so that its real part becomes close to that of $C_m$ (and even identic to it). Denote $D_0$ this “temporal retraction”, which is a “flexible” Riemannian surface, with fundamental class still of degree $m$, by homotopy-invariance of homology.

Now by total reality the first curve being of type I it admits a total pencil, all of whose members have Bézout saturated intersections with the curve. Taking a curve $P_k$ of the pencil passing through the additional oval of $D$ isotoped backward in time ($t=0$), create one extra intersection (that will count positively because the extended isotopy is orientation preserving). All other other intersections also counts positively if we are capable arranging the large isotopy $F_t$ to respect somehow the normal bundle of $\RR P^2$. Hence the pull-back $D_0$ will have excessive number of intersections with $P_k$. The homological Bézout (i.e. Poincaré, Lefschetz, etc.) is therefore corrupted. Rohlin’s maximality conjecture would be proved by “soft topology” plus some Ahlfors.

The critique to this argument however is that a priori it applies to any dividing curve supporting a total pencil and those can be of indefinite type (yet not maximal), cf. the case of degree $m=6$ where there is plenty of such examples (Fig.\[Gudkov-Table3:fig\]). So a serious gap requests to be filled. Maybe this will be an easy game for Alexis Marin?

To be optimistic, our argument looks so close to prove the big desideratum (=RMC) that it is certainly worth exploring further. In particular since the argument is spoiled by the objection of indefinite schemes there must be (for instance in degree $6$ where the Ahlfors total pencil are very easy to describe explicitly, e.g. for the scheme $9$ where we have the simple Fig.\[Fcubic:fig\]) there must be some obstruction to extend the Mangler isotopy to $\CC P^2$ (at least in a fashion that positivity of intersections are conserved).

Understanding this obstruction, and assuming that one capable to show that it vanishes if the scheme is of type I could afford a proof of Rohlin’s maximality conjecture. The proof is likely to involve some 4D-topology (say à la Marin-Siebenmann-Alexander).

Additional remarks on Rohlin-Le Touzé total reality for $(M-2)$-sextics of RKM-type
-----------------------------------------------------------------------------------

\[04.03.13\] We concentrate again on the case of a sextic $C_6$ of type $\frac{6}{1}2$. Before entering into the elusive technical details we recall the basic problematic yet quite elusive for the moment. We would like to show a phenomenon of total reality, yet its exact shape is still obscure to us.

Either we can use the RKM-congruence to infer a priori that the curve is of type I. Then we could apply the (abstract) result of either Ahlfors 1950, or Gabard 2006 and hope to effect a descent in the plane, to get a total pencil (hopefully of cubics). This descent probably requires some theory à la Brill-Noether as a Plato-cavern-style reflection of Riemann’s work. Alternatively, we can try to follow the (direct concrete) route proposed by Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics] depending upon a detailed analysis of pencils of cubics.

Finally we could dream to recover the lost proof of Rohlin, i.e. without assuming the dividing character while trying directly to ensure total reality of such a curve under a pencil of cubics with suitably assigned $8$ basepoints. Here the problem is that we know presently very little of how general the phenomenon of total reality is, i.e. which octuplets induce total reality. More about this soon.

A very first (extremely basic remark) that we all use subconsciously is a sort of closing lemma for algebraic curves:

\[Closing-lemma:lem\] [(Closing lemma)]{}.—Given any real plane projective curve, its real locus (if non empty) consists of closed circuits (Jordan curves in $\RR P^2$) or possible poly-cycles like figure 8, etc, or eventually an isolated singularity. Topologically it is always locally a multi-node consisting of a certain number of branches crossing transversally, or an isolated point. When the point of the real curve is non-isolated then there is at least one Jordan curve based on the given point.

This is a mixture of algebraic-geometry, implicit function theorem, and topological compactness of $\RR P^2$, and abstract classification of compact (Hausdorff) $1$-manifolds (just the circle), and some singular “graph” avatars.

This pertains to the very first step of our total reality story as follows:

Assume given 8 basepoints distributed on the empty ovals $E_i$ of our $C_6$. Let $\Pi$ be the pencil of cubics passing through the 8 points. Then all curves of $\Pi$ cut the curve $C_6$ in at least 16 real points (i.e. just 2 units less than the maximum permissible by Bézout). Say in that case that the pencil is quasi-total.

Our 8 basepoints forces 8 real intersections, but by the closing lemma each intersection has at least one companion (possibly the same yet then with a tangency and so counted with multiplicity 2). More precisely we look at one of the basepoint and choose any curve of the pencil. First note that the basepoint cannot be an isolated real point of the curve $C_3$, and so there is by the closing lemma a topological Jordan curve in the real locus of $C_3$ which has thus at least 2 intersections with the given oval.

So to reach Rohlin’s (lost) theorem just 2 real intersections are missing (quasi-total) but the gap toward total reality is still immense.

Let us first observe the following extension where basepoints are located inside the ovals, as opposed to the former case where they were directly imposed on the ovals themselves:

(Interior distribution quasi-total).— If the $8$ basepoints are assigned in the insides of the empty ovals $E_i$, then the pencil is also quasi-total, i.e. $C_3\cap
C_6$ has $16$ real intersection for all $C_3\in \Pi$.

Suppose given such a $C_3$, hence visiting the basepoints $p_i$ labelled as to be in the insides of the $E_i$. Each $p_i$ forces 2 intersections in $C_3\cap C_6$, except if the circuit of $C_3$ through $p_i$ is a small oval inside $E_i$. But then the residual pseudoline of $C_3$ has to visit all 7 remaining points, and so is forced to intercept $N$ the nonempty oval of $C_6$. We count then $14+2=16$ intersections.

[*Addendum*]{}.—Further in the above situation of a small oval of $C_3$ inside $E_i$, then for vibratory reasons the 9th basepoint of $\Pi$ has to be on it.

Hence if $p_i$ is an inner point of $N$, then (even) total reality is fulfilled (Lemmas \[9th-basepointI:lem\] and \[9th-basepoint-totalII:lem\]).

If $p_i$ is an outer point then the residual pseudoline of $C_3$ will intercept (twice) $N$, and we have again $14+2=16$ (real) intersections.

The above [*Addendum*]{} is not formally required for the proof of the lemma but we include it as it nearly give a hope to attack Rohlin’s total reality claim (abridged RTR in the sequel). Beware that Rohlin’s statement is very loose in the original paper (Rohlin 1978 [@Rohlin_1978]) and so RTR should not be given a too strong connotation from the scratch. Part of the problem is to decide with which level of generality Rohlin’s claim is correct.

With our zero knowledge, we can distinguish several layers of interpretation for RTR. In its strongest form this would be the assertion:

[(Inside total reality)=ITR]{} Denote by $E_i^{\ast}$ the (sealed) insides of the empty ovals $E_i$ of the $C_6$, then the pencil $\Pi$ of cubics through any points $(p_1,\dots, p_8)\in E_1^{\ast}\times \dots\times
E_8^{\ast}$ is totally real.

Then there are several weaker variants, namely the same conclusion under the assumption that the $p_i$ belong to the ovals $E_i$ themselves. The corresponding statement is called OTR, for oval total reality. Another weakening is to relax the conclusion by claiming only total reality of $\Pi$ for a suitable octuplet, as opposed to claiming it for all of them. This relaxed form induces statements called WITR resp. WOTR, where the “W” stands for weak total reality. Though being weak this would be enough to geometrize the degree $m=6$ case of the RKM-congruence. Le Touzé’s theorem is essentially WITR, i.e. weak inside total reality modulo the fact that Le Touzé assumes (or infers from the RKM-congruence) the dividing character of the curve.

Of course we have formal implications like the following commutative square $$\begin{array}{rcl}
\textrm{ITR} & \Rightarrow & \textrm{OTR}\\
\Downarrow \quad &      &\quad \Downarrow\\
\textrm{WITR} & \Leftarrow & \textrm{WOTR}.\\
\end{array}$$ Alas we know very little about those statements. We do not know if the strong versions (upper row) are true, and if foiled it could a priori still be the case that they hold true for special sextics $C_6$. To be factual at the time of writing (and modulo an understanding of Le Touzé’s proof) the only available knowledge is that the weakest form WITR holds true, and even as we said under the assumption that the curve is of type I as may be inferred from the RKM-congruence. Hence of course the whole square can be extended to a cube with another square face of statements assuming the dividing character of the curves. Modulo RKM-both squares are actually formally equivalent, but a very purist could prefer eliminating this dependency. As asserted (but never proved) by Rohlin 1978, one could hope to do more and prove one of the above statement [*ex nihilo*]{} (without reliance upon RKM).

As usual in mathematics (or in the world of bird of preys) one should always start attacking the weakest prey, namely WITR. This is a bit strange because quasi-total reality is slightly easier to establish when the basepoints are located on the ovals. Further keep in the subconscious part of the brain, that Rohlin’s hints are so vague that it is not even clear that our 4-fold strategy covers all what is permissible (for instance it could be useful to assign the basepoint not on the empty ovals but one also on $N$. This looks exotic, but perhaps useful in extreme case of desperation).

So what is a reasonable strategy toward WITR?

We may start from the observation that the pencil $\Pi$ is totally real iff the 9th basepoint $p_9$ of $\Pi$ is in the sealed inside $N^{\ast}$ of the $C_6$ (cf. Lemma \[9th-basepoint-totalII:lem\]). In reality this lemma holds true for basepoints assigned on the ovals, but probably extends to the broader setting. This leads to the following:

(Hypothetical lemma) Assume $p_9$ to be in $N^{\ast}$ and the $p_i\in E_i^\ast$ in the (sealed) insides. Then $\Pi$ is totally real (abridged total).

Let us show where the naive proof breaks down. Assume given any curve $C_3$ of the pencil. A priori $C_3$ may pass through an inner point $p_i$ (i.e. inside $N$) via a microscopic oval $E$ of $C_3$ entirely inside $E_i$, thereby creating no real intersections. Of course then the residual pseudoline of $C_3$ (i.e. $J=C_3(\RR)-E$) intercepts (twice) the nonempty oval $N$, but this affords altogether only $14+2=16$ real intersections. Note that $p_9$ has for vibratory reasons necessarily to be located on $E$, yet this is no contradiction. Perhaps I missed something and there is a more clever argument establishing this modest technical conjecture. (Le Touzé probably has some idea.)

Let us skip this conjecture, while attacking rather the stronger looking WOTR proposition, as in the latter case total reality is easier to ensure. Of course doing so we loose some freedom for the parameters as the large 16-dimensional cell $E^\ast=\times_{i=1}^8 E_i^\ast$ is traded against the 8-dimensional torus $E=\times_{i=1}^8 E_i$ but perhaps this suffices to conclude. Further the advantage would be to get a total map of lower degree, namely one corroborating Gabard’s bound.

So it is a delicate matter to decide which strategy “insides of the oval versus the ovals themselves” is more likely to give a proof of RTR (=Rohlin’s total reality claim).

For the moment we have no better idea than the topological approach sketched in one of the previous section, i.e. to ensure that the 9th basepoint lands in $N^{\ast}$, and so abort this delicate question.

Thom’s conjecture vs. Hilbert’s 16th {#Thom:sec}
====================================

\[21.03.13\] [*Warning.*]{}—All this Sec.\[Thom:sec\] is poorly organized for reasons to be soon explained. In particular it contains several mistakes, but also such fundamental results as Petrovskii inequalities, the strong-Petrovskii-Arnold inequalities. Some higher Gudkov tables of periodic elements (e.g. Fig.\[Degree10:fig\]) show the geographical impact of Petrovskii-Arnold as compared to Ragsdale’s conjecture (briefly discussed in Sec.\[Ragsdale-conj:sec\]). The importance of the Petrovskii-Arnold results was pointed out to me by Thomas Fiedler, who corrected several benign mistakes and one much more fatal bug of mine. This section should thus be read with extreme discernment, as it mixes both the best (Petrovskii-Arnold and even the marvellous construction of Itenberg-Viro) and the worst (Gabard). Several footnotes and WARNINGS should aid the reader to avoid going into the same pitfall as I did. All those WARNINGS are due to kind letters of Fiedler who fixed all my misconceptions and posed me challenging problems. We hope in the future to be able to reorganize the text in a more decent fashion after exploring in more depth a possible fascinating interplay between Hilbert, Ragsdale, Thom=Kronheimer-Mrowka 1994 [@Kronheimer-Mrowka_1994] (independently Morgan-Szabó-Taubes 1995/96 [@Morgan-Szabo-Taubes_1996]), and the work of Petrovskii-Arnold (1938–1971). Of course the interested reader is invited to consult more professional sources, notably Mikhalkin 1994 [@Mikhalkin_1994-adjunction-Thom].

\[21.03.13\] This section was built around the fundamental result $\chi\le k^2$ for any dividing curve of even degree $2k$ (Theorem \[Thom-Ragsdale:thm\]) directly inferred from the so-called [*Thom conjecture*]{} (which he humbly considered himself as rather belonging to the folklore, compare footnote in Lee Rudolph 1984 [@Rudolph_1984]). This should have implied a clear-cut impact of Thom upon Hilbert’s 16th problem. The summit of our fictional “Gabard-Thom” theory went so far as to establish one-half of Ragsdale’s conjecture, (still open for $M$-curves) (cf. Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]) and to show that René Thom was on his 31, i.e. can be stronger than the conjunction of all Russian estimates, congruences and formulas (due primarily to Petrovskii 1938, Gudkov 1969, Arnold 1971 and Rohlin 1972–74–78), cf. Theorem \[Alsatian-scheme-Thom-strong-Petrov-Arnold:thm\]. This would have refuted a belief of Th. Fiedler (cf. his letter ca. 13 March in Sec.\[e-mail-Viro:sec\]).

$\bigstar\bigstar\bigstar$ Fortunately, Fiedler brought us back to reality by showing that our reasoning is wrong as it overlooks the issue that despite being constructed by pasting two orientable pieces—namely Klein’s orthosymmetric half married with Miss Ragsdale’s membrane bounding the curve from inside—the so-called [*Arnold surface*]{} (1971) does [*not*]{} need to be orientable. My mistake is thus nearly as basic as having overlooked that one can create (like in Klein’s bottle) non-orientable objects merely by pasting a handle to itself in a twisted fashion (this reminds me some lovely pictures in the Fuks-Rohlin “Beginner’s course on topology”).

This being confessed, most of this section is foiled and we are much indebted to Th. Fiedler for having catched our mistake at the right moment and stimulated our investigations. Albeit much of the sequel is foiled we have decided to keep it for didactic reasons. In our case it was so pathetic to write ca. 40 pages based upon a misconception without noticing anything (prior to Fiedler’s correction) that we would by no mean that somebody else do the same mistake. More positively many questions arises through Fiedler’s correction.

\(1) Where (in particular in which degree) lives the first counterexample to the Gabard-Thom bound $\chi\le k^2$ (no false modesty in calling it so since it is false) for all dividing curves? \[The sole counterexample I know is the Itenberg-Viro curve corrupting Ragsdale. This is a beautiful picture, see Fig.\[Itenberg:fig\].\]

\(2) If “Gabard-Thom” is obviously false for theoretical reasons, why does it look nearly true as implying one-half of the vestiges of Ragsdale’s conjecture?

\(3) Under which condition is the estimate $\chi\le k^2$ still true? As noted by Fiedler, the answer seems rather clear namely iff the Arnold surface is orientable. This is in turn the case iff all primitive pairs are positive. The last condition is perhaps not an “iff”. In crude approximation one could say that the Arnold surface is orientable iff the Rohlin tree is positively charged throughout. At least if this is the case (a rather stringent condition) then Arnold’s surface is orientable, and the estimate $\chi\le k^2$ holds true (cf. the limpid proof of the erroneous Theorem \[Thom-Ragsdale:thm\]). Further it seems evident that when the tree is positively charged throughout then the Rohlin mass $\pi-\eta$ is maximized (We always set $\pi:=\Pi^+$, $\eta:=\Pi^-$ to abridge Rohlin’s notation). Alas even that is false (more details soon).

So Thom’s conjecture has still something to say on Hilbert’s 16th yet its impact is much more subtle than expected when doing the fundamental mistake. Finally, all this section on “Thom” contains above all comparative study with the strength of Rohlin’s formula $2(\pi-\eta)=r-k^2$ (see (\[Rohlin-formula:thm\])), hence a cuneiform formalism that really pertains to Rohlin’s complex orientations as it electrifies the Hilbert tree by putting charges (distributions of signs on the edge). Albeit our exploration of this topic was completely random (and biased by our erroneous Thom estimate), it should have some independent interest. Reorganizing all this material, without loosing any bit of information will take us several weeks, and cannot be done on the present edition. We hope in the future being able to give a more structured exposition of this cuneiform formalism (Hilbert’s tree with signs, alias Rohlin’s trees) and our messy account can motivate others to clarify this.

What can be salvaged after Fiedler’s earthquake?
------------------------------------------------

\[22.03.13\] As spotted by Fiedler, we overlooked that Arnold’s surface (arising by pasting Klein’s half with Ragsdale’s membrane) is not necessarily orientable though both its constituents are. Hence one cannot apply Thom so straightforwardly. Incidentally we hope that Thom applies without trouble coming from the necessity of rounding corners. This is folklore but it would be nice to find adequate reference (Thom?, Cerf?, Hirzebruch? Milnor?, Wall?, etc.)

Though Thom was a heuristic way to discover the inequality $\chi\le k^2$ it could be that it holds true for more elementary reasons directly rooted in Rohlin’s formula. Let us briefly explain how. \[$\bigstar$ Non-sense (!) by the Itenberg-Viro counterexample in degree $10$, Fig.\[Itenberg:fig\].\]

In the sequel we use the jargon of the [*Rohlin tree*]{} which is simply Hilbert’s tree (encoding the distribution of ovals via a POSET whose order relation comes from the inclusion of the insides of the ovals), plus a decoration of its edges by signs coming from Rohlin’s complex orientations of a curve of type I (in the sense of Klein, also called latter by him orthosymmetric curves). So if any plane real curve has a Hilbert tree, dividing curves have an extra distribution of signs on the primitive edges (those of length one) which by the signs-law (\[Signs-law:lem\]) propagates consistently along the whole injective pairs of the tree.

First if Rohlin’s tree is positively charged (i.e. if all primitive pairs of ovals are positive in the sense of Rohlin) then Arnold’s surface $A=C^+\cup R$ (=Klein’s half $C^+$ pasted with Ragsdale’s (orientable) membrane $R$) is orientable too! In that case via Thom we have $\chi\le k^2$ (cf. Theorem \[Thom-Ragsdale:thm\] and its simple proof). \[$\bigstar$ Okay, but as we shall soon see this may also be inferred from Rohlin’s formula!\]

\(1) Moreover for a positively charged tree we have $\pi-\eta=n$, i.e. the Rohlin mass $\pi-\eta$ is equal to the number $n$ of negative=odd ovals (compare Lemma \[Rohlin-mass-of-a-positively-charged-tree:lem\] below).

\(2) Rohlin’s formula reads $2(\pi-\eta)=r-k^2$, hence fixes the Rohlin mass $\mu:=\pi-\eta$ which is something coming from the “complexification” (i.e. the Riemannian) in terms of real characteristics ($r$ being the number of ovals and $k=m/2$ the semi-degree). This formula can be used to express the Euler-Ragsdale characteristic $\chi=\chi(R)$ as follows $$\begin{aligned}
\label{Rohlin-to-Arnold-tris:eq}
\chi=p-n=(p+n)-2n&=r-2n\cr
&=[2(\pi-\eta)+k^2]-2n\cr
&=k^2+2[(\pi -\eta) -n].\end{aligned}$$

Combining (1) and (2) gives the following lemma (incidentally remarked in Fiedler’s letter):

If all primitive pairs are positive (equivalently if the Rohlin tree is positively charged) then $\chi=k^2$.

Now it seemed to us realist to posit a sort of “positive mass conjecture” (POSMASS) stipulating that the Rohlin mass $\mu:=\pi-\eta$ (of a signed tree) is maximized when all primitive edges/pairs (of the tree) are positively charged. (We formulated this some 7 days ago, cf. optionally \[positive-mass-conjecture:conj\] dated \[15.03.13\].)

This looks quite appealing, albeit we have little evidence for the truth of this principle which if optimistic could be pure combinatorics (i.e. valid for any signed tree with charges respecting the signs-law) or be merely valid for such Rohlin signed trees arising via dividing curves. Of course it could be also false in this restricted case. Our evidence for POSMASS is presently only derived from the case of chains of length $\le 4$ (or so), compare the signs-law for dyads (Fig.\[Signs-law-dyad:fig\]), that for triads (Fig.\[Signs-law-triad:fig\]), and that for tetrads (Fig.\[Signs-law-tetrad:fig\]).

If POSMASS is true, then $\pi-\eta\le n$ and so the “Euler-Rohlin formula” implies $\chi\le
k^2$, i.e. the so-called Gabard-Thom (dubious) estimate. The striking issue is that the Gabard-Thom theorem would be still true but merely as a logical consequence of Rohlin’s formula (plus some combinatorics required to validate POSMASS). In particular the intervention of Thom could be completely dispensed.

This scenario looks risky, since Fiedler claims our Gabard-Thom theorem to be wrong. However, it is\[=was\] not clear to me if there is really a counter-example to the theorem, or if Fiedler just stated wrongness of our reasoning.

More ironically I forgot to remember that even 5 days before formulating POSMASS, I had a simple counterexample to it in the combinatorial setting (cf. Theorem \[Garidi-mass-conj-is-FALSE:thm\]). Hence there is no chance to prove the Gabard-Thom estimate via pure combinatorics.

Of course, it is also likely that the POSMASS conjecture is false also for Rohlin’s trees arising as dividing curves but that deserves an explicit example. Of course the method should be to ape algebraically the combinatorial structure of a batônnet that foils the mass conjecture (cf. Fig.\[Garidi-mass-false:fig\]a). A bâtonnet is merely a usual tree with a trunk that ramifies strongly into several branches at depth 2 (look at that picture Fig.\[Garidi-mass-false:fig\]a.)

Accordingly, my first idea was to look back in Gabard 2000 [@Gabard_2000 Fig.13, p.154] where is traced a classical (variant of) Hilbert’s construction of an $M$-curve of degree 8, which has nearly the required bâtonnet structure. We shall soon reproduce this and related pictures. Moreover why degree 8? Simply because in degree 6 the Thom-Gabard estimate $\chi\le k^2$ (for dividing curves) is trivially true as follows by glancing at Gudkov’s table (=Fig.\[Gudkov-Table3:fig\]) of which we merely use Hilbert’s intuition/theorem that the unnested $M$-scheme (symbol $11$) is not algebraic, plus the fact that the unnested $(M-1)$-scheme (symbol $10$) has no dividing realization (as follows from Klein’s congruence $r\equiv_2 g+1$).

This being said let us do an iterated Hilbert construction (Fig.\[HilbGab1:fig\]). This gives first the well-known $M$-sextic $C_6$ of Hilbert (symbol $\frac{9}{1}1$), and then an $M$-octic $C_8$ with $\chi=16=k^2$, and then an $M$-curve of degree 10, $C_{10}$ with only $\chi=9$, yet still $\le k^2=25$ (and congruent to it mod 8 as it should by virtue of Gudkov hypothesis). Of course the Gabard-Thom estimate $\chi\le k^2$ has little chance to be corrupted so, since it formally implies one-half of Ragsdale’s conjecture (which is still open for $M$-curves, cf. Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]). A more intrinsic reason is of course that Ragsdale conjectures were calibrated along a deep contemplation of the Harnack-Hilbert method. Hence historical continuity is fighting against our attempt to corrupt Gabard-Thom via Hilbert’s construct. As the (maximal) $M$-Ragsdale conjecture is still open (not succumbing even to the Viro-Itenberg method, cf. Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale]), it is much more likely to corrupt Gabard-Thom by using non-maximal curves. Keep this idea in mind for later.

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Before adventuring outside the realm of $M$-curves, let us do also a plate for Harnack $M$-curves constructed à la Hilbert (Fig.\[HilbGab2:fig\]). Those Harnack-style curves have a priori a larger $\chi$, so better suited to corrupt Gabard-Thom. Precisely Harnack’s $M$-sextic has $\chi=9=k^2$, then we get a $C_8$ with $\chi=16=k^2$, and then a $C_{10}$ with $\chi=25=k^2$. Our depicted $C_{12}$ (right) is not the most natural choice as we switched to an “internal vibration”, while in the first steps $C_6\rightsquigarrow C_{8}\rightsquigarrow C_{10}$ we consistently opted for an external oscillation (typical of Harnack’s curves reckoned à la Hilbert). It should be noted than even the natural choice of an external vibration does not lead to a $C_{12}$ with $\chi=36$, but we were only able to get one with $\chi=28$ (in accordance with Gudkov hypothesis). \[[*Added in proof*]{}.—THIS IS A MISTAKE, due to the fact that I reported ovals at the wrong place\] However Miss Ragsdale in 1906 was surely much more clever than we are \[THIS IS NOT PERTINENT ANY MORE\], and I presume she was able to reach always the “Gabard-Thom upper-bound” $\chi\le k^2$, since this really amounts to (one-half of) her conjecture (cf. again Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]).

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\[23.03.13\] Correcting my mistake, an extension of this figure (Fig.\[HilbGab2:fig\]) shows the following result (compare Hilbert 1891, and Ragsdale 1906):

There is an infinite series of $M$-curves of degree $2k$ with $\chi=k^2$. Hence if Ragsdale’s conjecture $\chi\le k^2$ (for $M$-curves) is true, then it is sharp.

Look at the first steps of Fig.\[HilbGab2:fig\] and do not commit the mistake of making an inner vibration, but choose always external vibrations as we did for $C_6, C_8$. On looking at the Hilbert trees of $C_6, C_8, C_{10}$ one easily derive the general evolution of the Hilbert tree of $C_{2k}$. Namely the number of outer ovals $9,17,27$ augments along the increment $+8, +10, +12, etc.$, while the tree itself is always pushed down one step deeper while acquiring new branches on its top, compare the windows on the figure for $C_6, C_8, C_{10}$. In view of the extreme regularity of the construction it is easy to extrapolate the nested structure of $C_{2k}$. Writing down a general formula looks not even necessary, and the Gudkov symbol will be something like $$(1,(2k-6) (1, 2k-8 (1, 2k-10) \dots )) (9+8+10+\dots+2k).$$ In more geometric terms this means that the Hilbert tree of $C_{2k}$ has $9+8+10+\dots+(2k-2)+(2k)$ outer ovals, and a trunk of length $2k-7$ with branches hanging on. It should be an easy matter to compute directly the Euler characteristic of $C_{2k}$ by the evolution rule for the tree.

\[23.03.13\] Perhaps instead of using Hilbert’s method one must really uses the more time-consuming Harnack original method which amounts to oscillate around a ground-line instead of the ellipse used in Hilbert’s method. (Incidentally, I wonder if one wants a fast-Hilbert method, if it is possible to vibrate across a split quartic, union of 2 ellipses.)

All this is good but will perhaps only confirm the intuition (of Ragsdale) that her estimate $\chi\le k^2$ is sharp for $M$-curves (if true at all). Our object is somewhat different namely to refute the Gabard-Thom estimate $\chi\le k^2$ for all dividing curves. Of course it could be possible to disprove the Ragsdale $M$-estimate $\chi\le k^2$ yet this deserves highbrow methods as even the powerful Viro-Itenberg method apparently failed as yet in that game (compare Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale]).

So how to construct non-Harnack-maximal dividing curves? As we know from degree 6 (Fig.\[GudHilb8:fig\]), the trick is just to disperse the vibratory energy on several ovals, as opposed to the monopole of Hilbert’s vibration where a single oval is oscillating. By Klein’s congruence we look at $(M-2)$-curves, and in degree 6 the specimen with largest $\chi$ is the scheme with symbol $9$, which can be obtained by (a variant of) Hilbert’s method where the vibration is dispatched on 2 ovals (cf. the earlier Fig.\[GudHilb8:fig\] or right below Fig.\[HilbGab3:fig\]). It is now hoped that when iterating the construction to higher degrees we get a refutation of Gabard-Thom.

Applying this idea we get the following series of $(M-2)$-curves (Fig.\[HilbGab3:fig\]), all of type I by Fiedler’s law of smoothing dictated by (and dictating) complex orientations. Of course at the higher steps we choose again a monopolized vibration as otherwise we descend further the energetic level and reach curves with less ovals that $(M-2)$. We use now a quicker depiction mode where vibration and smoothing are depicted on the same plate at each step of Hilbert’s inductive process. Further on the diagram standing right below each curve, we depict Hilbert’s nested tree encoding the distribution of ovals, and some easy calculation of topological characteristics of the curves so constructed. The conclusion is that we get an infinite series of $(M-2)$-curves with $\chi=k^2$ as shown by the figure up to $k=7$, and the regularity of the procedure is so evident that this property easily follows for all $k$. In particular it may be observed that the number of outer ovals evolves along the progression $9, 17,
27, 39, 53,\dots$, incrementing along the progression $+8,+10,+12, +14, \dots$, while the nested portion of the tree is simply pushed down at each step, with $(2k-6)$ new branches arising on the top.

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Hence, modulo some arithmetical nonsense, we have proved the:

There exists an infinite series of dividing $(M-2)$-curves $C_{2k}$ of degree $2k$ with $\chi=k^2$.

Alas, this does not refute the Gabard-Thom estimate, but rather show its sharpness in case it would be correct. (Another proof of the sharpness can be derived by perturbing ellipses, cf. Remark right after Thm \[Thom-Ragsdale:thm\]).

(The contrast between the present regularity and the lack thereof for $M$-curves was so striking that it permitted us to correct the earlier mistake that we did above.)

But where to find a counter-example to the Gabard-Thom-bound (as promised by Fiedler’s claim of erroneousness)?

Lacking some imagination let us redo the Hilbert vibration for $M$-curves more systematically by always vibrating from “outside”. This gives Fig.\[HilbGab4:fig\]. The same regularity is observed while the general pattern becomes evident after some few iterations. Hence one can reduce to the depiction of the trees. The latter get always more profound by one unit as $k$ increments, while the number of deepest ovals belongs to the series $9, 17,
27, 39, \dots$ which regularly increments by $+8,+10, +12, \dots$ so that one can predict the future evolution of the tree, compare the very bottom row. This shows that one times over two we will attain the Gabard-Thom bound $\chi=k^2$, while of course the sign of $\chi$ oscillates between negative and positive values. It is evident that we will not get a counterexample to Gabard-Thom in this fashion.

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Then we can still vary more the constructions, e.g. by starting with a Harnack-like outer vibration of the $C_6$ like on Fig.\[HilbGab2:fig\] and perform the dissipation (leading to $(M-2)$-curves) at the next step on the octic $C_8$. Alas we still found $\chi=k^2$ for the $C_8$ and even the $C_{10}$ (details of the picture on p.AR-114 of my hand-notes).

At this stage one gets a bit depressed. It seems hard to corrupt Gabard-Thom by construction à la Hilbert-Harnack. Maybe I missed something, or perhaps one should appeal to more sophisticated constructions like Viro-Itenberg. \[This turned out to be the good idea, more soon.\] At this stage I cannot therefore preclude the option that the Gabard-Thom estimate is true (of course for another reason than the gapped proof given in Theorem \[Thom-Ragsdale:thm\]).

So a priori 4 scenarios may happen:

(1).—Either there is an elementary counterexample to Gabard-Thom(=GT) via elementary constructions à la Harnack-Hilbert (and we missed it due to lack of cleverness).

(2).—There is a counterexample to Gabard-Thom via highbrow constructions à la Viro-Itenberg. In particular it is not impossible that the counterexample described by them to Ragsdale conjecture also supplies a counterexample to Gabard-Thom. This requires controlling the type in their construction (which is an issue known to them). More on this in the next Sec.\[Itenberg-Viro-disprove-Gab-Thom:sec\].

(3).—There is a counterexample to GT via another construction not covered by Viro-Itenberg. (To my knowledge there is no theorem stating that any algebraic curve is constructible via their method.)

(4).—GT is true for another reason than the one exposed in Theorem \[Thom-Ragsdale:thm\]. This would however conflict with Fiedler’s assertion that our theorem is wrong. (Again, it is not clear if Fiedler merely stated wrongness in the proof or of the statement.)

The next section gives a clear-cut answer to this puzzle.

A formal disproof of Gabard-Thom via Itenberg-Viro’s patchwork and Kharlamov-Marin {#Itenberg-Viro-disprove-Gab-Thom:sec}
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\[23.03.13\] Here we say “formal” just because we are not yet familiar with the patchwork method due to Viro, and elaborated by Itenberg later into the so-called $T$-curves context. Of course the method has nothing formal: it is crystallography of the best stock as we shall soon see.

As we failed along strategy (1) (cf. previous section), let us look at (2) which invites to take a better look on Itenberg-Viro seminal paper (1996 [@Itenberg-Viro_1996-disproves-Ragsdale]). There on Fig.2 (p.20) we find a remarkable picture reproduced below as Fig.\[Itenberg:fig\]. (Our sole change is to have traced the curve with less thickness to see better what happens than on the downloaded black-and-white pdf.) Note that the underlying triangulation (by triangles all of area one-half of the unit square) is symmetrical under the dihedral group (Vierergruppe $D_4\approx \ZZ_2\times \ZZ_2$, cf. Fig.a). I did not noticed this for a while when reproducing this figure! Then there is a signs-distribution that gives the red curve via a bisection procedure of all triangles. Interpret this merely as a piecewise-linear random walk if you have zero-knowledge like the writer. Tracing this curve is a miracle, very enjoying if one is computer assisted. Representing the curve alone gives Fig.b (a simple cut-and-past operation for the computer). As usual in projective geometry the boundary of the rhombs must be identified antipodically. Hence the 5 semi-ovals on the bottom left-side of the rhombs are really just capping off (closing) the long contorted oval occupying the oriental (Siberian) part of the rhombs. We count on Fig.c precisely 29 outer ovals. Hence the curve in question (which admits an algebraic realization by a deep theorem of Viro-Itenberg) has the scheme depicted on Fig.d, hence Gudkov symbol $(1, (1,2)2)29$. The total number of oval is $r=29+6=35$, i.e. 2 unit less than $M=37$ (temperature of the human body), so its an $(M-2)$-curve. Hence there no obstruction (via Klein’s congruence) for the curve being dividing. Looking optionally at the corresponding Hilbert tree (Fig.e) gives quickly $\chi=29+1-3+2=29$ (variant look at the scheme Fig.d and apply a Swiss cheese recipe à la Euler-Listing, etc.). Arnold’s congruence mod 4 ($\chi\equiv_4 k^2$ if type I) is verified and so there still no obstruction for the curve being dividing. In the fact, the wonderful RKM-congruence $\chi\equiv k^2+4 \pmod 8$ (Rohlin-Kharlamov-Marin, see (\[RKM-congruence-reformulated:thm\])) implies this scheme being of type I (and so is in particular any curve representing it). Hence the Itenberg-Viro curve is of type I, and it gives the long searched counter-example to our Gabard-Thom pseudo-theorem. Probably, there is a more elementary (organical) way to deduce the dividing character of the curve by an avatar of Fiedler’s signs-law in the realm of the Viro method. (This goes back to the early 1980’s, and perhaps in the present $T$-curve context is due to Itenberg. Parenti’s thesis 199X [@Parenti_199X] looks also involved in this topic.)

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So we are at this stage nearly in the paradise! To add some suspense note that this curve being of type I (even in the very strong sense that its scheme is) a vague conjecture of us (founded on Ahlfors) posits that there should be a total pencil of $(m-3)$-adjoint curves exhibiting the dividing character of the curve (as in the Rohlin-Le Touzé 1978–2013 theorem for sextics, cf. Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]). According to our (hypothetical) extension of the Le Touzé theorem, there should be a pencil of septics cutting only real points on the Itenberg-Viro curve. In general, the degree of such a pencil should be in accordance with Gabard’s bound $r+p$ on the degree of circle maps (compare Sec. \[census-and-extension-of-Rohlin-Le-Touzé:sec\], the numerology therein and especially Remark \[M-2-curve-degree-like-Gabard:rem\]). Pencil of septics have 34 basepoints freely assignable (in general this is $M-3$), whereas the Itenberg-Viro curve has 33 empty ovals. So where to assign the remaining basepoint? This is again a bit puzzling. However it is likely by the philosophy of dextrogyration that Rohlin’s complex orientation plays some rôle there. Rohlin’s formula applied to the Itenberg-Viro curve gives $2(\pi-\eta)=r-k^2=35-25=10$, hence $\pi-\eta=5$, while $\pi+\eta=7$ as is apparent from the Hilbert tree (Fig.e, where one counts 2 additional pairs of length 2). Hence $\pi=6$ and $\eta=1$. The presence of a negative pair is no surprise, as Fiedler noticed this being the gap in our (erroneous) proof of the Gabard-Thom estimate (Theorem \[Thom-Ragsdale:thm\]). By the signs-law (\[Signs-law:lem\]) it is clear that the (unique) negative pair must have length one (primitive pair), otherwise all primitive edges are positively charged but then there are 2 negative pairs (those of length 2). Since there is always a bijective correspondence between primitive edges of Hilbert’s tree and (non-maximal) ovals by taking the bottom of the edge which is always uniquely defined. So there is one negative oval (i.e. whose edge) and thinking more with the signs-law it seems that the negative pair is forced to be the trunk (i.e. the edge that ramifies at depth 2, cf. Fig.e and lemma below). So this ovals (negatively charged right above it) is perhaps the good candidate of where to assign the remaining basepoint. All this deserves of course to be better understood, and is digressed upon in the next section (\[Galton-brett:sec\]).

Let us verify the simple:

The Rohlin tree of the Itenberg-Viro curve has a unique negative charge that is forced to be on the trunk.

As noted above Rohlin’s formula implies that there is a unique negative pair ($\eta=1$). Since in Rohlin’s arithmetics $+\times +=-$ (i.e. consanguinity is bad) the deepest subtree of length 2 (“Y”-shaped modulo horizontal mirror) cannot be positively charged else there would be 2 negative pairs of length 2. So a primitive minus-charge must be located on the “Y”-subtree. If it is on one of the 2 branches (as opposed to being on the trunk at depth $0-1$) then by uniqueness, the trunk and the (other) branch are positively charged, so that it results a minus-charge on their concatenation (of length 2), violating $\eta=1$. So the negative charge must be located on the trunk as asserted.

Where to assign basepoints to ensure total reality: toward a Galton-Brett algorithm? {#Galton-brett:sec}
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\[24.03.13\] The above fantastic example of Itenberg-Viro (Fig.\[Itenberg:fig\]) raises again the general problem of deciding where to assign basepoints as to ensure total reality of an adjoint pencil. This means as usual a pencil of curves cutting only real points on a dividing curve. By general topology (the image of a connected set is connected) a dividing curve presents no obstruction to the existence of such a pencil. More than that, Ahlfors theorem says that there is no conformal obstruction to do this (being after all just an extended Riemann mapping theorem for bordered surfaces of higher topological structure that the disc). If one has a total map to the line $\PP^1$ then we have a branched cover taking boundary to boundary and interior to interior, and the map restricted to the real locus is an (unbranched) covering. Accordingly there is a phenomenon of dextrogyration, i.e. when the image-point circulates once around the fundamental circle $\PP^1
(\RR)$ the counter-images (fibre) circulate along the complex orientation of the abstract curve, i.e. as the boundary of the half. (This follows of course from the holomorphic, hence sense preserving, character of Ahlfors maps.)

If the curve is plane ($C\subset \PP^2$) we would like a procedure predicting where to assign basepoints. By the above dextrogyration principle, there should be some relation with Rohlin’s complex orientations, which measure merely the (abstract) complex orientations as compared with those of rings (annuli) bounding (injective) pair of ovals in the plane $\RR P^2$. By the signs law it suffices to determine Rohlin’s signs for primitive pairs of ovals.

A vague idea is as follows. Given a plane curve we have the Hilbert tree and Rohlin’s complex orientations decorate its edges with signs (pluses, minuses). We can imagine the resulting Rohlin tree as a “Galton Brett”, i.e. Galton’s table where billiard-balls fall downwards along an inclined table interspersed with a distribution of nails. Whenever meeting one of those nails the ball is deflected left or right with probability one-half. For the usual equilateral distribution we recover so the Chinese-Pascal binomial distribution.

Our naive idea is to interpret the Rohlin tree as a Galton-Brett, while putting balls at the top of Hilbert’s tree and looking where they stabilize to an equilibrium. It is imagined that negative pairs are inclined so that balls fall gravitionally along them. Consider the example of the deep nest. We know then either from Rohlin’s formula or via the dextrogyration argument applied to the obvious pencil of lines through the deep nest that all primitive pairs are negative. Here the Galton-Brett reduces to a simple track (without branching) always negatively charged, and the ball descends right up to its bottom. This is in agreement with the fact that total reality of a lines-pencil is ensured when assigning the basepoint in the deepest oval.

Similar considerations hold for the quadrifolium and its satellites, i.e. curves of degree divisible by four and totally really under a pencil of conics.

On the next example of Rohlin-Le Touzé’s sextics, e.g. that of type $\frac{6}{1}2$, the Rohlin tree has 6 branches emanating from an oval and Rohlin’s formula $2(\pi-\eta)=r-k^2=9-9=0$ shows that $\pi=\eta=3$ since we have a total $\pi+\eta=6$ of six pairs. So our tree has 3 negative and 3 positive edges. Our metaphor of the Galton-Brett already looks dubious on that example, since it would prescribe imposing basepoints only on the $3$ ovals surmounted by a negative charge (and of course the 2 outer ovals). So it remains to understand if an improved Galton-Brett principle permits to understand where to assign basepoints in function of a knowledge of Rohlin’s complex orientations.

Maybe an improved rule is to let balls fall-down regardless of signs along the Rohlin tree, and some few of them could stay in levitation (unstable equilibrium being blocked by a needle) with special signs-property, like being surmounted by a negative charge while branching down below (so-called hyperbolic ovals). This complicated condition comes to mind when looking at the deep nest plus the above Itenberg-Viro curve, where the trunk (of the tree) is negatively charged, which is the only reasonable signs-distribution compatible with Rohlin’s formula (cf. lemma above). So on the tree of Fig.\[Itenberg:fig\]e balls would fall along the Hilbert tree and stabilize of course in each “deep” ovals (aka empty ovals), but some nontrivial equilibrium arises at the vertex at depth 2 which branches further (alias hyperbolic oval). Of course hyperbolicity alone is not enough as shown by Rohlin-Le Touzé’s theorem. However hyperbolicity plus a negative charge above it could give an equilibrium, i.e. a place where to assign a basepoint. Though a bit complicated this looks even reasonable from the viewpoint of Galton’s Brett. Namely hyperbolic ovals are those where there is an indetermination (bifurcation) when falling down, while negativity of the edge above is a sort of kinetic impulse giving the particle some momentum forcing it to move against the bifurcation, whence an “unstable” equilibrium (crystallizing thereby in the formation of a basepoint). Of course all this need to be further explored, and to be related to more intrinsic properties of dividing plane curves. For the Viro-Itenberg curve this algorithm would assign basepoints of the septics-pencil on the 33 empty ovals (stable equilibrium) plus one unstable equilibrium materialized by the unique hyperbolic oval at depth 1. This would give the 34 basepoints required in a pencil of septics, and total reality could follow (assuming that our Galton algorithm is somehow compatible with dextrogyration or perhaps indexes formulae à la Gauss-Kronecker-Poincaré-von Dyck).

Let us look at more examples. For octics we have 4 basic schemes listed in Eq. \[octics-five-examples-RKM:eq\] which satisfy the RKM-congruence (cf. also the Gudkov table in degree 8, Fig.\[Degree8:fig\]). Those were $$\frac{16}{1}3,\quad \frac{12}{1}7,\quad \frac{8}{1}11,\quad
\frac{4}{1}15,\quad 20.$$ The last of which is precluded as it violates either Petrovskii’s inequality (\[Petrovskii’s-inequalities:thm\]) or the Thom estimate $\chi\le k^2$ which is valid when there is no nesting (or even the more elementary Rohlin’s formula). However for all other schemes it may be reasonable to expect total reality for a pencil of quintics which has 19 basepoints (recall that $B=M-3$ for the number of basepoints in terms of Harnack’s bound $M$) and all of them are ascribed on the empty ovals (in accordance with our Galton principle).

All those 4 RKM-schemes are just the top of the iceberg depicted on Fig.\[RKM-schemes-deg-8:fig\]. On that tabulation we find e.g. the scheme $\frac{3}{1}\frac{1}{1}14$. This has 18 empty ovals, and we need a 19th basepoint. Our algorithm of the negative hyperbolic oval fails to give it since looking at the Hilbert tree of the scheme we see a unique hyperbolic oval, and this has no edge above it! So our method fails and deserves to be further improved. Less likely, our method could be right and then it could preclude existence of those schemes in type I. (Note that our Galton method could have killed the scheme $20$, since we expect 19 basepoints but there are 20 stable equilibriums.) Of course all this must be further explored. Summarizing, a fundamental question seems to be:

Is there a general algorithm telling one where to ascribe basepoints in terms of the combinatorics of Rohlin’s tree encoding his complex orientations? If so then we get a mechanical device extending the Rohlin-Le Touzé phenomenon of total reality for $(M-2)$-sextics satisfying the RKM-congruence.

Viceversa, suppose zero-knowledge on the complex orientations one could argue that the principle of total reality via dextrogyration is a good recipe to infer a knowledge of them (e.g. as it is flagrant in the trivial deep nest case). Presently very little seems to be known in general, and this is of course much reminiscent of the lost proof of Rohlin’s (last) theorem.

When is Arnold’s surface orientable and Ragsdale via Bieberbach-Grunsky? {#Ragsdale-via-Riemann-Bieberbach-Grunsky:sec}
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\[27.03.13\] Here we propose a (naive) attack upon one half of the Ragsdale conjecture for $M$-curves. This may be translated as the condition $\chi\le k^2$ (cf. Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]). As shown by the proof of the erroneous Theorem \[Thom-Ragsdale:thm\], under the additional assumption that the Arnold surface is orientable, Thom applies and gives promptly the (upper) Ragsdale estimate $\chi\le k^2$. (Note: The full Ragsdale amounts to the pinching $-k^2\le \chi \le k^2$, equivalently $\vert \chi \vert \le k^2$.)

So the core of the question is to know when Arnold’s surface is orientable. We shall discuss this soon. The net impact could be as follows:

\[Arnold-surface-M-curves-orient:conj\] The Arnold surface of a (plane) $M$-curve is always orientable. If this is true then the upper-Ragsdale estimate $\chi\le k^2$ follows from Thom (cf. proof of (\[Thom-Ragsdale:thm\])). $\bigstar$ Alas, the sequel shows that this naive conjecture fails already for Hilbert’s $M$-sextic, cf. Fig.\[Ragsdale:fig\]c.

Recall that the [*Arnold surface*]{} of a dividing plane curve of even degree $2k$, is simply Klein’s half of the curve filled by Ragsdale’s membrane in $\RR P^2$ bounding the curve from inside. It will be orientable iff the orientation coming from the complexification and the real Ragsdale membrane match together in some sense made precise below.

First it is plain that if Rohlin’s tree is positively charged (on all its primitive edges) then Arnold’s surface is orientable (cf. Fig.\[Ragsdale:fig\]a). This positive-charge assumption is very stringent and implies actually much more, namely that the Rohlin mass $\pi-\eta$ equals $n$ (cf. Lemma \[Rohlin-mass-of-a-positively-charged-tree:lem\]). Via Rohlin’s formula rewritten as $\chi=k^2+2[(\pi-\eta)-n]$, this implies in turn that $\chi=k^2$ exactly.

However to derive the (elusive) upper-Ragsdale-estimate $\chi\le
k^2$ from Thom, it suffices that the Arnold surface is orientable. A small picture (Fig.\[Ragsdale:fig\]b) convinces one that this holds more generally whenever Rohlin’s tree is positively charged on odd edges. Here we always define the depth of an edge in reference to that of its bottom vertex (the latter being uniquely defined by—and defining uniquely—the given edge).

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So we get the:

\[Arnold-surface-orientable-iff-oddly-charged:lem\] The Arnold surface is orientable iff the Rohlin tree is positively charged on odd edges (say then that it is oddly charged).

This condition is much weaker, but still implies Ragsdale via Thom. So the upper Ragsdale $M$-conjecture (URMC) reduces to the:

\[oddly-charged-M-curves:conj\] Any $M$-curve is oddly charged.

A guess could be to use our translation (Theorem \[total-reality-of-plane-M-curves:thm\]) of the Bieberbach-Grunsky theorem (truly due to Riemann 1857). The idea is that for $M$-curves we have a fairly explicit way to construct a total pencil via curves of degree $(m-2)$ assigned to pass through any distribution of $g+1$ points (one on each oval) and then by looking at the residual group of points, while assigning them as basepoints. For more details cf. proof of Theorem \[total-reality-of-plane-M-curves:thm\], but we repeat the general recipe of the construction of a total series on a plane $M$-curve $C_m$ of degree $m$:

1.—Choose any distribution $D$ of $g+1$ points one on each oval.

2.—Let pass a curve $\Gamma_{m-2}=:\Gamma$ of degree $(m-2)$ through $D$.

3.—Consider $R$ the residual intersection $\Gamma\cap C$ less the points of $D$.

4.—Assign $R$ as the basepoints to the system of curves of degree $(m-2)$, and get (or choose) a pencil $\Pi$ putting the initial group $D$ into motion.

5.—By continuity the pencil $\Pi$ is total since there is only one point one each circuit hence no risk of collision. Total reality follows.

The dream would be that this procedure is sufficiently explicit as to control complex orientations, especially the issue that the Rohlin tree is oddly charged. If this is possible we get a proof of the upper-half $\chi\le k^2$ of Ragsdale’s conjecture.

Concretely, once the distribution $D$ is fixed we are assured that the curve $\Gamma$ will cut $C$ once more along each ovals (by the closing lemma for algebraic circuits \[Closing-lemma:lem\]). So we have $2(g+1)$ real intersections in $\Gamma\cap C$, i.e. $2(g+1)=2(\frac{(m-1)(m-2)}{2}+1)=(m-1)(m-2)+2=m^2-3m+4$. This is less than the $m(m-2)$ expected.

\[disproof-orientability-Arnold-M-curve:lem\] Alas, the oddly-charged conjecture (\[oddly-charged-M-curves:conj\]) or equivalently the orientability of the Arnold surface of an $M$-curve (\[Arnold-surface-M-curves-orient:conj\]) fails already in degree 6, e.g. for Hilbert’s $M$-sextic as shown on Fig.\[Ragsdale:fig\]c.

Indeed reporting complex orientation via Fiedler’s transmission-law we get Fig.\[Ragsdale:fig\]c. Here we report first the orientation induced on the quadrifolium quartic $C_4$ from the dashed ellipse, and then smooth $C_4\cup E_2$ (where $E_2$ is the thick-ellipse) along positive orientation and receive so the complex orientations of Hilbert’s sextic. We see that among the 9 nested ovals, 5 are dominated by a positive pair, while 4 are by a negative pair. This is in accordance with Rohlin’s formula, $2(\pi-\eta)=r-k^2=11-9=2$, i.e. $\pi-\eta=1$ while $\pi+\eta=9$. Hence $2\pi=10$, i.e. $\pi=5$ and $\eta=4$. However this refutes our very naive conjecture (\[oddly-charged-M-curves:conj\]), which diagrammatically amounts saying that the Rohlin tree is positively charged on edges at odd depths. Of course the equivalent formulation in terms of the orientability of the Arnold surface (\[Arnold-surface-M-curves-orient:conj\]) is killed in the same stroke.

So our naive strategy fails severely but of course Hilbert’s sextic has $\chi$ very negative ($\chi=2-9=-8$). Hence there is perhaps a refined argument, that can establish Ragsdale. Alas it seems that what we just did kill definitively an approach via Thom which requires orientability of the Arnold surface. Of course one could expect a tricky case distinction along the sign of $\chi$, and a strengthened conjecture stating orientability of the Arnold surface (of an $M$-curve) provided $\chi>0$. Even this is easily disproved, e.g. by looking at the $M$-curve $C_{10}$ with $\chi=9$ of Fig.\[HilbGab1:fig\], and reporting the complex orientations via Fiedler’s law. This is a bit tedious but straightforward and gives Fig.\[HilbGab1bis:fig\]. We find that the 11 edges at depth 3 splits into 7 positive pairs and 4 negative ones. Rohlin’s formula can be verified via the signs-law. However Rohlin’s tree is not positively charged at the odd depth 3.

-3.5cm0 -5pt0

-5pt0

Of course one could still expect that Rohlin’s tree is oddly-charged when $\chi>k^2$, and this would suffices via Thom to prove the upper Ragsdale conjecture, but we are obviously playing a sterile arithmetical game, without much geometrical penetration. Alternatively, if not via Thom one could hope to use directly Rohlin’s formula, but again some external information on complex orientations must be gained via some deep geometric procedure. As we said one can dream that a version of the Bieberbach-Grunsky theorem do the job, but that deserves investigating with much more care and patience than we are presently able to do. Good luck to anybody who still feel optimistic. Of course it may also be that Ragsdale’s upper bound is just false by a highbrow variant of the Itenberg-Viro construction (though since Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale] nobody apparently ever succeeded), and it is evident that we (personally) lack experimental data to feel really secure in claiming the Ragsdale bound. Hence we abort this problem for the moment.

A sporadic (?) obstruction via Thom (Kronheimer-Mrowka 1994)
------------------------------------------------------------

\[22.03.13\] It should be noted that the first examples of Thom’s conjecture as applied to Hilbert’s 16th where we fill an unnested curve by discs are not affected by Fiedler’s correction. So in particular the “elementary” degree 3 case of Thom due to Kervaire-Milnor 1961 (yet relying massively upon Rohlin’s early work ca. 1951–52) really implies e.g. a purely topological proof of Hilbert’s intuition of nesting for $M$-sextics. This is detailed below. More generally, Thom’s conjecture forbids in all (even) degrees $m=2k\ge 6$ the possibility of an unnested $M$-curve (symbol $M$). This was first proved by Petrovskii 1938, and can also be deduced from Rohlin’s formula $0=2(\pi-\eta)=r-k^2$, since $M$ is strictly larger than $k^2$ when $k\ge 3$.

\[05.03.13\] Now just a little remark along the Thom conjecture (=the Kronheimer-Mrowka theorem 1994 [@Kronheimer-Mrowka_1994], abridged as “Thom” in the sequel). If we look at the $(M-2)$-scheme $20$ of degree 8, and fill one half by the canonical orientable membrane we get a surface of genus $p=1$ whose homology class is $4H$ (where $H$ is the natural generator of $H_2(\CC P^2,\ZZ)=\ZZ$, the so-called hyperplane-section, here a line). By Thom the genus should be at least as big as that of a (smooth) quartic, hence 3. So we get the:

The scheme $20$ is not realized algebraically by a curve of degree $8$ (necessarily of type I by the RKM-congruence \[Kharlamov-Marin-cong:thm\]).

If we take the scheme $\frac{4}{1}15$ (cf. Fig.\[RKM-schemes-deg-8:fig\]), then the genus of the filled membrane will be $1+4=5$ and so Thom’s principle is not violated. No other scheme of that table are prohibited by Thom.

If we look at $m=6$, and the Gudkov table Fig.\[Gudkov-Table3:fig\], especially the seminal intuition of Hilbert ca. 1891–1900, that the unnested scheme $11$ does not exist algebraically, then again we see that this may be inferred from Thom’s conjecture (meanwhile a theorem). Indeed making the canonical filling of the half Riemann surface by the canonical (Ragsdale) membrane (often denoted $\RR P^2_+$) we get a surface of genus of 0 realizing the homology class $3H$, hence violating Thom’s conjecture.

[*Insertion*]{} \[21.03.13\].—As a matter of fact, this special degree 3 case of Thom’s conjecture was first established by Kervaire-Milnor 1961 [@Kervaire-Milnor_1961 Cor.2, p.1652], basing themselves much upon Rohlin’s work of 1951–52. Thom is alas not mentioned there (KM61). So historiographically, it is worth emphasizing that the Kervaire-Milnor paper afforded so-to-speak the first purely topological proof of Hilbert’s 1891 semi-intuition/semi-theorem that a Harnack-maximal sextic cannot have all its 11 ovals unnested! Prior to this we had only available: (1) the algebro-geometric (stratificational) proofs of Hilbert (1891 unpublished) and Rohn (1911–13), plus technical refinements of the same method by Gudkov ca. 1954 and (2) the proof of Petrovskii 1933/38 half analytical (Euler-Jacobi analytical interpolation) and half topological (Morse theory). Nowadays we have of course the proof via Rohlin’s formula (1974 [@Rohlin_1974/75]), which is more elementary and purely topological (or the related one via Arnold’s congruence mod 4).

Several ideas arise.

(1).—Of course this Thom argument is not the most elementary prohibition of the scheme $10$ of degree 6, but maybe it is a good way to prohibit the scheme $20$ in degree $8$ (at least I know no other method for the moment).

[*Update*]{} \[07.03.13\]: the prohibition of this scheme $20_{8}$ follows more elementarily from Rohlin’s formula $2(\Pi^+-\Pi^-)=r-k^2$ (\[Rohlin-formula:thm\]), since the absence of nesting implies vanishing of the left-hand side, hence $r=k^2$ has to be a square (even $16$ as $k=4$). Private anecdote, I missed this consequence of Rohlin, and noticed it while completing the Gudkov Table in degree 8 (cf. Fig. \[Degree8:fig\]). \[21.03.13\] Another way to prohibit this scheme $20$, I presume the first historically found, involves the Petrovskii inequalities, cf. (\[Petrovskii’s-inequalities:thm\]).

More generally what schemes can be prohibited by Thom, and did it affords new obstructions (not known before Kronheimer-Mrowka)? The first question is answered by Theorem \[Thom-Ragsdale:thm\] below \[alas erroneous!\], while the second is perhaps answered via Theorem \[Alsatian-scheme-Thom-strong-Petrov-Arnold:thm\].

(2).—Looking at this canonical membrane filling might be a good device toward understanding the RKM-congruence. (But this is merely a matter of reading once carefully the Kharlamov or Marin arguments.) At least it is tempting to calculate the self-intersection of this filled membrane with itself (or its companion) to get some numerical relation. Doing so we probably obtain nothing new but what exactly? (guess the Arnold congruence).

(3).—Despite having corners this filled Riemann surface is perhaps a good object to do conformal geometry with (compare especially works by H.A. Schwarz ca. 1870 and his student Koebe ca. 1906–07, also that of Hilb ca. 1907, NB: Hilb is not Hilbert misprinted but a less well-known conformal geometer of that period).

(4).—As this Thom argument prohibits the scheme $20$ \[true despite Fiedler’s correction\], which was an obstacle toward assigning the 19 basepoint, try to pursue the game of understanding the order 8 avatars of the Rohlin-Le Touzé theorem.

About (2), let $C_m$ be a dividing curve of even degree $m=2k$. Denote by $F$ the closed surface $C_m^{+}\cup R$, where $R:=\RR
P^2_+$ is the canonical “Ragsdale” membrane (my own jargon but historically justified I think after reading Viro’s admiration for Miss Ragsdale, as US-Studentin of Klein-Hilbert). On the one hand $F$ is homologous to $kH$. To compute the self-intersection $F^2$ we use a vector field on the canonical membrane $R$, which is either transverse or tangent along the boundary and with finitely many zeros inside $R$. We have by Kronecker-Poincaré’s index formula $\sum indices= \chi$, where $\chi$ is the Euler characteristic of the membrane $R$. Multiplying this vector field by $i=\sqrt{-1}$ permits to push one replica of $F$ in general position. Naively it seems to follow that $k^2=F^2=\chi$, but one needs to count better indices....

Another remark is to write down the Thom conjecture inequality for the filled surface, and this gives the following (which is certainly not new \[$\bigstar$ but alas false!!\], cf. maybe Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000], but after a rapid check it does not seem to be explicitly stated there):

\[Thom-Ragsdale:thm\]—[ERRONEOUS AT LEAST IN THE WEAK SENSE THAT THERE IS A BASIC BUG, YET NO CONCRETE EXAMPLE KNOWN TO ME[^79]]{} [(Thom applied to Klein-Hilbert-Ragsdale)]{}.—Let $C_m$ be a dividing plane curve of degree $m=2k$. Then $\chi \le k^2$, where $\chi$ denotes (as usual) the Euler characteristic of the Ragsdale membrane.

[*Insertion.*]{}\[14.03.13\].—For each $k$, it is a simple matter to convince that the estimate is sharp, compare Figs.\[CCCRoses:fig\] and \[CCCRoses2:fig\].

[*Inserted (optional reading).*]{} \[16.03.13\].—At first sight this result looks so limpid \[$\bigstar$ outdated now!\] that one may wonder if it extends to higher dimensions, e.g. to algebraic surfaces in $\PP^3$. First one requires an extension of the Thom conjecture for surfaces in $\CC P^3$. This is perhaps quite straightforward, by replacing the genus by the Euler characteristic and arguing experimentally that surgeries (aka spherical modifications increase the topological complexity, yet without changing the homology class). However the second step of the proof fails blatantly as we lack a natural extension of the concept of dividing curves to surfaces though several peoples (notably Viro) proposed extensions requiring e.g. that the homology class of the real part mod 2 vanishes in the complexification. Alas, the real locus of an algebraic surface, having (real) codimension 2 in its complexification, never divides. Even if it would the Ragsdale membrane has only (real) dimension 3, hence not ideally suited to cap off the 4D-half (if it existed). This could be remedied by looking at surfaces in $\PP^4$ instead, yet we still lack a way to split the complexification by the real locus. Of course all this failure is somewhat akin to the lack of a good extension of Rohlin’s formula to surfaces as deplored upon, e.g. in Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000]. Basically, the ideas behind Rohlin and the Thom estimate above are very similar, namely to fill the “imaginary” half by a “real” membrane coming from the real locus (either the Ragsdale membrane or bounding discs for ovals).

(of (\[Thom-Ragsdale:thm\])).—We paste to the (bordered) half $C_m^+$ of the dividing curve $C_m$ the Ragsdale membrane $R$ which is the orientable surface bounding $C_m(\RR)$. The resulting closed surface $F$ is orientable \[HERE IS THE MISTAKE (22.03.13)!!!\] and realizes the homology class $kH$ (of halved degree) in the group $H_2(\CC P^2, \ZZ)\approx \ZZ$. It is plain[^80] that we can smooth the “corners” arising along the “cut-and-paste-locus” to get a nearby smooth surface still denoted $F$. By Thom’s conjecture (=meanwile the Kronheimer-Mrowka theorem 1994 [@Kronheimer-Mrowka_1994]) we infer that the genus of $F$, say $f:=g(F)$, is at least as big as that of a smooth curve of the same degree, i.e., $$f\ge g(k)=\textstyle\frac{(k-1)(k-2)}{2}.$$ On the other hand we have by additivity of the characteristic $$\chi(F)=\chi (C_m^+)+\chi (R).$$ For the same reason $2\chi(C_m^+)=\chi(C_m)=2-2g(m)$, and so $$\begin{aligned}
\chi&:=\chi(R)=\chi(F)-\chi (C_m^+)=(2-2f)-1+g(m)=1-2f+g(m) \cr
&\le 1-(k-1)(k-2)+\textstyle\frac{(2k-1)(2k-2)}{2}\cr
&=1-(k-1)(k-2)+(2k-1)(k-1) =1+(k-1)(k+1)=k^2.\end{aligned}$$

As already discussed, this has some interesting applications, e.g. to the prohibition of Hilbert’s (unnested) scheme $11$ of degree 6, and to the scheme $20$ in degree 8. (However all this can be more elementarily deduced from Rohlin’s formula.) \[21.03.13\] Yet compare Theorem \[Alsatian-scheme-Thom-strong-Petrov-Arnold:thm\] below for an example showing that Thom’s estimate is sometimes stronger than the conjunction of several powerful prohibitions of the Russian school (Petrovskii 1938, Gudkov 1969, Arnold 1971, Rohlin 1972/74). Also pleasant is the direct link of this estimate with those conjectured decades prior to Thom by Virginia Ragsdale in 1906 (cf. Sec.\[Ragsdale-conj:sec\]). As I was informed by Th. Fiedler, it seems that it is Mikhalkin who first investigated systematically the repercussion of Thom-Kronheimer-Mrowka upon Hilbert’s 16th.

\[07.03.13\] When we look back at Gudkov’s Table (Fig.\[Gudkov-Table3:fig\]) we see that we get a nearly complete system of prohibition by using total reality and the Rohlin maximality conjecture (RMC), while combining it with the Thom obstruction. Remember that RMC ought to be a reliable principle whenever total reality is exhibited in some concrete fashion as in the Rohlin-Le Touzé theorem. Hence what misses is a prohibition of the scheme $\frac{10}{1}$ in degree 6. One may thus wonder if there is an avatar of Thom’s conjecture for non-orientable surfaces in $\CC P^2$, able to prohibit the sextic scheme $\frac{10}{1}$ (of Rohn). Cavalier, one could put forward something like the:

Every prohibition of Hilbert’s 16th problem, is either interpretable via total reality and the allied Rohlin maximality principle to the effect that a scheme flashed by a total pencil is maximal, or is a consequence of Thom’s conjecture plus an avatar thereof including non-orientable membranes.

Of course for the sextic scheme $\frac{10}{1}$, the idea would be to fill by the non-orientable membrane (residual to the Ragsdale membrane). Further our conjecture is certainly much premature unless it takes into account advanced Bézout-style prohibitions à la Fiedler-Viro (cf. Theorem \[Viro-Fiedler-prohibition:thm\]), and Petrovskii-Arnold style prohibitions (cf. Theorems \[Petrovskii’s-inequalities:thm\] and \[Strong-Petrovskii-Arnold-ineq:thm\]).

Ragsdale’s conjecture (Ragsdale 1906, Petrovskii 1938, Viro 1979/80, Itenberg 1993, Thom 19XX-Kronheimer-Mrowka 1995, and still open, Fiedler) {#Ragsdale-conj:sec}
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\[18.03.13\] As I was made (personally) aware by Fiedler (cf. his 9 March 2013 letter reproduced in Sec.\[e-mail-Viro:sec\]) there ought to be some connection between Thom’s and Ragsdale’s conjecture, which is still open for $M$-curves, despite the disproofs due to Viro 1979 (=Viro 1980/80 [@Viro_1980-degree-7-8-and-Ragsdale]) and Itenberg 1993 [@Itenberg_1993-ctrex-a-Ragsdale] (cf. also Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale]). I can also remember an oral discussion with Thomas Fiedler (Geneva ca. 2011), where Thomas alluded to all the effort he invested on the Ragsdale problem (for $M$-curves). At that time (and arguably still today), I could not appreciate the full swing of this investment.

This section makes no pretence of any breakthrough in the field. It is rather a humble attempt to get familiarized with the topic. Despite our incompetence, let us make some remarks. From our viewpoint of total reality much allied to Ahlfors theorem, which quite paradoxically seems more familiar to complex/conformal geometers than purely real ones (having in mind the antagonism between Riemann-Schottky-Klein-Bieberbach-Teichmüller-Ahlfors versus Harnack-Hilbert-Ragsdale-Rohn-Petrovskii-Gudkov-Arnold-Rohlin, etc.) we could expect a connection of Ragsdale’s conjecture to our paradigm of total reality, e.g. via Theorem \[total-reality-of-plane-M-curves:thm\] as a first basic step. This vague suggestion should probably not be taken too seriously. Another vague idea is to wonder if there is some connection of Ragsdale with the contraction conjectures of Itenberg-Viro, or perhaps our version thereof called CCC, cf. (\[CCC:conj\]).

After those abrupt remarks, let us be more pedestrian. First what is Ragsdale’s conjecture at all about? What is known on it and what is not? Especially does it connect to 4D-dimensional topology as the whole Hilbert problem was realized to be since Arnold’s breakthrough 1971 [@Arnold_1971/72] and the deeper investigations of Rohlin (e.g., the validation of Gudkov’s hypothesis $\chi\equiv_8 k^2$). In particular how does Ragsdale connect with Thom’s conjecture which is basically a problem of embedded differential topology of smooth surfaces in the complex projective plane $\CC P^2$ (arguably the 4-manifold simplest to visualize as the configuration space of all unordered pairs grooving on the 2-sphere). As we shall see, the link Thom-Ragsdale is very clear-cut, at least for one half of the Ragsdale conjecture (cf. Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]). \[$\bigstar$ Okay, but alas this is based on our erroneous estimate $\chi\le
k^2$!\]

First, Virginia Ragsdale, coming from the U.S. was a student of both Klein and Hilbert in Göttingen ca. 1906. Building upon a careful inspection of the features of Harnack’s and Hilbert’s constructions (of small vibratory perturbations), she posited a conjecture on the numbers $p,n$ of even resp. odd ovals of real plane algebraic curves. Although the chance of deriving any transcendental truth from such a specific mode of generation looks a priori very meagre, the conjecture in question turned out to be extremely robust requiring at least ca. 7 decades up to being disproved \[Viro, Itenberg\]. Yet some respectable vestiges remains open, and deserves further efforts.

Via some naive acquaintance with Gudkov’s Table in degree 6 (Fig.\[Gudkov-Table3:fig\]) (the little that we personally have at disposal) and the allied geometry of pyramids, it seems that the Harnack and Hilbert constructions explore only the superficies of the pyramid, while the profound part of the puzzle is cracked in Gudkov’s revolution (ca. 1969–72) constructing the scheme $\frac{5}{1}5$ (which is so-to-speak the Pharaoh chamber). The Ragsdale conjecture (in modernized shape) may be stated as the estimate $\vert \chi \vert \le k^2$, hence it is perhaps not too surprising that the particular methods of Harnack and Hilbert lead to sharp estimates at least for $M$-curves. What appears historically first is likely to be the most superficial objects, hence extremalizing the functional $\vert \chi\vert$ which roughly measures the level of superficiality in the pyramid.

We state now precisely Ragsdale’s original statement (compare Ragsdale 1906 [@Ragsdale_1906]).

\[Ragsdale-conj:conj\] [(Ragsdale 1906, disproved for the number $n$ of odd ovals by Viro 1979, and in general by Itenberg 1993)]{}—For any curve of even degree $m=2k$, we have $$p\le \textstyle\frac{3}{2}k(k-1)+1, \qquad n\le
\textstyle\frac{3}{2}k(k-1)$$

As explained in Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale] (especially p.24) this was refuted by Viro in 1979 for the number $n$ of odd ovals, and by Itenberg in general. Moreover it is explained (in ) that Petrovskii made similar conjectures, being apparently unaware of Ragsdale’s paper. In particular the so-called [*Petrovskii inequality*]{} (cf. (\[Petrovskii’s-inequalities:thm\])) is considered there as having been formerly conjectured by Ragsdale as a weak form of her conjecture. Finally it is remarked that Petrovskii himself (1938) formulated a version of Ragsdale’s conjecture (\[Ragsdale-conj:conj\]), yet more cautious by one unit than Ragsdale’s on the number $n$, so that both bounds are identic equal to $\textstyle\frac{3}{2}k(k-1)+1$.

Despite the disproof (by Viro-Itenberg) of both the Ragsdale and the weaker Petrovskii conjectures, the interesting quick is that the case of $M$-curves is still open (at least in the weaker formulation of Petrovskii). Precisely

[(Ragsdale’s conjecture on $M$-curves 1906, still open)]{}—For any $M$-curve of even degree $m=2k$, the Euler characteristic $\chi=p-n$ of the Ragsdale membrane is bounded by the square of the semi-degree $k$, i.e. $$\vert \chi \vert \le k^2.$$

We borrowed this from Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale p.24] (cf. also Kharlamov-Viro (undated) [@Kharlamov-Viro_XXXX-UNDATED p.15]). It should be remarked that one half of this conjecture (namely the estimate $\chi\le
k^2$) follows directly (cf. Theorem \[Thom-Ragsdale:thm\]) from Thom’s conjecture proved by Kronheimer-Mrowka in 1994 [@Kronheimer-Mrowka_1994]. Curiously, this is not pointed out in the Itenberg-Viro 1996 article (presumably due to backlog reasons). Actually as noted in Theorem \[Thom-Ragsdale:thm\], the estimate $\chi\le k^2$ holds more generally for dividing curves.

$\bigstar\bigstar$ [*Insertion.*]{} \[23.03.13\] This historical puzzle is now completely fixed by Fiedler’s correction of my mistake of overlooking that the Arnold surface is not necessarily orientable.

Some few words are required to understand why the above conjecture $\vert \chi \vert \le k^2$ is termed Ragsdale’s conjecture. (It seems to me that Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale p.24] contains a serious misprint at this place, specifically on p.24 in the statement of the Ragsdale conjecture on $M$-curves the equivalent conditions $p\ge \frac{(k-1)(k-2)}{2}$ and $n\ge\frac{(k-1)(k-2)}{2}$ looks to me erroneous; and the same misprint appears in Kharlamov-Viro [@Kharlamov-Viro_XXXX-UNDATED p.15]) Let us clarify this as follows:

\[Thom-implies-one-half-of-Ragsdale:lem\] For $M$-curves of degree $2k$, the condition $\vert \chi\vert\le
k^2$ is equivalent to Petrovskii’s cautious version of the Ragsdale conjecture, i.e. $$p\le \textstyle\frac{3}{2}k(k-1)+1, \qquad n\le
\textstyle\frac{3}{2}k(k-1)+1.$$ More precisely the upper estimate on $\chi$ (i.e. $\chi\le k^2$) is equivalent to the bound on $p$, while the lower estimate $-k^2\le \chi$ is equivalent to the bound on $n$. Further the first upper bound $\chi\le k^2$ follows from Thom’s bound (cf. [Theorem \[Thom-Ragsdale:thm\]]{} valid more generally for any dividing curve), while the other is perhaps still open, though one could dream reducing it to Thom too, after taking maybe an orientable cover (but looks dubious), or maybe by reducing it via differential geometry to Gauss-Bonnet and Wirtinger (as discussed below).

Start from the condition $-k^2\le\chi\le k^2$. By definition $\chi=p-n$, and $M=r=p+n$. As usual Harnack’s bound is $M=g+1=\frac{(2k-1)(2k-2)}{2}+1=(2k-1)(k-1)+1=2k^2-3k+2$. Adding $\chi=p-n\le k^2$ to $p+n=M=2k^2-3k+2$ gives $$2p\le k^2+2k^2-3k+2=3k^2-3k+2=3k(k-1)+2,$$ whence Ragsdale’s bound on $p$. On the other hand, the lower estimate on $\chi$, i.e. $-k^2\le \chi=p-n$, rewritten as $k^2\ge
n-p$, gives when added to $p+n=M$ the 2nd Ragsdale estimate on $n$.

Again the text of Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale p.24] which reads as follows, seems not perfectly up-to-date \[SORRY MY MISTAKE!\] in view of Kronheimer-Mrowka’s validation of Thom’s conjecture:

[“Which of Ragsdale’s questions are still open now? The inequalities[^81] $$p\le \textstyle\frac{3}{2}k(k-1)+1, \qquad n\le
\textstyle\frac{3}{2}k(k-1)+1.$$ have been neither proved[^82] nor disproved for $M$-curves.”]{}

As shown by Lemma \[Thom-implies-one-half-of-Ragsdale:lem\] (implicit in Itenberg-Viro’s article modulo the misprint), Thom’s conjecture implies one half of Ragsdale conjecture \[ALAS NOT TRUE\], namely the “positive” half concerning $p$ where it matches exactly with Petrovskii’s subsequent rediscovery of the conjecture. However the second half looks much out of reach, as it requires the estimate $-k^2\le\chi=p-n$ which seems to take care of the non-orientable (“anti-Ragsdale”) membrane (not ideally suited to Thom).

First without any idea the lower bound on $\chi=\chi(B^+)$ the characteristic of the (orientable) Ragsdale membrane $B^+$, i.e. $-k^2\le \chi$, can be transmuted using $B^+\cup B^-=\RR P^2$ into $k^2\ge\chi(B^-)-1$.

So one seems forced to study this non-orientable membrane $B^-$. One idea to explore is to arrange an orientable membrane via the usual trick of the double orienting cover (essentially due to Gauss, Möbius, Klein, Teichmüller 1939, etc.) but which have now to be implemented in some embedded fashion. This looks dubious as we do not know what to do along the boundary of $B^-$. At least the surface we get (granting that there is some natural way to construct an oriented Verdoppelung=double) would have tripodal singularities along the boundary. This is common in soap film experiment, yet a priori outside the tolerance permitted in Thom’s conjecture.

As a completely different strategy there could be a result (dual to Thom’s) stating that for smooth surfaces the genus cannot be too big when attention is confined to smooth surfaces arising by rounding corners of the half of a dividing curve capped off by the Ragsdale membrane. Of course in general the genus of a smooth surface of prescribed (homological) degree can be made as large as we please (just attach small handles), yet perhaps the surfaces that arise by the “Ragsdale filling procedure” of Klein’s orthosymmetric half (a method truly inaugurated by Arnold 1971, and Rohlin 1974, etc.) are of a special type subsumed to an upper-bound upon the genus in terms of the degree.

Recall the Wirtinger’s inequalities stating that complex projective varieties (or even Kähler manifolds) minimize the volume among differential-geometric submanifolds in a given homology class. Perhaps this combined with Gauss-Bonnet can supply the required upper-estimate upon the genus dual to Thom’s estimate, hence validating the remaining half of Ragsdale’s conjecture (as modified by Petrovskii 1938).

\[19.03.13\] In fact I do not know if the lower estimate of the pinching $-k^2\le \chi\le k^2$ holds true more generally for dividing curves as do the upper estimate $\chi \le k^2$ by virtue of Thom’s (genus) bound (Theorem \[Thom-Ragsdale:thm\]) \[FALSE, cf. Itenberg-Viro’s curve on Fig.\[Itenberg:fig\]\]. It is worth first noting that the Petrovskii jargon (1938 [@Petrowsky_1938]) of $p$ and $n$ as positive and negative ovals is quite good as they contribute positively resp. negatively to $\chi$ of the Ragsdale membrane. \[Of course the sign of $\chi$ is a matter of convention that varied through the ages, but at least now we seem to all agree about its sign ($\chi(pt)=+1$).\] Then it is also useful to keep in mind the geography of the generic Gudkov pyramid (cf. Fig.\[Pyramidragsdale:fig\]) though this is a coarse simplification of the real one which is a multidimensional (non-planar) object as soon as $m\ge 8$. Now the point is that as shown by Lemma \[Thom-implies-one-half-of-Ragsdale:lem\] the Thom estimate $\chi\le k^2$ and its dual $-k^2\le\chi$ formally implies for $M$-curves the Ragsdale-Petrovskii estimates $p\le P$, and $n\le P$ respectively, where we set $P:=\frac{3}{2}k(k-1)+1$ (for Petrovskii’s bound). Diagrammatically, this amounts saying that [*Ragsdale’s zone*]{} ($p,n\le P$) arises from Thom’s vertical strip $\vert\chi\vert\le k^2$ by reflecting vertical rays at angle of 60 degrees (cf. Fig.\[Pyramidragsdale:fig\]). We get so a pentagonal diamond (the Ragsdale diamond) that was supposed to contain all algebraic schemes by virtue of the Ragsdale-Petrovskii conjecture, which alas turned out wrong by Itenberg-Viro \[Fig.\[Itenberg:fig\]\]. However on the top face of the diamond ($M$-curves) the conjecture is still robust. Further, Thom’s strip $-k^2\le \chi \le k^2$ is perhaps a container for all dividing curves \[FALSE, cf. again the Itenberg-Viro Fig.\[Itenberg:fig\]\]. This holds true on the right positive side (still Theorem \[Thom-Ragsdale:thm\]) \[FALSE!\] but the lower estimate is presently more dubious. A look on Gudkov’s table(=Fig.\[Gudkov-Table3:fig\]) shows that $-k^2\le\chi$ holds true in degree $6$ (actually for all curves regardless of being dividing), yet this low-degree case is probably atypical.

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By analogy with Thom’s conjecture, let us put forward the:

\[anti-Thom-conj:conj\] [(Anti-Thom conjecture, due to Gabard hence probably not serious at all)]{} Any plane dividing curve of even degree $m=2k$ respects a Thom-style lower bound $$-k^2\le \chi.$$

If true this would imply the remaining half of the Ragsdale conjecture for $M$-curves (via Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]). It would be interesting to know if the Viro-Itenberg method (viz. counter-examples) already disproves this naive conjecture. \[QUITE PROBABLE!\] If not, one would like to imagine a proof along the above sketched line (Gauss-Bonnet-Wirtinger), or via an oriented double cover of the non-orientable Ragsdale membrane $B^-$. Another third strategy would be to use an eversion (cf. Sec. \[Eversion:sec\]) to reduce to the case of Thom, yet this looks hazardous as it requires a large deformation (for which very little is known apart vague speculation of us that differential-geometric flows could do such jobs).

At this stage it is wise to contemplate the higher Gudkov’s pyramids in degree 8 and 10 (cf. resp. Figs.\[Degree8:fig\] and \[Degree10:fig\]). In degree 8 (Fig.\[Degree8:fig\]) we see that the anti-Thom line $-k^2\le \chi$ is well adjusted to the blue-rhombs materializing Arnold’s congruence mod 4 for dividing curves. Hence there is little chance to corrupt our anti-Thom conjecture (\[anti-Thom-conj:conj\]). In contrast in the degree 10 table (Fig.\[Degree10:fig\]) there is a myriad of 7 schemes adventuring outside the anti-Thom line. Those are given by the symbols $\frac{34}{1}2$, $\frac{33}{1}1$, $\frac{32}{1}$ and $\frac{31}{1}3$, $\frac{30}{1}2$, $\frac{29}{1}1$, $\frac{28}{1}$. If any one of those schemes admits a type I(=orthosymmetric) realization our conjecture is faulty. This problem can either be approached by Harnack or Hilbert’s method of vibrations or by the Viro-Itenberg patchworking. Note that it is unlikely that those schemes (in type I) are prohibited by Rohlin’s formula. \[21.03.13\] However the first 3 listed (with $\chi=-31$) are prohibited by Petrovskii’s inequality (\[Petrovskii’s-inequalities:thm\]).

For the 4 remaining schemes (with $\chi=-27$) one can expect to do naive Hilbert constructions like on Fig.13 of Gabard 2000 [@Gabard_2000], and look what happens. That requires some concentration and is differed to latter.

\[21.03.13\] It may be noted that the strong Petrovskii inequality (\[Strong-Petrovskii-Arnold-ineq:thm\]) $n-p^- \le
\frac{3}{2}k(k-1)=30$ specialized to the range of our diagram (Fig.\[Degree10:fig\]) involving simple symbols of the form $\frac{x}{1}y$ where $p^-=1$ (one hyperbolic oval, i.e. which ramifies in Hilbert’s tree) implies that $n\le 31$ and so Ragsdale line is corroborated. This does not kill any of the 4 schemes in candidature above. Note further that it must be a general issue that the strong Petrovskii-Arnold estimate gives Ragsdale in the “planar” range of the pyramid involving symbols $\frac{x}{1}y$. On the right-side of Fig.\[Degree10:fig\], the (other dual) strong Petrovskii-Arnold estimate $p-n^-\le
\frac{3}{2}k(k-1)=30+1$ (with now $n^-=0$) also kills all schemes lying “above” Ragsdale’s line. This implies a severe crumbling in the corners of the pyramid (cf. Fig.\[Degree10:fig\]) on both the right and left side of it, yet note that a priori the scheme $\frac{31}{1}3$ (if it exists) seems to imply some mysterious asymmetry in the architecture. Further this scheme, being of type I, could be an interesting place to look for a counterexample to Rohlin’s maximality conjecture.

Thom versus Rohlin
------------------

\[07.03.13\] As noticed above the Thom obstruction (\[Thom-Ragsdale:thm\]) is at least for degree 6 (and to some extend in degree 8) subsumed to Rohlin’s formula (\[Rohlin-formula:thm\]). The latter also prohibits the scheme $\frac{10}{1}$ (cf. Fig.\[Gudkov-Table3:fig\]) of degree 6 without that we have to worry about a dubious non-orientable extension of Rohlin’s formula. One can wonder if in general the information derived from Thom (\[Thom-Ragsdale:thm\]) is always subsumed to Rohlin’s formula.

Let us notice the following:

The scheme $20$ cannot be realized in degree $8$.

Let $C_8$ be a (hypothetical) octic of type $20$. By the RKM-congruence (\[Kharlamov-Marin-cong:thm\]) or better its reformulation as (\[RKM-congruence-reformulated:thm\]) the curve has to be of type I, but then its existence is ruled out by Rohlin’s formula (\[Rohlin-formula:thm\]).

\[21.03.13\] Another proof of the lemma (historically sharper) is to appeal to Petrovskii’s inequality (1933/38), cf. (\[Petrovskii’s-inequalities:thm\]).

It is worth then comparing the Gudkov table in degree 8 (Fig.\[Degree8:fig\]), which shows that several obstructions are not readily interpreted via total reality and Rohlin’s allied principle of maximality (look especially at the upper-right corner of that figure).

Now we turn to the question of deciding if Thom is subsumed to Rohlin (at least in the realm of Hilbert’s 16th problem). Glancing at Fig.\[Degree8:fig\] the answers seems to be yes for degree 8 (at least for schemes of the form $\frac{x}{1}y$ as those represented on that figure). If true in general this should follow from a simple combinatorial argument.

If we look at the degree $m=10$ table, we find the following structure (Fig.\[Degree10:fig\]). This picture is built as usual. First one compute Harnack’s bound $M=g+1=\frac{(m-1)(m-2)}{2}+1=\frac{9\cdot 8}{2}+1=36+1=37$. So one extends the previous pyramid in degree 8, up to that level $37$. Then there is the sawtooth broken line à la Gudkov. Its upper undulations have to be adjusted at the Gudkov-Rohlin congruence $\chi\equiv k^2 \pmod 8$, here $k^2=25$. Remind that on such pictures Euler-Ragsdale’s $\chi$ may (always) be interpreted as the abscissa (“$x$-axis”, i.e. horizontal axis). So we have the $M$-schemes lying at the top of the sawtooth broken line, while in their depressions (“creux”) we have the RKM-schemes with $\chi\equiv k^2+4 \pmod 8$, that are forced being of type I. All this can be extended into the lattice of blue-rhombs where type I curves are forced to live (Arnold’s congruence mod 4). \[[*Warning.*]{}—On our Fig.\[Degree10:fig\] this lattice of blue rhombs is correct on the upper half, but need to be adjusted (mentally) on the lower part, where we just copied the pyramid in degree 8. However we thought it would be more instructive to see this lower object intact as to appreciate better the growing mode of pyramids.\]

By Thom (\[Thom-Ragsdale:thm\]) we have $\chi\le k^2=25$ for curves of type I. It seems (at first sight) that several schemes permissible for Rohlin are prohibited by Thom, yet the ultimate answer will be nearly the opposite one.

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First the $M$-scheme $\frac{2}{1}34$ is prohibited by Thom \[DUBIOUS INFERENCE OF THOM!\] though it is not by the Gudkov-Rohlin congruence. (Incidentally this scheme is prohibited by Petrovskii (\[Petrovskii’s-inequalities:thm\]).) Is this scheme prohibited by Rohlin’s formula $2(\Pi^+-\Pi^-)=r-k^2$? Here we have two nested pairs hence $\Pi=\Pi^+ +\Pi^-=2$, and $r-k^2=37-25=12$. So $\Pi^+-\Pi^-\le \Pi=2$ and Rohlin’s formula cannot be fulfilled. So Thom brings nothing new.

Then there is the scheme $\frac{1}{1}33$, here $\Pi=1$, so $\Pi^+-\Pi^-\le \Pi=1$, but $r-k^2=35- 25=10$ and Rohlin’s formula cannot hold.

For the scheme $33$, Rohlin’s formula cannot hold as well (here $\Pi=0$ so $\Pi^+-\Pi^-=0$) and $r$ has to be a square.

Next we have the RKM-scheme $\frac{3}{1}31$. Here $\Pi=3$ and so $\Pi^+-\Pi^-\le 3$, while $r-k^2=35-25=10$ and again Rohlin’s formula cannot be verified.

Below the former we have $\frac{2}{1}30$. Here $\Pi=2$ and so $\Pi^+-\Pi^-\le 2$, while $r-k^2=33-25=8$ and Rohlin cannot be fulfilled.

Below, we have $\frac{1}{1}29$, where $\Pi=1$ and so $\Pi^+-\Pi^-\le 1$, while $r-k^2=31-25=6$ and Rohlin cannot be fulfilled.

Below, we have $29$ which is not realized in type I, as $r$ is not a square.

Conclusion: [*all schemes prohibited by Thom are actually also prohibited by Rohlin.*]{} (at least within the range of Fig.\[Degree10:fig\]).

Perhaps Rohlin even prohibits more than Thom. The next boy is the scheme $\frac{6}{1}30$. Here $\Pi=6$, and so $\Pi^+-\Pi^-\le 6$, while $r-k^2=37-25=12$ and Rohlin can by now be fulfilled. So no obstruction.

All this little experiments points out to a subsumation of Thom to Rohlin (at least for schemes of the form $\frac{x}{1}y$).

For schemes of the type $\frac{x}{1}y$, Thom’s inequality is subsumed to Rohlin’s formula.

For a scheme of this form the total number of pairs (denoted $\Pi$) is $\Pi= x$, hence $\Pi^+-\Pi^-\le x$. By Rohlin’s formula (\[Rohlin-formula:thm\]) we infer $$2x\ge 2(\Pi^+ -\Pi^-)=r-k^2=(1+x+y)-k^2.$$ Calculating $\chi$ gives $$\chi=1-x+y\le k^2,$$ by the above estimate.

Perhaps this is even true in general, but this deserves another argument. (Update the sequel, will show that the contrary is true, cf e.g. Theorem \[Alsatian-scheme-Thom-strong-Petrov-Arnold:thm\].)

First the argument extends to schemes of the form $\frac{x}{1}\frac{y}{1}z$. Then $\Pi=x+y$, and so $\Pi^+-\Pi^-\le
\Pi=x+y$. Then writing down Rohlin’s formula $$2(x+y)\ge 2(\Pi^+-\Pi^-)=r-k^2=(2+x+y+z)-k^2,$$ from which it is inferred that $$\chi=1-x+1-y+z=2-(x+y)+z\le k^2,$$ i.e. Thom’s estimate.

And so on, it seems that the passage from Rohlin to Thom will always succeed as long as the depth is at most one. So for a counterexample we shall investigate deeper schemes.

For instance in degree 8, we can look at an extension of the $3$-nest (of depth 3). For instance the $M$-scheme $(1,1,1)19=(3\times 1) 19 $ (in our satellite notation). Here $\Pi=3$, and Rohlin’s formula $2(\Pi^+ -\Pi^-)=r-k^2=22-16=6$ is verified for $\Pi^+=3$, and $\Pi^-=0$. But Thom’s estimate $\chi\le k^2=16$ is not (as $\chi=1-1+1+19=20$). So here we get an example where Thom is not subsumed to Rohlin. However our example is artificial being prohibited by the Gudkov-Rohlin congruence $20=\chi\equiv
k^2=16 \pmod 8$. Yet our example makes unlikely a general subordination of Thom to Rohlin’s formula alone.

A basic idea is to adjust $\chi=20$ at $\chi=16$ to make it Gudkov-Rohlin compatible. Starting from the above scheme, we may trade an outer oval for one at depth 1 in the $3$-nest. Each such trading diminishes $\chi$ by 2, and so two trades are required to adjust $\chi=16$. Doing this we get the scheme $(1,\frac{1}{1}2)17$ but alas now Thom’s inequality is verified. (It is also easy to check that Rohlin’s formula is satisfied.)

So our game becomes: find a [ *French scheme*]{}, i.e. one prohibited by Thom yet not succumbing under the armada of Russian prohibitions (Gudkov, Arnold, and above all Rohlin, and its companions especially Kharlamov-Marin).

Let us look at degree 10, and to a deep scheme, say extending the $4$-nest. As the Harnack bound is $M=g+1=9\cdot 4+1=37$, we look at the scheme $(1,1,1,1)33=(4\times 1)33$. Now $\chi=(1-1+1-1)+33=33\equiv k^2=25 \pmod 8$, i.e. the Gudkov-Rohlin congruence is fulfilled. However the scheme is prohibited by Thom’s estimate $\chi \le k^2$. Further Rohlin’s formula $2(\Pi^+ -\Pi^-)=r-k^2=37-25=12$, and $\Pi=\binom{4}{2}=6$ (count all pair in the $4$-nest) so that $\Pi^+=6$ and $\Pi^-=0$. So Rohlin’s formula affords no prohibition. We have proven the:

\[French-scheme:thm\] [(ERRONEOUS, cf. Corrigendum right below, and for a corrected version cf. Theorem \[French-scheme-corrected:thm\] below)]{} There exists a French scheme, i.e. where Thom is not subsumed to Rohlin’s formula nor to the Gudkov-Rohlin congruence mod $8$. The scheme in question is even an $M$-scheme of degree $10$, namely $(1,1,1,1)33$. However for schemes of depth $\le 2$, Rohlin’s formula is as strong (and of course stronger) than the Thom obstruction.

\[10.03.13\] [*Corrigendum*]{}.—Th. Fiedler objected as follows to the above theorem (compare his letter dated \[09.03.13\] in Sec.\[e-mail-Viro:sec\]):

“The $M$-curve of degree 10 mentioned in your Thm 28.11\[=\[French-scheme:thm\]\] is in fact ruled out by Rokhlin’s formula. I think that you have mixed $\Pi^+$ with $\Pi^-$. In a positive couple the orientations are just opposite. So, four nested ovals can contribute at most $+2$ to Rokhlin’s formula.”

After some hesitation, Gabard realized of course that Fiedler is perfectly right. Let me paraphrase his explanation differently. This is of course allied to the signs-law of Fig.\[Signs-law-dyad:fig\], but let us be more specific. If we consider a 4-nest and choose on it complex orientations forming positive pairs at each immediately successive nested ovals then we get Fig.\[Fiedler-correction:fig\]a. Pairs of length 2 becomes negative, while the unique pair of length 3 is positive again. (This is seen either by looking at the picture or if one like extracting an arithmetical law one finds the twisted signs-law of (\[Signs-law:lem\]) akin to usual arithmetics modulo a twisted sign, so $+\times
+=-$, $+\times -=+$, $-\times +=+$, $-\times -=-$. This is best memorized by saying that “mixing the genes is good, while consanguinity is bad”!) So the contribution to $\Delta
\Pi:=\Pi^{+}-\Pi^{-}$ is at most 2, and certainly never equal to $6$ (though it can be $-6$ as on Fig.\[Fiedler-correction:fig\]b).

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It remains now to see how Fiedler’s remark generalizes as to see if Thom’s estimate $\chi\le k^2$ is a formal consequence of Rohlin’s formula. If not then it remains to find another (more serious!) French scheme.

The naive scenario would be that Rohlin always implies Thom (at least within the realm of Hilbert’s 16th). The only chance to prove this seems to involve an estimation of the corrector term in the “Rohlin-to-Arnold formula” , which we reproduce for convenience $$\begin{aligned}
\label{Rohlin-to-Arnold-bis:eq}
\chi=p-n=(p+n)-2n&=r-2n\cr
&=[2(\Pi^+-\Pi^-)+k^2]-2n\cr
&=k^2+2(\Pi^+ -\Pi^- -n).\end{aligned}$$ So setting $\Delta \Pi:=\Pi^+ -\Pi^-$ we would like to show the:

[(Garidi[^83] mass conjecture.)]{}—It holds universally $\Delta \Pi \le n$.

Of course the conjecture is true (at least if one believes in the Kronheimer-Mrowka validation of the Thom conjecture) \[THIS REASONING IS DUBIOUS BEING BASED ON OUR FALSE THOM ESTIMATE\], so that the true meaning of our conjecture is an independent derivation of the estimate via pure combinatorics. Let us be more precise.

[A signed or [*Rohlin tree*]{} is a combinatorial object consisting of a (finite) directed set plus a distribution of signs $\pm$ on its edges such that the signs-law (of Lemma \[Signs-law:lem\]) is verified. By a directed set we mean a finite POSET such that each element as at most one superior, i.e. an element larger and minimal with this property. Recall also that the signs-law can be easily remembered by saying that consanguinity is bad, i.e. $+\times +=-$, $-\times -=-$, while mixing the genes is good $+\times -= +$ and $-\times += +$ (this exotic signs-law is the exact opposite of the usual convention).]{}

Of course this concept arises naturally when taking a smooth dividing plane (algebraic) curve $C_m$ of even degree $m=2k$ and assigning to it its Hilbert tree (encoding the distribution of ovals), while decorating the edges with signs coming from the complex orientations as in Rohlin’s formula (\[Rohlin-formula:thm\]).

More precisely, given an oriented real scheme (i.e. an isotopy class of embedding of a disjoint union of circles in $\RR P^2$ supplied with an orientation), we can assign to it a Rohlin tree. Conversely it is clear that any Rohlin tree arises in this fashion. To a real plane dividing curve is assigned a complex orientation (uniquely defined up to reversal of all orientations), yet this leaves invariant the concept of positive or negative pairs as defined by Rohlin. Hence to be very formal, we have first the map taking a dividing real curve to its real scheme with complex orientation, which is a “projectively” oriented real scheme (weel-defined up to reversing all orientations), which in turn defines unambiguously a Rohlin tree. Diagrammatically, $$\textrm{dividing plane}  \to \textrm{oriented real schemes } \to
\textrm{Rohlin trees}.$$ -23pt $$\hskip-80pt\textrm{curves ($m=2k$) \hskip0.5cm (mod reversion) }$$

So the precise meaning of the above conjecture is that any Rohlin tree (not necessarily induced by a real algebraic curve) satisfies the above estimate $\Delta \Pi \le n$. After several hours of attempting to prove this “Garidi mass conjecture”, one finds a simple counterexample as follows.

To keep $n$ small (say $n=1$), we consider a tree with only one vertex at depth 1, but ramifying (violently) at depth 2 (cf Fig.\[Garidi-mass-false:fig\]a).

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Let us introduce the sign distribution of Fig.a, and denote by $p_2$ the number of (even) vertices at depth 2. We find by using the twisted signs-law: $$\Delta \Pi= (-1)+p_2+p_2=2p_2-1.$$ (Here the first $-1$ comes from the top edge (visible on Fig.a), the second term $p_2$ is the contribution of the $+$-signs visible on Fig.a), while the 3rd term $p_2$ comes from the $p_2$ pairs of length 2 obtained by concatenation of elementary edges. The signs-law in question (based on Fig.\[Signs-law-dyad:fig\]) is the same as the usual one modulo a twist by $-1$. So here $+\times
-$ gives $+$ (the opposite of the usual sign rule!). The displayed formula is justified.)

Now as soon as $p_2\ge 2$, the above $\Delta \Pi $ will be $\ge
3$, foiling thereby the mass conjecture.

[*Insertion*]{} \[22.03.13\] For later reference, let us state this as a:

\[Garidi-mass-conj-is-FALSE:thm\] The Garidi mass conjecture is false, and therefore the positive mass conjecture (\[positive-mass-conjecture:conj\]) is erroneous too. For a simple counterexample cf. Fig.\[Garidi-mass-false:fig\]a right above.

This basic corruption aids us to detect a more serious French scheme (where Thom is not subsumed to Rohlin). As above we consider curves of degree $m=10$. Harnack’s bound is $M=g+1=\frac{9\cdot 8}{2}+1=9\cdot 4 +1=37$ (temperature of the human body). Our tree converts then to the scheme of Fig.\[Garidi-mass-false:fig\]b, where we see 2 nested ovals containing 35 unnested ones. The characteristic of the “Ragsdale membrane” is $\chi=(1-1)+35=35 \nleqslant 25=k^2$, so that the scheme is prohibited by Thom. Is this scheme (with Gudkov symbol $(1,1,35)$) prohibited by Rohlin’s formula?

Remember that Rohlin’s formula implies Arnold’s congruence (cf. (\[Rohlin-implies-Arnold:lem\])), so the answer is an (indirect) yes since $\chi\equiv k^2=25\neq 35 \pmod 4$.

As we have an $M$-scheme, let us even adjust to the (stronger) Gudkov congruence mod 8: $\chi\equiv k^2=25=33 \pmod 8$. So in order to diminish $\chi$ by 2 (from 35 to 33), let us trade a deep oval (at depth 2) against one at depth 1 (cf. transition from Fig.b to Fig.d). We have now $\chi=33$, and so Thom is still violated. Is this new scheme (symbol $(1,1(1,34))$) prohibited by Rohlin’s formula?

We count (e.g. via Fig.e) that the total number of pair is $\Pi=2+34+34=70$ (this can be viewed as an application of the formula $\Pi=n_1+2p_2+3n_3+4p_4+ etc$, cf. proof of (\[Stalin:lemma\])). To abridge Rohlin’s (heavy “Cyrillic”) notation let us set $\pi:=\Pi^+$, and $\eta:=\Pi^-$, and Rohlin’s formula then reads $2(\pi-\eta)=r-k^2=37-25=12$. So we get the pair of equations $\pi-\eta=6$ and $\pi+\eta=\Pi=70$. Adding them gives $2\pi=76$, whence $\pi=38$, and $\eta=32$. So Rohlin’s is (formally) soluble but is there a distribution of signs compatible with the signs law?

To answer this let us consider a “variable” distribution of signs like on Fig.f with $x$ many $+$ and $y$ many $-$ for the edges rooted “at depth 1”, while both edges rooted at depth 0 have signs $-$. Of course we assume $x+y=34$. Counting the number of positive pairs $\pi$ we find $\pi=x+x=2x$ (were the second $x$ term comes from the sign-law $+\times -= +$ the opposite of the usual convention!). For the number $\eta$ of negative pairs we find $\eta=2+y+y=2+2y$, where the 3rd $y$ term comes again from the exotic sign-law (“of Rohlin”). Combining with the previous paragraph, gives $x=19$ and $y=15$. All equations are then verified! Conclusion there is a distribution of signs on the tree which satisfies Rohlin’s formula, which therefore does nor prohibit the scheme under examination, i.e. $(1,1(1,34))$. The latter is therefore a French scheme. So we hope to have this time proven the:

\[French-scheme-corrected:thm\] There exists a “French” $M$-scheme of degree $10$, namely that with Gudkov symbol $(1,1(1,34))$ (cf. [Fig.\[Garidi-mass-false:fig\]d]{} above), i.e. which is prohibited by Thom but not by the armada of Russian congruences (especially Gudkov’s) nor by Rohlin’s formula.

[*Insertion*]{}—\[17.03.13\] Thomas Fiedler kindly reacted as follows to this statement, cf. his \[12.03.13\]-letter in Sec.\[e-mail-Viro:sec\] reproduced below for convenience (our brackets are just automatized updates of labels):

“sorry, but all your $M$-schemes of degree 10 in Thm 30.14\[=\[French-scheme-corrected:thm\]\] and 30.15\[=\[French-scheme-corrected-bis:thm\]\] have $n=2$ and are ruled out simply by Petrovskis inequality. I don’t think that genus bounds give anything new for real schemes alone[^84] but they definitely do so for configurations of several real curves. Just take a look on Mikhalkin’s paper.”

Though a pertinent remark (since Petrovskii is not a French guy), Fiedler’s remark does not affect the modest truth of our statement but points to the Petrovskii inequality as another sharp Russian weapon. (Shamefully, I confess to have not properly appreciated this fundamental statement prior to Fiedler’s comment.) The latter states:

[(Petrovskii 1933/38 [@Petrowsky_1938])]{} \[Petrovskii’s-inequalities:thm\] For any real plane smooth curve of even degree $m=2k$ we have the (so-called) Petrovskii inequalities (which are pure jewels nearly coming out of the blue safe for having been apparently anticipated conjecturally by Miss Ragsdale in 1906) $$- \textstyle\frac{3}{2} k(k-1)\le \chi \le \textstyle\frac{3}{2}
k(k-1)+1.$$ (An analogous but more complicated statement holds for curves of odd degrees.)

We make just some few remarks.

[*Historical substance.*]{}—Petrovskii 1938 [@Petrowsky_1938 p.191] comments that his method of proof is based on two ingredients:

\(1) a formula of Jacobi-Euler 1768–70 (and also cite en passant Kronecker, which as we know is also one of the forerunner of Poincaré’s index theory via Hermite’s transmissive rôle), and,

\(2) on the consideration of the deformations of lines $F(x,y)=C$ when $C$ crosses the critical values of $F(x,y)$. These last investigations being identified as analogous to those of Morse (1925 [@Morse_1925?]) on the critical points of a function.

[*Neo-expressionist proofs*]{}.—Another proof of Petrovskii’s inequalities namely the “Preuve d’Arnol’d dans une présentation de A. Marin” is given in A’Campo 1979 [@A'Campo_1979 p.537–17], where the result is stated as: $$\vert 2\chi-1 \vert \le \textstyle\frac{3m^2}{4}-\frac{3m}{2}+1,$$ which (after setting $m=2k$) is readily seen to be equivalent to the above formulation. Indeed $\vert 2\chi-1 \vert \le
\textstyle\frac{3(2k)^2}{4}-\frac{3(2k)}{2}+1=3k^2-3k+1=3k(k-1)+1$. Hence $3k(k-1) \le 2\chi\le 3k(k-1)+2$, and the equivalence is now obvious.

[*The original statement differs slightly.*]{}—On adapting to our (fairly standard modern) notations, Petrovskii’s original result is stated as follows (cf. p.190 of Petrovskii 1938 [@Petrowsky_1938]) $$\vert p-n \vert \le \textstyle\frac{3m^2-6m}{8}+1.$$ As $m=2k$ and $\chi=p-n$, this gives indeed $$\vert \chi \vert\le 3
\textstyle\frac{(2k)^2-2(2k)}{8}+1=\textstyle\frac{3}{2}(k^2-k)+1
=\textstyle\frac{3}{2}k(k-1)+1,$$ which is essentially the announced bound modulo a discrepancy on the lower-bound by one unit. In fact we copied the stated lower bound from Rohlin 1978 and hope that there is no misprint there.

Let us look at the example $k=3$ of sextics. Then $\frac{3}{2}
k(k-1)=\frac{3}{2} 3\cdot 2=9$, and so $-9\le \chi $, hence even the stronger version written down by Rohlin 1978 (also in Wilson 1978, p.55) does not prohibit “Rohn’s scheme” $\frac{10}{1}$ (cf. Gudkov’s Table=Fig.\[Gudkov-Table3:fig\]).

Applied to our situation $k=5$, Petrovskii’s theorem shows that $\chi \le \frac{3}{2} 5\cdot 4+1=31$ and so our scheme with $\chi=33$ is prohibited by Petrovskii.

In general, for an even degree $m=2k$ we have Harnack’s bound $M=g+1=(2k-1)(k-1)+1=2k^2-3k+2\approx 2k^2$, the universal Petrovskii’s bounds $-P\le \chi\le P+1$, where $P=\frac{3}{2}
k(k-1)\approx \frac{3}{2} k^2$, and finally Thom’s bound $\chi \le
k^2 $ for dividing curves (only). So when $k$ is large the “Hilbert-Petrovskii-Gudkov” pyramid looks as follows (Fig.\[PyramidPetrov:fig\]), and of course Thom will asymptotically be stronger than Petrovskii (at least for dividing curves and on the right-wing of the pyramid where $\chi$ is positive).

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So I am not sure not adhere completely with Fiedler’s illuminating comment, because if we take a larger $m=2k$ than $10$ then it will be possible to arrange Petrovskii’s bound yet not Thom’s one, while further taking care of respecting Gudkov’s congruence and Rohlin’s formula.

Let us first take $m=12$ (so $k=6$) then Petrovskii’s upper-bound is $P+1=\frac{3}{2} 6\cdot 5+1=46$, while Thom’s is the sharper $k^2=36$. Arranging Gudkov’s hypothesis $\chi\equiv k^2 \pmod 8$ permits to take $36+8=44$ which is still lower than Petrovskii’s upper-bound. Now $M=g+1=\frac{11\cdot 10}{2}+1=56$. Hence to arrange $\chi=44$, we transplant 6 ovals of the unnested configuration at depth 1 (each such move drops $\chi$ by 2 units) to get the (12)-scheme[^85] $(1,6)49$ (with $\chi=44$). By Rohlin’s formula $2(\pi-\eta)=r-k^2=56-36=20$, hence $\pi-\eta=10$. As there are no deep nesting the signs-law is negligible and we merely have $\pi+\eta=6$, so that $2\pi=16$, whence $\pi=8$ and we get an obstruction.

\[17.03.13\][*Optional reading (skip if you do not want to loose the main-flow and move to $\clubsuit\clubsuit$)*]{}.—This argument extends to the following formal consequence of Rohlin (probably subsumed to Thom, yet much more elementary).

\[Rohlin-consequence-for-M-curves:lem\] An $M$-curve of degree $2k$ and of type $\frac{x}{1}y$ with few nested ovals in the sense that $x<\frac{(k-1)(k-2)}{2}$ is prohibited by Rohlin’s formula. In particular there is no unnested $M$-curve provided $k\ge 3$.

First recall that Harnack’s bound is $r=M=g+1=
(2k-1)(k-1)+1=2k^2-3k+2$. By Rohlin’s formula $2(\pi-\eta)=r-k^2=k^2-3k+2=(k-1)(k-2)$. So $\pi-\eta=\frac{(k-1)(k-2)}{2}=:\binom{k-1}{2}$. But $\pi+\eta=x$, so that $2\pi=x+\binom{k-1}{2}$, and hence $\pi=(x+\binom{k-1}{2})/2$. Yet the equation $\pi+\eta=x$ is impossible whenever $\pi>x$, that is when $\pi=(x+\binom{k-1}{2})/2>x$, i.e. as $(x+\binom{k-1}{2})>2x$, so when $x<\binom{k-1}{2}$, which is the asserted condition.

$\clubsuit\clubsuit$ \[18.03.13\] Back to our main object of the (12)-scheme $(1,6)49$, our idea is to remove this Rohlin obstruction by injecting more freedom gained by transferring some ovals at depth 2 (leaving thus $\chi$ unchanged). So starting from the (12)-scheme $(1,6)49$, whose tree is depicted as Fig.\[Fied3:fig\]a, we transplant ovals at depth 2 to get Fig.b with a certain quantity $x+y$ of ovals at depth 2. By Rohlin’s formula $\pi-\eta=10$, and from Fig.b we have $\pi+\eta=6+2(x+y)$. Adding the last equations gives $2\pi=16+2(x+y)$, whence $\pi=8+(x+y)$. The condition $\pi\le
\pi+\eta$ becomes so $8+(x+y)\le 6+2(x+y)$, i.e. $x+y\ge 2$, which is necessary to solve Rohlin’s equation.

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Now introduce signs as on Fig.b by putting $a$ many pluses on the 5 edges at depth 1 and $b$ many minuses (so $a+b=5$), and a $+$-sign on the trunk ([*Warning.*]{}—This choice is fatally bad as we shall see, and a minus sign is more fruitful as we shall experiment soon). Applying the signs-law (cf. Fig.\[Signs-law-dyad:fig\]) to Fig.b we find by sorting out contribution according to their edge-length (which appears underbraced) $$\pi-\eta=\underbrace{a-b+1+x-y}_1+\underbrace{(-x+y)}_2=a-b+1.$$ But as $\pi-\eta=10$, we deduce the system $a-b=9$, $a+b=5$, whence $2a=14$ and $a=7$ which is incompatible with $a+b=5$.

(Then we repeated such calculation in higher degrees $14, 16$ finding always the same obstruction, though for $m=16$ there are even two possible values of $\chi$ permissible under Gudkov and Petrovskii).

However if we take a $-$-sign on the trunk (Fig.c), then the signs-law gives $$\pi-\eta=\underbrace{a-b-1+x-y}_1+\underbrace{(x-y)}_2=
a-b-1+2(x-y).$$ As by Rohlin we still have $\pi-\eta=10$, this gives the system $a-b=11-2(x-y)$, $a+b=5$, whence $2a=16-2(x-y)$, i.e. $a=8-(x-y)$. Let us fix $x-y=3$, so that $a=5$, $b=0$. Since $x+y\ge 2$ (cf. necessary condition discussed few lines above), we may choose $x=3$ and $y=0$. It is worth at his stage checking that the resulting signs distribution indeed solves Rohlin’s equation, and we have proven:

\[Alsatian-schemes:thm\] There exists an Alsatian scheme, i.e. a French scheme which furthermore respects Petrovskii’s inequalities. More precisely, there is an $M$-scheme of degree $12$ for instance $(1, 5(1,3))46$ (cf. [Figs.\[Fied3:fig\]d,e]{}) which respects both the Petrovskii bound, the Gudkov congruence and Rohlin’s formula, yet which is prohibited by Thom’s bound.

[*Insertion*]{}.—\[20.03.13\] Thomas Fiedler was kind enough to object (once more) to this result as follows (cf. his message dated 19 March 2013 in Sec.\[e-mail-Viro:sec\], yet reproduced here for convenience):

“sorry again, but your curve has $p=50$ and is ruled out by Arnold’s inequality : $p\le 3/2 k(k-1) + 1 + n_-$[^86], which is $47$ in this case. In fact Arnold’s inequalities are by fare the strongest result in the whole field.”

We will try to react to this objection right below the proof, cf. $\bigstar\bigstar$ below.

The assertion is clear by our search, but as it is easy to make mistakes, let us do an ad hoc self-contained verification. The scheme in question is depicted on Fig.\[Fied3:fig\]e. It has $\chi=(1-6+3)+46=44$. Petrovskii’s inequalities says $-P\le\chi\le P+1$, where $P:=\frac{3}{2}k(k-1)=\frac{3}{2}6\cdot 5=45$, and this is satisfied by our scheme. Gudkov’s congruence $\chi\equiv_8
k^2=6^2=36$ is also verified. Finally Rohlin’s formula is fulfilled when like on Fig.\[Fied3:fig\]d all signs are positive safe that on the trunk at depth 1 which has further ramifications at depth 2. (The depth of an edge is that of the unique vertex below it.) In that case the signs-law gives $$\pi-\eta=\underbrace{5-1+3}_1+\underbrace{3}_2,$$ where contributions are underbraced along the length of the pairs. Hence $\pi-\eta=10$ in accordance with Rohlin’s formula $2(\pi-\eta)=r-k^2=56-6^2=20$. However the scheme in question is prohibited by Thom’s estimate $\chi\le k^2=36$.

It should be easy to extend the result to other schemes but it looks artificial to strive toward maximum generality as our purpose was merely to find an example where Thom affords valuable information.

$\bigstar\bigstar$ [*Trying to fix Fiedler’s new objection based on Arnold’s strong Petrovskii inequalities*]{}.—The result mentioned by Fiedler is the following sometimes called the [*strong Petrovskii inequalities*]{}. Those are really due to Arnold 1971 [@Arnold_1971/72], and sharper than Petrovskii’s original inequalities of 1933/38. Apparently (cf. Rohlin 1978 [@Rohlin_1978], p.87, footnote), Arnold’s original statement contained further unnecessary restrictions that were relaxed in Rohlin 1974 [@Rohlin_1974/75]. The final shape of the result is as follows:

\[Strong-Petrovskii-Arnold-ineq:thm\] [(Strong Petrovskii inequalities, aka Arnold inequalities).—(Arnold 1971, Rohlin 1974)]{}.—For any curve of even degree $m=2k$, $$n-p^- \le \textstyle\frac{3}{2} k(k-1), \qquad p-n^- \le
\textstyle\frac{3}{2} k(k-1)+1,$$ where $p^-, n^-$ are the number of positive=even resp. negative=odd ovals which are hyperbolic (cf. [Definition \[hyperbol-ovals:def\]]{} right below).

\[hyperbol-ovals:def\] $\bullet$

An oval of a plane real algebraic curve (or a scheme=distribution of ovals) is [*elliptic*]{}, [*parabolic*]{} or [*hyperbolic*]{}[^87] depending on whether its poros[^88] (cf. below) has positive, zero or negative Euler characteristic[^89].

$\bullet$ The [*poros*]{} of an oval of a plane curve $C_m(\RR)$ is the inside of the oval minus the insides of all ovals immediately nested in the given one (equivalently remove the insides of all subordinated ovals).

An oval is hyperbolic iff the Hilbert tree of the scheme ramifies at the corresponding vertex. This basic remark is the key to the little problem suggested by Fiedler’s objection.

Now our Alsatian question is again: is there a scheme where Thom is stronger than the conjunction of strong-Petrovskki=Arnold, Gudkov’s hypothesis and Rohlin’s formula. Our basic algorithm to do this is always same:

\(1) Start from the unnested configuration of $M$-ovals.

\(2) Then adjust $\chi$ to $k^2$ as to verify Gudkov’s hypothesis. (This can be done by transplanting outer ovals at depth 1, dragging them say inside a fixed oval.)

\(3) Then Thom’s estimate $\chi\le k^2$ is verified, but corrupt it by incrementing $\chi$ by 8. Do this as many times as Petrovskii’s bound $\chi\le \frac{3}{2}k(k-1)+1$ permits.

\(4) Next without changing $\chi$ transplants outer ovals at depth 2 (as many as you want), while introducing a branched structure making Rohlin’s equation soluble. (This being essentially inspired by our counter-example to the Garidi mass conjecture discussed above.)

Let us be more explicit. Suppose $k=6$ (so $m=2k=12$). Harnack’s bound is $M=g+1=\frac{11\cdot 10}{2}+1=55+1=56$. Adjust to $\chi=k^2=36$ (Gudkov) and increment by 8 to get $44,52$. Petrovskii 1938 says $\chi \le \frac{3}{2}k(k-1)+1=3\cdot 3\cdot
5+1=46$. So we consider $\chi=44$. Hence we transplant from the unnested scheme $M=56$, precisely $(M-\chi)/2=(56-44)/2=6$ ovals at depth 1 to get adjusted at $\chi=44$. This gives the scheme on Fig.\[Fied4:fig\]a with symbol $(1,6)49$. This scheme is prohibited by Rohlin’s formula $2(\pi-\eta)=r-k^2=56-36=20$, hence $\pi-\eta=10$, while $\pi+\eta=6$. This is of course soluble as $\pi=8$ and $\eta=-2$ (ruled out as $\eta\ge 0$ is a cardinal!). Variant: imagine signs on the edges of the tree (of Fig.a) we can have at most 6 pluses and so $\pi-\eta\le 6$ hence cannot be $10$. To arrange Rohlin the idea is to transplant ovals at depth 2 creating thereby more freedom to solve Rohlin’s equation.

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The extra challenge is to take care of the (strong) Petrovskii-Arnold estimate. Note first that our configuration $(1,6)49$ has $p=50$ and this will not change under transfers at depth 2. Hence to respect Arnold’s estimate $$p\le \textstyle\frac{3}{2}k(k-1)+1+n^-=46+n^-,$$ it suffices to arrange $n^-=4$. Remember that $n^-$ counts the number of hyperbolic negative(=odd) ovals, so we just have to transplant ovals at 4 different places of Fig.a to get a tree like Fig.b with at least two extra branches growing at each 4 places. To ensure hyperbolicity it is sufficient to have branchings of “order 2”, so we look to Fig.c where the 3 first branches ramifies by 2, while the fourth by a magnitude $x+y$ (yet undetermined) safe for $x+y\ge 2$. On the left of the tree (still Fig.c), we put $+$-signs at $a$ many places (w.l.o.g. on the “left” though this has no intrinsic meaning here), and $-$-signs at $b$ many places. Hence $a+b=2$. Likewise on the right of the tree we introduce $x$ many $+$ resp. $y$ many $-$ as depicted on Fig.c. On the “center” of the tree where we have 3 branches like “Y”-letters inverted, we plug everywhere $-$-signs as those depressive guys are simplest to calculate by the signs-law ($-\times -=-$). This rigidification looks a reasonable Ansatz for there should be already enough free parameters available with $x,y$ and $a,b$ (subjected to $a+b=2$). Once the combinatorics of the tree is fixed and the signs-distribution too (modulo the free parameters) we can compute the Rohlin mass $\pi-\eta$ according to the signs-law. First note that each 3 central subtrees (“Y-shaped”) contributes for 5 pairs (3 visible, plus 2 concatenation) each being negatively charged. Hence the contribution of each such subtree is $-5$. Globally on the whole tree, we find therefore (upon remembering the signs-law (\[Signs-law:lem\]) saying that mixing the genes is good so $+\times -=+$, while consanguinity is bad, e.g. $-\times -=-$) $$\pi-\eta=a-b-15-1+2x-2y=a-b-16+2(x-y).$$ By Rohlin’s formula $\pi-\eta=10$, and we get the system $a-b=26-2(x-y)$, $a+b=2$. Hence $2a=28-2(x-y)$, i.e. $a=14-(x-y)$. Choose (freewill vs. predestination!) $x-y=12$, so that $a=2$, $b=0$, and choose again $x=12$, and $y=0$ as a special solution. This proves the:

\[Alsatian-scheme-Thom-strong-Petrov-Arnold:thm\] There exists an Alsatian $M$-scheme where Thom is stronger than the conjunction of (strong) Petrovskii-Arnold 1971, Gudkov hypothesis 1969–72 (proved by Rohlin-Marin), and Rohlin’s formula. Specifically, there is such a scheme in degree $12$, namely the one allied to [Fig.\[Fied4:fig\]c]{} for $x+y=12$, whose scheme is depicted on [Fig.d]{} called “René Thom sur son $31$” since the nested portion of the tree involves (counting along increasing depths) $p_0+n_1+p_2=1+6+18=25$ ovals so that it remains left $56-25=31$ outer ovals. The $M$-scheme in question has Gudkov symbol $(1,2(1,2)(1,2)(1,2)(1,12))31$.

Let us do again an ad hoc verification. From the scheme (Fig.c with $x+y=12$ or Fig.d) we have $\chi=1-6+18+31=44$. So Gudkov $\chi\equiv_8 k^2$ is happy. Petrovskii 1938, i.e. $\chi \le \frac{3}{2}k(k-1)+1=45+1=46$ is also satisfied. Now the strong version of Petrovskii due to Arnold, reads $$p-n^-\le \textstyle\frac{3}{2}k(k-1)+1=46.$$ But our scheme has (cf. Fig.c or d) altogether $r^-=5$ hyperbolic ovals (i.e. containing immediately at least 2 other ovals in their insides), yet only 4 of them are at odd depth 1, so $n^-=4$. Though equivalent, working with the tree (as opposed to the scheme) looks more convenient to see this (at least when one is tired). On the other hand our scheme has either by its construction (cf. Fig.a) or by counting $p=50$, since $p=p_0+p_2=(1+31)+(6+12)=32+18=50$. Hence the Petrovskii-Arnold estimate is verified.

Note also that the other strong Petrovskii inequality $n-p^-\le
\frac{3}{2}k(k-1)=45$ is verified, as $n=6$ (either from Fig.c or from $p+n=r=M=56$, where $r$ is as usual in our notation the number of “reellen Züge”, denoted $l$ in Russia). Although not needed it may be noted that $p^-=1$ either via Fig.c or via the relation $p^- + n^-=r^-$ (splitting hyperbolic ovals according to their parities).

Finally Rohlin’s formula is verified for the signs-distribution of Fig.c where $x=12$, $y=0$ and $a=2$. Indeed the signs-law gives $\pi-\eta=2-3\cdot 5+(-1)+12+12=2-16+24=10$, in accordance with Rohlin’s formula $2(\pi-\eta)=r-k^2=56-36=20$. The verification is complete.

Old material (to skip or reorganize)
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[*Sequel of my text (prior to Fiedler’s objection(s) via Petrovskii, and then via Arnold)*]{}.—One can paraphrase the statement (\[French-scheme-corrected:thm\]) by saying that there is a complex scheme (i.e. with orientation) which satisfies Rohlin’s formula, but which is not realized algebraically (being ruled out by Thom’s $\chi\le k^2$). Our example is of degree 10, and is simply the complex scheme associated to the Rohlin tree of Fig.\[Garidi-mass-false:fig\]f for $(x,y)=(19,15)$, hence is representable as the complex scheme of Fig.\[Garidi-mass-false:fig\]g. In fact such schemes already exist in degree 6 (cf. optionally Theorem \[no-chance-to-reduce-Gudkov-to-Rohlin:thm\]), where it is just a matter of solving Rohlin’s equation for the two $M$-schemes of degree 6 which are prohibited by Gudkov’s hypothesis.

Of course the example proposed probably belongs to a larger list of such French schemes. More about this soon. On the other hand it could be nice to know if there is a French scheme in degree 8 already.

[*Degree $m=8$*]{}.—Then the Harnack bound is $M=g+1=\frac{7\cdot 6}{2}+1=22$. Applying the same method, we start with the $M$-scheme $(1,1,20)$ with $\chi=20$. This has to be adjusted to the Gudkov hypothesis $\chi\equiv_8 k^2=16$, but then Thom’s inequality $\chi\le k^2$ is satisfied, except if we could move up to $\chi=24$ but this violates the basic estimate $\chi\le M$. \[Proof: $\chi=p-n\le p+n=r\le M$ by Harnack.\] So it seems that there is no French in degree 8, but a more systematic study is required. As a loose evidence for the absence of French scheme in degree 8, we note that the RKM-congruence for $(M-2)$-schemes $\chi\equiv k^2+4 \pmod 8$ (ensuring type I) forces under Thom’s inequality ($16\le \chi \le M=22$ and Harnack’s bound) to have $\chi=20$. But then $p-n=20$ and $p+n=r=20$, so that $2p=40$, hence $p=20$ and $n=0$. So our scheme is forced to be unnested and is $20$, which is prohibited by Rohlin’s formula.

[*More French schemes in degree $10$*]{}.—By the above method we now proceed to find more French schemes. The idea is merely that starting from the scheme on Fig.d we may move innermost ovals at depth 2 outside at depth $0$ without changing $\chi=33$ (as forced by the Gudkov hypothesis). So we consider a scheme whose tree is like Fig.\[Fiedler3:fig\]a where there are $z$ outer ovals which are empty. We introduce $x$ and $y$ many free signs plus and minus resp. as on Fig.\[Fiedler3:fig\]a, with both top signs negative. To get an $M$-scheme we impose $x+y+z=34$. As before we seek to solve the Rohlin’s equation. Recall that we abridge $\pi:=\Pi^+$, $\eta:=\Pi^-$. By Rohlin’s formula $2(\pi-\eta)=r-k^2=37-25=12$ so $\pi-\eta=6$. But on the other hand by the signs-law, we have (by looking at Fig.a) $$\pi=2x \qquad \textrm{and} \qquad \eta=2+2y.$$ If $z$ is given (a priori in the range $0\le z \le 34$, but we shall soon see that some more constraint are required), we solve in $x,y$ as to satisfy Rohlin’s formula. This gives $\pi-\eta=6$ and $\pi+\eta=2(1+x+y)$, so adding $2\pi=8+2(x+y)$, hence $\pi=4+x+y=38-z$. So $\eta=32-z$. Finally we find $x=\frac{\pi}{2}=19-\frac{z}{2}$. This requires so the assumption $z$ even ($\bigstar$!!!), which looks anomalous but more about this soon. And finally $y=\frac{\eta-2}{2}=\frac{30-z}{2}$, so that we must assume $z\le 30$. ($\bigstar$ HYPOTHESIS to add!)

Now when $z$ is odd we proceed similarly, but the trick is to change one of the top sign as on Fig.b into a plus. We still have $x+y+z=34$, but now by the signs-law applied to the new diagram (Fig.b): $$\pi=1+2x \qquad \textrm{and} \qquad \eta=1+2y.$$ By Rohlin’s formula we still have $\pi-\eta=6$, and now $\pi+\eta=2(1+x+y)$ (actually like above!) and so repeating the above $\pi=38-z$, $\eta=32-z$. Solving gives $x=\frac{\pi-1}{2}=\frac{37-z}{2}$ and $y=\frac{\eta-1}{2}=\frac{31-z}{2}$. So we assume $z\le 31$ ($\bigstar$ Hypothesis!).

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All this should prove the admittedly insignificant following result:

\[French-scheme-corrected-bis:thm\] For any integer $0\le z \le 31$, the $M$-scheme $(1,1(1,34-z))z$ (compare Fig.\[Fiedler3:fig\]c) is a French scheme, i.e. it is prohibited by Thom but not by Gudkov’s hypothesis nor by Rohlin’s formula.

From the proof $z=31$ is sharp with this property, but it is perhaps tranquilizing to check this more experimentally. Then the tree reduces to Fig.d, and the total number of pair is $\Pi=4+2=6$, but as $\pi-\eta=6$ (by Rohlin’s formula) we are forced to have only positive pairs, but this is impossible by the signs-law (since in Rohlin’s arithmetics $+$ times $+$ is minus!)

What to do next? A naive game would be to classify all French scheme in degree 10. Another more serious problem would be to detect some universal rule as to understand better the prohibitions given by Rohlin’s formula, or the lack thereof when the Hilbert tree can be given a sign distribution so that Rohlin’s formula is satisfied. All this looks a bit unappealing combinatorics, yet the lack of conceptualization in our above account surely ask for a better understanding of the Rohlin tree. One would like to understand all Rohlin trees of dividing curves. Any such must satisfy the Rohlin formula, but as we saw this is not the sole obstruction (as sometimes Thom imposes additional constraints).

As a more specific goal we could try to find a French $(M-2)$-scheme in degree 10. As noted earlier this is impossible to do on the “planar” face of the Gudkov pyramid (cf. our Fig.\[Degree10:fig\]). There Rohlin’s formula is always as strong as Thom. (This follows also from the truth of the Garidi mass principle for such simple schemes. We leave as a loose-end exercise to exhibit classes of schemes for which the mass conjecture holds true, albeit disproved in general for “batônnet” like schemes, cf. Fig.\[Garidi-mass-false:fig\].)

We now proceed to find $(M-2)$-schemes of degree 10 where Thom is stronger than Rohlin’s formula. The method is similar as above, but we repeat the detail by unifying somewhat the proof. The batônnet structure of schemes violating the mass conjecture suggests looking at the scheme $(1,1,33)$ (cf. Fig.\[Fiedler4:fig\]a). Here $\chi=33$, but in order to apply Thom we have to ensure type I, and the best known recipe to do this is to adjust to the RKM-congruence $\chi\equiv k^2+4=25+4=29
\pmod 8$. So as $\chi=p-n\le p+n=r=35$, we cannot move up to 37, but instead lower down $\chi=33$ to $29$. This is achieved by delocalizing two ovals at depth 2 toward ovals at depth 1, and we get the scheme $(1,2(1,31))$ (cf. Fig.\[Fiedler4:fig\]b).

The scheme $(1,2(1,31))$ and more generally its companions $(1,2(1,31-z))z$ (cf. Fig.\[Fiedler4:fig\]c) are by RKM of type I, but prohibited by Thom $\chi\le k^2=25$. However, provided $z\le 29$, all these schemes are not prohibited by Rohlin’s formula.

As above we prove that there is a distribution of signs on the Hilbert tree of the scheme compatible with the signs-law and with Rohlin’s formula. For this we consider the diagram of Fig.\[Fiedler4:fig\]d, where we have free parameters $x,y$ counting the number of positive resp. negative edges at depth 2. We introduce also $\epsilon, \delta$ counting positive resp. negative signs on the only 2 available edges at depth 1 (lacking prolongation). So $\epsilon+\delta=2$. The edge at depth 1 prolonging to depth 2 is given the sign $-1$. Further we have $z$ isolated vertices at depth 0 corresponding to the outer empty ovals of the scheme of Fig.c. We have thus $x+y+z=31$. .

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By Rohlin’s formula $2(\pi-\eta)=r-k^2=35-25=10$, we find $\pi-\eta=5$. By applying the signs-law to Fig.d, we find $$\pi=2x+\epsilon \qquad \textrm{and} \qquad \eta=2y+\delta+1.$$ So $\pi+\eta=2(x+y)+3$, and thus $2\pi=8+2(x+y)$, whence $\pi=4+(x+y)=35-z$ and $\eta=\pi-5=30-z$. So we solve $$x=\frac{\pi-\epsilon}{2}=\frac{35-z-\epsilon}{2} \quad
\textrm{and} \quad
y=\frac{\eta-\delta-1}{2}=\frac{29-z-\delta}{2}.$$ It remains just to ensure integrality by choosing appropriately the free parameters. Specifically,

$\bullet$ if $z$ is odd, then for $x$ to be integral choose $\epsilon$ even (and then $\delta$ is even and so $y$ as well, but sorry this parenthetical stuff is automatic!);

$\bullet$ if $z$ is even, choose $\epsilon$ odd.

Note that $z\le 29$ (to ensure $y\ge 0$), while for $z=29$ the system of equations is soluble for $\delta=0$, $y=0$ and $\epsilon=2$, whence $x=\frac{35-29-2}{2}=2$. (Optionally, one can check diagrammatically that Rohlin’s formula is verified on this extremal example, cf. Fig.e. where we have $\pi=4+2=6$ and $\eta=1$, so $\pi-\eta=5$ as it should by Rohlin.)

What is the moral of all these messy calculations with this exotic signs-law of Rohlin? Is there some respectable way to extract a general result. A vague moral is that it is always quite boring to apply Rohlin’s formula and it is quite easy to make mistakes, as was pointed out by Fiedler. Perhaps some higher intelligence than me is able to discern some order in this chaos of Rohlin trees and proves valuable corollaries of Rohlin’s formula. However it is also clear that the latter has some limitation in failing to prohibit schemes ruled out by Thom.

In the same spirit we wondered (earlier in this text, Sec.\[Gudkov-hyp-via-Rohlin’s-formula:sec\]) if Rohlin’s formula could imply the Gudkov hypothesis. Perhaps a variant of the method used right above can detect a counterexample, i.e. a scheme prohibited by Gudkov but not by Rohlin. Actually we have candidates in degree 6 already, namely the schemes $\frac{7}{1}3$ and its mirror $\frac{3}{1}7$. Taking the first its diagram-tree is like on Fig.\[Fiedler4:fig\]f, where we have $x$ many positive edges and $y$ many negative ones. Of course $x+y=7$. Since here we have no adjacent edges to compose, and we simply have $\pi=x$ and $\pi=y$ and Rohlin’s equation $2(\pi-\eta)=r-k^2=11-9=2$ is trivial to solve. Indeed, $x-y=1$ and $x+y=7$ gives $2x=8$, whence $x=4$ and $y=3$. This proves the:

\[no-chance-to-reduce-Gudkov-to-Rohlin:thm\] There is no chance to reduce Gudkov’s hypothesis to Rohlin’s formula, unless one is able to put more stringent restriction on the complex orientations (via geometric procedures).

$\bigstar$—[*Old stuff, pre-Fiedler’s correction, hence to be read with discernment.*]{}—Several questions arise:

1.—On working out more carefully the combinatorics, try to understand if a French scheme already exists in degree 8.

2.—It seems that as the degree increases French schemes will be more and more frequent and so the rôle of Thom increases more and more and becomes a valuable complement to the Russian congruences, and Rohlin’s formula.

3.—Glancing at the Gudkov table in degree 10 (Fig.\[Degree10:fig\]), we see that the famous Gudkov broken line is subjected to a severe deformation. More precisely, the $M$-schemes $\frac{2}{1}34$ is prohibited by either Rohlin’s formula or by Thom \[or even by Petrovskii’s inequality (\[Petrovskii’s-inequalities:thm\])\], and so is the $(M-2)$-scheme $\frac{3}{1}31$ for the same reasons plus (the full punch of) the RKM-congruence (\[RKM-congruence-reformulated:thm\]). The diagrammatic consequence is a distortion of the Gudkov front-line which is penetrated from above to become the lilac line on Fig.\[Degree10:fig\].

Hence directly below $\frac{3}{1}31$ we have 2 schemes (whose symbols are not even worth writing down), and which according to the diagrammatic of Fig.\[Degree10:fig\] are not maximal schemes. In contrast below the scheme $\frac{2}{1}34$ we have the two schemes $\frac{2}{1}33=:S_1$ and $\frac{1}{1}34=:S_2$. Those guys are interesting birds because they are maximal (at least in the planar model of the Hilbert-Gudkov pyramid, as follows from Rohlin’s formula), yet they are certainly not of type I (by Klein’s congruence $r\equiv_2 g+1$ of 1876). So there is some little opportunity here to corrupt Rohlin’s maximality conjecture (RMC). Of course as all what we are using here was well-known to Rohlin, our expectation will certainly quickly turn to disillusion.

[*Insertion.*]{} \[19.03.13\]—Apart from the fact that (as discussed in the sequel) it is hard to prohibit all extensions of any one of those schemes $S_1,S_2$, a sharper look at the Russian architecture of Fig.\[Degree10:fig\] shows that both those schemes are simply ruled out by Petrovskii’s inequality (\[Petrovskii’s-inequalities:thm\]) telling us that $\chi\le
\frac{3}{2}k(k-1)+1=\frac{3}{2}5(5-1)+1=3\cdot 5\cdot 2+1=31$.

A first question is whether those two $(M-1)$-schemes $S_1,S_2$ are realized[^90]. If so, then it remains to check that they are maximal. So we have to list all their possible enlargements (extensions).

Let us do this for $S_2=\frac{1}{1}34$ (which looks more appealing as it is “less nested”), we have (up to isotopy) 4 possible extensions depending upon the additional oval is added:

\(1) outside the nest, which option leads to $\frac{1}{1}35$, which is prohibited by the Gudkov-Rohlin congruence;

\(2) inside the nest at depth 1, which option leads to $\frac{2}{1}34$, which is prohibited by Rohlin’s formula (or Thom);

\(3) inside the nest at depth 2, which option leads to $(1,1,1)34$. Then $\chi=(1-1+1)+34=35\neq k^2=25 \pmod 8 $ so that the scheme is prohibited by the Gudkov-Rohlin congruence, while one can even note that its forerunner namely the Arnold congruence mod 4 suffices;

\(4) inside an outer oval, which option leads to $\frac{1}{1}\frac{1}{1}33$. Then $\chi=(1-1)+(1-1)+33=k^2=25\pmod
8$ so that Gudkov-Rohlin is not violated. However Thom’s inequation is violated, and so this scheme is not realized. As to Rohlin’s formula we have $\Pi=2$, and so $2(\Pi^+-\Pi^-)=r-k^2=37-25=12$ cannot be verified as $(\Pi^+-\Pi^-)\le(\Pi^++\Pi^-)=\Pi$. Intuitively we see that when $\chi$ is large Rohlin’s formula forces some nesting.

At this stage we have clearly explored all possible extensions of $S_2$ and prohibited all of them via the classical congruence (Gudkov-Arnold-Rohlin) or the Rohlin formula.

Yet even that is wrong as the additional oval needs not to be a small one injected as above, but it can also surround other ovals. Hence we have at least the following two families of schemes enlarging $S_2$:

\(5) $(1,\frac{1}{1}x)y$ where $x+y=34$, and

\(6) $\frac{1}{1} \frac{x}{1} y $, where $x+y=34$.

This can of course be much diminished by the Gudkov-Rohlin congruence, as follows. For (5) and $x=0$, we have the primitive scheme $(1,1,1)34$ with $\chi=(1-1+1)+34=35$ which 2 unit above $33=25 \pmod 8$. Hence by trading outer “$y$” ovals against “$x$” ovals at depth 1 we decrease $\chi$ by 2, so that the first scheme with correct $\chi$ is $(1,\frac{1}{1}1)33$, and then the list extends by using 4-fold periodicity as $$(1,\frac{1}{1}1)33, \quad, (1,\frac{1}{1}5)29, etc.$$ with resp. $\chi=33, 25, etc.$ (descent by 8). Hence the first is still prohibited by Thom, yet the subsequent schemes are unlikely to be prohibited by Thom nor by Rohlin’s formula. At this stage our naive project to corrupt Rohlin’s maximality conjecture breaks down. For instance the second listed Thom compatible scheme $(1,\frac{1}{1}5)29$ has $\Pi=\binom{3}{2}+5=8$ and so Rohlin’s formula $2(\Pi^+-\Pi^-)=r-k^2=37-25=12$ is soluble for the pair $\Pi^{\pm}=(7,1)$.

At this stage the only vestiges of expectance to foil RMC via our naive strategy would be that all those extended schemes are prohibited by a degree-10-extension of the Fiedler-Viro theorem (cf. Theorem \[Viro-Fiedler-prohibition:thm\] in degree 8), but that looks a dubious expectation.

Of course it is much more likely that either Harnack, Hilbert or Viro’s method of construction realize one of those schemes, and so that our scheme $S_2$ (and likewise $S_1$) are not maximal.

Some intuition behind Hilbert, Thom, Rohlin?
--------------------------------------------

\[12.03.13\] The theory of abstract real algebraic curves is essentially a closed chapter of mathematics. More precisely since Riemann 1857 [@Riemann_1857] paper on Abelian integrals, we have a clear-cut vision of all complex curves and their moduli. Avatars thereof in the bordered setting or even nonorientable realm were worked out by Klein 1882 [@Klein_1882] (upon the heritage of Gauss, Möbius, Listing the discoverers of “il nastro di Möbius”). Teichmüller 1939 [@Teichmueller_1939] put the nearly final touch by approaching the moduli problem via quasiconformal maps along the philosophy developed by Grötzsch 1928, Lavrentieff 1929, Ahlfors ca. 1930–35. Notwithstanding our thesis is that from the viewpoint of total reality (or what amounts to the same branched covers of the disc), some quantitative aspects have not yet attained their ultimate perfection and sharpness, though it may be only a matter of assimilating the heritage of previous generations. This is discussed at length in the Introduction of this text, but briefly we may recall that in the closed case the ultimate perfection regarding Riemann surfaces expressed as branched cover of the line(=Riemann sphere) is achieved in Riemann 1857, Brill-Noether 1874 [@Brill-Noether_1874], and especially Meis 1960 [@Meis_1960] (which I had never the occasion to consult). When it comes to bordered surfaces (equivalently orthosymmetric real curves), the first steps belongs to Riemann 1857 (Nachlass) [@Riemann_1857], Schottky 1875–1877, Bieberbach 1925, Grunsky 1937, Ahlfors 1950, etc., up to perhaps Gabard 2004 [@Gabard_2004], 2006 [@Gabard_2006], whose result still deserves better introspection, but whose sharpness is adhered upon (perhaps too hastily) in several works (e.g. Fraser-Schoen 2011 [@Fraser-Schoen_2011], Coppens 2011 [@Coppens_2011]. Actually Coppens’ result although adhering to the truth of Gabard’s is so logically independent that its truth may not be jeopardized by a disproof of Gabard’s bound $\gamma\le
r+p$.

Next we may move in the Plato cavern of plane curves and contemplate the so-called [*Hilbert’s problem*]{} on the topology of real plane algebraic curves. Here most of the difficulty is allied to a certain combinatorial mess arising from the nested structures of ovals and their distribution. The whole point is to look at a certain hierarchical structure (POSET) arising from the inclusion between the insides of ovals. We call it the [*Hilbert tree*]{}.

As early as 1891, Hilbert was the first to formulate the intuition that this Hilbert tree has a certain verticality, in the sense that an algebraic curve cannot reduce to an unnested collection of ovals. He formulates this for $M$-curves of degree 6, and of course the assertion holds only for maximal curves (cf. the Gudkov table=Fig.\[Gudkov-Table3:fig\] or for a specific example Fig.\[KleinRo-sextic:fig\]). The Hilbert tree of such an unnested scheme (Gudkov’s symbol $M$, where $M=g+1$ is Harnack’s bound) just reduce to a collection of vertices at depth 0, without nesting (hence without edges).

At some stage Thom had the following intuition.

[(Thom conjecture, Kronheimer-Mrowka theorem 1994, and independently Morgan-Szabó-Taubes 1995/96 [@Morgan-Szabo-Taubes_1996])]{}.—The genus of a smooth embedded oriented surface in $\CC P^2$ is at least as large than that of “the” algebraic smooth curve realizing the same homology class, alias the degree.

This prompts a wide extension of Hilbert’s verticality principle \[or nesting Ansatz\] that the Hilbert tree cannot be to flat. This is alas a bit like in human feodal systems where a certain concentration of power, slavery and subjection is observed. Precisely this is given by Theorem \[Thom-Ragsdale:thm\] stating that any dividing curve of degree $m=2k$ satisfies $\chi \le k^2$. As the degree increases to 8, or 10, etc. this prohibits more and more schemes on the right wing of the Gudkov pyramid (cf. e.g. Fig.\[Degree10:fig\]), yet to some extend those schemes can also be more elementarily ruled out by Rohlin’s formula.

Since the Harnack bound is $M=g+1=\frac{(2k-1)(2k-2)}{2}+1=(2k-1)(k-1)+1=2k^2-3k+2$, the asymptotic location as $m=2k\to \infty$ of Thom’s bound $k^2$ \[ALAS INCORRECT, MY MISTAKE CORRECTED BY FIEDLER\] is the half value of Harnack’s bound $M\approx 2k^2$, and so nearly one quarter of the $M$-curves are prohibited by Thom (cf. e.g. Fig.\[Degree10:fig\]). However this figure only represents schemes with one nonempty oval so that the real pyramid is much less amputated than its planar sheet leads one to suspect.

On the other hand the Hilbert hierarchies cannot be too deep. Indeed one cannot observe in degree $m$ a nested chain of ovals of depth larger than $k=m/2$ as follows directly from Bézout for lines. More generally using conics through 5 points there cannot be schemes with 5 chains of total length larger that $2m$. This is the well-known topics of Hilbert’s bound on the depth of nests.

Further when those deepest configurations are attained then the curve is necessarily of type I via the phenomenon of total reality.

Summarizing we see that the Hilbert tree cannot be too deep (as follows from reality consideration and Bézout) nor can it be too superficial (as follows from the complexification, e.g. Thom’s principle or the Petrovskii inequalities \[Petrovskii’s-inequalities:thm\]). Further the total reality phenomenon ought to play some big rôle as envisioned by Rohlin. Of course all this needs to be made more precise (further explored).

On an arithmetical problem valorizing Thom in the detriment of Rohlin (and sometimes Gudkov): yet another numerical coincidence regarding Hilbert’s 16th
--------------------------------------------------------------------------------------------------------------------------------------------------------

\[13.04.13\] This section (and its title!) is somewhat naive and misleading (as I failed to exploit the full punch of Rohlin’s formula), but diverges to a lovely basic arithmetic problem (on which I have little grasp, but must be very classical, surely Gauss? or probably much older Diophante?). Hence it was not censured but can safely be omitted.

\[16.03.13\] As we said, but repeat it once more, Hilbert arrived ca. 1891 (vgl. Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege]) at the intuition (at least for sextics) that ovals of algebraic curves necessarily exhibit some feodal structure of nesting impeding all the ovals lying outside each others. Is this really true in general? \[OF COURSE, E.G., VIA ROHLIN’S FORMULA, AND FIRST KNOWN TO PETROVSKII 1933/38.\] Of course Hilbert posited this for $M$-curves (of order $m\ge 6$) and nearly proved it only for $m=
6$, yet presumably we are allowed to extrapolate his thoughts. In fact, the original text proceeds more carefully and reads as follows (cf. Hilbert 1891, , Fussnote, S.418, in Ges. Abhandl., Bd.II, Algebra, Invariantentheorie, Geometrie):

“Diesen Fall $n=6$ habe ich einer weiteren eingehenden Untersuchung unterworfen, wobei ich—freilich auf einem au[ß]{}erordentlich umständlichen Wege— fand, da[ß]{} die elf Züge einer Kurve 6-ter Ordnung keinesfalls sämtlich au[ß]{}erhalb und voneinander getrennt verlaufen können. Dieses Resultat erscheint mir deshalb von Interesse, weil er zeigt, da[ß]{} für Kurven mit der Maximalzahl von Zügen der topologisch einfachste Fall nicht immer[^91] möglich ist.”

Hilbert addressed this again, and more generally the question of elucidating the isotopic classification of sextic curves (or even quartics surfaces), into his well-known 16th problem at the Paris Congress of 1900. Circa 7 decades came the deep semi-experimental work of Academician Dimitrii Andreevich Gudkov solving Hilbert’s isotopic problem for sextics curves, though according to Arnold (e.g. 1997/00 [@Arnold_1997/200X-Symplectization-Complexification]) this left the supervising teacher Petrovskii quite dubitative, not to say skeptical). Later V.A. Rohlin made a series of jokes by noticing that his 1952 theorem on the divisibility by 16 of the signature (coined by Weyl 1923, in Analisis Situs Combinatorio written in Spanish with the assistance of his linguist wife) of a spinorial smooth $4$-manifold turns out to imply the Gudkov hypothesis $\chi\equiv k^2 \pmod 8$ for $M$-curves (extrapolating widely the phenomenology observed in degree $6$), where $\chi$ denotes as usual the Euler characteristic of the “Ragsdale” orientable membrane bounding the ovals from “inside”.

In the same elan, Rohlin 1974 (and 1978) wrote down the [*Rohlin formula*]{} $2(\Pi^+-\Pi^-)=r-k^2$ (cf. \[Rohlin-formula:thm\]) which is so fundamental that it seems worth (to save ink) to abridge notation by letting $\pi:=\Pi^+$ and $\eta:=\Pi^-$. When the curve has no nesting this formula implies that $r=k^2$ is a square (whenever the curve is dividing, as it is automatically the case for $M$-curves).

[*Warning.*]{}—\[17.03.13\] The sequel is much ill-posed as I (very shamefully) missed to notice that $r$ is not any square, but that of the semi-degree $k$. Yet since it seems to involve pleasant arithmetics, hence we did not censured it!

Restricting attention to $M$-curves of even degree $m=2k$, we are therefore invited to study the following arithmetical problem as a way to corrupt Hilbert’s feudalistic intuition (forced presence of nesting for $M$-curves of high-degrees $m\ge 6$).

[(Quadrature of the Harnack bound).]{}—Given any integer $k\ge$, set $m=2k$ (interpreted as the order of the curve) and let $M=g+1=\frac{(2k-1)(2k-2)}{2}+1=(2k-1)(k-1)+1=2k^2-3k+2$ be the corresponding Harnack bound. For which values of $k$ is $M$ predestined to be a square, so that Rohlin does not prohibit the unnested $M$-scheme (THIS IS FALSE), and additionally try to arrange the Gudkov congruence $M=\chi\equiv k^2 \pmod 8$.

Assume that there is such an integer $k$ then the unnested $M$-scheme (Gudkov symbol $M$ also!) is not prohibited by Gudkov nor by Rohlin and so constitutes a potential violation of Hilbert’s principle. To kill the suspense, as far as I know this scenario could not have been precluded until the Kronheimer-Mrowka validation of Thom’s conjecture. Recall (from \[Thom-Ragsdale:thm\]) that Thom implies $\chi\le k^2$ and so $\chi$ cannot be as large as $M$ which is asymptotically twice so big (except of course for low degrees $m\le 4$).

So curiously I would say (personal feeling probably foiled due to a lack of Russian wisdoms) that before Thom-Kronheimer-Mrowka 1994 [@Kronheimer-Mrowka_1994] Hilbert’s intuition could have been completely wrong depending upon a resolution of the above arithmetical problem.

Of course I arrived at the problem by modest acquaintance with the geometry of Gudkov’s pyramid which look basically as follows (Fig.\[Pyramid:fig\]):

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Now being as bad in arithmetics than in combinatorics, we propose to tackle the arithmetical problem naively. We want to know first of all when Harnack’s bound is a square. We know that Harnack’s bound evolves like the genus of plane curves as a quadratic function (Gauss freshman calculus at 5 years old) namely $g=g(m)=1+2+3+\dots+(m-2)=\frac{(m-1)(m-2)}{2}$. Here we concentrate upon even degrees $m=2k$ and we may tabulate the values of $M$ by adding a linear progression of increment 4 (cf. Fig.\[Pyramid:fig\] right-side table). Then we compare this with the list of all squares $1,4,9, 16\dots$. Apart from the trivial values $k=1,2$, the Harnack bound is never a square until we reach the value $k=17$ where $M=529$ is the square of $23$. Since $17=16+1$ this is yet another numerological coincidence in Hilbert’s 16th (yet surely not as deep of those of Gudkov-Rohlin). In view of this I naively expected that the next quadrature of Harnack’s bound occurs at $k=2\cdot 16+1=33$. But a simple calculation shows this to be a wrong intuition. What about $k=4
\cdot 16+1=65$, also not good! At this stage we stopped guessing and continued the tabulation by hand, and found the next square Harnack bound at $k=46$, namely $M=\frac{91\cdot 90}{2}+1=91\cdot
45+1=4096$ which is $64^2$.

It is now time looking at the Gudkov congruence. For $k=17$, the Harnack bound $M=529$ (which is also $\chi$ the characteristic of the Ragsdale membrane as we suppose no nesting) reduces modulo 8 to $529\equiv_8 49 $ (after removing $480$) and then to $1$ (after removing 48). On the other hand $k^2\equiv_8 17^2\equiv_8 1^2
\equiv_8 1$. Hence Gudkov’s congruence is satisfied. So the $M$-scheme (with Gudkov symbol $529$) could be a potential counter-example to Hilbert’s intuition, if Thom would not salvage it!

For the next value $k=46$, $\chi=M=4096\equiv_8 96 \equiv_8 16
\equiv_8 0$, whereas $k^2=(46)^2\equiv_8 (-2)^2=4$. So Gudkov hypothesis is not verified and here Hilbert’s intuition is already vindicated by Gudkov (and Rohlin who proved it). Notice that Arnold’s congruence would be not enough not rescue Hilbert.

This is essentially all what we have to say on this problem. Though the arithmetical problem looks attractive, I do not know how to solve it in general. Of course it looks an easy Diophantine equation $$x^2=2k^2-3k+2,$$ of degree 2 but this does not boils down to studying the rational points on a conic (via the usual method of sweeping like for Pythagorean triplets), as we are here really concerned with integral solutions of an equation laking homogeneity. Of course I presume it is a well-known and easy problem of arithmetics, yet actually its solution has little impact upon Hilbert’s sixteenth problem, since the corresponding (unnested) schemes are defacto prohibited by Thom. Still the particular solution exhibited above $(k,x)=(17, 23)$ offers another instance of French scheme where Thom is stronger that the conjunction of Gudkov and Rohlin. I do not if prior to Kronheimer-Mrowka one was able to prohibit the existence of the corresponding $M$-scheme of degree $34$ with $M=529$ unnested ovals.

\[17.03.13\] In fact the answer is a trivial consequence of Rohlin’s formula, since $0=2(\pi-\eta)=r-k^2$, but $r=529$ is not the square of $17$ (as $17^2=289$). More generally, all the problematic of this section is spoiled since I omitted at the beginning of the discussion to notice that Rohlin’s formula does not merely implies in the unnested case that $r$ is the square, but is indeed the square of the semi-degree $k$. In fact as shown, e.g. by Lemma \[Rohlin-consequence-for-M-curves:lem\] it is a trivial consequence of Rohlin’s formula that Hilbert’s nesting intuition occurs for $m\ge 6$. The simplest way to argue is as follows.

\[Hilbert’s-nesting-intuition:lem\] [(Hilbert’s nesting intuition, validated via Rohlin 1974 or via Petrovskii 1938)]{} Any $M$-curve of even degree $m=2k$ with $m\ge 6$ cannot be unnested, i.e. have all its ovals outside another. Actually, a dividing unnested curve is forced to have $r=k^2$ ovals.

A simple way to argue is via Rohlin’s formula 1974 (\[Rohlin-formula:thm\]). We have $0=2(\pi-\eta)=r-k^2$, hence $r=k^2$. But by the $M$-curve assumption $r=M=g+1=(2k-1)(k-1)+1=2k^2-3k+2$, which is strictly larger than $r=k^2$ as soon as $k\ge 3$.

The first clause also follows by specializing Petrovskii’s upper-bound $\chi\le \frac{3}{2}k(k-1)+1=:P$ valid for all curves. It is plain indeed to check that $P<M$ (“Petrovskii is sharper than Harnack”) for $k\ge 3$. Indeed the difference $M-P=(2k^2-3k+2)-(\frac{3}{2}k(k-1)+1)=\frac{1}{2}k^2
-\frac{3}{2}k+\frac{1}{2}=\frac{1}{2}\underbrace{(k^2-3k+1)}$. The roots of the underbraced quadratics are $k_{1,2}=\frac{3\pm
\sqrt{9-4}}{2}=\frac{3\pm \sqrt{5}}{2}\le \frac{3+
\sqrt{9}}{2}=3$. Variant: $\sqrt{5}\approx 2.24$, so $k_1\approx
2.62$.

\[17.03.13\] Misha Gabard (my father) and his skills in Excel calculated the next values of $k$ making $M=2k^2-3k+2$ into a square. The complete list for $M\le 1'000'000=10^6$ is $$k=1,2,17,46,553,1538,18761,52222,637297,$$ where $M$ is resp. the square of $$x=1,2,23,64,781,2174,26531,73852,901273.$$ On looking at the successive ratio of $k$, one finds the list $$R:=\frac{k_{n+1}}{k_n}=2, 8.5, 2.71, 12.02, 2.78, 12.20, 2.7835,
12.2036.$$ So naively the progression oscillates between a factor of about 2.78 and one about 12.20. So there seems to some extreme regularity, and one can predict in advance the size of the solutions.

The long quest of a Caucasian scheme, i.e. where Rohlin is stronger than Thom and Gudkov (united)
-------------------------------------------------------------------------------------------------

\[12.03.13\] The following section has been written in realm time out of paper notes, and we have reproduced our long march toward the trivial truth. The pressed reader can directly jump to Theorem \[Caucasian-scheme:thm\]. (WHICH IS ACTUALLY FALSE) So one must really read the first few sections which collects loosely organized thoughts on the problem of finding a scheme prohibited by Rohlin but not by Thom. Of course to avoid trivialities one must add more assumptions like the curve being an $M$-curves satisfying the Arnold or even Gudkov congruence.

\[12.03.13\] As a consequence of Thom’s estimate $\chi\le k^2$ (\[Thom-Ragsdale:thm\]), it seems that the Gudkov pyramids are much amputated on their right wings causing thereby a certain asymmetry of them. It is at this stage of some interest to wonder what happens on the opposite left-wings where Thom becomes useless but perhaps Rohlin’s formula has still some prohibitive impact.

In view of Fig.\[Degree10:fig\] let us look at the scheme $\frac{34}{1}2$ (not prohibited by Gudkov’s hypothesis). Here $\chi=1-34+2=-31$ and we have no prohibition by Thom. Is this scheme prohibited by Rohlin’s formula? Certainly not because as the Rohlin tree is simple there is no signs-law and the Rohlin equation $2(\pi-\eta)=r-k^2=37-25=12$ is soluble under the obvious additional relation $\pi+\eta=34$. Indeed $2\pi=40$, so $\pi=20$ and $\eta=14$. (Optional question: Is this scheme realized algebraically?) (\[21.03.13\] Answer: negative as follows from Petrovskii 1933/38 (\[Petrovskii’s-inequalities:thm\])!)

Without changing $\chi$ we may delocalize the 2 outer islands to make them islands in a lake. We find so the scheme $(1,(1,2)33)$ (cf. Fig.\[Signs-law-triad:fig\]b). Is this schemes prohibited by Rohlin? We think the answer is no and the proof proceeds along the usual algorithm of solving the Rohlin equation under the signs-law. Recall that the latter can be easily remembered by saying that consanguinity is bad, i.e. $+\times +=-$, $-\times
-=+$, while mixing the genes is good, i.e. $+\times -= +$ and $-\times += -$. This exotic signs law is the exact opposite of the usual convention.

Fig.c depicts the Hilbert tree of the scheme of Fig.b, and we decorate it with a sign distribution as depicted, i.e. with $x$-many plus, and $y$-many minus, and likewise $\epsilon$ plus and $\delta$ minus at the indicated place. We have thus $x+y=33$, and $\epsilon+\delta=2$. Using the signs-law we find for the number of positive $\pi:=\Pi^+$ resp. negative pair $\eta:=\Pi^-$ the following expressions: $$\pi=x+2\epsilon, \quad \textrm{and} \quad \eta =y+2\delta+1.$$ Adding gives $\pi+\eta=(x+y)+4+1=38$. By Rohlin’s formula $2(\pi-\eta)=r-k^2=37-25=12$, so $\pi-\eta=6$. Hence $2\pi=44$, so $\pi=22$ and $\eta=\pi-6=16$. Solving finally in $x$ gives $x=\pi-2\epsilon=22-2\epsilon$. We are free to choose say $\epsilon=2$, then $x=22-4=18$ and $y=15$. So Rohlin’s equation is soluble.

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At this stage the question becomes under which condition is Rohlin’s equation not soluble? A priori it may be remarked that Rohlin’s formula $2(\pi-\eta)=r-k^2$ is coupled with the formula $\pi+\eta=\Pi$ the total number of pair. Both right-hand sides of this pair of equations are entirely determined by the real scheme without having to worry about orientations. So under reasonable hypothesis, like $r-k^2$ even which is a disguised (Russian) version of Klein’s congruence $r\equiv g+1 \pmod 2$—as follows from the boring calculation $$r\equiv_2 g+1=(2k-1)(k-1)+1=2k^2-3k+2\equiv_2 3k \equiv_2 k
\equiv_2 k^2,$$ —we could expect that Rohlin’s equation is always soluble, especially if the (concomitant) Arnold congruence is satisfied.

However this is not the case as shown by Fiedler’s corrigendum. There we considered the $M$-scheme $(1,1,1,1)33$ (of degree 10) which is prohibited by Rohlin’s formula since $\pi-\eta\le 2$ by the signs-law for triad (compare Fig.\[Signs-law-triad:fig\]d). But on the other hand Rohlin’s formula forces $2(\pi-\eta)=r-k^2=37-25=12$, hence $\pi-\eta=6$.

Hence the philosophy behind Fiedler’s corrigendum seems to be that Rohlin’s formula imposes restriction, when there is much predestination forced by the signs-law, and this is naively speaking the case when there is much nesting like on the example just given. It seems of course to be of some interest to generalize the estimate $\Delta \Pi=\pi-\eta\le 2$ for triads to tetrads, etc.

So the scheme $(1,1,1,1)33$ is prohibited by Rohlin (in its strong form of the signs-law), but it is also by Thom as $\chi=33
\nleqslant k^2=25$. Now however we would rather be interested in schemes prohibited by Rohlin but not by Thom.

A first idea we had was to start from the scheme on Fig.b, and move $z$-many ovals inside to get Fig.e. As to maximize the “predestination” it seems wise to move the empty oval outside, and to assume $z=33$. But still under these circumstances it turned out that the Rohlin equation is soluble for a suitable distribution of sign, cf. Fig.f, from which we infer by the signs-law the following (where underbraced is the length of the corresponding edges=pairs) $$\pi-\eta=\underbrace{x-y+2}_{1}+\underbrace{(-x+y-1)}_{2}
+\underbrace{(+x-y)}_{3}=x-y+1.$$ (In this calculation it is useful to remind the consanguinity law, and the Fig.d for a triad.) Hence as $\pi-\eta=6$ (by Rohlin), we $x-y=5$, and $x+y=33$. So $2x=38$, whence $x=19$, and $y=14$. So Rohlin’s equations is soluble.

Philosophically it seems to be that whenever we have the free parameters $x,y$ then we can solve despite the predestination forced by the signs law.

Keep in mind that our question is whether Rohlin implies prohibitions on the left wing of the pyramid (i.e. for $\chi \ll
0$ much negative) where Thom tells nothing. In fact $\chi \le k^2$ is enough for Thom to be non-prohibitive. Hence our next idea was to start from Fiedler’s example $(1,1,1,1)33$ with $\chi=33$, and lower down to $\chi=25$. This lowering may be achieved by trading the (oceanic) outer islands against lakes, i.e. ovals at odd depths, and this requires to be done four times. (This can be done in several ways, cf. Fig.g.) However on applying the usual algorithm of sign distributions, we were always able to solve the Rohlin equation in a way compatible with the signs-law. Details on p.AR91 of my hand-notes, but the philosophy seems to be basically that Rohlin’s formula gives one equation and the signs-law another but as soon as there free-parameter $x,y$ counting the number of signs on branches there is enough freedom to solve all equations consistently.

In fact let us write down the argument. Starting from Fiedler’s example $(1,1,1,1)33$ with $\chi=33$, we lower down to $\chi=25$ by dragging 4 outer ovals at odd depth. This can be done in several fashions as we said, for instance like on Fig.\[Fied9:fig\]a by putting the 4 ovals at depth 1. However this new $M$-scheme is not prohibited by Rohlin’s formula. The latter says that $2(\pi-\eta)=r-k^2=37-25=12$, $\pi-\eta=6$. On the other hand by using the signs-law, and splitting the 4 empty ovals at depth 1 into $4=x+y$, where $x$ are positive, and $y$ are negative (w.r.t. the sign of the edge right above it), and assuming further for simplicity that the trunk is everywhere positive, we can still solve the equation. Indeed by the signs-law we have $\pi-\eta=2+x-y$, and so $x-y=4$, $x+y=4$ whence $x=4$, $y=0$.

If instead we drag the 4 ovals at depth 3 we get Fig.\[Fied9:fig\]b. Now the signs-law becomes more involved, but we find splitting according to the length of the pairs (underbraced index) the following expression (assuming for simplicity $+$-signs already fixed on the trunk and as above there are $x$ many $+$ and $y$ many $-$): $$\pi-\eta=\underbrace{3+x-y}_{1}+ \underbrace{(-2-x+y)}_{2}+
\underbrace{(+1+x-y)}_{3}=2+x-y,$$ where we used the signs-laws for triad (Fig.\[Signs-law-triad:fig\]d) As $\pi-\eta=6$, this gives $x-y=4$, $x+y=4$, which is soluble.

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Then we can also disperse the 4 ovals at different heights like on Fig.\[Fied9:fig\]c where 2 are at depth 1 and the 2 others at depth 3. Instead of applying the method of indeterminate signs, we content to give a solving sign distribution as depicted with only $+$-signs. One checks that $\pi-\eta=\underbrace{7}_{1}+\underbrace{(-4)}_{2}+
\underbrace{(+3)}_{3}=6$, in agreement with Rohlin’s formula.

Finally it remains to analyze the case of Fig.d, where we rigidify already some signs. Here $x+y=3$. Calculating via the signs-law gives $$\pi-\eta=\underbrace{4+x-y}_{1}+\underbrace{(-3)}_{2}+
\underbrace{(+2)}_{3}=3+x-y,$$ whence, as $\pi-\eta=6$, the system $x-y=3$, $x+y=3$, which is soluble (integrally) as $x=3$, $y=0$. (One checks mentally that this everywhere positive distribution works, as one do quickly mistakes in such calculation!)

Last the case of Fig.\[Fied9:fig\]e, is also handled by the same method. Here $$\pi-\eta=\underbrace{4+x-y}_{1}+\underbrace{(-2-x+y)}_{2}+
\underbrace{(+1+x-y)}_{3}=3+x-y,$$ like above (!) hence soluble.

Some further idea would be to increase the “predestination” by adding one or more triads as on Fig.f,g, yet such schemes are already prohibited by Bézout. Actually to lower $\chi$ down to Thom’s range $\chi\le k^2$ we look at Fig.g, but the latter is not even prohibited by Rohlin as $\pi-\eta=T_1+T_2+T_3$ is contributed by 3 trunks each $T_i\in\{2,0,-6\}$ (by Fig.\[Signs-law-triad:fig\]d) hence soluble as by Rohlin $\pi-\eta=6$. On Fig.f instead we have only 2 trunks so the scheme is prohibited by Rohlin (of course more elementarily by Bézout), yet it is also by Thom. So it does not solve our problem of finding a scheme where Rohlin is stronger than Thom.

Hence Fiedler’s example looks a typical case of predestination of the Rohlin mass $\Delta \Pi=\pi-\eta$. Of course it may be generalized in higher degrees than $10$, by looking at $M$-schemes of the form $(1,1,1,1)M-4$. Then by Rohlin $2(\pi-\eta)=r-k^2=M-k^2=(g+1)-k^2=(2k-1)(k-1)+1-k^2
=k^2-3k+2=(k-1)(k-2)$. But on the other hand by the signs-law for triad (Fig.\[Signs-law-triad:fig\]d) we have $\Delta\Pi:=\pi-\eta\le 2$. It follows that the scheme considered is prohibited as soon as $\pi-\eta=(k-1)(k-2)/2$ is $\ge 3$ that is for $k\ge 4$. (The case $k=4$ being quite stupid for Bézout would have sufficed.)

Okay, but such schemes are also prohibited by Thom. One could also try to deepen the nest as the degree increase. Yet our goal is really to find a “Caucasian” scheme, i.e. obstructed by Rohlin, but not by Thom (nor by Gudkov or Arnold).

Here is an example that I discovered later and of degree 8 already. It is the $M$-scheme of degree 8 with symbol $\frac{1}{3}\frac{1}{17}$. Now Rohlin’s formula says $2(\pi-\eta)=r-k^2=22-16=6$, so $\pi-\eta=3$. On the other hand $\pi+\eta=20$, so $2\pi=23$ which is insoluble. Besides we have $\chi=(1-3)+(1-17)=-2-16=-18$ SO GUDKOV NOT VERIFIED

\[14.03.13\] So let us approach this problem more systematically. First when $m=6$ it is clear that there is no Caucasian $M$-scheme as follows from Gudkov’s table (=Fig.\[Gudkov-Table3:fig\]). Indeed all the Thom permissible $M$-schemes are prohibited by Gudkov.

So we move to $m=8$, so $M=22$. Here we start with the $M$-scheme $(1,1,1)19$ with $\chi=20$ (cf. Fig.\[Fied1:fig\]a). As the 3-nest corresponds to a dyad (2 pairs of length 1) they contribute by the signs-law to at most 1 to $\pi-\eta$. Hence Rohlin’s formula $2(\pi-\eta)=r-k^2=22-16=6$ (i.e. $\pi-\eta=3$) cannot be solved. Of course the scheme in question is also prohibited for deeper reasons (at via deeper results) like Gudkov hypothesis $\chi\equiv k^2=16\pmod 8$, or Thom’s inequality $\chi \le k^2$. Our goal is to find an $M$-scheme where the “elementary” Rohlin formula becomes stronger than the conjunction of 2 deep results (Gudkov hypothesis proved by Rohlin-Rohlin/Atiyah-Singer-Marin and Thom proved by Kronheimer-Mrowka).

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Can we adjust the invariants as to neutralize Gudkov and Thom? A first idea is to drag one outer oval at depth 1. Then the contribution to the Rohlin (signed) mass $\pi-\eta$ is still $\le 2$, and so his formula cannot be solved. Yet doing so we have $\chi=18$ and the scheme is also prohibited by Gudkov or Thom.

If we delocalize one more outer oval at depth 1 we get Fig.c, where now Rohlin’s formula can be satisfied, and as $\chi=16$ both Gudkov and Thom are happy. So we see some annoying (as far as our Caucasian policy is concerned) concomitance between the 3 forces involved Rohlin, Gudkov, Thom.

This phenomenon is not specific to degree 8 and repeats itself in degree $m=10$, where $M=37$ (temperature of the body of a primate). Let us experiment this concretely. As Bézout now permits, we start now with a deep nest of profundity 4 (4 nested ovals) and add 33 outer ovals to reach Harnack’s bound $M=37$ (cf. Fig.d, which involve a “triad” chain with 3 consecutive edges). By Rohlin’s formula $2(\pi-\eta)=r-k^2=37-25=12$, so $\pi-\eta=6$. However by the signs-law for triad the contribution of the triad is at most $2$ (cf. Fig.\[Signs-law-triad:fig\]), and thus the scheme $(1,1,1,1)33$ is prohibited by Rohlin. As $\chi=33\equiv_8 k^2=25$, the scheme is not prohibited by Gudkov, but it is by Thom. So our Caucasian goal is not achieved.

To improve the situation we have to lower $\chi$ down to $k^2=25$. As far as the signs-law is involved we can transfer at most 3 oval at depth 1 (like on Fig.e), so as to have a contribution to $\pi-\eta$ still $\le 2+3=5<6$. Doing so $\chi=27$, and the scheme is still prohibited by Thom (and anew by Gudkov). Another idea is to use 3 dyads as on Fig.f as the latter also contribute to at most $1$ to Rohlin’s mass $\pi-\eta$. Alas doing so does not diminish $\chi$ in the Thom range, and actually violates Bézout (cf. Fig.g). This can be remedied if we abort the triad, and look at a configuration with 5 dyads (the maximum possible while still taking care to making Rohlin’s equation $\pi-\eta=6$ impossible). This gives Fig.h with alas still $\chi=27$. So this schemes is prohibited by both Rohlin, Gudkov, and Thom (but as far as I see not by Bézout even for conics).

If we nest one more outer oval we may get Fig.i with $\chi=25$, but suddenly Rohlin’s equation is now soluble (choose e.g. $+$-signs throughout). Likewise we may consider Fig.j, but Rohlin is likewise soluble. Considering Fig.k still calibrated as to make Rohlin’s equation impossible (as each triad contribute for $\le 2$), we only reach $\chi=27$ (of course this configuration is anti-Bézout). We can push it further to Fig.l, which despite being prohibited by Bézout it is not by Gudkov nor by Thom, yet alas not by Rohlin since $2+2+2=6$ and so Rohlin’s equation is soluble taking $+$-signs on all edges.

Our naive cuneiform construction can still be more varied, yet it seems unlikely that we will ever find a Caucasian scheme by this method. We still consider Fig.m (not interesting). Next look at Fig.n with $\chi=25$. Denoting by $x$, $y$ the number of $+$’s resp. $-$’s on the edge right above the corresponding letter, the signs-law gives (after fixing $+$ on the trunks) $$\pi-\eta=\underbrace{4+x-y}_{1}+\underbrace{(-3)}_{2}
+\underbrace{(+2)}_{3}=3+x-y,$$ whence the system $x-y=3$, $x+y=3$ soluble as $(x,y)=(3,0)$. So the scheme is not obstructed by Rohlin.

We can also transmute Fig.d into Fig.o, where again 3 branches are added so as to keep Rohlin’s formula impossible, but again $\chi$ drops only to $27$ (and not $25$). Of course such a scheme is defacto prohibited by the Bézout-Hilbert bound on the depth of nests (cf. Fig.p).

If it is not possible to find a Caucasian scheme in degree $10$, what about degree 12. First we need to extend the signs-law to tetrads. While the latter involves for dyads a square (4 possible products of two signs), and for triads a cube (with 8 possible signs combinations), we have now a 4D-hypercube with $2^4=16$ combinations. The signs-law for tetrads is depicted below (Fig.\[Signs-law-tetrad:fig\]). It corroborates the a-priori reasonable expectation that the maximum contribution arises when all 4 signs are $+$, in which case the contribution to Rohlin’s mass $\pi-\eta$ is 2. Still a priori we may expect that our (Caucasian) game will not become easier since Harnack’s bound $M=g+1=(2k-1)(k-1)+1=2k^2-3k+2$ increases much faster than Thom’s bound $k^2$ and so we will have more pain to lower down $\chi$ in Thom’s range. Despite these objections [*a priori*]{} let us track down our prey more slowly.

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\[15.03.13\] Hence:

The contribution to Rohlin’s mass $\pi-\eta$ of a deep nest is:

$\bullet$ for a dyad either $+1$ or $-3$,

$\bullet$ for a triad either $+2,0$ or $-6$,

$\bullet$ for a tetrad either $+2,0,-2$ or $-10$.

With some combinatorial ingeniousness it should be easy to extend to the general case. However let us first tackle our Caucasian problem in degree 12. Again we resort to the cuneiform formalism used above. When $m=12$, $M=g+1=\frac{11\cdot 10}{2}+1=55+1=56$. We consider first Fig.\[Fied2:fig\]a. Here $\chi=52$, while Gudkov says $\chi\equiv_8 k^2=36=44=52$, which is verified. By Rohlin’s formula $2(\pi-\eta)=r-k^2=56-36=20$, so $\pi-\eta=10$, but the contribution of the tetrad is at most 2 by the signs-law, and thus the scheme is prohibited by Rohlin. Of course it is also by Thom $\chi \le k^2$. As above the game is to lower $\chi$, by transferring outer ovals at depth 1. As to keep Rohlin in defeat, we may add at most 7 branches as on Fig.b, and obtain a scheme with $\chi=52-2\cdot 7=52-14=38$, again 2 unit above Thom’s bound. The next idea is to let branches of dyads (dyadic branches) hang on. Each contributes at most $1$ to Rohlin’s mass $\pi-\eta$, and so keeping Rohlin’s formula in check we may add 7 of them, but of course $\chi$ remains unchanged to $\chi=38$ (as we merely traded outer ovals at depth 0 for ones at depth 2). Next Fig.d involves only dyads contributing for at most 1, so keeping Rohlin check-mate we can plug 9 of them, and the resulting $\chi$ is still $38$. Using instead triadic branches as on Fig.e which contributes for 2 (at most) we may plug 4 of them and the remaining unit is consumed by inserting a dyadic branch, and we find of course again $\chi=38$. Considering only monadic branches as on Fig.f (und zwar 9 of them to Rohlin in check) yields again $\chi=38$. If like on Fig.g the monadic branches are not subsumed to a single dominator we find again introducing 9 of them, $\chi=38$. Using instead 9 dyadic branches of the same sort, we get Fig.h, where still $\chi=38$.

So it looks again hard to find a Caucasian scheme where Rohlin is stronger than Thom (and Gudkov united). Either we are looking at the wrong place or there is some subsumation of Rohlin to Thom, for some trivial arithmetical reasons. That is assume you have an $M$-scheme with $\chi\equiv_8 k^2$ (Gudkov) and $\chi\le k^2$ then Rohlin’s equation is always soluble. Indeed write formally $2(\pi-\eta)=r-k^2$. By the $M$-curve assumption $r-k^2$ is $r-k^2=(2k-1)(k-1)+1-k^2=k^2-3k+2$, etc.

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One could also ask if for $M$-curves the Thom and Gudkov obstructions are the sole one, but this is probably corrupted by Bézout-style obstruction à la Fiedler-Viro (in degree 8 already).

Some weak evidence against Caucasian schemes
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\[15.03.13\] In the previous section we tried (hard) to find a “Caucasian” scheme, i.e. prohibited by Rohlin’s formula $2(\pi-\eta)=r-k^2$ but not by Thom $\chi\le k^2$, but failed. Of course as Rohlin’s formula formally implies Arnold congruence it is actually a simple matter to find such a scheme, e.g. any $M$-scheme violating Arnold’s congruence but not Thom do the job. For instance in degree 6, the $M$-scheme $\frac{2}{1}8$ is prohibited by Rohlin, but not by Thom. Likewise for the $(M-2)$-scheme $\frac{1}{1}7$ in type I.

However if we add the Gudkov hypothesis as a side condition (or perhaps just the Arnold congruence) then it seemed difficult to find a Caucasian scheme where Rohlin is stronger than Thom. The sequel tries to give some evidence that it is impossible to find a Caucasian scheme, but our argument will be somewhat loose. The main difficulty is the mess arising with the signs-law and so the difficulty looks merely combinatorial. Of course it is not impossible that we missed a trivial counter-example that impedes the completion of the present programme.

First we described in the previous section the mess arising form the signs-law applied on dyad, triads, up to tetrads, which are totally ordered chains. In general the situation is more tricky as the Hilbert tree of the scheme may be highly branched.

Recall that to a dividing curve is assigned complex orientations (up to global reversion of all of them), which in turn decorates the Hilbert tree (of the curve) with a signs-distribution (abridged charge) making it into what we call the Rohlin tree. (Of course the tree can be a “forest”, i.e. have several components, and it really “branches downwards” as to look more like Arnold’s paradigm of the mushrooms.)

Each such (Rohlin) signed tree is completely determined by the signs ascribed to the edges of length 1 as it then extends to longer edges by the signs-law. Further each such tree has a Rohlin mass $\mu=\pi-\eta$ which is the difference between $\pi:=\Pi^+$ the number of positive pairs and $\eta:=\Pi^-$ the number of negative pairs.

When the Rohlin tree is the one induced by a curve of degree $m=2k$, (the marvellous) Rohlin’s formula $2(\pi-\eta)=r-k^2$ fixes the mass in function of $r$ the number of ovals and $k$ the semi-degree. Further $\pi+\eta=\Pi$ is the total number of pair, and so one gets the wrong impression that Rohlin’s equation is always soluble but there is some hidden rôle played by the mass $\mu:=\pi-\eta$ which for certain configuration can only be very low (especially on chains), cf. Fiedler’s example with $(1,1,1,1)33$ in degree 10.

So the core of the problem is to understand the behavior of the Rohlin mass $\mu$. In the previous section we nearly understood this for vertical chains. Especially easy, is the case where all signs are positive, in which case it is a simple matter to show the:

Given an $n$-chain of $n$ consecutive edges all positively charged then the Rohlin mass of the chain is the integer part $[\frac{n+1}{2}]$

Look and see (i.e. make pictures). Indeed looking at the Signs-laws for triad ($n=3$) or tetrad ($n=4$), we get resp. $\mu=3-2+1=1+1=2$ and $\mu=(4-3)+(2-1)=1+1=2$. For a 5-chain this extends as $\mu=(5-4)+(3-2)+1=3$, for a 6-chain as $\mu=(6-5)+(4-3)+(2-1)=3$, and so on.

In general for a signed tree there ought to be a sort of skein relation permitting an iterated evaluation of the Rohlin mass $\mu$, based on the formula that $\mu$ of an inverted “Y” looking like a $\Lambda$ surmounted by a chain is equal to $\mu$ of the left maximal chain in the inverted “Y” plus the right chain, minus $\mu$ of the common trunk. If this is not clear, please compare Fig.\[Fied2:fig\]i. This formula is of course a formal consequence of the inclusion-exclusion principle in combinatorics or measure theory.

Now in sloppy fashion first, the idea is that under certain assumptions (like the conjunction of Gudkov and Thom’s $\chi\le
k^2$) one could show the existence of a charge (=distribution of signs) solving Rohlin’s equation. Perhaps this can be done via a sort of linear algebra modulo 2.

First the charge in question is merely a $\pm 1$-valued function on the set $E$ of all edges of the tree. The set of all such distributions denoted $\cal E$ can be turned into a vector space over the field ${\Bbb F}_2$ with 2 elements. Define indeed the sum of two charges $\epsilon, \delta$ as $(\epsilon+\delta)(e)=\epsilon(e)\times \delta(e)$, where $\times$ is the Rohlin product given by the signs-law (i.e. the opposite of the usual sign convention for products). The neutral element is $\epsilon_0$ the minus distribution, as $(\epsilon+\epsilon_0)(e)=\epsilon(e)\times (-1)=\epsilon(e)$. Also each element has order 2, as it should. (More boring details in p.AR99=hand-notes, especially the inversed charge where all signs are switched is not the inverse charge!) The multiplication by a scalar is naturally defined. The Rohlin mass is the function $\mu\colon \cal E \to {\Bbb Z}$.

Now that we have a good vector space, we could hope our problem reducible to linear algebra! Intuitively if $\chi\le k^2$ then there must be enough edges as to solve Rohlin’s equation. More precisely Rohlin’s formula fixes the mass via $2\mu =r-k^2$, and via the skein relation the mass of the tree reduces to that of chains which in turn can be reduced to that of edges via the signs-law, i.e. the knowledge of the charge $\epsilon$ itself. If one is good in combinatorics there is a little hope to show that each skein relation induces a linear equation and count that there is enough free parameter as to solve the equation. [*Warning.*]{}—As remarked in more details latter, already in degree 6 we have the scheme $5$ whose type I realization is prohibited by Rohlin but not by Thom, so there is no chance to complete this programme, unless extra assumptions are added, e.g. that of being an $M$-curves (which further must satisfy the Arnold congruence, else prohibited by Rohlin but not by Thom).

As a trivial example we may extend the observations of the previous section. Consider an $M$-scheme without nesting. Then $\chi=M$. Now to diminish $\chi$ we introduce edges (i.e. nested pairs), cf. Fig.\[Fied2:fig\]g for an example. We have $r=M=g+1=(2k-1)(k-1)+1=2k^2-3k+2$, so $r-k^2=k^2-3k+2=(k-1)(k-2)$. By Rohlin’s formula $\mu:=\pi-\eta=(k-1)(k-2)/2$. Hence to keep Rohlin’s formula in default, we introduce only $\mu-1$ edges. Then we compute $\chi$, and find $\chi=r-2(\mu-1)$ and a boring calculation shows this to be $k^2+2$. So if obstructed by Rohlin then also by Thom. (More details in p.AR.99 of the hand-notes.) Of course this is not the general case as the scheme has a very specific structure akin to Fig.\[Fied2:fig\]g. Can we generalize, perhaps but requires to work out some messy combinatorics.

A somewhat more appealing idea is that if the scheme is obstructed by Rohlin then it is because its mass $\mu$ is strictly less than $\pi-\eta=(r-k^2)/2$ even when the tree is positively charged, and conjecturally this should maximize the mass. All this is vague but points to the right direction. Namely it gives the idea of computing the Rohlin mass of a tree with positive charges only. The answer turns out to be simple and elegant:

\[Rohlin-mass-of-a-positively-charged-tree:lem\] The Rohlin mass $\mu$ of a positively charged (signed) tree $T$ is equal to $$\mu(T)=n_1+n_3+n_5+\dots=n,$$ the number of ovals at odd depths, where $n_1$ counts those at depth $1$, $n_3$ at depth $3$, and so on.

Make a picture of a tree with possibly several components. Put plus signs everywhere as stipulated. By additivity we may focus attention on a single component of the tree. Each vertex at depth $\ge 1$ has exactly one edge above it. All pairs are enumerated by starting from vertices at depth 1 and looking at edges above them gives the contribution $n_1$. Next we look at the $p_2$ many vertices at depth 2, each inducing two pairs above it (of length 1 and 2 resp.). The first contribute for $+1$, while the other has sign $-1$ by the signs-law $+\times +=-$ (recall that consanguinity is bad). So ovals at depth 2 contributes for $p_2-p_2=0$. Continuing in this fashion we find: $$\mu=n_1+(p_2-p_2)+(n_3-n_3+n_3)+(p_4-p_4+p_4-p_4)+\dots,$$ which implies the announced formula.

As implicit above we posit the:

\[positive-mass-conjecture:conj\] The Rohlin mass of a signed (Rohlin) tree is maximized when the tree is positively charged throughout.

Some evidence comes from the case of chains (as tabulated on the signs-law tables, e.g. Fig.\[Signs-law-tetrad:fig\]). There is perhaps a simple argument.

But do we really need this? Let us make another observation based on the lemma.

Assume that Thom holds, i.e. $\chi\le k^2$. In general, we have: $$\chi=p_0-n_1+p_2-n_3+p_4-\dots,$$ where each symbol $p_i, n_i$ counts the number of ovals at depth $i$, where $p,n$ are just “residue” of Petrovskii notations for positive and negative but best interpreted in terms of even or odd depth resp. (The notation are nearly consistent in French-Swiss-German, where “even=pair” and “odd=uNgerade”.)

Besides, the total number of ovals, denoted $r$, is expressible as $$r=p_0+n_1+p_2+n_3+p_4+\dots,$$ so that subtracting the double of the Rohlin mass $\mu$ of the positively charged Rohlin tree (as calculated in the lemma) gives the relation: $$r-2\mu=\chi,$$ which holds universally when the tree is positively charged.

So if Thom is verified, i.e. $\chi\le k^2$, we find $2\mu=r-\chi\ge r-k^2$. This means that there is no obstruction [*a priori*]{} to solve Rohlin’s equation, since Rohlin’s mass is as large as it should by virtue of Rohlin’s formula $2\mu=r-k^2$. Paraphrasing, Rohlin’s equation is virtually soluble.

If Thom’s estimate $\chi\le k^2$ is fulfilled, then there is no “quantitative” obstruction to solve Rohlin’s equation. Yet beware that there may of course be finer arithmetical reasons impeding solubility as with the scheme $5$ of degree $6$ which has no type I realization.

This prompts some evidence toward the:

There is no Caucasian scheme, where Rohlin is stronger than Thom (at least modulo adding some suitable hypotheses, e.g. that of an $M$-scheme).

Can we find a formal proof? A crudely idea is to notice that if we charge the tree negatively throughout then by the signs-law all pairs are negative as $-\times -=-$. So the Rohlin mass of the negatively charged tree is $-\Pi$, where $\Pi$ is the total number of pairs. Hence by a dubious mean-value theorem (in the discontinuous realm of the arithmetics of quanta) we would like to infer existence of a charge fulfilling Rohlin’s formula.

Another idea is to introduce indeterminate signs and try to solve a system of linear equations. We did this frequently formerly, but we had some grasp on the geometry of the tree. Whether this can be done in abstracto is not clear to me, and may of course converge to the first strategy using linear algebra on the spaces of all charges (plus the skein-relation).

Of course recall that we have the Rohlin-Marin inequality that a dividing curve $C_{m=2k}$ has $r\ge m/2=k$, i.e. at least as many ovals as the semi-degree. This is a formal consequence of Rohlin’s formula and precludes in degree $6$ a type I incarnation of the scheme $1$ (unifolium), which is not prohibited by Arnold’s congruence mod 4 (cf. the Gudkov-Rohlin table=Fig.\[Gudkov-Table3:fig\]). This example (or also the scheme $5$ in degree 6) are of course trivial counterexamples to the above conjecture (freed from the parenthetical proviso). The latter gains however some more credibility when the curve is assumed to be an $M$-curve (or perhaps even an $(M-2)$-curve).

It is evident that our whole problem is somewhat ill-posed, yet we hope to have demonstrated that some complicity between Rohlin and Thom requires to be elucidated.

The simple example of the scheme $5$ in degree 6, where Thom’s estimate $\chi\le k^2$ as well as Arnold’s congruence are fulfilled, but whose realizability in type I is precluded by Rohlin’s formula (as $r$ is not a square and there is no nesting hence $\Pi=0$, and so a fortiori $\pi=\eta=0$) shows that our above desideratum of solving Rohlin’s equation under the sole assumption of being in Thom’s range is not realistic. So one must really add some extra assumptions, typically that of being an $M$-curve, which can perhaps be somewhat relaxed.

Let us close the discussion via precise conjectures:

$\bullet$ An $M$-scheme verifying the Gudkov congruence $\chi\equiv k^2 \pmod 8$ and the Thom estimate $\chi\le k^2$ is never prohibited by Rohlin’s formula.

$\bullet$ An $(M-2)$-scheme verifying the RKM=Rohlin-Kharlamov-Marin congruence $\chi\equiv k^2 +4 \pmod 8$ (ensuring the scheme to be of type I) and the Thom estimate $\chi\le k^2$ is never prohibited by Rohlin’s formula.

A basic mistake in the search of a Caucasian scheme
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DO NOT READ THE SEQUEL IT IS FALSE!

Our next idea was to look at an $M$-scheme extending 5 nests of depth 2 (cf. Fig.\[Fiedler6:fig\]a,b). By Rohlin’s formula we still have $\pi-\eta=6$, but $\pi-\eta\le \pi+\eta=\Pi=5$ so Rohlin is violated, but alas $\chi=27$ hence the scheme is also prohibited by Thom. Can we diminish $\chi$? Yes as usual by trading an outer oval at depth 0 against one at depth 0, cf. Fig.c. But then the corresponding tree (Fig.d) has $\Pi=6$ pairs and so Rohlin’s formula is soluble with $\pi=6$ (i.e. all edges positive).

Finally, we started from the scheme $(1,1)35$ (cf. Fig.e) with $\chi=35$ and to lower to Thom’s range $\chi=25$, we make 5 moves to get Fig.f and its allied tree on Fig.g. But again we have $\Pi=6$ and so can solve Rohlin’s equation by putting only positive signs.

From this last configuration, we decided to drag one of the ovals at depth 2 to get Fig.h, which has still $\chi=25$. By Rohlin $\pi-\eta=6$. Now introducing free variables $x,y$ counting positive resp. negative signs on the 5 edges and choosing any distribution of signs on the trunk of length 2, we know that the latter will contribute for at most $\le 2$ to $\Delta\Pi:=\pi-\eta$ (by Fig.\[Signs-law-triad:fig\]d) and its contribution $T$ is either $2,0,-6$, which is at any rate even. WARNING: HERE I MADE A BASIC CONFUSION IN THE LENGTH OF THIS CHAIN!!! On calculating $\pi-\eta$ by the signs-law we get $$\pi-\eta=T+x-y,$$ so $x-y=6-T$ and $x+y=5$ so that $2x=11-T$ which is impossible modulo 2! So we found our first scheme prohibited by Rohlin but not by Thom, an therefore:

\[Caucasian-scheme:thm\] (WARNING=ERRONEOUS) There exists a “Caucasian” scheme[^92], where Rohlin is stronger than Gudkov and Thom. More precisely the $M$-scheme of degree 10 of Fig.h that is $(1,(1,1)5)29$—in Gudkov’s notation—is prohibited by Rohlin’s formula, but not by Gudkov’s hypothesis $\chi=k^2 \pmod 8$ nor by Thom’s inequality $\chi\le k^2$. The example has $\chi=25$ (by construction).

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Of course the above argument extends to the case where we drop 3 ovals at depth 2 (Fig.i) so that we have $x+y=3$ still odd. (This could even be $x+y=1$.) Indeed denoting by $T_1,T_2,T_3$ the contributions of the 3 trunks of length 2 on Fig.i and by $T$ their sum (which is even by signs-law for triad), we find by the signs-law $$\pi-\eta=T+x-y,$$ and therefore as $\pi-\eta=6$ (by Rohlin’s formula) we have $x-y=6-T$ and $x+y=3$, so that summing $2x=9-T$, which is impossible modulo 2.

The phenomenon just discovered is probably not new and perhaps related to Slepyan’s law (also a formal consequence of Rohlin’s formula), cf. perhaps Rohlin 1978 or Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000].

\[13.03.13\] It is quite evident that we may generalize somewhat the result. What seems essential to the argument is that the Hilbert tree of the scheme as the structure of Figs.h.i with an odd number $x+y$ of empty ovals at depth 1, so to be like Fig.\[Fiedler7:fig\]a with $x+y$ odd.

In fact let us be more general and leave degree 10. So suppose to have an $M$-scheme of (arbitrary) degree $2k$ (or more generally a scheme of type I, but we reserve this for latter) so that the dividing character is granted and therefore Rohlin’s formula applies.

We suppose additionally that the scheme is like Fig.a with $x+y$ denoting the number of empty ovals at depth 1, partitioned into $x$ many positive pairs and $y$ many negative pairs (when looking at the unique edge right above those ovals).

The argument is then to compute $\pi-\eta$ in two fashions. One way involves Rohlin’s formula $2(\pi-\eta)=r-k^2$, while the other route involves the signs-law for triad (cf. Fig.\[Signs-law-triad:fig\]d) and gives $$\pi-\eta=T+x-y,$$ where $T$ is the contribution of the trunks of length 2 which is necessarily even (again by Fig.\[Signs-law-triad:fig\]d). Now if $r-k^2\equiv 0\pmod 4$, by Rohlin $\pi-\eta$ is even and the signs-law equation is corrupted if $x-y\equiv_2 x+y$ is odd. Viceversa if $r-k^2$ is not divisible by 4, then Rohlin says that $\pi-\eta$ is odd, but the signs-law that it is even provided $x+y$ is even.

Finally for $M$-curves it is a simple matter to check that $r-k^2\equiv 0\pmod 4$ iff $k\equiv 1,2 \pmod 4$. Indeed $$r-k^2=(g+1)-k^2=(2k-1)(k-1)+1-k^2=k^2-3k+2,$$ which is mod 4 for $k=1$, $1-3+2=0$ and for $k=2$, $4-6+2=0$, while for $k=3$, it is $9-9+2=2$ and for $k=0=4$, it is $2$.

Hence we have proved the:

Define a dendritic scheme as one like depicted on Fig.\[Fiedler7:fig\]a, i.e. with Gudkov symbol of the form $(1,(1,1)\dots(1,1) x+y ) z$.

If $k\equiv 1,2 \pmod 4$, there is no $M$-curve of degree $2k$ with dentritic scheme having an odd number of empty ovals at depth 1.

If instead $k\equiv 1,2 \pmod 4$, there is no such curve having an even number of empty ovals at depth 1.

Some additional remarks are in order, which we detail right after.

1.—First it is a simple matter to see that Caucasian schemes exists already in degree 8.

2.—Second it seems clear that we may formulate of the theorem for $(M-2)$-schemes satisfying the RKM-congruence ensuring type I.

3.—Third, we could expect to extend the result to the case where there are several dendrites, or deeper nests.

1.—Indeed for definiteness we may assume that there is a single trunk of length 2 (like on Fig.b). For $m=8$, $M=22$ and so for an $M$-curve we have $2(\pi-\eta)=r-k^2=22-16=6$, so that $\pi-\eta$ is odd. Yet calculating via the signs-law prompts that $\pi-\eta=T+x-y$ which is even provided the number $x+y$ of branches of length 1 is even. (THIS IS AGAIN A MISTAKE CAUSED BY CONFUSION IN THE LENGTH OF THE CHAIN: BASICALLY IT INVOLVES 3 OVALS BUT THE LENGTH IS 2!!!) So we get schemes prohibited by Rohlin along the series depicted on Fig.c, which traduced in Gudkov’s symbols gives the list $$(1,1,1)19, (1,(1,1)2)17, (1,(1,1)4)15, (1,(1,1)6)13, (1,(1,1)8)11,
\dots, (1,(1,1)18)1,$$ where $\chi$ is first $\chi=(1-1+1)+19=20$ (hence the scheme is also prohibited by Gudkov or Thom) and then successively drops by 4 units, so that the second listed scheme $(1,(1,1)2)17$ has $\chi=16$ (hence not prohibited by Gudkov nor by Thom, but prohibited by Rohlin), and so on. At $\chi=8$ we find another Caucasian scheme $(1,(1,1)6)13$.

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2.—For $(M-2)$-schemes satisfying the RKM-congruence $\chi\equiv
k^2+4 \pmod 8$ we are ensured to be of type I. We consider again a dentritic scheme like on Fig.\[Fiedler7:fig\]a and repeat the above argument. By Rohlin $2(\pi-\eta)=r-k^2$, while on the other hand $\pi-\eta=T+x-y$ where $T$ is the contribution from the trunks which is even. As $x+y\equiv_2 x-y$, we get a contradiction as soon as $x+y$ and $(r-k^2)/2$ have opposite parities. When $r=M-2$, $r-k^2$ is congruent to 0 mod 4 precisely when $k\equiv
0,3 \pmod 4$ (just shift by 2 the previous calculation).

To get an example, consider $m=8$ and take just one trunk like on Fig.c, but now consider the $(M-2)$-scheme $(1,1,1)17$. It has $\chi=18$. But we impose the RKM-congruence, hence adjust $\chi$ to $12$. Hence we consider $(1,(1,1)3)14$, and this is prohibited by Rohlin. Indeed his formula becomes $2(\pi-\eta)=r-k^2=20-16=4$, which is divisible by 4 (as predicted above), while by the signs-law $\pi-\eta=T+x-y$ is odd since there are $x+y=3$ little branches hanging on. \[21.03.13\] THIS IS ERRONEOUS BUT MAYBE CAN BE CORRECTED BY CHANGING OF PARITY, as the trunks contributes for an odd number.

So the real outcome of this method of prohibition based on Rohlin’s formula and the signs-law seems to be a powerful tool for prohibition. It remains of course to examine its exact significance, and how it generalize to nest of deeper structure.

It may be noted that earlier in this text we attempted a complete classification of RKM-schemes of degree 8. This was rather a census, i.e. a weak form of combinatorially possible schemes yet without any claim of realizability. Now with the present method we see that some of them are prohibited. It remains to understand which of them are prohibited by Rohlin enhanced by the signs-law. Clearly the argument given above extends and implies the following lemma.

\[RKM-scheme-ruled-out:lem\] (ERRONEOUS) The four primitive types of RKM-schemes, i.e. $$(1,(1,1)3)14, (1,(1,1)7)10, (1,(1,1)11)6, (1,(1,1)15)2,$$ already listed in Equation \[RKM-scheme-deg-8-four-primitive-type:eq\], are prohibited by Rohlin’s formula.

However their derived products looks harder to prohibit as they ramify and do not anymore belong to the dendrite type.

For instance for the scheme $(1,(1,2)3)13$ one can easily solve Rohlin’s equation with a distribution of sign, and so probably for all other derived products. That remains to be checked.

On the other hand, it is clear that our census of 100 schemes was far from exhaustive. Indeed we may consider a dendrite with 2 trunks like on Fig.d. This has $\chi=16$, and to adjust the RKM-congruence $\chi\equiv_8 k^2+4=20\equiv_8 12$ we move down to $\chi=12$ by transferring 2 outer ovals at depth 1 to get Fig.e. This scheme is not prohibited by Rohlin. Indeed $2(\pi-\eta)=r-k^2=20-16=4$ (so $\pi-\eta=2$), and by the signs-law $\pi-\eta=T_1+T_2+x-y$ where each trunks contributes to $T_i\in \{2,0,-6\}$ by Fig.\[Signs-law-triad:fig\]d. Even if we impose $T_1=T_2=2$, the system is still soluble, being $x-y=2$ and $x+y=2$, hence $(x,y)=(2,0)$.

As usual, the RKM-scheme of Fig.e whose (Gudkov) symbol is $(1,\frac{1}{1}\frac{1}{1}2)13$ has a myriad of companions. One can either:

$\bullet$ without changing $\chi$ transfer outer ovals at depth 2 (in various ways);

$\bullet$ drop $\chi$ by 8 units by transfer quanta of 4 ovals at depth 1.

All this is a bit messy to write down, and this will still not be exhaustive as we can start from the configuration with 3 trunks (Fig.f) which has $\chi=14$, and to adjust to RKM we make one transfer at depth 1 (Fig.g) and get so another RKM-scheme. This times the scheme is prohibited by Rohlin as $2(\pi-\eta)=r-k^2=20-16=4$ (so $\pi-\eta=2$), but by the signs-law $\pi-\eta=T+x-y$ where each 3 trunks contributes evenly while $x-y\equiv_2 x+y=1$ is odd. Likewise if from Fig.g we transfer quanta of 4 outer ovals at depth 1 the number of branches $x+y$ is still odd, and so those schemes are prohibited too. However those schemes derived from Fig.g by transferring outer ovals at depth 2 are probably not prohibited.

Then there is still the cases of 4, 5, etc. trunks and the classification looks quite messy to obtain. Fortunately the story the story as soon as the tree contains 4 disjoint edges, since this correspond to 4 nest of depth 2 (a configuration which is saturated by Bézout or better Rohlin’s maximality principle). Note incidentally that this principle also prohibits the schemes like Fig.g with 3 trunks since there are extra branches, so that the above prohibition via Rohlin’s formula can be subsumed to total reality. \[Warning this last sentence looks dubious!\] Albeit messy, it could be of primary importance to get a good view of what happens along the way to extend the Rohlin-Le Touzé phenomenon of total reality from degree 6 to degree 8.

So a pivotal question is whether there is any reasonable way to list all RKM-schemes of degree 8? If so then make some cleaning by ruling out those prohibited by Rohlin’s formula (enhanced by the signs-law), and finally try to understand which are realized algebraically (simplified form of Hilbert’s 16th nearly solved by the experts, but by far by myself, as one requires certainly the Viro method). Once this s achieved try to understand if all those schemes are subjected to the phenomenon of total reality (probably under pencil of quintics, as we discussed earlier). If so then there is some chance that Rohlin’s maximality conjecture holds true in degree $8$.

All this requires either Herculean forces or some good idea.

Toward a complete census of RKM-schemes of degree 8
---------------------------------------------------

\[13.03.13\] Our goal is to list all RKM-schemes of degree 8. Those are $(M-2)$-scheme satisfying the RKM-congruence $\chi\equiv
k^2+4=20 \pmod 8$. Of course there is a menagerie of them, but we have also upper bound given by the saturation principle of Rohlin allied to total reality, once the depth is 4 the configuration is saturated and cannot develop further. Actually the nest of depth 4, is not an $(M-2)$-scheme and so the depth is at most 3 (or 2 depending on the way you count). Likewise the pencil of conics shows that that there can be at most 4 nests of depth 2, or when translated in the cuneiform language of Hilbert’s tree there is at most 4 edges which are disjoint.

With this upper bound in mind, there is some little hope to make a complete classification. Further as the number of ovals $r$ is fixed to $M-2=20$, we may kill all empty ovals lacking a superior (so-called outer ovals) and thus condense a bit notation. So to each scheme is assigned a “skeleton” (kill the outer ovals) from which we may recover the scheme unambiguously.

We abort this project as it is quite overwhelming.

Trying to corrupt Rohlin’s maximality conjecture (RMC)
------------------------------------------------------

WARNING DO NOT READ: FULL OF MISTAKES. But try to correct at the occasion. Compare p.AR95–96 for the original, and keep in MIND the example of p.AR96. This is the $M$-scheme of degree 8 with symbol $\frac{1}{3}\frac{1}{17}$ which is prohibited by Rohlin, since $\pi-\eta=3$ (via Rohlin), but $\pi+\eta=20$, so $2\pi=23$ which is not soluble integrally. BUT yet another MISTAKE, this scheme is already prohibited by Gudkov (or even Arnold!) as $\chi=(1-3)+(1-17)=-18$

\[14.03.13\] Another basic question is whether from all schemes (in particular $M$-schemes) that we prohibited via Rohlin (and the signs-law), if it is not possible to corrupt RMC by finding an $(M-1)$-scheme right below which is maximal, but of course not of type I by Klein’s congruence $r\equiv g+1 \pmod 2$ for dividing curves.

One such scheme in degree 8 was given by $(1,\frac{1}{1}2)17$ (cf. Fig.\[Fiedler7:fig\]c) with $\chi=16$. WARNING THIS IS A MISTAKE, as Rohlin’s equation can be solved with all signs positive!!! Of course there is a myriad of other such $M$-schemes verifying the Gudkov congruence $\chi\equiv k^2 \pmod 8$ but prohibited for different reasons (Rohlin with signs-law, or by Thom (\[Thom-Ragsdale:thm\])). So the vague idea is that if the scheme is French or Caucasian (i.e. prohibited by Thom resp. Rohlin but not by Gudkov) then the scheme is nearly realized in the sense that killing one of its oval then the GKK-congruence $\chi\equiv k^2\pm 1 \pmod 8$ (cf. \[Gudkov-Krakhnov-Kharlamov-cong:thm\]) is satisfied and so there is some hope to construct some $(M-1)$-curve. Further, and this is the most dubious part of the game, we would like that the resulting $(M-1)$-curve cannot be enlarged, which requires to inspect a menagerie of schemes.

Of course we play this game [*à contre coeur*]{} as it is against our philosophy that the phenomenon of total reality is ubiquitous, and as posited by Rohlin 1978, that it governs the saturation principle (alias Rohlin’s maximality conjecture) saying that a scheme of type I is maximal in the hierarchy of all schemes.

Note actually our logical MISTAKE, namely our project only disproof the half of RMC already disproved by Shustin, as it will exhibit a maximal $(M-1)$-scheme which is not of type I. However the harder game is to find a scheme of type I which is not maximal.

Let us however work out an example of this disproof strategy for RMC as a potential application of the Rohlin formula and the signs-law obstruction.

We start with an $M$-scheme which is prohibited by Rohlin though not by Gudkov. Our (fairly random) candidate is, as said above, the $M$-scheme in degree 8 given by $(1,\frac{1}{1})$ (cf. Fig.\[Fiedler7:fig\]c or Fig.\[Fiedler8:fig\]) with $\chi=16$. The latter can be diminished to an $(M-1)$-schemes, 3 typical ways being depicted on Fig.b, where either the trunk is killed, or a branch or an outer oval. Of course one could also kill the maximal nonempty oval like on Fig.c, but then the GKK-congruence is not verified as $\chi=19$.

The 3rd specimen of Fig.b...

WARNING A THIS STAGE I HAD TO STOP AS I NOTICED AN EARLIER MISTAKE!!!!

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E-mail correspondence {#e-mail-Viro:sec}
=====================

\[09.01.13\] This section gathers responses given by experts (Viro, Marin, Orevkov, Kharlamov, Shustin, Le Touzé, Fiedler, etc.) to some naive questions of mine about the work of Rohlin. Here are the original messages in chronological order (inserted with the tacit approval of their authors). I acknowledge most sincerely their authors for the stimulating atmosphere it created and their generous answers. Messages are left in their original shapes safe for adding brackets \[ \] supplying electronically-updated label-links to the present text.

\[09.01.13\] Two naive questions on Rohlin 1978

Dear Viatcheslav, Alexis, Oleg, Stefan and Grisha,

Sorry for disturbing so many experts among yours with some little aspect of the work of academician Vladimir Abramovich. (I should have written this message in French, yet cannot remember exactly about Oleg’s progresses over the last 6 years in that language.) I was those last days quite fascinated by reading Rohlin’s 1978 survey on complex topological characteristics of real curves in some more detail. As you all know, he gave a quite spectacular enhancement of Gudkov’s pyramid for all schemes of sextics by enriching it with the data of Klein’s type I/II (1876). (Compare optionally the attached pdf file giving a graphical snapshot view of Rohlin’s classification.)

My two questions are as follows.

\(1) First Rohlin (1978) claims to have a certain synthetic argument (via pencils of cubics) able to show the type I of the schemes 6/1 2 and 2/1 6. He confesses however his argument to be a complicated one. Let me cite Rohlin exactly:

“...when we apply it to curves of degree 3, we can establish (in a rather complicated way) that the schemes $\frac{6}{1}2$ and $\frac{2}{1} 6$ of degree 6 belong to type I. However, all the schemes that we have so far succeeded in coping with by means of these devices are covered by Theorem 3.4 and 3.5.”

My first question is whether Rohlin’s synthetic argument has ever been published (assuming its truth of course)? I suspect the proof to be quite beautiful, but I am myself not able to prove it for the moment. Did one of you ever worked out the argument in detail, or remember about some exposition during Rohlin’s lectures? Is it of the same order of difficulty as the Hilbert-Rohn method, requiring “roughness” á la Andronov-Pontrjagin to turn round? Would it be didactically useful to publish (on the arXiv) an account of Rohlin’s argument if one is able to reconstruct it?

\(2) The second question is of course the general Rohlin’s maximality conjecture (a scheme is of type I iff it is maximal in the hierarchy of all real schemes of some fixed degree). As reported in Viro’s survey (1986 Progresses over the last 6 years) it seems that one implication was disproved by Polotovskii and Shustin (combined efforts ca. 1982, 1985). Yet one implication looks still possible, namely type I implies maximal (if I am not wrong). It seems to me that this (last vestige of the) Rohlin conjecture could be proved somewhat eclectically in two lines via Ahlfors theorem (1950) on the total reality of orthosymmetric curves (alias type I). Namely if the curve is of type I, then there is a pencil of curves cutting only real points on the curve, so its real scheme cannot be enlarged without violating Bezout. q.e.d. Some more thinking shows of course this argument to be insufficient but maybe there is a (clever) way to complete it. Qu’en pensez-vous[^93]?

Many thanks for your attention, and also for all your fantastic papers (I am presently trying to digest, so do not take the pain to answer me properly if my questions look too naive.) I apologize again for this collective message, but as the material is quite old, most of you probably forgot some details. So I hoped to maximize some chance of getting an answer from a collective chat room.

Best regards, Alex

PS: In attachment I send you a copy of an informal text of mine on the Ahlfors map. Section 24 (pp. 205–229) is more specifically devoted to Rohlin’s conjecture, yet contains nothing original (while being quite poorly organized).

$\bullet$ On Wed, 9 Jan 2013 13:33:23 +0100 (alexandregabard@hotmail.com) wrote to Kharlamov, Marin, Viro, Fiedler, Orevkov, and Mikhalkin a collective e-mail titled “Two naive questions on Rohlin 1978”:

Dear Viatcheslav, Alexis, Oleg, Thomas, Stepan and Grisha,

Sorry for disturbing so many experts among yours with some little aspect of the work of academician Vladimir Abramovich. (I should have written this message in French, yet cannot remember exactly about Oleg’s progresses over the last 6 years in that language.)

I was those last days quite fascinated by reading Rohlin’s 1978 survey on complex topological characteristics of real curves in some more detail. As you all know, he gave a quite spectacular enhancement of Gudkov’s pyramid for all schemes of sextics by enriching it with the data of Klein’s type I/II (1876). (Compare optionally Fig.71\[=\[Gudkov-Table3:fig\]\] on page 208 of the attached pdf file giving a graphical snapshot view of Rohlin’s achievement.)

My two questions are as follows.

\(1) First Rohlin (1978) claims to have a certain synthetic argument (via pencils of cubics) able to show the type I of the schemes 6/1 2 and 2/1 6. He confesses however his argument to be a complicated one. Let me cite Rohlin exactly:

“...when we apply it to curves of degree 3, we can establish (in a rather complicated way) that the schemes $\frac{6}{1}2$ and $\frac{2}{1} 6$ of degree 6 belong to type I. However, all the schemes that we have so far succeeded in coping with by means of these devices are covered by Theorem 3.4 and 3.5.”

My first question is whether Rohlin’s synthetic argument has ever been published (assuming its truth of course)? I suspect the proof to be quite beautiful, but I am myself not quite able to write it down for the moment. Did one of you ever worked out the argument in detail, or remember about some exposition during Rohlin’s seminar? Is it of the same order of difficulty as the Hilbert-Rohn method, requiring “roughness” à la Andronov-Pontrjagin to turn round? Would it be didactically useful to publish (on the arXiv) an account of Rohlin’s argument if one is able to reconstruct it? Many thanks if you have some ideas (or recent references) on those or related questions...

\(2) The second question is of course the general Rohlin’s maximality conjecture (a scheme is of type I iff it is maximal in the hierarchy of all real schemes of some fixed degree). As reported in Viro’s survey (1986 Progresses over the last 6 years) it seems that one implication was disproved by Polotovskii and Shustin (combined efforts ca. 1982, 1985). Yet one implication looks still possible, namely “type I implies maximal” (if I am not wrong). It seems to me that this (last vestige of the) Rohlin conjecture could be proved (somewhat eclectically) in two lines via Ahlfors theorem (1950) on the total reality of orthosymmetric curves (alias type I). Namely if the curve is of type I, then there is a pencil of curves cutting only real points on the curve, so its real scheme cannot be enlarged without violating Bézout. q.e.d. Alas, some more thinking shows of course this argument to be insufficient but maybe there is a (clever) way to complete it. Qu’en pensez-vous?

Many thanks for your attention, and also for all your fantastic papers (I am presently trying to digest). So do not take the pain to answer me properly if my questions sound too naive. I apologize again for this collective message, but as the material is quite old, most of you probably forgot some details. So I hoped to maximize some chance of getting an answer from a collective chat room.

Best regards, Alex (Gabard)

PS: The attachment[^94] is a copy of an informal text of mine on the Ahlfors map. Section 24 (pp. 205–229) is more specifically devoted to Rohlin’s conjecture, yet contains nothing original (except being poorly organized).

$\bullet$ \[Viro’s answer the same day (09.01.13) ca. 20h00, additional footnotes are mine (Gabard)\]

Dear Alexandre,

Thank you for your message and manuscript. I was not aware about the Ahlfors theorem[^95]. It seems to be very interesting.

I doubt though if it can be used for proving the half of Rokhlin conjecture. It gives a proof for impossibility of raising the number of components of a type I curve by a single algebraic Morse modification (what I called Klein’s thesis).

I do not remember if I even ever heard about Rokhlin’s proof that you ask about, but the fact follows from the congruence. Slava[^96] did not mention it when he proved the corresponding congruence (at the moment the type was not yet considered). I learned this theorem from Slava in September 1977 and wrote down Slava’s proof to my notebook then. I guess the first proofs was[^97] published by Slava Nikulin (among many other statements) and Alexis Marin. Marin’s proof looks simpler, but requires Pin- structures.

Best regards, Oleg

$\bullet$ Gabard’s reply \[Same day (09.01.13) ca. 21h00\]

Dear Oleg,

Many thanks for your rapid and illuminating responses, plus all the historical details. If you see no objection, I would be very happy to cut-and-paste them in my survey. I still need to assimilate some congruences of the early phase (Rohlin, Gudkov-Krakhnov-Kharlamov, etc.) Hence you cannot imagine how your hints are illuminating my modest understanding of that golden period. Regarding Ahlfors, as you say, there is little hope to crack the big fish, yet of course I shall keep you informed if I get not too depressed by the immense difficulty.

All the best, and so many thanks again, Alex

$\bullet$ 10 Jan 2013 (Marin’s answer)

Cher Gabard

En plein déménagement, je met un peu plus de temps à vous répondre que Viro.

Comme Viro, je ne connais pas la preuve de Rohlin pour votre première question (c’est pourquoi j’avais imaginé la preuve dont parle Viro qui est dans “Quelques remarques sur les courbes algébriques planes réelle”, votre référence 742\[=[@Marin_1979]\]) Cependant ce séminaire de Paris VII est dans un carton et y restera tant que je n’aurai pû trouver un nouvel appartement assez grand pour contenir ma bibliothèque et, n’ayant le temps d’aller à la bibliothèque, ma mémoire ne me permet pas de vous en dire plus que Viro.

Pour la seconde question par contre je peux vous répondre, c’est à dire lever votre aveux d’incompréhension en fin (p. 226) de preuve du Lemme 24.20[^98].

Soit une courbe séparante gagnant un ovale de plus après franchiment d’un point quadratique ordinaire. Un argument de congruence (utilisant $d > 2$ dans le cas plan ou une hypothèse dans le cas général donnant que la désingularisée de cette courbe de franchiment est irréductible : l’ensemble de ses points complexe est connexe) donne que cette désingularisée de la courbe de franchiment est non séparante.

Ainsi deux points non réels conjugués de la courbe de franchiment sont lié par un arc évitant la partie réelle, en particulier le point singulier, et par extension des isotipie[^99] un tel arc subsite dans toute déformation vers l’un des des deux côtés du discriminant, en particulier avant le franchiment la courbe est non séparante ce qui contredit l’hypothèse.

Par contre si le franchiment du discriminant se fait en un point singulier plus compliqué il me semble que l’on peut augmenter le nombre de composantes connexes d’une courbe séparante. Je crois me souvenir que selon les constructions de Viro (ou peut être seulement après avec la présentation Itenbergienne de cette méthode de Viro) il y a une courbe singulière de degré 6 dont tout voisinage contient tous les types.

N’étant plus familier du sujet depuis plus de 20 ans je ne peux vous en dire plus, par contre pour les surfaces de degré 3 vous trouverez dans le second tome des oeuvres de Klein un magnifique article illustré de non moins magnifiques figures où il établi que tous les types de surface cubique s’obtiennent par déformation de la (unique à changement projectif de coordonnées) surface cubique qui a 4 points quadratiques ordinaire.

Merci de votre long article que j’essayerai de lire quand déménagement, vente,.... seront terminés.

Bien cordialement et bonne année.

Alexis Marin

PS 1 Je trouve Viro un peu “oublieux” d’écrire “ (at the moment the type was not yet considered)”: en parcourant le second tome des oeuvres de Klein vous vous appercevrez qu’un sciècle avant Viro “tout” était chez Klein!

2 Vous trouverez un article historique, beaucoup plus court\* et sur un autre sujet en mettant dans la boite de recherche d’Arxiv le mot clef “troupeau”

\*il fait 6 pages table des matières comprise et tout est dit (de façon “autocontenue”) dans le résumé en français de la première page, mais si vous remontez à toutes les références\*\* des commentaires bibliographiques celà peut vous prendre un peu de temps.

\*\*accesibles à travers la "bibliothèque des sophomores http://alexis.marin.free.fr/BIB/

$\bullet$ Gabard’s answer \[12.01.13 ca. 23h00\]

Cher Alexis, C’est avec une immense joie que j’ai reçu votre message. N’ayant pas d’internet à la maison, je l’ai seulement découvert ce soir en visitant mon père, qui lui est connecté. Je vais donc tenter d’assimiler toutes vos remarques savantes, et si vous le permettez, de les intégrer dans mon survey, en spécifiant bien sûr qu’il s’agit de vos contributions. De mon côté, je me demande si une courbe non-séparante peut toujours acquérir un point double ordinaire solitaire. (C’est semi-implicite dans Klein 1876 qui écrivait “noch entwicklungsfähig”, mais il me semble que ça contredit le résultat de Shustin 1985 (contre-exemple à la conjecture de Rohlin), dont la logique m’échappe quelque peu, mais j’ai sûrement raté une subtilité).

Grâce à vos commentaires je devrais pouvoir produire prochainement une version plus solide et limpide de la section correspondante du survey, que je vous enverrai dès que possible. L’interaction avec Ahlfors me semble aussi prometteuse...

Amitiés, et merci infiniment pour votre message, Alex

PS J’espère que le déménagement se passe bien. Restez-vous à Grenoble, ou bien s’agit-il d’une opération plus conséquente?

PPS: J’ai bien à la maison votre article de Paris VII, qui a toujours été mon meilleur compagnon (en 1999-2000), et je suis content de le retrouver pour ce point encore plus profond.

PPPS: je me suis procuré une copie de l’article sur “il capo”, qui me semble fabuleux. Merci beaucoup. C’est exactement l’analyse que l’on rencontre à proximité de Dirichlet, etc jusqu’à Ahlfors, et Rogosinski, et que je dois essayer à l’occasion d’apprivoiser...

PS 1 Je trouve Viro un peu “oublieux” d’écrire “ (at the moment the type was not yet considered)”: en parcourant le second tome des oeuvres de Klein vous vous appercevrez qu’un sciècle avant Viro “tout” était chez Klein!

Vous avez parfaitement raison, et je suis moi même très “spécialisé ” dans l’oeuvre de Klein. Cependant le gros quiz, c’est l’assertion de Teichmüller 1941, qui prétend que Klein 18XX? anticipe Ahlfors 1950, de 70 ans environ. Toute courbe séparante (ou surface de Riemann orthosymmétrique, pour reprendre le jargon kleinéen) admet un morphisme réel vers la droite dont les fibres au dessus des points réels sont toutes exclusivement formées de points réels. C’est cet énoncé fondamental qui me semble être sous-exploité! Evidemment[^100] comme la noté Viro, il implique la partie facile de l’assertion de Klein (1876): une courbe séparante ne peut gagner un ovale spontanément comme une bulle de champagne surgit du néant.

$\bullet$ Réponse de Marin (le lendemain 13 Jan 2013 ca. 09h00)

de les intégrer dans mon survey, en spécifiant bien sûr qu’il s’agit de vos contributions.

A part l’explication de votre doute (où relativement à l’article que vous citez il n’y a que les mots “extension des isotopies” en plus) ce ne sont que de très vagues souvenirs que je vous conseille de vérifier (éventuellement auprès de plus compétent : Viro, Itenberg,... avant de les intégrer)

De mon côté, je me demande si une courbe non-séparante peut toujours acquérir un point double ordinaire solitaire.

voulez-vous dire dont les deux directions tangentes sont complexes conjuguée? celà me parait très très optimiste.

(C’est semi-implicite dans Klein 1876 qui écrivait “noch entwicklungsfähig”,

Êtes vous sûr que c’est ce que pensait Klein, ou incluait-il dans ce terme les modification par franchiment d’une courbe ayant un unique point double qui est quadratique ordinaire à tangentes réelles “apparu en rapprochant deux points d’un même ovale”?

mais il me semble que ça contredit le résultat de Shustin 1985 (contre-exemple à la conjecture de Rohlin), dont la logique m’échappe quelque peu, mais j’ai sûrement raté une subtilité).

PS J’espère que le déménagement se passe bien.

oui mais c’est long, à ce propos, vous trouverez sur

http://alexis.marin.free.fr/BIB/papier/

la liste des livres que j’ai en plusieurs exemplaires et (sauf ceux dont la colonne “héritier” est remplie (par Vinel et/ou Guillou)) qui sont à la disposition de qui (en particulier vous) les demande. Restez-vous à Grenoble, ou bien s’agit-il d’une opération plus conséquente?

Je reste près de Grenoble (mon adresse est dans la signature électronique ci-dessous

PPPS: je me suis procuré une copie de l’article sur “il capo”,

Voulez vous dire “Le capo”?

Cependant le gros quiz, c’est l’assertion de Teichmüller 1941, qui prétend que Klein 18XX? anticipe Ahlfors 1950, de 70 ans environ. Toute courbe séparante (ou surface de Riemann orthosymmétrique, pour reprendre le jargon kleinéen) admet un morphisme réel vers la droite dont les fibres au dessus des points réels sont toutes exclusivement formées de points réels.

Voulez-vous dire revêtement d’espace total l’ensemble des ovales? Il y a-t-il quelque chose de plus précis sur le degré et sa répartition parmis les ovales? Les références sont-elles dans votre article?

C’est cet énoncé fondamental qui me semble être sous-exploité! Evidemment comme la noté Viro, il implique la partie facile de l’assertion de Klein (1876): une courbe séparante ne peut gagner un ovale spontanément comme une bulle de champagne surgit du néant.

Soyez plus précis pourquoi un tel morphisme admettrait-il une déformation le long de la modification d’adjonction d’un ovale?

Amitiés.

Alexis

– http://le-tonneau-de-thales.tumblr.com/

Alexis Marin, chez Danielle Bozonat 6 Allée de la roseraie, 38240 Meylan fixe : 04 76 00 96 54 port. : 06 38 29 33 99, 00351925 271 040

$\bullet$ Gabard 13 Jan 2013 ca. 13h30

Cher Alexis, Merci pour votre message. Je vais en effet essayer d’intégrer vos commentaires de manière ciblée et prudente. De toute manière avant d’arXiver une nouvelle version d’ici six mois environ, j’aurai l’occasion de vous montrer précisement la prose que je vous aurez emprunté. J’essaye maintenant de répondre à vos questions:

De mon côté, je me demande si une courbe non-séparante peut toujours acquérir un point double ordinaire solitaire.

voulez-vous dire dont les deux directions tangentes sont complexes conjuguée? celà me parait très très optimiste.

REPONSE: Oui, exactement à tangentes imaginaires conjuguées. Cela me parait aussi très optimiste. Klein semble le prétendre semi-implicitement (du moins qu’il n’ y a a priori pas d’obstruction topologique à la formation de telles bulles de champagne). Cependant si ce truc fou (“Klein-vache”) est vrai alors un des sens de la conjecture de Rohlin 1978 (type I iff maximal real scheme) est vérifié. Malheureusement, ce que donne “Klein-vache” est le sens de Rohlin détruit par Shustin 1985 (dont je n’ai cependant pas compris l’argument). Mais vous avez surement raison “Klein-vache” est probablement beaucoup trop optimiste...

Êtes vous sûr que c’est ce que pensait Klein, ou incluait-il dans ce terme les modification par franchiment d’une courbe ayant un unique point double qui est quadratique ordinaire à tangentes réelles “apparu en rapprochant deux points d’un même ovale”?

REPONSE: je pense que oui, car Klein précise “isolierte reelle Doppeltangente”, comparez ma Quote 24.2[^101] page 205 de mon survey (si vous n’avez pas le volume 2 de Klein sous la main). Ainsi il me semble que votre interprétation moderne (Marin 1988) diffère un peu de l’original Kleinéen, en étant toutefois plus puissant que l’assertion d’origine.

PS J’espère que le déménagement se passe bien.

oui mais c’est long, à ce propos, vous trouverez sur

http://alexis.marin.free.fr/BIB/papier/

la liste des livres que j’ai en plusieurs exemplaires et (sauf ceux dont la colonne “héritier” est remplie (par Vinel et/ou Guillou)) qui sont à la disposition de qui (en particulier vous) les demande.

C’est une magnifique liste de trésor. Je voudrais bien les acquérir, mais je me demande si mon hygiène de vie (overwork) rend une telle acquisition raisonable...(Il faudrait que je passe à Grenoble avec la camionnette de mon oncle pour récupérer les “invendus”. Il est préférable en effet de trouver des preneurs plus compétents que moi. Si en dernier recours, vous ne trouvez pas de preneurs je pourrais récupérer les volumes restants en vrac...Merci infiniment pour cette généreuse proposition. Moi même je suis très marginal financièrement et spatialement, petit appartement à Genève partagé avec ma mère (avec environ 8 tonnes de littérature mathématique), mais dans le futur je pourrai peut être m’installer dans une ferme fribourgoise, où il reste de l’espace pour expandre la bibliothèque...)

PPPS: je me suis procuré une copie de l’article sur “il capo”,

Voulez vous dire “Le capo”? Oui, j’essayais d’improviser en italien, mais c’est une langue plus subtil que vous utilisez...

Cependant le gros quiz, c’est l’assertion de Teichmüller 1941, qui prétend que Klein 18XX? anticipe Ahlfors 1950, de 70 ans environ. Toute courbe séparante (ou surface de Riemann orthosymmétrique, pour reprendre le jargon kleinéen) admet un morphisme réel vers la droite dont les fibres au dessus des points réels sont toutes exclusivement formées de points réels.

Voulez- vous dire revêtement d’espace total l’ensemble des ovales? Il y a-t-il quelque chose de plus précis sur le degré et sa répartition parmis les ovales? Les références sont-elles dans votre article?

OUI, toute surface de Riemann à bord (=membrane compacte) s’exprime comme revêtement holomorphe ramifié du disque. C’est juste une version relative (à bord) du théorème d’existence de Riemann qui concrètise toute surface de Riemann close comme revetement conforme de la sphère (ronde). Il a fallut toutefois attendre la contribution d’Ahlfors 1950 qui donne en plus un contrôle sur le degré d’un tel revêtement conforme, à savoir r+2p, où r est le nombre d’“ovales” (mieux le nombre de contours de la membrane), et p son genre. La Thèse de moi-même (Gabard 2004, et l’article de 2006 au Commentarii Math. Helv.) donne un meilleur contrôle, à savoir $r+p$, en économisant donc une cartouche pour chaque anse. Les références précises sont dans le survey. L’énoncé d’Ahlfors était vachement anticipé dans le cas $p=0$ (membrane planaire ou schlichtartig pour reprendre la terminologie de Paul Koebe) par la grande lignée Riemann 1857 (Nachlass), Schottky 1875-77, Bieberbach 1925 et Grunsky 1937. Lorsqu’on passe au double de Schottky-Klein de la surface à bord on obtient (via Ahlfors) une courbe séparante avec un morphisme totalement réel vers la droite projective. Inversement toute courbe séparante est totalement réelle, puisqu’il suffit d’appliquer Ahlfors à une des moitiés orthosymétrique de Klein.

\[Gabard\] C’est cet énoncé fondamental qui me semble être sous-exploité! Evidemment comme la noté Viro, il implique la partie facile de l’assertion de Klein (1876): une courbe séparante ne peut gagner un ovale spontanément comme une bulle de champagne surgit du néant.

\[Marin\] Soyez plus précis pourquoi un tel morphisme admettrait-il une déformation le long de la modification d’adjonction d’un ovale?

\[Gabard\] Je pense que ça marche car lorsque la courbe est plongée dans le plan, le morphisme total d’Ahlfors admet une réalisation projective comme un pinceau de courbes planes dont tous les membres découpent seulement des points réels sur la courbe orthosymmétrique (=séparante). Par conséquent, en traçant la courbe du pinceau total qui passe par un point de l’oval spontanément créé, on obtient une contradiction avec Bézout. Donc Ahlfors 1950 implique Klein 1876, mais votre démonstration de 1988$-\varepsilon$ (votre preuve est déjà mentionnée dans Viro 1986) est surement plus intrinsèque et voisine de l’argument d’origine de Klein (s’il en avait un au delà de la pure contemplation empirique des quartiques notamment...)

Merci infiniment pour vos messages, et d’ici tout bientôt (3-4 jours) je vous enverrai une version mise-à-jour du survey qui clarifiera peut-être les assertions précédentes. Toutefois les grands problèmes et plein de détails m’échappent encore dans la pyramide Gudkovo-Rohlinienne. Quelle splendide pyramide qui joint à la perfection Klein et Hilbert! Un détail qui m’échappe, c’est le fait que le discriminant est de degré $3(m-1)^2=75$ pour $m=6$, tandis que que du point de vue des chirurgies “de Morse” il y a des cycles de longueur 4 dans la pyramide de Gudkov. Donc il y a un problème de parité si on déforme le long d’un pinceau générique (transverse au discriminant)...Désolé, de vous embêter avec ces détails que j’ai honte de ne pas réussir à clarifier depuis quelques jours.

Amitiés, et bon courage pour la suite du déménagement, Alex

$\bullet$ \[16h40 15.01.13\] Cher Alexis, Merci encore pour vos messages et vos remarques fascinantes que je dois encore bien digérer. De mon côté, j’ai fait de minimes progrès, et vous envoie malgré votre déménagement une version ajournée de mon survey. Il me semble que le truc fou dont nous parlions il y a quelques jours, que j’appele depuis “Klein-vache”, i.e. la possiblilité de faire naitre un noeud solitaire (à tangentes conjuguées) depuis n’importe quelle courbe diasymétrique est vrai pour les sextiques. Pour cela j’utilise un argument qui combine Rohlin 1978, Klein-Marin 1988, et Nikulin 1979 (classification isotopique) et un résultat relié de Itenberg 1994 (possibilité de contracter n’importe quel ovale vide, i.e. sans autre ovales dans son intérieur, sur un tel noeud isolé). Les détails de la preuve sont exposés dans la Prop.24.24\[meanwhile this is \[Klein-vache-deg-6:prop\]\], page 235 du fichier ci-joint. ? Evidemment, en principe “Klein-vache” n’a aucune chance d’être vrai en degré supérieur. Cependant la seule obstruction que je connaisse est ce résultat de Shustin 1985, dont je ne comprends toujours pas la logique de base (sans même parler du fait que c’est fondé sur la méthode de Viro, dissipation de singularités tacnodales..., une technologie que je n’ai jamais maitrisée). Mes objections naives à l’argument de Shustin se trouvent en page 248 (dans le paragraphe qui précède la Figure 94\[=meanwile Fig.\[Shustin:fig\]\]). Dans cette figure, je ne sais pas comment prohiber le $(M-1)$-schémas encadré par le carré vert (à mi-hauteur de la figure), et dans son article de 1985 Shustin n’est pas trés explicite. Mais bon, il s’agit la d’une question assez ennuyeuse et en fait je vais peut-être prendre l’initiative d’écrire un nouveau message collectif pour clarifier ce point d’ici quelques heures. Merci infiniment encore pour vos messages, et meilleurs voeux de courage pour la suite du déménagement, Amitiés, Alex PS: Pour l’instant j’ai inégré en vrac tous nos échanges e-mail dans le survey (p.219 et suivante), mais bien entendu dès que possible je censurerai les remarques plus confidentielles..., et masquerai les répététitions, voire l’intégralité de la discussion si je parviens bien à résumer votre apport malgré mon anglais catastrophique. Cependant en relisant vos remarques, elles apportent une prose substantielle que je ne saurais jamais reproduire en anglais, donc je trouverais très dommage de censurer vos souvenirs en vracs!!! Evidemment rien ne presse et je suis désolé de vous avoir dérangé durant cette délicate opération du déménagement inter-grenoblois. Amitiés, encore, et je vous tiens au courant d’éventuelles progrès...Je suis surtout curieux des réponses de Shustin (et Viro) s’ils parviennent à éclairer ma lanterne. PPS: Je joins une copie de la note de Shustin, si jamais, mais je ne veux pas vous distraire de votre tâche prioritaire...

$\bullet$ \[15.01.13–18h30\] Dear Evgenii, Ilia, Oleg and Alexis (and Felix Klein),

I was much fascinated those last days by Evgenii’s counterexample to (one part of) Rohlin’s maximality conjecture to the effect that a real scheme is of type I iff it is maximal in the hierarchy of all schemes. Quite interestingly this work of you (Shustin) also destroys an old (semi-)conjecture of Klein (1876) positing that any nondividing plane curve can acquire a solitary node by crossing only once the discriminant (the resulting Morse surgery then sembling like the formation a champagne bubble arising like a blue sky catastrophe of little green men’s coming with flying saucers).

Alas from Shustin’s note of 1985 (in its English translation), I was not quite able to understand your proof (compare optionally the attached file, on page 248, in the paragraph right before Figure 94\[=Fig.\[Shustin:fig\]\]). In fact I do not know how to prohibit the $(M-1)$-scheme $4/1 2/1 1/1 11$ enlarging Shustin’s (M-2)-scheme. Alas I am not an expert in the field and I feel quite shameful disturbing you with such a detail. Despite having myself full Leningradian origins (through my father), I do not master the Russian language so that it may well be the case that the original Russian text is more detailed than its translation. Of course it is much more likely that I missed something well-known, that you perhaps may not have made completely explicit in the note? (Incidentally I send you a copy of Shustin’s note for convenience!)

I apologize for this question of detail, yet it seems quite important to me for your result of 1985 is the only obstruction (I am aware of) to the naive desideratum of truth about Klein’s conjecture. Klein himself is extremely clever and quite ambiguous about stating this as a conjecture or as a result (compare optionally Klein’s original quote reproduced on page 206 of the attachment). Today I managed as a simple exercise to check the truth of Klein’s hypothesis in degree 6, via an armada of Russian results (especially Itenberg 1994 contraction principle for empty ovals), plus the Klein-Marin theorem (for the details of this exercise cf. optionally Prop.24.24\[=\[Klein-vache-deg-6:prop\]\] on page 235 of the attached text).

You, Oleg Viro, in the preface of that volume presenting Itenberg’s article (1994) advanced the (crazy?) conjecture that one might always be able to contract empty ovals!!! Do you know if there is meanwhile some counterexample (in high degrees)? Of course there is some vague parallelism between Itenberg’s contraction and the one required to implement Klein’s hypothesis (which must amount shrinking an anti-oval, i.e. an invariant circle acted upon antipodically by conj).

Sorry again for disturbing you with all these naive questions, and do not take the pain answering me properly if you are overwhelmed by other more important duties. Many thanks for all your attention. Sincerely yours, Alex (Gabard)

$\bullet$$\bullet$$\bullet$ \[16.01.13–02h57: Oleg Viro\]

Dear Alexandre,

I do not mind to pose crazy conjectures. I do not mind if my crazy conjecture would be disproved.

However, I suspect that my conjecture is not as crazy as possibility of shrinking of an anti-oval. The difference between the oval and an anti-oval is that the oval is assumed to exist and be empty, i.e., not linked with the complex curve in whatever sense, while the anti-oval apparently has none of these properties.

I am not aware about any counter-examples that you ask about. I do not bet that they do not exist, but find the question stimulating, and better motivated than the conjecture that was proven to be wrong.

Best, Oleg

$\bullet$$\bullet$$\bullet$ \[16.01.13–14h56: Stepan Orevkov\]

A small remark:

It is wrong that $11 U 1<1> U 1<2> U 1<4>$ is not a part of an $(M-1)$-scheme. It is[^102]. Moreover, there is no known example of $(M-2)$-curve of type II which cannot be obtained from an $(M-1)$-curve by removing an empty oval.

In contrary, there are $(M-1)$-curves of degree $8$ (which are necessarily of type II) which do not come from any $M$-curve. These are:

$3<6>$

$4 U 1<2> U 2<6>$

$8 U 2<2> U 1<6>$

$12 U 3<2>$

Constru\[r\]ction (inspired by Shustin’s construction of $4 U
3<5>$):

Consire\[der\] a tricuspidal quartic $Q_{sing}$ symmetric by a rotation $R$ by $120$ degree and perturb\[e\] is\[=it\] so that each cusp gives an oval (we assume that this perturbation is very small). Let $Q$ be the perturbed curve. Two flex points appear on $Q$ near each cusp of $Q_{sing}$. We chose flex points $p_0, p_1,
p_2$ (one flex point near each cusp) so that $R(p_0)=p_1,
R(p_1)=p_2, R(p2)=p_0$. We choose homogeneous coordinates $(x_0 :
x_1 : x_2)$ so that the line $x_i = 0$ is tangent to $Q$ at $p_i$ $(i = 0,1,2)$.

Let $C$ be the image of $Q$ under the Cremona transformation $(x_0
: x_1 : x_2) \mapsto (x_1x_2 : x_2x_0 : x_0x_1)$. Then $C$ has 3 singular points, each singular point has two irreducible local branches: a branch with $E6$ and a smooth branch which cuts it “transversally”. By a perturbation of $C$ we obtain all the four curves mentioned above.

The fact that these curves cannot be obtained from $M$-curves immediately follows from the fact that, for any $M$-curve of degree 8 of the form $b U 1<a_1> U 1<a_2> U 1<a_3>$, all the numbers $a_1$, $a_2$, $a_3$ are odd[^103].

Best regards Stepa O

$\bullet$ \[17.01.13 ca. 23h00\]

Dear Oleg and Stepa,

Many thanks for all your fascinating remarks and detailed answers. I look forward digesting them carefully tomorrow.

Sorry for my late reply as I have no internet at home and was quite busy trying to understand some basic facts, notably that one may have some “eversion” of a real scheme when the oval explodes at infinity undergoing a Morse surgery not affecting its connectedness. This implies that there is some hidden passages in the Gudkov-Rohlin pyramid of all sextics changing a Gudkov symbol $k/l \ell$ to its mirror $\ell/1 k$. The resulting combinatorics of this graph looks quite formidable and I wonder if it is known whether each of those secret edges corresponding to eversions (except those linking $M$-curves) can be explored algebraically. Perhaps the problem is related to Ilia’s shrinking process for empty ovals, but seems to involve yet another species of “anti-ovals”, namely those with two fixed points under conj, yet located on the same oval. All what I am saying is for sure well-known to you since time immemorial, yet I was very happy to understand this point which solved several paradoxes of mine, notably those related to the degree of the discriminant and the contiguity graph between chambers residual to the discriminant under elementary algebraic Morse surgeries, as Oleg says. Of course, I shall send you an updated version of my file, when I manage to reorganize slightly the exposition.

Many many thanks for all your excellent answers! All the best, Alex

$\bullet$ \[18.01.13 ca. 10h00, Viatcheslav Kharlamov\]

Dear Alex,

I followed rather attentively the discussions, but kept silence since had no much to add to the reaction of the others.

This “eversion”, as you call it, played some important rôle in the prehistory of the Gudkov conjecture. As you probably know, the first classification declared by Gudkov was wrong, and it is one of his “thesis referees”, Prof. Morosov, who had objected the first classification exactly because of a small irregularity with respect to “eversion” of the answer. Repairing this asymmetry Gudkov came to his final result, and, if my memory is correct, in particular, at this stage discovered the missing $M$-curve.

If honestly, I don’t remember did somebody ever before discussed seriously any conceptual explanation to this “eversion”. However, it was implicitly present in all results obtained through $K3$ and their lattices. Recently, studying the shadows of cubic surfaces with Sergey Finashin and having proven, to our own surprise, for them a very similar “symmetry”, which we have called “partners relation”, we have formalized it as follows.

First level of explanation is coming from lattices of double coverings: the partner relation consists indeed in transferring an $U$-summand (unimodular even lattice of rang $2$ and signature $0$) from one eighenlattice to another. Second level of explanation is coming from moduli in terms of periods: each partner in the partner pair can be deformed to a triple conic, near the triple conic the family looks as $Q^3+tbQ^2+t^2cQ+d=0$, and switching of the sign of $t$ (passing through the triple conic) replace curves of one deformation class by curves from the partners class: moreover, such degenerations are deformationally unique. Literally the same explanation (and with much easier proofs at the both levels) works for nonsingular sextics (the shadows are sextic curves with $6$ cusps on a conic; remarkably, in many respects they behave in a way more similar to that of nonsingular sextics, than other sextics with singularities).

Yours, Viatcheslav Kharlamov

$\bullet$ \[18.01.13, Kharlamov, title of message=Correction\] Writing the message a bit in a hurry I did not describe fully and appropriately the partner relation at the lattice language. The summand $U$ does play a crucial rôle, and it should be moved from one eighenlattice to another, but then additionally one should exchange the eighenlattices. In fact this $U$ contains indeed the $2$-polarization vector, $h¨2=2$, and thus the eighenlattice containing this distinguished U is aways $(-1)$-eighenlattice.

The existing exception to the partner relation (as I remember, in the nonsingular case, there is only one) is the case when the $(-1)$-eighenlattice does not contain such a pair $(U,h)$.

Sorry, for being in a hurry, but I should stop at this point. Hope that now it is more clear.

$\bullet$ \[18.01.13, Gabard, ca. 21h00\] Dear Viatcheslav, Oleg and Stepa (and all the others),

So many thanks for all the excellent comments, especially on Morosov. There was some allusion to this issue in Viro’s survey from 2006, in Japanese Journal of Math, as to the lack of symmetry in Gudkov’s initial answer. Yet Morosov was not mentioned if I remember well...

On my side I was quite stimulated by the last letter from Oleg, about the contraction conjecture, as looking indeed much more realist than Klein’s Ansatz on the champagne bubbling in any nondividing curve. I attempted today to imagine what sort of proof one could expect to find for this fascinating Itenberg-Viro contraction conjecture of empty ovals.

After some trials with orthogonal trajectories to the functional computing the area of the empty oval, I arrived at some sort of strategy (probably completely fantasist) consisting in using the Riemann mapping theorem as applied to the interior of the empty oval. Naively as the contour is algebraic so is the Riemann map and hence the concentric sublevels of it ought to be algebraic curves of the same degree!!??? This would give the shrinking.

I am sure that tomorrow while checking more carefully the details all this argument will crash down. Hence sorry for this premature message. Some more details about this and my naive understanding of “eversions” are in the attached file, especially Section 24.15 (p.251) and p.242 (Section 24.12 for eversions). Regarding eversions I wonder which edges in the Gudkov pyramid are actually realized algebro-geometrically? All, except those connecting the $M$-schemes is my naive guess, yet it is probably too optimistic...

Many thanks again for sharing all your knowledge on that fascinating topic, and all your exciting letters. All the best, Alex

\[21.01.13, ca. 20h00\]

Dear real geometers,

Thank you again, Oleg, Alexis, Stepa, and Viatcheslav, for all your messages which I have carefully integrated in my TeX-notes, and to which I frequently refer for citation in my text. Your messages suggested me several ideas I would never have explored without your precious hints.

On my side, I noticed of course that the cavalier Riemann mapping strategy toward the (Itenberg-Viro 1994) contraction conjecture (CC) of empty ovals fails blatantly (cf. Section 25.7(=\[CC-via-Riemann:sec\]), pages 255-258, roughly even if the Riemann map of an algebraic oval would be algebraic then its degree seems to be twice as big as it should, or better the polynomials arising as norms of algebraic Riemann maps are not the most general representatives of their degree!!!). Perhaps the Riemann method works for special ovals, but of course they are unlikely to be interspersed in all chambers of the discriminant! This failure drifted me toward another formulation of the contraction conjecture which I call CCC, for collective contraction conjecture. This posits that all empty ovals of a real algebraic curve can be contracted simultaneously toward solitary nodes (by a path having solely its end-point in the discriminant). This looks even more “crazy” than CC, but I found no counterexamples (in my pockets). I would much appreciate if you already thought about this natural variant, especially if you detected some counterexample (perhaps arising from the Viro-Itenberg patchworking method or the dissipation of higher singularities, with which I am alas still unfamiliar with, like in Shustin’s counterexample to Rohlin’s maximality conjecture).

Here are the trivialities I managed to prove. Via Brusotti 1921, it is plain that CCC implies the usual contraction conjecture (CC) (cf. details in Lemma 25.22(=\[CCviaCCC-Brusotti:lem\]) on page 263 of the attached file). On the other hand CCC implies (as a large deformation principle) several well-known prohibitions. E.g. a two-seconds proof of the Hilbert-Rohn-Petrovskii prohibition of the sextic $M$-scheme $11$ (eleven ovals without nesting), as well as Rohlin’s prohibition of the sextic scheme $5$ of type I (by the way causing the unique asymmetry in the Gudkov-Rohlin table of sextics). Under CCC, all these facts appear as trivial consequences of Bézout (compare Section 25.8(=\[CCC:sec\]) on pages 258-259). I found this simplicity quite exciting (even though it leads to nothing new as compared to Arnold-Rohlin). One can wonder if Hilbert already used this, at least as a heuristic tool??? Philosophically, I found also interesting that such large deformation conjectures produce prohibitions, in contradistinction to small perturbations as being primarily a method of construction (Harnack-Hilbert-Brusotti, etc.). There is accordingly some nice duality between Luigi Brusotti and Ilia-Oleg’s contraction conjecture. Of course you surely noted this issue a long time ago, yet for me it was a happy discovery (yesterday).

Perhaps CCC and CC are actually equivalent, yet this looks more hazardous but maybe not completely improbable... (One would just have to synchronize the death of all ovals posited by CC.) This is all the modest news I have collected during the week-end. Of course I still have some naive hope that CCC (hence CC) could be attacked via some gradient flow, but it looks quite difficult to locate the right functional (or Morse function). Looking at the area (or length) of all empty ovals is probably too naive...Perhaps some “degree of roughness” à la Gudkov could be projectively more intrinsic and useful...

Thank you so much for your attention and all your brilliant letters and answers, while apologizing me for sending you only easy doodlings. Best regards, Alex

\[26.01.13, ca 20h00\]

Dear Oleg, Stepa, Viatcheslav, Evgenii, Alexis, Thomas, etc.

I continued my naive investigations of real plane curves. What a beautiful story! I finally “understood” and studied in detail the marvellous construction of Gudkov $\frac{5}{1} 5$ (ca. 1971-73), as to understand the more tricky (but related) construction proposed by Stepa, which I attempted to depict on Fig. 111(=\[Orevkov2:fig\]) of the attached file. (I did not as yet assimilated the full details but feel on the good way. In fact I tried to use the dissipation of $Z_15$ in Viro’s survey from 1989/90 in Leningrad Math. J., which I hope is the same as the $E_6$ advocated by Stepa. Sorry for being very ignorant about singularities...)

Yesterday, I also finally understood the correctedness of Evgenii’s argument. (As helped by Stepa’s e-mail, the point which I missed is this obstruction of Viro extending that of Fiedler) for $M$-schemes of degree 8 as having necessarily “odd content”.

On the other hand, I was scared (since three days) by the fact that something which I subconsciously thought as evident (or rather which I was sure to have read somewhere) is perhaps not true. My (naive) question is whether two empty curves are necessarily rigid-isotopic? This looks at first between metaphysical nonsense and “triviality”? Maybe it is unknown, when $m$ is large enough. (m=6 follows from Nikulin 1979, and as far as I know there is not a simpler proof, say valid for all (even) degrees). So I am quite shameful asking you about this point: Is the empty chamber always connected?

I tried a dynamical approach (to this problem) in Section 25.12(\[rigidity-empty-scheme-via-dyna:sec\]), but it is not very convincing. On the other hand, if the empty room is connected, then maybe the space of all curves with one component is also connected? (Naively one would apply the Itenberg-Viro contraction conjecture, to reduce to the empty case, move there for a while to resurface at the other curve (the contraction thereof). Perturbing this path in the “visible world” would conclude the proof modulo some difficulties...) Again, you Oleg, in your wonderful survey of 2008 (in Japanese J. Math) lists as an open problem the question of deciding the rigid-isotopy of curves of odd degree having a unique real circuit. As you emphasize the word “odd degree”, I wondered if the case of even degree (again with only one oval) is already settled?

In Section 25.10(=\[CCCviaDynamics:sec\]), I have attempted a naive dynamical approach to the collective contraction conjecture(CCC). This states that we can shrink simultaneously all the empty ovals toward solitary nodes. This is a bit like a perfect landing in flight simulator where all wheels touch the ground simultaneously. My naive strategy is just to study the gradient lines of the functional measuring the total area of all empty ovals, but it is surely not serious. It would be exciting, in my opinion, to describe a counterexample to CCC if there is one.

Many thanks for the attention, all your patience about my naive reasonings, and above all for the brilliant answers you already gave me. Best regards, Alex

$\bullet$$\bullet$$\bullet$ samedi 26 janvier 2013 20:15:54, the prompt response of Eugenii Shustin:

Dear Alex,

The chamber of empty curves of a given (even) degree is indeed connected: two such curves can be defined by homogeneous polynomials, positive for any real not all zero variables, and their linear homotopy $(1-t)P+tQ$, $0\le t\le 1$, gives a path in the chamber of empty curves.

By the way, another (well) known connected chamber consists of hyperbolic curves (i.e. those which have totally real intersection with lines of certain pencil) - this is a consequence of Nuij W. A note on hyperbolic polynomials. Math. Scandinavica 23 (1968), no. 1, 69–72.

With best wishes, Eugenii

$\bullet$$\bullet$$\bullet$ samedi 26 janvier 2013 21:08:27, Oleg Viro:

Dear Alex,

The counterpart of the Rokhlin conjecture[^104] about rigid-isotopy of any two curves of odd degree with one component is the obvious observation described by Evgenii, about empty curves of even degree. The question about curves of even degree with a single oval is equivalent to the question about removing this single oval by an algebraic Morse modification. I don’t think it was ever discussed, but I could miss it.

$Z_{15}$ is not $E_6$. The easiest way to construct the Gudkov $M$-curve is by perturbing two $J_{10}$ singularities of the union of 3 non-singular conics tangent to each other at 2 points.[^105]

Best, Oleg

$\bullet$ \[28.01.13, lundi 28 janvier 2013 20:03:58\] Gabard wrote euphorically[^106] an e-mail titled “Some more metaphysical non-sense about the rigid-isotopy of empty curves?”:

Dear Eugenii, Oleg, Viatcheslav, Alexis, Stepa, etc.

So many thanks, Eugenii, for putting me again on the right track, and recalling me the argument which I shamefully forgot about. Yesterday, I was quite excited by trying to digest your argument (albeit it seems so simple). In fact the little detail that worried me is that I do not know why during the linear homotopy $(1-t)P+tQ$ the variable curve could not acquire (while staying of course empty if $P,Q$ have both the same sign) a pair of conjugate nodal singularities. This puzzled me for a while, and then using systematically your argument, I arrived at the somewhat opposite conclusion that the empty (smooth) chamber must be disconnected (for all even degrees $m\ge 4$)!!! This violates all what we know since Rohlin 1978 (and surely Gudkov as well??), while the former refers directly back to the argument of Klein 1876 based on Schläfli cubics surfaces $F_3$’s and Zeuthen correspondence between cubic surfaces and quartics (via the apparent contour). Klein’s proof is a bit tricky and uses as well his rigidification (Klein 1873) of Schläfli’s isotopic classification. Needless to say I could not follow Klein’s reasoning completely, as I just studied it today for ca. 2 hours. Marin informed me recently that he, in contrast, was able to digest all of those Kleinian works!

So using your method of linear homotopy, one sees quickly that the (cone) space $C^+$ of positive anisotropic (=not representing zero) forms is contractile (convex actually) hence simply-connected. Its projection in the space of curves is the invisible locus $I$ consisting of all empty curves. Since the latter is merely a quotient of $C^+$ (by positive homotheties) it follows that it is also simply-connected (via the exact homotopy sequence of a fibering). But the discriminant is visible inside this invisible locus $I$, since it is a simple matter via Brusotti (1921) to construct empty curves with a pair of conjugate nodes. Thus we see inside the simply-connected manifold $I$ a certain hypersurface (namely a portion of the discriminant), which by Jordan-Brouwer (or a slight extension thereof) should separate this manifold $I$ in pieces (at least so is my naive intuition). It follows that our empty chamber (consisting of smooth curves) is disconnected!!!! This is my proof in its broad lines (for more details, compare Section 25.13, page 282-283 of the attachement, Theorem 25.29 and its proof on page 283). This is just one page long...

Since this conclusion contradicts violently what is asserted by Klein 1876 (and approved by Rohlin 1978), it is of course very likely that my proof contains a serious flaw, or at least that I am confusing somehow the basic conceptions. However presently I do not see where is my mistake! Of course, my pseudo-theorem also violates the part of Nikulin 1979 concerned with the rigid-isotopy of the empty chamber of sextics.

Many thanks again for your attention, and sorry for overflowing your mail boxes with my naive questions (and dubious reasonings). Thank you again so much for all your excellent and detailed responses (especially on $E_6$ and $Z_15$).

Best wishes, Alex PS: I send you a copy of my TeX-file in case someone would like to work out a specific passage. At the occasion I would also be happy to send you my figures in zipped format so that one of you can continue the project in case I make a fatal bicycle accident (like Academician V.I. Arnold?)

$\bullet$ \[30.01.13, 18h10\]

Dear Oleg, Eugenii, and the other experts,

I think that I found the mistake in my “proof” of the disconnection of the empty locus (that you certainly noticed meanwhile in case my explanation is the correct one). The reason seems to be simply that the discriminant inside the invisible locus has only real codimension 2, hence cannot separate anything. I have attempted to explain this in Section 25.14 on page 288. If this is not wrong it seems that the next natural question is to decide which chambers residual to the principal stratum of the discriminant contains such smaller pieces of the discriminant shrunk to codimension 2. I think to have found a topological obstacle for $M$ and $(M-1)$-curves, and conjecture (very naively) this to be the sole obstruction. In more geometric terms, this amounts essentially to decide which smooth curves can acquire a pair of imaginary conjugate nodes.

Many thanks, Eugenii and Oleg, for your detailed answers. As you said, it seems that the (Itenberg-Viro) contraction conjecture of empty ovals implies the rigidity conjecture for even order curves with a unique oval. However I should probably still try to understand this implication in some more details. Perhaps it is somehow related to the previous codimension 2 phenomenon inside the “invisible” chamber.

Sorry for all my confusing messages, and many thanks again for all your kind efforts in trying to educate myself. All the best, Alex

$\bullet$ \[01.02.13, ca. 20h00\] Obstruction to rigid-isotopy (strictly) below height DEEP+2?

Dear Oleg, Eugenii, Alexis, Thomas, Stepa, etc.

Many thanks for all your brilliant messages and articles I am still slowly trying to assimilate properly. I hope not taking too much of your precious time. Albeit I met all of you only rarely, I remind very accurately your brilliant talks (in Geneva or Rennes), and so it is a special pleasure to remind each of yours while trying to explore this fantastic topic.

On my side I was those last two days fascinated by the conjecture that the one-oval scheme ought to be rigid, as Oleg or Rokhlin conjectures. (Let me say that a scheme is rigid, if all the curves representing it are rigid-isotopic.) Given a degree $m$, one may wonder what is the smallest height $r(m)$ at which there is a non-rigid scheme. (For me the height of a scheme just means its number of components.)

For any degree $m$, there is of course the deep nest with $r=[(m+1)/2]=:DEEP$ real branches. Two units above the latter’s height, it is easy to construct (for each $m$) curves having the same real scheme yet different types (I vs. II) hence not rigid-isotopic. (This is a simple iteration of Rohlin’s construction in degree $5$, cf. Figs. 102, 103 in my file). Using the Marin-Fiedler method of the lock it is even possible to exhibit at this height $DEEP+2$ curves of degree 7 or 9 having the same real scheme and the same type II, yet not rigid-isotopic (Figs. 105, 104). (Probably the method extends to all other odd degrees.) However, it seems much more tricky (and the lock-method seems ineffective) to detect obstruction below this height $DEEP+1$ (i.e. one unit above the height of the deep nest). Could it be the case that all schemes at or below this height are rigid? Of course this looks super-optimistic as we do not even know rigidity at height one, but I was unable to find a counterexample. I would be very happy if you know one? If there is a simple candidate, I hope to detect it alone during the next few days…. So do not take care answering me if my question is trivial. (As I just work on this since two days, I probably missed something accessible.)

Paraphrasing slightly, I found quite puzzling, that the very explicit function $r(m)$ measuring the smallest height of a non-rigid scheme is only subsumed to the large pinching $1 \le
r(m)\le [(m+1)/2]+2=DEEP+2$. Of course a better lower bound seems out of reach, but perhaps you know better upper bounds.

I also wondered if there is an extension of the Nuij-Dubrovin rigidity of the deep nest to, say, the totally real scheme of degree 8 consisting of 4 nests of depth 2. I should think more seriously on this at the occasion.

Many thanks for your attention, and sorry again for all my enthusiastic and naive e-mails. All the best, Alex

$\bullet$ \[written 08.02.13 and sent 09.01.13\]

Dear Oleg, Eugenii, Stepa, Viatcheslav, etc.

I still continued my trip through real plane curves and cannot say that my curiosity is starting to fade out. I tried for several days to find a counter-example to the conjecture (of mine so probably quite wrong) that all schemes below height $DEEP+2$ are rigid, where $DEEP=[(m+1)/2]$ is the number of branches of the deep nest of degree $m$. At least the method of the lock (Fiedler-Marin) seems quite inoperant to detect an obstruction to rigid-isotopy at such low altitudes. If true, the proof probably involves a geometric flow collapsing either the pseudoline to a line (by shortening its length like a systole) or improving the rotundity of some oval to a circle (via an isoperimetric functional?). If all this works, it would reduce the low-altitude rigidity conjecture to Nikulin’s theorem (or maybe even Klein’s on $C_4$) as the starting step of a big recursive process. Of course this seems still quite out reach (canary music) unless one feels very motivated!

Next I tried to corrupt the truth of Slava’s remarkable rigid-isotopic classification (Nikulin 1979) of sextics via the Marin-Fiedler locking argument using Bézout saturation. Of course I have nothing against Slava, but this was rather intended to test experimentally the power of Nikulin’s result. Specifically I looked at sextic schemes of the form $3/1 \ell$, and wondered if for some specific curves the distribution of the $\ell$ outer ovals away the fundamental triangle traced through the 3 inner ovals (those enveloped by the unique nonempty oval) could be different for different curves. On all examples I tested it seems that the outer ovals are never separated by the “deep” triangle. So we find no violation of Nikulin’s theorem, and the latter rather implies that as soon as we are able to visualize the distribution for a single curve it will be the same for all curves belonging to this scheme. The case most tricky to understand is the maximal permissible, namely $3/1 5$. I managed to construct it à la Harnack (as preconized in Gudkov 1974 or 1954). But being quite unable to decide from this model the distributional question of the outer ovals past the fundamental triangle, I decided to switch to Oleg’s method of construction via dissipation of the singularities of a triplet of coaxial ellipses. I played this game yesterday but could not decide the distributional question for this Viro curve (cf. especially Fig.126 on page 320 and the hypothetical Theorem 26.29 on page 322). In fact today I tried again to inspect directly Harnack construction and found Lemma 26.27 on page 319 whose proof seemed to me very transparent until I found the little warning, which I think is not fatal. In conclusion I believe now that there is no separation by the fundamental triangle!!!??

Of course I imagine that, if I am not completely wrong, what I am investigating must be quite familiar to you. I would much appreciate if you know if this hypothetical theorem (26.29 page 322) is true. It amounts essentially to check whether in Viro’s construction of $3/1 5$ the triangle through the 3 deep inner ovals does not separate the 5 outer ovals. I find this question quite attractive as it seems to require some understanding of the geometric location of the microscopic ovals arising in Viro’s method (optionally compare Fig. 127 (page 322) which shows a scenario with the two bottom micro-ovals aligned vertically in which case the fundamental triangle would separate the outer ovals). This scenario seems to me quite unlikely but it does not seem to be impeded by naive Bézout obstructions.

Many thanks for your attention, and sorry again for all my naive and confuse questions. Thank you very much again for your precious guidance and answers.

All the best, Alex

$\bullet$$\bullet$$\bullet$ (10.02.13) Bonjour Alexandre, Thomas m’a transmis ta question. La réponse est toute simple: soient $A$, $B$, $C$ trois ovales intérieurs et $D$, $E$ deux ovales exterieurs de ta sextique. Le triangle fondamental $ABC$ est entiérement contenu dans l’ovale non-vide. Si $D$ et $E$ sont dans deux triangles $ABC$ (non-fondamentaux) différents, alors la conique passant par $A$, $B$, $C$, $D$, $E$ coupe la sextique en $14$ points, contradiction. Avec des coniques, on montre plus généralement que: Les ovales vides de la sextique sont distribués dans deux chaines (int, ext), l’ordre cyclique est donné par les pinceaux de droites basés dans les ovales interieurs. Les ovales interieurs sont disposés en position convexe dans l’ovale non-vide. Bon dimanche,   Séverine

$\bullet$ \[12.02.13\] Is the Gudkov chamber simply-connected?

Dear Séverine, Viatcheslav, Ilia, Oleg, and all the other experts,

First many thanks, Séverine, for your excellent answer on my distribution question of ovals of sextic, and sorry for my late reply on it as I lack an Internet connection at home.

I tried today to understand when a dividing (plane) curve admits a transmutation, i.e. a rigid-isotopy permuting both halves of the curve. I also studied the weaker notion of mutation of when there is a linear automorphism of the plane permuting both halves. Using the Kharlamov-Itenberg calculation of the monodromy of sextics I think that I managed to get some obstruction to mutability, especially for the 3 dividing curves which have trivial monodromies (compare Lemma 26.6, page 288, which is hopefully correct). However I don’t know if the Gudkov chamber (or the 2 other related “antidromic” chambers, i.e. having trivial monodromies) is simply-connected. I hoped to detect some non simple-connectivity by looking at the monodromy induced on the halves instead of the ovals. At least this works of course for the deep-nest chamber which is not simply-connected since there is a symmetric model which can be mutated. So my (hopefully not too naive) question is the following: is it known whether or not the Gudkov chamber is simply-connected? (equivalently is the Gudkov curve transmutable?) The same question looks attractive for the other 2 antidromic curves, i.e. the left wing “Rohlin curve” $6/1 2$ and $4/1 4$ in type I.

Thank you so much for all your attention and patience, and in advance for your answer if it is known. Best regards, Alex

$\bullet$$\bullet$ $\bullet$ mercredi 13 février 2013 04:34:12

Dear Alexandre, If I understand correctly the question then the answer is not, if I state the question appropriately then the answer is yes. I mean the following precise statements. Let consider the part of the projective space of real sextics that is represented by maximal sextics of Gudkov’s type. Then the fundamental group of this part is $Z/2$. It becomes simply connected after taking quotient by the natural action of $SL(3,R)$. In fact, before factorization it is a fibration over contractible base with the fiber $SL(3,R)$. These results (and there analogs for other maximal sextics and certain curves of lower degree) are contained in my talk On monodromies of real plane algebraic curves at one of Petrovsky seminars in 80th, I guess (short summary should be found in Russian Surveys). The proof (in the case of sextics) is rather straightforward as soon as based on the $K3$ surfaces periods uniformization. As it happens rather often with this approach, to treat the maximal curves is extremely easy, since the corresponding eighenlattices become unimodular. In general the period domain, which is the product of two polyhedra in the real case, represents the studied sextics (or associated K3 surfaces) only up to codimension 2. Which makes laborious to treat the fundamental group. But, surprise, in the case of maximal curves there are no codimension 2 phenomena, since such holes appear only as traces of $(-2)$-cycles having nontrivial components in the both eighenspaces, which is impossible since in the maximal case the components are integral and the eighenlattices are even. I don’t remember by heart the final result for other maximal sextics. It should be pointed in the same summary and by the way easy to get following the same approach I have pointed. The key is that even if it is no more a pure fibration - it has special fibers which are quotients of $SL(3,R)$ by the corresponding monodromy group (which indeed coincides with the maximal possible group of symmetries for the given type of sextics) - its fundamental group is exactly the fundamental group of this special quotient. Yours VK

$\bullet$ mercredi 13 février 2013 11:46:20

Dear Colleagues, Am I alone who did not receive a copy of Severine’s letter? I would be happy to know its content :) Yours VK

$\bullet$ mercredi 13 février 2013 13:43:10

Dear colleagues, I had written only to Alexandre, sorry! My answer was this: let $A, B, C$ be three inner ovals, and $D, E$ be two outer ovals of the sextic. The fundamental triangle $ABC$ is entirely contained in the nonempty oval. If $D$ and $E$ are in two different (non-fundamental) triangles $ABC$, then the conic through $A, B, C, D, E$ cuts the sextic at 14 points, contradiction. Using conics, one proves more generally that there is a natural cyclic ordering of the empty ovals, given by the pencils of lines based at the inner ovals. The empty ovals are distributed in two consecutive chains (inner, outer). The inner ovals lie in convex position in the nonempty oval. Best regards, Séverine

$\bullet$ \[14.02.13\]

Dear Viatcheslav, Séverine and all the other colleagues,

Thank you very much for this beautiful answer on the Gudkov chamber. I look forward to digest properly all that incredible technology that you and Nikulin developed. Again many thanks also to Séverine for the clever argument which I digested yesterday with great pleasure, and integrated in my notes in Section 26.10(=\[LeTouze:sec\]) pages 332–334. This gave me yesterday some motivation again to attack the very first question of all our chat room, namely Rohlin’s claim that the pencil of cubics through the 8 deep basepoints located inside the 8 empty ovals of any sextic curve of type $6/1 2$ or its mirror $2/1 6$ is totally real, hence of type I (also called orthosymmetry by Klein ca. 1881-82 and his student Weichold 1883).

In fact I (naively) hoped to prove this Rohlin claim via Poincaré’s index theorem, yet the qualitative picture (Fig. 133 on page 337) rather inclined me to believe that the proof cannot reduce to mere combinatorial topology of foliations (i.e. Poincaré’s index formula of 1885). So I am still puzzled, but perhaps an argument like Séverine’s one do the job. At any rate I would be very excited if someone manages to reconstruct this proof asserted by Rohlin (1978) if it is not too tantalizing for the brain.

Otherwise I am also much frustrated by failing to visualize totally real pencil on the three $M$-sextics, whose existence is I think predicted by Ahlfors theorem of 1950 (or better the special zero-genus case thereof known to Riemann 1857, and reworked by Schottky 1875-77, or even Bieberbach 1925 and his more respectable student Grunsky 1937). Marin warned me recently that the transition from the abstract Riemann surface viewpoint to the planar context “of Hilbert’s 16th problem” may be not so easy as I always assumed subconsciously. (If necessary, all the correspondence I received from all the colleagues is gathered in Section 24.6, p.221). Overpassing this difficulty (which I hope is not fatal) there should be on all $M$-curves (more generally dividing curves) auxiliary pencils which are totally real. Alas for $M$-sextics (even $M$-quintics), I am completely unable to trace them and know nothing about the degree of the curves involved (in the pencil). I hope to be able to tackle such questions in the future, but perhaps you have better ideas (or motivations) than I do have.

Thank you very much again for all your brilliant answers, and kind messages. All the best, Alex

$\bullet\bullet\bullet$ samedi 16 février 2013 17:54:55

Dear Alexandre, dear other colleagues,

I have managed to prove that a pencil of cubics with eight base points distributed in the eight empty ovals of a sextic $2 \cup 1(6)$ is necessarily totally real. Details will follow soon in a paper. Yours,

Séverine

$\bullet$ \[16.02.13,19h41\]

Dear Séverine and colleagues,

Congratulations for this fantastic achievement. I am sure the proof must be very beautiful. On my side I tried to work out for all sextics of type I an optical recognition procedure of the type by some synthetical procedure akin to Rohlin’s claim, yet this is still much in embryo. In particular the case of $(M-4)$-sextics is quite puzzling as it seems to contradict the version of Ahlfors theorem due to myself (existence of a totally real map of degree the mean value the number of ovals and Harnack’s bound). I hope to send you more palatable material soon, but confess that the questions look quite hard and I seem much less efficient than Séverine. So I suppose that Rohlin’s claim is one among several other (less pure) total reality result. So I look forward with great interest to see Séverine’s article.

All the best, Alex

\[19.02.13\] Dear colleagues,

Many congratulations again to Séverine for your fantastic achievement. Sorry to have been brief in my last letter, as I wrote (lacking an internet connection at home) from a friend of mine who had a romantic party with his girlfriend, and I do not wanted to disturb too long his romantic evening. Meanwhile I also tried hard to concentrate on a proof of the Rohlin-le Touzé’s theorem, which still overwhelms my intelligence. The last things that I have written are on pages 336–352 (Sections 27.1, 27.2, and 27.3), but this is poorly organized and supplies no serious proof of the Rohlin-Le Touzé’s theorem.

Some few days ago, I got Theorem 27.5 (on page 346), which (if it is correct) answers one of the question I asked in my penultimate e-mail (as well as desideratum of Alexis), namely the question of estimating the order of curves involved in a total pencil on an $M$-curve. It seems that there is always such a pencil of order $(m-2)$, i.e. two units less than the given degree $m$ of the $M$-curve. In fact, the proof is a nearly trivial adaptation of the abstract argument going back to several peoples (in chronological order Riemann 1857, Schottky 1875, Enriques-Chisini 1915, Bieberbach 1925, Grunsky 1937, Courant 1939, Wirtinger 1942, Ahlfors 1947, 1950, a myriad of Japaneses, a myriad of Russians including Golusin 1953/57, etc....., up to Huisman 2000, and Gabard 2001/2006, who else?). The point is that total reality is trivial in the case of $M$-curves since we have one point circulating on each oval (such a group moves by Riemann-Roch!!!) and so we have like a train-track with only one train on each track, hence no collision can occur and total reality is automatic. If we work with plane curves we only need to take curves of order $(m-2)$ which have enough free parameters to pass through any given distribution of $M$ points (one on each oval), and this works by looking at the residual group of points (details in the proof on page 346). So this is quite interesting but probably only a first step toward deeper things. (One could dream to recover all the Gudkov-Rohlin/Arnold congruence via this method but that looks hard work...)

After this little discovery I focused again on the Rohlin-Séverine theorem, yet without any success. So I have not more to report for the moment.

Thanks a lot for the attention, and all my congratulations again to Séverine for your deep advance. Best wishes, Alex

$\bullet$ 19.02.13 Dear Alex,

Let me ask you a question from your previous field of interest. Do you know any example of a non-Hausdorff 1-manifold which does not admit a differential structure? I heard about existence and could easily construct examples of exotic, i.e., homeomorphic but not diffeomorphic non-Hausdorff 1-manifolds. See

http://www.map.mpim-bonn.mpg.de/1-manifolds

Sincerely, Oleg

20.02.13

Dear Oleg, David and Mathieu,

Many thanks, Oleg, for your lovely question, and best greetings to the other friends. Alas my memory is failing quite dramatically, so my answer will be of poor quality. If I remember well I asked myself the same question some 3-4 years ago, but I cannot record to have ever found an answer. Thus I forward your question to David and Mathieu, the leading experts of non-metric surfaces who perhaps will supply a better answer. On my side I hope to think more seriously to your question when I see clearer with Rohlin-Le Touzé’s sextics.

Maybe a first idea is that there ought to be a (non-canonical) “twistor construction” assigning to each non-Hausdorff curve a Hausdorff surface fibered by (real) lines. This construction should go back to Haefliger’s very first note in the colloque de topologie de Strasbourg ca. 1955-1956 (yet it is not very detailed). In substance it is like a train-track construction à la Penner-Thurston…(some intuition about this is given in my article ‘Ebullition and gravitational clumping, arXiv, 2011). Do not worry if you don’t understand me, as I myself remember only vague souvenirs and are not so convinced by what I am saying!!! In fact Haefliger (ca. 1956) claims this construction only for second countable curve (even with a proviso on the fundamental group), but when I was in touch with the subject I was fairly convinced that it must work universally.

OPTIONAL REMARK: Haefliger, and Haefliger-Reeb 1957 use this construction to prove that any simply-connected curve (second countable) arises as the leaf space of a foliation of the plane. (Sketch of proof: take the twistor of the given curve which is by the exact sequence of a fibering 1-connected and (by Poincaré-Volterra) second countable, hence it is the plane, q.e.d)

So the idea would be to descend a smooth structure on the surface to get one on the curve. Alas, it is a well-known open problem whether any (non-metric but Hausdorff) surface admits a smooth structure (Spivak 1971, Nyikos, etc.) However quite puzzlingly Siebenmann 2005 (Russian Math Surveys) claims (and even prove in some details) that a PL structure exists universally on all such surfaces, merely as a consequence of Schoenflies theorem. So perhaps Siebenmann argument work as well for DIFF structures, and the metaphysical problem of Spivak-Nyikos is cracked. If this works (ask maybe Siebenmann, or an Indian in the States(=Ramachandran) who albeit not an expert was fairly convinced that there should be no asymmetry between PL and DIFF in dimension 2), then there is perhaps some chance to get a smooth structure on all non-Hausdorff curves. Of course there is perhaps a more direct strategy without transiting through surfaces.

Otherwise, regarding exotic smooth structures on curves the original reference is Haefliger-Reeb 1957 article in L’Enseignement Math. Perhaps you could quote this in your brilliant web-page.

Sorry for this vague answer, but at the moment my brain is much concentrated on this Rohlin-Le Fiedler total reality claim which still puzzles me a lot!!!

Best greetings to all, as well as to Rachel and Chiara. All the best, Alex

$\bullet$ \[22.02.13\] Dear colleagues (especially Séverine),

I worked hard (but without success) on the Le Touzé’s theorem, at least for 8 basepoints assigned on the nonempty ovals of a sextic of type $6/1 2$. If I understood well Séverine’s announcement, you rather handle the case of $2/1 6$ and assign more generally the points in the insides of the empty ovals (but of course I suppose that your argument adapts to $6/1 2$). Even in my weaker form I am not really able to conclude but send you my last thinking on the question (Section 27.4, p.352–356, esp. Fig.141). Ultimately I found a method which I call “barrages”. A special rôle is played by nodal cubics of the pencil, and I try to get a corruption with Bézout by looking at nodal curves with a barrage, i.e. such that 4 arcs of some other cubic joins pairwise the 8 basepoints distributed on the loop of the original cubic. (By the loop of a nodal cubic, I mean the unique path from the node to itself which is null-homotopic in the plane $RP^2$.) Of course I am not sure that details can be decently completed, but for the moment it is the only reasonable strategy I could imagine. I am sure that Séverine’s argument is much more elegant and convincing. My reasoning is completely conditioned by Fig.141, and I am probably too naive in believing that it reflects the general situation.

Sorry for sending you this very coarse material, and of course do not take the pain to react to this message. Many thanks again a lot to all for sharing so generously your knowledge and for all your answers. Best regards, Alex

\[25.02.13\] Dear real geometers,

I was still much fascinated by the Rohlin-Le Touzé theorem (RLT) albeit still not able to prove it. Being frustrated by my failing attempts (probably due to a lack of stubbornness and competence in algebraic geometry) I decided to speculate a bit of why it is so important or at least to explore how the statement could generalize.

In its most elementary incarnation involving pencil of lines and conics, the phenomenon of total reality occurs along infinite series stable under the operation of satellite of a real scheme (of even order). Satellite just amounts to trace each oval with a certain multiplicity $k$ (jargon obviously borrowed from knot theory). So the unifolium scheme of degree 2 (allied to a conic) gives rise to the deep nests, and the quadrifolium scheme of degree 4 gives rise by taking its satellites to an infinite series of schemes of order multiples of 4 which are totally real under a pencil of conics (assigned to pass through the deepest ovals). It seems therefore natural to ask if the satellites (e.g. the second satellite) of the Rohlin’s scheme $6/1 2$ (or its partner $2/1 6$) are also totally real (and hence of type I) under the “same” pencil of cubics as posited by the Rohlin-Le Touzé theorem. Alas I was not even able to settle this question. (Of course this seems evident (granting RLT) for a small perturbation of the algebraic double (essentially $F \cup F+\epsilon$), since total reality forces transversality of the foliation induced by the pencil with the curve.)

Next, I tried to understand what are the higher order avatars of the RLT-theorem (in the hope that it is not an isolated phenomenon as vaguely suggested by Ahlfors theorem). I found using the Rohlin-Kharlamov-Marin congruence ensuring the type I-ness (=orthosymmetry) of some $(M-2)$-schemes an (obvious) infinite series of avatars of the Rohlin’s $(M-2)$-schemes of degree 6 . Those are also $(M-2)$-schemes and total reality seems to be possible for a pencil of curves of order $(m-3)$, exactly like for the Gürtelkurve of Zeuthen-Klein (bifolium quartic with 2 nested ovals totally flashed by a pencil of line through the deep nest) or for the Rohlin’s sextic (flashed by a pencil of cubics). So it seems that the theory of adjoint curves of order $(m-3)$ plays some special rôle in this question of Rohlin-Séverine. I would be very happy if one of you knows if it is reasonable to expect an extension the RLT total reality theorem to all this schemes whose type I ness is ensured by Rohlin-Kharlamov-Marin congruence (sorry if I am not hundred percent right in crediting as I could not extract the exact history of this subliminal result). Specifically I have Conjectures 27.17 and 27.18 (page 365 and 367 resp.) which list some candidate-schemes for total reality in degree 8 and 10. If the conjectures are right, it would be of great interest to know if Séverine’s proof adapts to them. Sorry if I am too naive about the real difficulty of such problems, but I found exciting to wonder if there is something more general behind the cryptical allusion of Rohlin. Of course I presume that he derived the synthetic result a posteriori from highbrow topology (or Kähler geometry in Kharlamov’s case?), but perhaps there is a simple explanation with (“basic”) algebraic geometry and total reality as Séverine was able to do? As Oleg knows my problem is that I wasted too much time with non-metric manifolds and so forgot all the little I ever knew about algebraic geometry.

During the way, I think to have found a counterexample to the conjecture of mine (inspired by the Itenberg-Viro contraction conjecture of empty ovals), and according to which all empty ovals could be contracted simultaneously to solitary nodes. This counter-example is Thm 27.16 on page 364 (which I hope is correct and sharp as far as the degree is concerned).

Thanks a lot for the attention, and sorry for all the modest news (you surely thought about in sharper form already). All the best, Alex PS: The material summarized in this message occupies page 357-367 (Sections 27.5, 27.6, 27.7), as usual I had not much time to polish, but I hope it is still readable.

\[27.02.13\] A census of 100 octic $(M-2)$-schemes of type I satisfying the RKM-congruence, plus a little addendum for Oleg’s non-Hausdorff curves

Dear colleagues,

I have pursued some preliminary study toward the total reality phenomenon, yet merely in its combinatorial aspect prompted by the modulo 8 RKM-congruence (for Rohlin-Kharlamov-Marin) ensuring the type I of $(M-2)$-schemes of degree $2k$ with $\chi = k^2+4 \pmod
8$. Accordingly, I call an RKM-scheme any $(M-2)$-scheme satisfying this congruence. While any RKM-scheme is of type I, I do not know alas whether the converse statement is true. If it is known I would be extremely grateful if someone can tell me (and our collective chat room) the answer. Further I noticed that the list given in my previous e-mail of RKM-schemes of degree 8 can be much enlarged. If I am not too bad in combinatorics, there are precisely 100 such schemes in degree 8, all of them being potentially subsumed to the phenomenon of total reality under a pencil of quintics akin to the Rohlin-Le Touzé theorem (for sextics flashed by cubics). This modest material is to be found in Section 27.8, p.368-373 (especially Fig. 146 page 370 and Lemma 27.24, p.372, plus all the 36 Gudkov symbols on page 372). I hope of course that I missed nobody in this catalogue. Extrapolating a bit using the (hypothetical) converse statement to RKM, I would say that there are precisely 100 schemes of type I which are $(M-2)$-schemes. Is this well-known and correct?

Actually, I do not really know if all these 100 schemes are realized algebraically, but presume that most of them (all?) are. Possibly I am much too naive. Of course it is quite amazing to see that the only two RKM-schemes of degree 6 (namely $6/1 2$ and $2/1
6$) demographically explodes to a menagerie of 100 such schemes in degree $8$, but that should be no surprise for you much acquainted with the higher cases of Hilbert’s 16th problem. It would be even more crazy if all those 100 schemes (or at least a good portion thereof) are subsumed to the phenomenon of total reality. If you have some ideas on those circle of ideas, I would be extremely thankful.

Many thanks again for the patience and attention, and I hope that what I am telling is nearly correct (not too surrealist). Very best regards, Alex

PS: For Oleg, regarding my loose answer on smooth structures on non-Hausdorff 1-manifolds, I would like to add another philosophical remark related to the method of Haefliger’s “twistor”. This is of course like a thickening along a normal bundle except that there is no ambient manifold (safe the ether) and so the construction must be intrinsic. To my knowledge it was never exposed in details (albeit Haefliger’s 1st article ca. 1955-56 in Colloque de Topologie de Strasbourg uses implicitly this construction). Now my point is that albeit the twistor method looks somewhat indirect, I think that it is fairly useful. For instance, I was since 2006-07 puzzled by the naive question if the fundamental group of a one-manifold is always a free group. (Of course such non-Hausdorff curves resemble somehow graphs, whence some intuition). For instance the line with 2 origins has $\pi_1=Z$ as follows quickly from Seifert-van Kampen (and if 3 origins or 2 doubled origins then $\pi_1=F_2$ is free of rank 2). Ultimately in 2011 I found a general answer to this “freeness” puzzle by using the Haefliger twistor construction, while showing first that all open (non-metric) surfaces have free fundamental groups. (This is actually a very modest extension of the metric case, which to my knowledge is first treated in Ahlfors-Sario book of 1960, albeit it may have belonged to the folklore much earlier, say Kerekjarto, H. Kneser, Rado, in the 1920’s, Papakyriakopoulos in the 1940’s???). This material is exposed in some details in my arXiv note of ca. 2011 (Ebullition in Foliated surfaces versus gravitational clumping). I hope that those results are nearly correct but they certainly require more professional treatments and exposition than I was able to do. I hope this little remark makes perhaps more plausible that the approach via the (Hausdorffizing) Haefliger twistor is also reasonable for your problem of DIFF structures.

$\bullet\bullet\bullet$ vendredi 1 mars 2013 19:07:12

Dear Alexandre, dear other colleagues, here is the note I had promised to send you. There are still many open questions, as Alexandre wrote. It would be also interesting to know whether one could find a totally real pencil with respect to the dividing $M-2$-sextics with real scheme of indefinite type. I will think about it when I have more time. Best regards,

Séverine

(01.03.13, 22h15)

Dear Séverine and the other colleagues,

So many thanks Séverine for sending us your splendid article. I am much excited to read the details tomorrow, as myself started today to doubt about the whole result (at least in the strong form that any points 8 points distributed on the empty ovals ensures total reality). (If I am not wrong the whole phenomenon depends upon the location of the 9th base point, namely the pencil is totally real iff the 9th base point lands in the inside of the nonempty oval.) So I was much depressed and lost in my poorly organized thoughts. So your sending arrives as a true deliverance for my brain.

Many congratulations again to Séverine for this fantastic work. Very best regards, Alex

\[02.03.13\] Can total reality fail for a distribution of 8 points on the empty ovals?

Dear Séverine and the other geometers (especially Professor Nikulin),

I enjoyed much a detailed look at your splendid article full of illuminating remarks. I will probably need much more time to digest the impressive technology you use, and need to print the material to make a deeper reading (especially of the former works upon which your argument seems to depend). So many thanks again for sending us your work in so rapid delay. I wrote some naive reactions in Section 27.11, where I mostly copied your sayings, and tried to add hopefully pertinent footnotes.

Regarding your question “Can conversely any dividing curve be endowed with some totally real pencil?”, I still wonder if a positive answer is not a trivial consequence of Ahlfors theorem (compare very optionally Gabard’s Thesis 2004, page 7). However since Marin warned me in January 2013 (cf. Section of e-mails) it may be the case that the transition from the abstract conception of Riemann-Schottky-Klein to the embedded viewpoints of Hilbert-Gudkov-Rohlin is not so easy. Yet I am still confident (or naive enough) to believe that it holds true. The point seems to be primarily a matter of projective algebraic geometry, namely the question if any abstract morphism on a concrete plane curve to the line $\PP^1$ is induced by a (linear) pencil of ambient curves. This is either trivially true or trivially wrong, but alas I do not know the answer due to my failing memory about the foundations of algebraic geometry.

Your article already helped much as I suffered under the misconception that your result states that any distribution of 8 points on the empty ovals induces a totally real pencil. Your statement is much more subtle, yet personally I do not know if this stronger (universal) form of total reality is wrong! If you know a counterexample foiling universal total reality I would be very happy. It could then still be the case that there is some special sextics for which universal total reality holds true, i.e. for all octuplets distributed on the empty ovals. (Perhaps reading more carefully your article, especially the aspect related to Nikulin-Kharlamov’s rigid-isotopic classification already answers those questions?)

(The newest material of mine (as usual confusing and poorly organized) occupies Section 27.9–27.11 on pages 373–378. Here I attempted a topological approach to the existence of octuplets inducing a totally real pencil, but alas was not able to conclude, presumably because I know too little on the predestination process creating the 9th basepoint as a function of the 8 assigned ones.)

Many congratulations again to Séverine for this breakthrough. Best regards, Alex

dimanche 3 mars 2013 18:07:57 a new version with small corrections?

Dear Alexandre, dear other colleagues, I owe you some apologies: the Theorem was slightly incorrect, as Alexandre pointed out. I let you discover this new version, where I have reformulated the Theorem, and added a few words in the end of the proof. Best regards, Séverine

(04.03.13) Dear Séverine and the other colleagues,

Many thanks for the new version. In fact, it seems that the main change is that you now assign the 8 basepoints ON the empty ovals instead of IN their insides. Rereading my previous message, I realize that I misstated your original statement and so it is pure chance that assignation on the ovals turned out to be “more correct”.

Your fascinating article gave me new forces to think about the problem, but alas still without success. For instance, I still do not know if there exist octuplets (on the empty ovals) failing to induce a totally real pencil. Of course assigning them in the insides gives more freedom, but presently it looks to me harder to ensure total reality. So despite your correction, it could still be the case (in my modest understanding) that the pencil is total for all octuplets chosen in the insides of the empty ovals. Perhaps you know a counterexample to this strongest form of the statement?

Many thanks again for the article, which guided much my thinkings. I hope to send you more exciting news soon, but the whole problem which you call “the lost proof of Rohlin” seems to me still much out of reach. All the best, Alex

$\bullet$$\bullet$$\bullet$ answer to Alexandre’s questions (mardi 5 mars 2013 13:30:42)

Dear Alexandre, dear other colleagues, let me try to answer the question with a new formulation.

Assume first that the base points are distributed [*inside*]{} of the empty ovals. Applying your nice “dextrogyration argument” to all nine ovals gives the following lemma:

[*The pencil is totally real iff 9 lies inside of the non-empty oval $O$ and outside of the empty ovals.*]{}

If 9 is outside of $O$, the bad cubics are as shown in Figure 2 of the paper. If 9 is inside of an empty oval $X$, the bad cubics have an oval passing through the two base points 9 and X only, and this oval is entirely contained in the empty oval denoted also $X$. To get rid of this latter possibility, it suffices to take the base points [*on*]{} the empty ovals.

In ii), I give an explicit description of the pencil, valuable for any generic choice of the eight base points [*inside*]{} of the eight empty ovals. (It turns out that the only possible non-generic situation is that of a pencil with a double base point $9=2$, this means that the points 1, ..8 lie on a nodal cubic with node at 2.) Recall that 2 is the base point chosen in the extreme inner oval forming a positive pair with $O$. For this pencil, the only possibly bad cubics are those with an oval passing through 9 and 2 only. To grant total reality, it suffices to choose the base point 2 [*on*]{} the corresponding empty oval, the other base points lie arbitrarily in the inside discs of the other empty ovals. Thus, your conjecture 27.29\[=\[SRLT:conj\]\] is true, and an even stronger result holds for the sextic with six inner ovals. Best regards, Séverine

\[07.03.13\] Little news from Alex, and so many thanks to Séverine for the answer

Dear Colleagues,

First many thanks to Séverine for your very detailed answer (which I will study in detail tomorrow). Sorry for being always a bit differed in time due to my lack of internet at home.

I added some material in my loose notes. In Section 28.1–28.2 (pp.384–392), I tried once more to explore the grand programme that Rohlin might have had in mind, namely total reality and its connection with his maximality conjecture. As I often said it seems to me that the missing link could be played by Ahlfors theorem, or perhaps Rohlin had a grand vision that he could arrange total reality by purely synthetical processes extending in all degree the already tricky theorem of Rohlin-Le Touzé in degree $m=6$. This idea when explored in full looks to me extremely vertiginous, but its net impact would be a sort of upper bound upon the complexity of Hilbert’s 16th problem, and in some sense subsume all prohibitions (à la Gudkov et cie.) to the paradigm of total reality. All this necessitates to be made much more precise, but I \[have\] attempted to make a psychoanalysis of what Rohlin may have had in the brain, without that he himself ventured to put it on the paper due to his own modesty and pragmatism.

Next I discovered the little Theorem 28.7\[=\[Thom-Ragsdale:thm\]\] (p.393), which is just a matter of making explicit the consequence of Thom’s conjecture (=Kronheimer-Mrowka theorem) as it pertains to Hilbert’s 16th problem. The result is the lovely estimate[^107] $\chi \le k^2$ for a curve of type I and degree $2k$. With this I realized that my former counterexample (with the scheme $20$ in degree 8) to CCC(=collective contraction conjecture) is actually killed by Thom, and realized (later only!!) that it is also killed by Rohlin’s formula. So CCC is again resuscitated but probably not for long!?

Then I tried to make a comparative study of Rohlin’s formula versus the Thom obstruction. It seems that the latter is often implied by Rohlin’s formula, but not always. More in Section 28.4 (p.393). It seems however that at least for degree $m \ge 10$ there is some cases where Thom really affords new information not covered by Russian congruences or Rohlin’s formula (cf. Thm 28.11, p.396). Finally using the Gudkov table in degree 10 (=Fig.148 on page 395), I got some naive hope to disprove the Rohlin maximality conjecture, but this quickly turned into disillusion (cf. Point 3 on p.396–397).

Sorry for all these messy remarks, yet I found the rôle of Thom quite pleasant. I am sure that this is not new, and that I read it somewhere, but again could not recover where precisely. (I thought it was in Degtyarev-Kharlamov 2000’s survey but apparently not, though Kronheimer-Mrowka is alluded to.) If you remember some anecdotes about the rôle of Thom’s conjecture in Hilbert’s 16th problem, and who puts it first into action as a such, I would be extremely happy to insert your remarks in my (messy) survey.

Thanks a lot for the attention, Best regards, Alex

$\bullet\bullet\bullet$ Thomas Fiedler wrote (samedi 9 mars 2013 17:32:59)

Dear Alexandre,

I am no longer in business in this field, but let me just make some remarks which could be perhaps helpful.

The $M$-curve of degree 10 mentioned in your Thm 28.11\[=\[French-scheme:thm\]\] is in fact ruled out by Rokhlin’s formula. I think that you have mixed $Pi^+$ with $Pi^-$. In a positive couple the orientations are just opposite. So, four nested ovals can contribute at most $+2$ to Rokhlin’s formula.

It is an interesting idea to apply the Thom conjecture to real algebraic curves. To my knowledge the only new result obtained this way is contained in G. Mikhalkin “Adjunction inequality for real algebraic curves”.

Let me formulate the problem (which exists certainly already somewhere).

GENERALIZED THOM PROBLEM.

Let $X$ be a simply connected smooth closed $4$-manifold and let $h$ be a non trivial integer $2$-dimensional homology class. Let $F$ be a smoothly embedded oriented surface which represents $h$ and such that the components of $F$ represent classes which are linearly independent over $Z/2Z$. What is the maximal Euler characteristic of $F$?

In the complex projective plane this boils down to the Thom problem, because evidently the surface F has to be connected in this case. However, it becomes interesting in a more general complex surface. It is an easy matter to make a non connected surface $F$ connected but the opposite is quite hard. It is equivalent to finding an embedded “membrane” with trivial normal Euler number. I don’t know wether Seiberg-Witten theory nowadays can give a sufficient criterium to ensure the existence of such a membrane (as stretching the neck of the surface $F$). But it seems to me that this is the place to look at.

If one considers as $X$ the double cover of the projective plane ramified in the complexification of an $M$-curve of even degree then one can consider as $F$ the fix point set of one of the two induced anti-holomorphic involutions on $X$. We know $2b_0(F)+b_1(F)$ from Harnacks equality. Hence the Euler characteristic $2b_0(F)-b_1(F)$ is maximal when $b_1(F)$ is minimal, i.e. the numbers $p$ or $n$ of the real curve are maximal. So this problem is closely related to the still open Ragsdale conjecture for $M$-curves.

Best regards, Thomas

$\bullet$ \[Gabard, 09.03.13, ca. 21h00\] Does the Gudkov hypothesis (mod 8) reduces to Rohlin’s (complex orientation) formula

Dear Colleagues,

On reading recently the Degtyarev-Kharlamov 2000 survey, I learned the (simple) fact that the Arnold congruence mod 4 (weak Gudkov hypothesis) can be reduced to Rohlin’s formula. I wrote down a proof of this simple issue in Lemma 24.16\[=\[Rohlin-implies-Arnold:lem\]\], on page 219 of the messy survey.

Of course then I wondered if a sharpened combinatorial argument taking into account the signs distribution on the edges of the “Hilbert tree” (encoding the distribution of ovals) prompted by Rohlin’s complex orientations could likewise subsume the Gudkov hypothesis to Rohlin’s formula. The combinatorics becomes much more messy and I lack a good idea on how to exploit the $M$-curve assumption (apart from the dubious idea of using the dextrogyration allied to a totally real pencil, but this looks somewhat ad hoc!!!?). If feasible (so or otherwise), the net impact would be that the highbrow topology used in the Rohlin-Rohlin/Atiyah-Singer-Marin proofs could be replaced perhaps by basic algebraic geometry. This issue is merely didactic of course, yet it looks perhaps technically challenging to implement this modest “dream”. My (unsuccessful) attempt to tackle this reduction is given in Section 24.5, pages 220–223.

Of course, I am sure that you already tried hard along this way and that this is a pot-pourri naive problem. So I write you the letter, only in the hope that someone already worked this reduction successfully, though I doubt (as otherwise it would certainly have been mentioned in the Degtyarev-Kharlamov 2000 survey).

Many thanks again for all your attention and all your precious hints, Alex

PS: I apologize much to Séverine for not having yet found the time to study carefully enough the last brilliant explanations, but look forward doing so in the best delay.

PPS: Many thanks also to Thomas for the brilliant answer on Thom and Mikhalkin, Seiberg-Witten, etc. I will include your letter in (the next version of) my notes so that anybody can contemplate it.

$\bullet$ \[11.03.13\]

Dear Thomas and the other Colleagues,

So many thanks to Thomas for having catched my mistake regarding Thom versus Rohlin (existence of schemes prohibited by Thom but not by Rohlin’s formula). I have written down a corrected version of the Theorem (whose first clause is I think still true despite Thomas’s corrigendum) as Thm 30.14 on page 382. I hope this time it is correct! The subsequent Thm 30.15 and Lemma 30.16 describe larger family of such schemes. Alas, what I have written is not extremely appealing, but I hope still readable for such experts as you are (though I confess that it is not extremely exciting!). The philosophy is just of course that there is no subsumation of Thom’s estimate $\chi\le k^2$ to Rohlin’s formula.

Many thanks again to all for all your kind answers, and especially to Thomas for taking care to bring me on the right track!

Best regards, Alex

PS: I have drifted the Section of your e-mails at the end of the text, in particular the last letter of Thomas is to be found on page 435. Alas, I had not yet the time to digest its full swing, but the idea looks very promising!

$\bullet\bullet\bullet$ mardi 12 mars 2013 13:56:45

Dear Alexander,

sorry, but all your $M$-schemes of degree $10$ in Thm 30.14 and 30.15 have $n=2$ and are ruled out simply by Petrovskis inequality. I don’t think that genus bounds give anything new for real schemes alone but they definitely do so for configurations of several real curves. Just take a look on Mikhalkin’s paper.

Best regards Thomas

\[16.03.13\]

Dear Geometers,

Many thanks again to Thomas for his former correction. I got so drifted in a sort of cuneiform formalism of trees with signs but mostly lost myself into dubious combinatorics of (what I call) the Rohlin tree (=Hilbert’s nested tree with signs materializing of course Rohlin’s complex orientations). Ultimately after several basic combinatorial mistakes, I did NOT even succeeded in finding a Caucasian $M$-scheme where Rohlin’s formula is stronger than Thom $\chi \le k^2$. This problem is discussed in Sections 30.7 and 30.8 (pages 393–402). It seems to me that there is (for $M$-curves at least) a certain concomitance between Rohlin and Thom, i.e. you cannot corrupt Rohlin’s formula without corrupting simultaneously Thom’s bound. So maybe it is reasonable to conjecture that if Thom’s estimate is fulfilled then Rohlin’s equation is always soluble for a suitable distribution of signs on the edges of the tree. Sorry that my summary is vague as I do not myself understand properly what happens.

Today, I switched on a somewhat more pleasant arithmetical problem exposed in Section 30.6, p.391–392. This is hopefully more readable, and you surely studied this a long time ago. Here the question is to find an $M$-scheme without nesting of high-degree (say $m\ge 6$ assuming zero-knowledge). As we know, Hilbert posited the intuition that $M$-curves are forced to exhibit nesting. (In fact on reading this afternoon more carefully Hilbert’s text, he is not so categoric but let us assume so, to add some suspense to our story!) In view of Rohlin’s formula (which forces the number of ovals of an unnested dividing curve to be a square) and the Gudkov congruence mod 8, one is invited to ask when the Harnack bound $M=g+1$ of a degree $2k$ curve is a (perfect) SQUARE? This leads to a little arithmetical problem which admits a nontrivial solution at $k=17$ for $M=529$ which is by a lucky stroke equal to $23$ squared. (Of course I found this just by an “exhaustive” tabulation.) (Note at this stage that $17=16+1$ yet another subconscious coincidence in Hilbert’s numbering!!??) It turns out moreover that the Gudkov congruence is then fulfilled! So this leads me to ask if, you in Russia, knew a way (prior to Thom-Kronheimer-Mrowka 1994) to prohibit this $M$-scheme of degree $2k=34$ with 529 ovals (all lying outside another, i.e. no nesting). Personally, I would be very happy if someone can tell me an answer (in case I did not foiled the arithmetics!!!) Incidentally, I am so ignorant in that field that I do not know how to solve in general the Diophantine problem of the quadrature of Harnack’s bound, i.e. for which $k$ is Harnack’s bound $M=g(2k)+1$ a square (=Problem 30.19, on page 391).

Finally, this morning, building upon Séverine’s (elementary) remark on $M$-quintics totally real under a pencil of cubics (nothing so hard as the Rohlin-Le Touzé theorem for sextics), I had the idea to extend the construction of satellites to curves of odd degrees, and so found a scheme of degree 10 (the 2nd satellite of Harnack $M$-quintic). This schemes, according to the philosophy of total reality, should be of type I (stability under satellites). This material is exposed in Section 29.8 page 352–353. The scheme in question has 13 ovals one of them enclosing 6 nests of depth 2 (cf. Fig 149 p.353). Assuming that this scheme is of type I, it could be a counterexample to Rohlin’s maximality conjecture, in case someone ever saw a curve of degree 10 enlarging it. Otherwise, more in line with Rohlin’s philosophy, this scheme being totally real in some explicit way it should be maximal and it results a myriad of prohibitions on all schemes enlarging it! If you know some experimental construction à la Viro-Itenberg that may help to see clearer, I would be extremely thankful. Of course, it would also be very interesting to know if Séverine thinks that there is some good chance that total reality holds true for this satellite.

Many thanks for your attention, and I hope my questions are not too ill-posed for such experts as you are! All the best, Alex

PPS: Many thanks for the message meanwhile received from Thomas which I hope make my message not to obsolete. Best regards.

PPPS: Dear Viatcheslav, I try twice to send you my message as I received delivery failure notification! I hope I am not overloading your mail-box. I try now with a zipped file hopefully toujours lisible pour vous. Amitiés. Alexandre

$\bullet\bullet\bullet$ mardi 19 mars 2013 17:54:39

Dear colleagues, my note “totally real pencils. . .” is now available on the archiv, here is the reference: http://arxiv.org/abs/1303.4341 Best regards, Séverine

$\bullet$ \[19.03.13, ca. 19h30\] An Alsatian scheme and planning to put a version on arXiv in ca. 10 days (please confirm me if you accept the insertion of your kind letters in my messy survey)

Dear Colleagues,

Many thanks to Thomas again for pointing out the marvellous Petrovskii’s inequalities (which I confess I had not assimilated properly before, despite all the brilliant surveys available). Shame on me! So of course most of the questions of my previous message were rather stupid. In particular that relating to the arithmetical problem was completely ill-posed as I missed to use the full punch of Rohlin’s formula. I apologize much for all these inconsistencies.

Meanwhile I have attempted to find what I call an Alsatian scheme where Thom is stronger than the conjunction of Petrovskii 1933/38, Gudkov hypothesis (1969–72=Rohlin’s semi-proof) and Rohlin’s formula (1974–78). I think that such an Alsatian scheme do exist (cf. Thm 30.22 on page 392). This answers (I hope correctly) a question raised by Thomas in the message reproduced right below, where if I understand it well Thomas expected that Thom says nothing new already known earlier in Russia.

Then in view of Thomas’ stimulating messages and also the marvellous Itenberg-Viro 1996 Math. Intelligencer article, I adventured slightly in the province of Miss Ragsdale. Here first I learned that the Ragsdale conjecture for $M$-curves really boils down to Thom (at least one-half thereof)[^108]. I found strange that Kronheimer-Mrowka 1994, was not cited in Itenberg-Viro 1996 probably due to backlog reasons (by Intelligencer)[^109]? Further it seems to me that there is a misprint in the statement of the RAGSDALE CONJECTURE ON $M$-CURVES (on p.24 of Itenberg-Viro 1996). At any rate, I tried to write down my own naive account on Ragsdale in Sect. 30.4\[=\[Ragsdale-conj:sec\]\] (p.380–384) which details more slowly what I understood (hopefully correctly). In fact, I wonder at my (premature) stage if the full Ragsdale conjecture could not follow à la Thom via the lower estimate $-k^2 \le \chi$ which I very naively conjecture to be true for all dividing curves? (I confess that I made no experiments even with the methods of constructions I am aware of, i.e. Hilbert-Harnack).

So as you see, life really starts becoming exciting. Alas I fear that I will be strongly interrupted due to editorial reasons of our Journal L’Enseign. Math. in Geneva of which I am the TeX-editor. So perhaps I should stop thinking and try to polish a bit the big mess I produced (during ca. 10 days), before putting it on the arXiv (prior to my long editorial job). Alas, I had already great difficulties to submit the previous version of my file (due to size limitation policy of arXiv), but received an exceptional derogation to do so. Now my text on Ahlfors + Rohlin is twice as large as it was before (600 vs. 300 pages) so there is little chance that I get accepted. I will try to see if an arXiv administrator looks optimistic. Meanwhile, I would be very happy if you all confirm me that you accept the integration of your marvellous letters in my modest text. Of course answer me only in case of objection. I plan to submit the new version ca. the 1 March 2013. Of course my text is so messy that your letters alas are not properly pushed into evidence, but I expect to produce a better text in the next months or years.

Many thanks again to all for your letters, indulgence, patience and kind answers!

Best regards, Alex

$\bullet$$\bullet$$\bullet$ [*Reproduction of Fiedler’s former letter (already above but the other colleagues did not saw it)*]{} mardi 12 mars 2013 13:56:45

Dear Alexander,

sorry, but all your $M$-schemes of degree $10$ in Thm 30.14 and 30.15 have $n=2$ and are ruled out simply by Petrovskis inequality. I don’t think that genus bounds give anything new for real schemes alone but they definitely do so for configurations of several real curves. Just take a look on Mikhalkin’s paper.

Best regards Thomas

$\bullet\bullet\bullet$ mardi 19 mars 2013 22:17:27

Dear Alexandre,

sorry again, but your curve has $p=50$ and is ruled out by Arnold’s inequality: $p\le 3/2 k(k-1) + 1 + n_-$, which is $47$ in this case. In fact Arnold’s inequalities are by fare the strongest result in the whole field.

Best regards Thomas

$\bullet$ \[20.03.13\] “René Thom sur son 31?”

Lieber Thomas (and the other colleagues),

Es wird jadoch immer spannender, oder wat? Many thanks for the new challenge raised by Thomas (which I received as a special gift on my birthday date, today 20st of March). As Thomas demonstrated yesterday evening (cf. message reproduced below), my example of Alsatian scheme (where Thom is stronger than the Soviet Red army) (cf. Thm 30.22\[=\[Alsatian-schemes:thm\]\], p.392) collapses under the strong Petrovskii inequality of Academician Vladimir Igorevich Arnold. It took me just some few hours (as I was quite tired) to find a stronger candidate of Alsatian scheme where Thom looks stronger that the conjunction of (strong) Petrovskii-Arnold 1971, Gudkov 1969 (proved by Rohlin 1972-Marin ca. 1977), and Rohlin’s formula (1974–78). For the exact statement cf. Thm 30.25\[=\[Alsatian-scheme-Thom-strong-Petrov-Arnold:thm\]\] on page 395. I hope I made no mistake (I checked the details twice). By (3-fold) experience, I am quite confident that Thomas will find a new obstruction in his pockets killing this new example.

Hence many thanks again to Thomas for all this precious guidance that oriented much my modest working. As I said I am now under strong temporal constraints, and will not be able to pursue any thinking for a long period of circa 1 month. So please feel free to elaborate more upon the direction indicated by Thomas if it looks hard stopping the inertia. Myself find the Alsatian topic quite pleasant, yet it would (I presume) be interesting to penetrate deeper in the geographical question (by really understanding the diagrammatic impact of Petrovskii-Arnold, Ragsdale, etc. upon the higher Gudkov’s tables of periodic elements) I made several pictures of big pyramids (e.g Fig. 153\[=\[Degree10:fig\]\] p. 386) and that could be a first step toward understanding better what happens. Of course all this must be familiar to you, yet personally I still lack a good algorithm to make good (color-plates) maps evidencing “all” obstructions.

My main worry now, is how to publish (and polish) the 600 pages long article that I have produced, especially in view of stringent size restrictions imposed by the arXiv administrators.

Many thanks that you seem to all approve the insertion of your letters in my survey, which looks to me essential as you influenced much my (chaotical) trajectory. Of course in the future we will have the occasion to clean better in case some bugs are detected, which is quite unlikely (apart of course in my own letters full of inconsistencies).

All the best, Alex

$\bullet\bullet\bullet$ jeudi 21 mars 2013 07:51:45

Dear Alexandre,

unfortunately your Theorem 30.7\[=\[Thom-Ragsdale:thm\]\] is wrong. Half of the curve together with R is usually called Arnold’s surface. It is an orientable surface iff R has an orientation which induces the complex orientation on its boundary, i.e. the real curve. Hence if there is a negative pair of ovals in in the boundary of one component of R then Arnold’s surface is not orientable.

By the way it is well known and easy to prove that if Arnold’s surface is orientable then $p-n=k^2$.

Best regards, Thomas

$\bullet$ \[21.03.2013, ca. 22h00\] Dear Thomas and the other Colleagues,

Many thanks to Thomas for having spotted out my fundamental mistake. I apologize much hence for all the dubious letters that I sent you the former days. I hope I will still be able to repair a bit the situation in the next weeks, though this will require dramatic changes in my messy text! So many thanks again to Thomas for having detected this great Harnaque!!!

All the best, Alex

$\bullet\bullet\bullet$ vendredi 22 mars 2013 07:54:47

Dear Alexandre,

don’t worry. Just read Mikhalkins paper and find other interesting applications of his method. Having a reducible dividing curve you can switch the canonical orientation of exactly one component. This allows to construct immersed surfaces which are far from being complex curves. Then indeed genus bounds start to work.

Best regards, Thomas

$\bullet$ \[24.03.13\] Is there a simpler counter-example to “Gabard-Thom” than via Itenberg-Viro mirabilis $(M-2)$-curve of degree 10 disproving Ragsdale

Dear Colleagues,

As pointed out by Thomas (cf. my last message) my proof of the “Gabard-Thom” estimate $\chi \le k^2$ for all dividing curves of degree $2k$ was highly fraudulent as it was based on the (erroneous) assumption that the Arnold surface (=Klein’s half married with Miss Ragsdale) is always orientable. Of course Thom has still something to say on Hilbert’s 16th e.g. in the very special case of no nesting. As a historical curiosity one can notice that the elementary case due to Kervaire-Milnor (1961) of Thom in degree 3 gives in my opinion the first purely topological proof of Hilbert’s Ansatz of nesting for $M$-sextics. Prior to that we had only Hilbert-Rohn (stratificational as explained by Eugenii), and Petrovskii 1933/38 which involves the Euler-Jacobi-Kronecker stuff (interpolation formula). Of course Kervaire-Milnor is merely Rohlin’s early work 1951 disguised!

Next I was a bit puzzled by Thomas’s claim that my “Gabard-Thom” theorem is wrong, since I was not able to find an explicit counter-example. I tried a while with elementary constructions à la Harnack-Hilbert but could not find a single counterexample. So I found the pleasant plates Fig. 152-153-154-155\[=\[HilbGab1:fig\]–\[HilbGab4:fig\]\] on pages 382 and ff. Here we get nice infinite families of curves with $\chi=k^2$ that were surely known to Hilbert and Ragsdale. Alas this inclined more toward thinking that Gabard-Thom is sharp, rather than disproving it. Then of course I had the idea to take a closer look to Itenberg-Viro 1996’s article (disproof of Ragsdale), which came like a deliverance and killed in the same stroke the Gabard-Thom dubious estimate. I used the Kharlamov-Marin congruence to check type I of the Itenberg-Viro curve (which I reproduced on Fig. 156\[=\[Itenberg:fig\]\], p.387), though there is surely a more elementary argument à la Fiedler-Viro-Itenberg-Parenti. Many congratulations by the way to Ilia and Oleg for this geometric paradise, which I contemplated with much pleasure and extreme admiration yesterday evening!!! A naive question of mine, is whether we really need to resort to the patchwork method to disprove Gabard-Thom, in the sense that perhaps I missed (a non-maximal) counterexample via Harnack-Hilbert.

Finally this marvellous curve (Itenberg-Viro’s) poses again the question of total reality, this time under a pencil of septics (in general I conjecture total reality for adjoint curves of order $m-3$ when it comes to $(M-2)$-curves). The question is where to assign exactly the anchor basepoints! I tried some guesses in Sec. 31.3\[\[Galton-brett:sec\]\] (p. 388). This deserve perhaps much deeper investigations than what I am presently able to do. Perhaps Séverine and Thomas already have some good ideas. My own dream is that there should be a combinatorial recipe telling one from the sole knowledge of the Rohlin tree (=Hilbert’s one with signs given by complex orientations) where to assign basepoints. One should perhaps imagine the Rohlin tree as a Galton-Brett=table with billiards balls falling downwards to the empty ovals and perhaps stabilizing at some other (unstable) equilibriums when they meet a “nail”, which in first approximation could be a hyperbolic oval negatively charged on the edge right above it.

All my thanks again to Thomas for having catched my fundamental mistake at the right moment. I hope to polish a bit the text during the next days before submitting it to the arXives. Then I must move toward doing more boring editorial duties for the Swiss journal L’ Ens. Math. (close to collapse by the way).

Good Sunday to all, and best regards, Alex

PS: Many thanks also for Thomas’s last letter which I just discover now. It is an invitation to read Mikhalkin’s article, which alas I had not yet the time to do properly...

Synoptic tabulations
====================

This is an attempt to gather information scattered through the literature. The first synoptic project compiles a list of nomenclatures. A second tabulation reflects how Ahlfors work (existence of circle maps) has been appreciated by subsequent workers of a slightly dissident nature in the sense that they cite conjointly other sources.

Nomenclature project
--------------------

This section tries to get sharp lower bounds on the basic nomenclature of our topic. As Poincaré tried to convince Felix Klein “[*Name ist Schall und Rauch*]{}” (cf. e.g. Klein 1923 [@Klein-Werke-III_1923 p.611]), but it is somehow pleasant to investigate the historical background of some jargons to use them hopefully appropriately.

$\bullet$ (Gauss 1825/1844, F.T. Schubert earlier?) [**Conformal mapping=konforme Abbildung**]{}, maybe the first non-trivial result is to be found in Gauss 1825 [@Gauss_1825], yet the word “conformal” itself appears in Gauss 1844 in the first paper on higher geodesy: “[*ich werde daher dieselben conforme Abbildungen oder Übertragungen nennen, indem ich diesem sonst vagen Beiworte eine mathematisch scharf bestimmte Bedeutung beilege*]{}” \[Werke IV, p.262\]. As noted in Struik 1933 [@Struik_1933 p.164] (via Cantor), the word “conformal” is already used prior to Gauss by F.T. Schubert in “De projectione sphaeroidis ellipticae geographica”, [*Nova Acta Petr.*]{}, p.130–146.

$\bullet$ (1865) [**Riemann surface**]{}, maybe first coined by C. Neumann 1865 [@Neumann_1865], followed by Lüroth 1871 [@Lueroth_1871], Clebsch 1872 [@Clebsch_1872], Klein 1874–76 [@Klein_1876], [@Klein_1876], Clifford 1877 [@Clifford_1877] and then too many to record.

$\bullet$ [**Berandete (Riemannsche) Flächen, Compact bordered Riemann surfaces, finite Riemann surface, membranes**]{}. The first appellation appears often in Klein 1882 [@Klein_1882] (reprint in Klein 1923 [@Klein-Werke-III_1923 p.569,§23]) and others. The second appellation is coined and popularized in Ahlfors-Sario’s 1960 book [@Ahlfors-Sario_1960], whereas the third competing name is used in Schiffer-Spencer’s book of 1954 [@Schiffer-Spencer_1954]. The term membrane also occurs (in this context) by Klein in his lecture notes.

$\bullet$ (1907?) [**Uniformization**]{} probably a coinage of Poincaré. In 1883, just the word “fonction uniforme” appears and the word “uniformization” as a such, came in vogue ca. two decades latter in Poincaré 1907 [@Poincare_1907] and Koebe 1907 [@Koebe_1907_UbaK1].

$\bullet$ (1908) [**Kreisnormierungsprinzip**]{} coined and proved (in fairly general special cases: finite connectivity and symmetric under complex conjugation) by Koebe in 1908 [@Koebe_1908_UbaK3].

$\bullet$ (1912) [**Schwarz’s lemma**]{}. The coinage as a such appears first in Carathéodory 1912 [@Caratheodory_1912], but already published in the modern fashion in 1907 by the same writer [@Caratheodory_1907], acknowledging the argument of E. Schmidt.

$\bullet$ (1916) [**Extremal problems=Extremalprobleme**]{} used in function theory by Bieberbach 1916 [@Bieberbach_1916].

$\bullet$ (ca. 1914) [**Circle mapping=Kreisabbildung.**]{} This is used (at least) since Bieberbach 1914 [@Bieberbach_1914 p.100], Koebe 1915 [@Koebe_1915], Bergman\[n\] 1922 [@Bergman_1922 p.238], Bochner 1922 [@Bochner_1922 p.184], with the English translation appearing first in Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950].

$\bullet$ (ca. 1975–1977) [**Total reality, “total reell”, etc.**]{} The adjective “total reell” (totally real) is first used (in the generality) in Geyer-Martens 1977 [@Geyer-Martens_1977 p.101, p.103], where the connection with Ahlfors theorem is made explicit along the line already suggested in Alling-Greenleaf 1969 [@Alling-Greenleaf_1969]. Geyer-Martens ascribe (cf. p.101) the concept of total reality (when paraphrased in the language of field extensions) to J.T. Knight 1969 [@Knight_1969]. Somewhat earlier in lesser generality of Galois coverings “total reell” appears already in Martens 1975 [@Martens_1975]. Meanwhile I think that “totally real” is quite widespread, especially in the growing field of real enumerative algebraic geometry (e.g. works by Sottile).

$\bullet$ [**Erster Art=Type I, orthosymmetric and dividing curves**]{} All this jargon is due to Klein. More precisely, Erster Art appears in Klein 1876 [@Klein_1876] and is much used in Russian literature (meanwhile diffusing in the west), cf. e.g. Rohlin 1978 [@Rohlin_1978] (and Gudkov 1974 [@Gudkov_1974/74]???). I remember some irony of Grisha Mikhalkin during a talk by Orevkov, where he found Klein’s subsequent jargon “orthosymmetric” (first in print in Weichold 1883 [@Weichold_1883]) quite awkward. Yet, Klein himself turned to be quite proud of this more intrinsic coinage. This turned to be quite influential, adhered by eminent workers like Koebe, J. Douglas, etc. albeit quite in desuetude today. The reason is mostly due to synonyms like dividing (or separating) curves.

Summarizing, the following words are used resp by:

$\bullet$ “Erster Art=Type I” first coined in Klein 1876 [@Klein_1876], and adhered to by Rohlin 1978 [@Rohlin_1978 p.90], and then much of the subsequent Russian literature,

$\bullet$ orthosymetrisch Klein ca. 1882 (lectures), adhered to Weichold 1883 [@Weichold_1883] (first occurrence in print), then Klein 1891/92 [@Klein_1891--92_Vorlesung-Goettingen] (Vorles. Göttingen), and followed by Koebe 1907 [@Koebe_1907_UrAK] (etc.), Fatou 1930 (in Appel-Goursat 1930 [@Appell-Goursat-Fatou_1930]), Julia 1932 [@Julia_1932], Douglas 1936 [@Douglas_1936-Some-new-results]–1939 [@Douglas_1939-The-most-general], etc.

$\bullet$ “zerteilend vs. nichtzerteilend” in Fiedler 1981 [@Fiedler_1981 p.7]

$\bullet$ “divide” alone is briefly mentioned in Arnold 1971 [@Arnold_1971/72] (yet only as a property of $M$-curves)

$\bullet$ “dividing curves” is used by Wilson 1978 [@Wilson_1978 p.66], Viro 1986/86 [@Viro_1986/86-Progress p.58] Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91 p.359], Gilmer 1991 [@Gilmer_1991], Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.736,737].

$\bullet$ “separating curves”, occurs in Fiedler 1982/83 [@Fiedler_1982/83-Pencil p.162], Dubrovin 1983/85 [@Dubrovin_1983/85], Nikulin 1983/84 [@Nikulin_1983/84], Benedetti-Risler 1990 [@Benedetti-Risler_1990], Natanzon 1990 [@Natanzon_1990/90] or 1999 [@Natanzon_1999-Moduli-real-alg-surf.superanal-differ-spinors], Coppens 2011 [@Coppens_2011],

$\bullet$ “courbes séparantes” in Marin 1979 [@Marin_1979], “courbe qui sépare sa complexifiée in Marin 1988 [@Marin_1988], Gabard 2006 [@Gabard_2006] (alas in French it sounds strange to say “courbe divisante”).

The following concept is a priori foreign to our survey, albeit it would be interesting to see if the methods of Grötzsch-Teichmüller are of some relevance to the Ahlfors mapping of 1950. This is another mathematical question, but here we content ourselves with a point of terminology:

$\bullet$ (1928/1935) [**Quasiconformal mappings=quasikonforme Abbildungen**]{}. This nomenclature is usually ascribed to Ahlfors 1935, who however could not remember precisely from where he borrowed the jargon, according to Kühnau 1997 [@Kuehnau_1997 p.133]), which is worth quoting:

Der Name Grötzsch ist wohl bei vielen vor allem mit der Theorie der quasikonformen Abbildungen verbunden, die er ab 1928 begründete. Die Bezeichnung “Quasikonforme Abbildungen” wurde allerdings erst später von L.V. Ahlfors eingeführt. (Freilich sagte mir Ahlfors Februar 1992 in Oberwolfach, da[ß]{} er diese Bezeichnung bei jemandem “gestohlen” habe, er wisse nur nicht mehr bei wem.)

Maybe it contributes to the question to remember that the jargon “[*quasikonform*]{}” appears already in 1914, und zwar bei Carathéodory 1914 [@Caratheodory_1914 §16](=page 294 in the pagination of the Ges.Math.Schriften,Bd.3).

Dissidence from Ahlfors {#dissident:sec}
-----------------------

\[31.08.12\] Sec.\[Ahlfors-proof:sec\] attempted to present Ahlfors’ proof in full details, but failed to digest the details. This deplorable issue motivated us to tabulate a list of “dissident” authors, who instead of quoting the original source Ahlfors 1950 [@Ahlfors_1950] adhered to subsequent treatments. Two accounts emerge with high rating, namely:

$\bullet$ Heins 1950 [@Heins_1950]

$\bullet$ Royden 1962 [@Royden_1962]

Of course, our “dissident” writers (quoting beside Ahlfors some derived product) never (as far as I know) criticizes directly the 1950 work of Ahlfors. At least there dissidence may suggest that themselves were not completely happy with (resp. convinced by) the original text finding more convenient another implementation. Albeit nobody ever expressed frontal objections against Ahlfors 1950 [@Ahlfors_1950], it is not to be excluded (yet of very low probability ca. $10^{-14}$) that somebody once detected some little bug, explaining perhaps the numerous initiatives to reprove Ahlfors’ result from different viewpoints. (We mention again the articles by Mizumoto 1960 [@Mizumoto_1960] and Kuramochi 1952 [@Kuramochi_1952] (undigest?), and refers for a extensive tabulation of such initiatives to the circled item of Fig.\[Map:fig\]).

Here is a sample of dissident authors (grouped according to their preferred source) with relevant extracts in “…”:

$\bullet$ Stout 1972 [@Stout_1972 p.345]: “…a theorem of Ahlfors \[2\](=Ahlfors 1950 [@Ahlfors_1950]) shows that $\cal H(R)$ contains many inner functions. (See also the elegrant \[sic!\] construction of Heins \[15\](=Heins 1950 [@Heins_1950]) as well as the earlier paper of Bieberbach \[3\](=Bieberbach 1925 [@Bieberbach_1925]) which deals with the case of planar domains.)”

$\bullet$ Khavinson 1984 [@Khavinson-Dimitri_1984 p.377]: “The following theorem is a classical result of Bieberbach and Grunsky (see \[6\](=Golusin 1952/57 [@Golusin_1952/57]), \[8\](=Grunsky 1978 [@Grunsky_1978])). For a different approach due to L. Ahlfors, see \[1\](=Ahlfors 1950 [@Ahlfors_1950]). Our proof, although discovered independently, is almost the same as that due to M. Heins in \[11\](=preprint=now published as Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF]) or H. Grunsky in \[8\](=Grunsky 1978 [@Grunsky_1978]). [Theorem 3.]{} [*Let $\zeta_1, \dots, \zeta_n$ be arbitrary fixed points on $\gamma_1, \dots,\gamma_n$ respectively. Then, for each $j$, $\phi(z)$ is the unique function giving a conformal mapping of $G$ onto an $n$-sheeted right half-plane such that $\phi(\zeta_j)=\infty$, for all $j$, $\phi(z_0)=1$.*]{}

$\bigstar$ admittedly, this Khavinson’ extract in not hundred percent pertinent to our present purpose inasmuch as the Bieberbach-Grunsky theorem is confined to the planar case.

$\bullet$ Stout 1965 [@Stout_1965]: “In order to establish our result, we shall need to make use of a result of Ahlfors \[1\](=Ahlfors 1950 [@Ahlfors_1950]). (For an alternative proof, one may consult Royden \[15\](=Royden 1962 [@Royden_1962].)

Theorem 3.1 [*There exists a function $P$ holomorphic on a neighborhood of $\bar R$ which maps $R$ onto the open unit disc in an one-to-one manner for some $n$ and which satisfies $\vert P \vert =1$ on $\partial R$.*]{}”

$\bigstar$ Of course the above “one-to-one” is a typo to be read as “$n$-to-one”.

$\bullet$ Alling 1966 [@Alling_1966 p.346]: “Finally, I am indebted to Professor Royden for his excellent paper, [*The boundary values of analytic and harmonic functions*]{}, \[24\](=Royden 1962 [@Royden_1962]), which not only gave a new proof of the existence of the Ahlfors’ map, but also gave generalizations of the classical boundary value theorems over the disc. …”

$\bullet$ Stout 1966/67 [@Stout_1966/67 p.366]: “Let $R$ be a finite open Riemann surface whose boundary $\Gamma$ consists of $N$ analytic, pairwise disjoint, simple closed curves. Let $\eta$ be an analytic mapping from $R$ onto $U$, the open unit disc which is holomorphic on a neighborhood of $\overline R$ and which is of modulus one on $\Gamma$. That such functions exists was first established by Ahlfors \[1\](=Ahlfors 1950 [@Ahlfors_1950]); another proof of their existence is in the paper \[12\](=Royden 1962 [@Royden_1962]).”

$\bullet$ Stout 1967 [@Stout_1967-Interpolation]: “It is convenient to make use of an [*Ahlfors map*]{} for $R$, i.e., a function continuous on $\overline R$ and holomorphic in $R$ which is constantly of modulus one on $\Gamma$. The existence of such function was established by Ahlfors in \[1\](=Ahlfors 1950 [@Ahlfors_1950]); an alternative proof of their existence is in \[4\](=Royden 1962 [@Royden_1962]).”

$\bullet$ O’Neill-Wermer 1968 [@O'Neill-Wermer_1968]: “Let $W$ be a region on some Riemann surface whose boundary is the union of a finite number of analytic simple closed curves and with $W$ having compact closure. In “Open Riemann surfaces and extremal problems on compact subregions”, (1950), L. Ahlfors considers the following extremal problem:

[*Problem*]{} I. [*Let $a,b$ be points of $W$. among the functions $F$ analytic on $W$ with $\vert F(z)\vert\le 1$ on $W$ and $F(a)=0$, it is required to find the one which makes $\vert F(b)\vert$ a maximum.*]{}

He shows that this problem has a unique solution[^110] $f$ which maps $W$ in an $n$-to-$1$ fashion[^111] onto the unit disk, for some $n$. His method of proof depends on a certain associated extremal problem introduced by P.R. Garabedian in his Thesis. (See Garabedian 1949 [@Garabedian_1949]). Another proof is given by H. Royden, “The boundary values of analytic and harmonic functions,” Math. Z. 78 (1962), 1–24.”

$\bullet$ Stanton 1971 [@Stanton_1971 p.293]: “Our argument rests on the following theorem of Ahlfors \[1\](=1950). [Theorem.]{} [*There is a function $f$ which is analytic on $W\cup \Gamma$ and which maps \[the interior\] $W$ onto $U$ and $\Gamma$ onto $T$.*]{} This theorem is also proved in Royden \[7\](=1962). A function $f$ of the kind described in this theorem is called an [*Ahlfors mapping*]{}.”

$\bigstar$ Upon recalling, that Stanton is a Royden student this may eventually be counted as a self-voting.

$\bullet$ Hejhal 1972 [@Hejhal_1972 p.119]: “Suppose first of all that $W$ is the interior of a compact bordered surface $\overline W$. L. Ahlfors \[2\](=1950) and H. Royden \[24\](=1962) have studied the present linear extremal problem on such $W$ at least for the case $\chi \equiv {\rm constant}$ and $\frak L [f]=f(b)$ with $b\in W$. …”

$\bullet$ Gamelin 1973 [@Gamelin_1973-Extremal-I p.3]: “…the paper of H.L. Royden deals with finite bordered Riemann surfaces.”

$\bullet$ Gamelin 1973 [@Gamelin_1973-BAMS p.1105]: “For dual extremal problems on Riemann surfaces, see \[2\](=Ahlfors 1950 [@Ahlfors_1950]) and \[36\](=Royden 1962 [@Royden_1962]).”

$\bullet$ Fisher 1973 [@Fisher_1973 p.1183]: “A similar problem \[…\] has been investigated by L. Ahlfors \[A1\], H. Royden \[R\], and others. In that case, the class of competing function is convex, the solution is unique, is analytic across the boundary $\Gamma$, and has modulus one on $\Gamma$.” And further on page 1187: “Let $F$ be the solution to the Ahlfors-Royden extremal problem described in the introduction. …”

$\bullet$ Lund 1974 [@Lund_1974 p.495]: “Let $U$ be the open unit disk in ${\Bbb C}$. We call $F$ an unimodular function if $F$ is analytic in a neighborhood of $\overline{R}$ and maps $\overline R$ onto $\overline U$ so that $F$ is $n$-to-one if we count the multiplicity of $F$ where $dF$ vanishes. If $T$ is the unit circle, then $F$ maps $\Gamma$ onto $T$. The existence of such a function was first proved by Ahlfors \[1\](=1950). Later, Royden \[4\](=1962) gave another proof of this result.”

$\bullet$ Kirsch 2005 [@Kirsch_2005]: “Ahlfors generalized Garabedian’s result to regions on Riemann surfaces \[2\](=Ahlfors 1950 [@Ahlfors_1950]); see Royden’s paper \[159\](=Royden 1962 [@Royden_1962]) for another treatment as well as further references to the literature.”

$\bullet$ Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000 p.240]: “It has been shown by Ahlfors \[4\](=Ahlfors 1950 [@Ahlfors_1950]) that such a function \[=ramified $n$-sheeted covering of the unit disk\] always exists, and it may be chosen to have the minimal possible degree $g+1$; see also \[5\](=Alling-Greenleaf 1971 [@Alling-Greenleaf_1971]), \[19\](=Fay 1973 [@Fay_1973]), and \[21\](=Fedorov 1991 [@Fedorov_1991]).”

Apart from the fact that the writer (Gabard) does not adhere with Alpay-Vinnikov’s claim about $g+1$ being the minimal possible degree for such a mapping ($g$ is the genus of the double, cf. op.cit. p.230), the three proposed references are in our opinion not perfectly adequate as substitute to Ahlfors 1950 [@Ahlfors_1950]. Alling-Greenleaf [@Alling-Greenleaf_1971 p.16, Thm1.3.6] only states Ahlfors’ result yet without reproving it, whereas both Fay and Fedorov recover the result in the planar case only.

Acknowledgements
----------------

The author wishes to thank the following long list of geometers (in chronological order of their interaction with the writer in connection to the present text)

$\bullet$ Felice Ronga (ca. 1997/98 for his explanation of Brusotti’s theorem),

$\bullet$ Claude Weber, Michel Kervaire (for their explanations on how to classify Klein’s symmetric surfaces by looking at the quotient bordered surface)

$\bullet$ Frédéric Bihan for pleasant discussions about real algebraic geometry,

$\bullet$ Lee Rudolph (ca. 1999 for explaining to us what is the natural topological model for a real elliptic curve with only one “oval”, namely just a torus $S^1\times S^1$ acted upon by factor permutation $(x,y)\mapsto (y,x)$ fixing thereby the diagonal circle),

$\bullet$ Alexis Marin, Viatcheslav Kharlamov, Oleg Viro, Jean-Jacques Risler, Thierry Vust, Michel Kervaire, Pierre de la Harpe, John Steinig (for their comments and corrections improving the shape of the article Gabard 2000 [@Gabard_2000])

$\bullet$ Ragahavan Narasimhan, Jacek Bochnack (ca. 1999 for [*not*]{} having been in touch with Ahlfors’ result of 1950 [@Ahlfors_1950] enabling me some free gestation about thinking on the problem)

$\bullet$ Manfred Knebusch for his kind interest in the modest work Gabard 2000 [@Gabard_2000],

$\bullet$ Johannes Huisman for his constant interest (2001–04–06), and his care about correcting bugs in both my Thesis and the article Gabard 2006 [@Gabard_2006],

$\bullet$ Sergei Finashin for an exciting discussion in Rennes 2001,

$\bullet$ Jean-Claude Hausmann (ca. 2000/01) for telling me about the standard surjectivity criterion via the Brouwer degree, which was decisive to complete Gabard 2006 [@Gabard_2006],

$\bullet$ Antonio Costa, for his fascinating talks in Geneva,

$\bullet$ Bujalance for his surely over-enthusiastic Zentralblatt review of my article (Gabard 2006 [@Gabard_2006]),

$\bullet$ Fraser-Schoen, whose brilliant work revived my interest in the theory of the Ahlfors’ mapping (ca. the 13 March 2011) at a stage were I was mostly sidetracked by “non-metric manifolds” thanks to efforts of Mathieu Baillif and David Gauld.

$\bullet$ Stepan Orevkov, Oleg Viro (2011) for their talks and pleasant discussions,

$\bullet$ Marc Coppens (2011–12) for e-mails, and his work on the separating gonality (2011 [@Coppens_2011]) adumbrating sharper insights on the degree of the Ahlfors function (or rather the more general allied circle maps). His turning-point result appeals to a better conciliation of the analytic theory of Ahlfors with the algebro-geometric viewpoint.

$\bullet$ (2011/12) Hugo Parlier, Peter Buser, Alexandre Girouard, Gerhard Wanner und Martin Gander are acknowledged for their recent e-mail exchanges.

$\bullet$ (Oct. 2012) Daniel Coray for enlarging the capacity of my TeX-compilator, and for his lovely (Cambridge-style) teaching about the geometric Galois action in ca. 1999.

$\bullet$ (16 Nov.2012) Franc Forstnerič for pointing out his text with Wold (2012 [@Forstneric-Wold_2012]) showing the state of the art on the proper holomorphic embedding problem.

$\bullet$ (09 Jan. 2013) Oleg Viro for his excellent answers on some naive questions on Rohlin’s paper (cf. Sec.\[e-mail-Viro:sec\]).

$\bullet$ (10 Jan. 2013) Alexis Marin for his invaluable insights on Klein’s intuitions and more (cf. Sec.\[e-mail-Viro:sec\]).

$\bullet$ (January-February-March 2013) Viatcheslav Kharlamov, Stepa Orevkov, Eugenii Shustin, Séverine Le Touzé, Thomas Fiedler for all their letters reproduced in Sec.\[e-mail-Viro:sec\].

Bibliographic comments
======================

The writer does not pretend that the following bibliography is complete (nor that he absorbed all those fantastic contributions in full details). More extensive bibliographies (overlapping ours), but covering more material include those of:

$\bullet$ Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] (ca. 40 pages times 25 items per pages=1000 entries covering such topics as the Dirichlet problem, extremal problems, the type problem, the allied classification theory, etc.);

$\bullet$ Grunsky 1978 [@Grunsky_1978] (=562 refs, including 48 Books).

Most entries of our bibliography are followed by some comment explaining briefly the connection to our primary topic of the Ahlfors map. The following symbolism is used:

$\clubsuit$ serves to point out a special connection to Ahlfors 1950 (especially alternative proofs).

$\spadesuit$ gives other comments (attempting to summarize the paper contents or to explicit the connection in which we cite it).

$\P$ signals papers not quoted in the main-body of the text but connected to our topic.

$\bigstar$ marks sources, I could not as yet procure a copy.

47

50

60 78

$\bullet$ the stickers/sigles 60, 78 are assigned when the source has already been cited in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] resp. Grunsky 1978 [@Grunsky_1978].

$\bullet$ 50 designates those references citing the paper Ahlfors 1950 [@Ahlfors_1950] (there represents circa 106 articles on “Google”), and occasionally 47 those quoting Ahlfors 1947 [@Ahlfors_1947].

[**$\heartsuit$n**]{} is something like the indicator of the US rating agency (to be read “liked by $n$”). It indicates the cardinal number [**n**]{} of citations of the paper as measured by “Google Scholar”. The latter machine often misses cross-citations, especially those in old books, or old articles with references given in footnotes format. Many sources cited in Grunsky’s book (1978 [@Grunsky_1978]) are never cited electronically. Accordingly, those rating numbers only supply a statistical idea of the literature ramifications lying beyond a given entry. Also low-citation articles are sometimes the most polished product ripe for museum entrance. Forelli 1979 [@Forelli_1979] is typical: self-contained, elegant and polished proof of Ahlfors result, yet only rated by 3. Our bibliography is somewhat conservative with comparatively few modern references. Our excuse is two-fold: modern expressionism is sometimes harder to grasp, and recent references are usually well detected through computer search. (Papers are listed in alphabetical, and then chronological order, regardless of shared co-authorship.)

The primary focus is on the Ahlfors map and the weaker (but more general) circle maps. As a such the topic overflow slightly over the territory of real algebraic geometry. Ahlfors-Sario’s book 60 address Riemann surfaces, whereas Grunsky’s book 78 focuses to the case of planar domains. Hence both bibliographies 60, 78 are quite complementary, and ours is essentially a fusion of both, but we gradually included more and more recent contribution. Still additional references are welcome. For conformal maps, it is helpful when browsing the vast literature to keep in mind the basic question: [*what result through which method?*]{}

[**Results.**]{} Objects traditionally range along increasing order of generality through: simply-connected regions, multiply-connected ones and finally Riemann surfaces. We often add a humble compactness proviso, as the passage to open objects is traditionally achieved through the exhaustion trick (going back at least to Poincaré 1883 [@Poincare_1883], and see also Koebe 1907 [@Koebe_1907_UbaK1]), and active in recent time (e.g. Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950].)

As to the mappings, they may all be interpreted in some way or another as ramification of RMT (Riemann’s mapping to a circle=disc). We distinguish primarily:

$\bullet$ CM=circle maps (usually not univalent, but multi-sheeted disc with branch, or winding points=Windungspunkte)

$\bullet$ KNP=Kreisnormierung(sprinzip) (univalent map to a circular domain)

$\bullet$ SM=slit mappings for various types of them (parallel, circular, radial, logarithmic spiral, etc.). Those are all allied to certain natural foliation of the sphere, and some extreme generality in this respect is achieved in Schramm’s Thesis where any foliation is permitted as support for the slits.

[**Methods.**]{} They may be classified in two broad classes quantitative vs. qualitative (each having some branchings):

$\bigstar$ (Quantitative) variational methods, including:

$\bullet$ DP=Dirichlet principle (or more broadly speaking, potential theory=PT, centering around such concepts as the Green’s function, harmonic measures (i.e. harmonic function with special null/one boundary prescription of the various contour), etc. Of course, there is a standard yoga between Dirichlet and Green, so all this is essentially one and the same method.

$\bullet$ IM=Iterative methods (originators: Koebe and Carathéodory), and by extension this may proliferate up to including the circle packings.

$\bullet$ EP=extremal problems (e.g. the one of maximizing the derivative amongst the class of function bounded-by-one) and leads to the Ahlfors map.

$\bullet$ BK=Bergman kernel (or Szegö kernel), here the fundamental ideas rest upon Hilbert’s space methods, and the idea of orthogonal system. Initially, the method is also inspired by Ritz, and Bieberbach extremal problem (1914 [@Bieberbach_1914]) for the area swept out by the function. Since the middle 1940’s, there were found several conformal identities among so-called domain functions (Green’s, Neumann’s, etc.) and the kernel functions so that virtually this is now highly connected to DP$\approx$PT. Also the Ahlfors map is expressible in term of the Bergman kernel (cf. e.g., Nehari 1950 [@Nehari_1950]) so that this heading is strongly connected to EP.

$\bullet$ PP=Plateau problem style methods (for RMT, this starts with the observation of Douglas 1931 [@Douglas_1931-Solution]). This strongly allied to DP, albeit some distinction is useful to keep in mind just for cataloguing purposes.

$\bigstar$ (Qualitative) topological methods:

$\bullet$ the [*continuity method*]{}, as old as Schläfli, (as Koebe notices somewhere) is involved in the accessory parameters of Schwarz-Christoffel, in Klein-Poincaré’s uniformization through automorphic functions, Brouwer (invariance of the domain), Koebe, etc., e.g. Golusin 1952/57 [@Golusin_1952/57])

$\bullet$ Brouwer topological degree and the allied surjectivity criterion (cf. e.g., Mizumoto 1960 [@Mizumoto_1960], Gabard 2006 [@Gabard_2006]). Here the idea is that there is some topological stability of the embedding of a curve into its Jacobian via the Abel mapping in the sense that its homological feature are unsensitive to variation of the complex (analytic) structure (moduli), and this enables one to draw universal statement by purely topological considerations.

Finally we have attempted to manufacture a genealogy map showing the affiliation between the authors. The picture turned out to be so large that TeX prefers reject it at the very end of the file.

-1.2cm0 -4.2cm0 -5pt0

-5pt0

\[15.10.12\] When I reached 884 references, I unfortunatel met the so-called “TeX capacity exceeded, sorry.” obstruction (cf. Knuth’s “The TeX Book”, p.300 for more details). Thus I had to deactivate some references which are not used for cross-citation, albeit they clearly belong to our topic. \[16.10.12\] This problem was ultimately solved by my advisor Daniel Coray, to whom I express my deepest gratitude for enlarging the TeX capacity of my compilator.

[9]{}

H. Abe, [*On some analytic functions in an annulus*]{}, Kodai Math. Sem. Rep. 10 (1958), 38–45. \[$\spadesuit$ quoted in Minda 1979 [@Minda_1979] in connection with the theta function expression of the Ahlfors function of an annulus\] 

N.H. Abel, [*Mémoire sur une propriété générale d’une classe très étendue de fonctions transcendantes*]{}, présenté à l’Académie des Sciences à Paris le 30 octobre 1826; published (only) in: [*Mémoires présentés par divers savants*]{}, t. VII, Paris, 1841. Also in [Œ]{}uvres, t. I, 145–211. \[$\spadesuit$ first occurrence of Abel’s theorem, which in Gabard 2004/06 [@Gabard_2006] is used as the main weapon toward proving existence of Ahlfors circle maps\] 

N.H. Abel, [*Remarques sur quelques propriétés générales d’une certaine sorte de fonctions transcendantes*]{}, Crelle J. Reine Angew. Math. 3 (1828). \[$\spadesuit$\] 

N.H. Abel, [*Démonstration d’une propriété générale d’une certaine classe de fonctions transcendantes*]{}, Crelle J. Reine Angew. Math. 4 (1829), 201–202. \[$\spadesuit$\] 

W. Abikoff, [*The Real Analytic Theory of Teichmüller spaces*]{}, Lecture Notes in Math. 820, Springer, 1980. \[$\spadesuit$\] 

M.B. Abrahamse, [*Toeplitz operators in multiply connected domains*]{}, Amer. J. Math. 96 (1974), 261–297. \[$\spadesuit$ extend to finite Riemann surfaces? try Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000]\] 

M.B. Abrahamse, R.G. Douglas, [*A class of subnormal operators related to multiply connected domains*]{}, Adv. Math. 19 (1976), 106–148. \[$\spadesuit$ for extension to finite Riemann surfaces? try Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000], Yakubovich 2006 [@Yakubovich_2006]\] 

M.B. Abrahamse, J.J. Bastian, [*Bundle shifts and Ahlfors functions*]{}, Proc. Amer. Math. Soc. 72 (1978), 95–96. 47 \[$\spadesuit$ the Ahlfors function of a domain $R$ with $n$ contours is applied to the calculation of a bundle shift (that is, a pure subnormal operator with spectrum contained in the closure of $R$ and normal spectrum contained in the boundary of $R$)\] 

M.B. Abrahamse, [*The Pick interpolation theorem for finitely connected domains*]{}, Michigan J. Math. 26 (1979), 195–203. \[$\spadesuit$ extend to finite Riemann surfaces? try Heins 1975 [@Heins_1975], Jenkins-Suita 1979 [@Jenkins-Suita_1979], both works subsuming in principle Ahlfors 1950 [@Ahlfors_1950]\] 

N. A’Campo, [*Sur la première partie du seizième problème de Hilbert*]{}, Sém. Bourbaki 537 (1979). \[$\spadesuit$ contains a nice picture of Hilbert’s method in degree 6 $\spadesuit$ an exposition of Gudkov’s construction of the $M$-scheme $\frac{5}{1}5$ omitted by Hilbert\] 

R. Accola, [*The bilinear relation on open Riemann surfaces*]{}, Trans. Amer. Math. Soc. ?? (1960), ??–??. 50 \[$\spadesuit$\] 

J. Agler, J. Harland, B.J. Raphael, [*Classical function theory, operator dilation theory, and machine computation on multiply-connected domains*]{}, Mem. Amer. Math. Soc. 191 (2008), 159pp. 78 \[$\spadesuit$ cite Grunsky 1978 [@Grunsky_1978] and gives via the Grunsky-(Ahlfors) extremal function an interpretation of the Herglotz integral representation via the Kreĭn-Milman theorem $\spadesuit$ \[06.10.12\] contains also a nice desription of circle maps (in the form of half-plane maps, which seems to be directly inspired from Heins’ treatment (1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF]) who probably offers an alternative derivation of Ahlfors’ bound $r+2p$) $\spadesuit$ “In three chapters the authors first cover generalizations of the Herglotz representation theorem, von Neumann’s inequality and the Sz.-Nagy dilation theorem to multiply connected domains. They describe the fist through third Herglotz representation and provide an …”\] 

D. Aharonov, H.S. Shapiro, [*Domains on which analytic functions satisfy quadrature identities*]{}, J. Anal. Math. 30 (1976), 39–73. 78 \[$\spadesuit$ includes the result that the Ahlfors map of a quadrature domain is algebraic, see also papers by Gustafsson, Bell, etc.\] 

P.R. Ahern, D. Sarason, [*On some hypo-Dirichlet algebras of analytic functions*]{}, Amer. J. Math. 89 (1967), 932–941. \[$\spadesuit$\] 

P.R. Ahern, D. Sarason, [*The $H^p$ spaces of a class of function algebras*]{}, Acta Math. 117 (1967), 123–163. \[$\spadesuit$ “This paper is a study of a class of uniform algebras and of the associated Hardy spaces of generalized analytic functions. It is a natural continuation of a number of similar studies which have appeared in recent years; see Bochner \[7\], Helson and Lowdenslager \[15\], ...”\] 

P.R. Ahern, [*On the geometry of the unit ball in the space of real annihilating measures*]{}, Pacific J. Math. 28 (1969), 1–7. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited on p.4, yet not exactly for the result we have in mind, but see also the related paper Nash 1974 [@Nash_1974] where the fascinating study of the geometry of the convex body of representing measures is continued $\spadesuit$ \[13.10.12\] it could be fascinating to penetrate the geometry of this body in relation to the “link” (collection of circles in the “same” Euclidean space ${\Bbb R}^g$, where $g$ is the genus of the double), as it occurs in Ahlfors original proof 1950 [@Ahlfors_1950]). Understanding this is probably the key to a sharper understanding of circle maps, in particular of their lowest possible degrees\] 

L.V. Ahlfors, H. Grunsky, [*Über die Blochsche Konstante*]{}, Math. Z. 42 (1937), 671–673. \[$\spadesuit$ not directly relevant to this text, except for the hardness of some extremal problems. Compare the front cover of Grunsky’s Coll. Papers for a depiction of the hyperbolic tessellation allied to the conjectured extremal function. Basically one consider in the Euclidean plane ${\Bbb C}$ the triangulation by equilateral triangles and above it in the hyperbolic disc one manufactures a equilateral triangle of angle the half value namely $\pi/6$. Mapping conformally (via Schwarz-Christoffel) this hyperbolic triangle to the fatter Euclidean triangle, while reproducing this map by the (Riemann)-Schwarz principle of reflection yields a function conjectured to have extremal Bloch constant. The latter amounts to have the largest schlicht disc avoiding the ramification. In our case the densest packing (à la Kepler) is clearly the equilateral triangulation of ${\Bbb C}$, so the above Ahlfors-Grunsky function is the natural candidate for having the maximum Bloch radius, alas nobody ever succeeded to show this. Similar problems make sense for other geometries, e.g. planar to spherical, in which case the Bloch constant was estimated in the breakthrough of Bonk-Eremenko, ca. 2002\] 

L.V. Ahlfors, [*Bounded analytic functions*]{}, Duke Math. J. 14 (1947), 1–11. 60, 78 \[$\clubsuit$ the planar case of Ahlfors 1950 [@Ahlfors_1950], cite Grunsky 1940–42 [@Grunsky_1940; @Grunsky_1942] as an independent forerunner $\spadesuit$ albeit less general than the next item (Ahlfors 1950 [@Ahlfors_1950], which includes the case of positive genus) this article is more quoted that its successor essentially due to the intense activity centering around analytic capacity and the Painlevé null-sets implying a super vertical series of workers like Vitushkin, Melnikov, Garnett, Calderón 1977 [@Calderon_1977], David, etc. culminating to Tolsa’s 2002/03 [@Tolsa_2003] resolution of the Painlevé problem $\spadesuit$ as a detail matter, it may be recalled that the present article contains a minor logical gap fixed in Ahlfors 1950 [@Ahlfors_1950] (cf. the later source, especially footnote p.123) and also the “Commentary” in the Collected papers Ahlfors 1982 [@Ahlfors_1982_Coll_papers p.438]: “When writing the paper I overlooked a minor difficulty in the proof. This was corrected in \[36\](=Ahlfors 1950 [@Ahlfors_1950]).” $\spadesuit$ compare also the comment in the German edition of Golusin 1952/57 [@Golusin_1952/57 p.415, footnote 2]: “Der biesherige Beweisgang erlaubt es nicht, zu schlie[ß]{}en, da[ß]{} keine der $n-1$ Nullstellen eine mehrfache ist. Diese Lücke des [Ahlfors]{}schen Beweises wurde von P.R. Garabedian in seiner Dissertation (=1949 [@Garabedian_1949]), beseitigt., Anm.d.Red.d.deutschenAusgabe.”\] 

L.V. Ahlfors, [*Open Riemann surfaces and extremal problems on compact subregions*]{}, Comment. Math. Helv. 24 (1950), 100–134. 60, 78 \[$\clubsuit$ the central reference of the present article $\clubsuit$ contains the first existence-proof of a circle map on a general compact bordered Riemann surface $\clubsuit$ in fact both a qualitative existence result as well as a quantitative extremal problem are presented\] 60, 78 

L.V. Ahlfors, A. Beurling, [*Conformal invariants and function-theoretic null sets*]{}, Acta Math. 83 (1950), 101–129. 60, 78 

L.V. Ahlfors, [*Development of the theory of conformal mapping and Riemann surfaces through a century*]{}. In: [*Contributions to the Theory of Riemann Surfaces. Centennial Celebration of Riemann’s Dissertation*]{}, Annals of Math. Studies 30, 3–13, Princeton 1953; or [*Collected Papers*]{}, Vol.1, 1929–1955, Birkhäuser, 1982. \[$\spadesuit$ a colorful historical survey of Riemann, Schwarz, Poincaré, Koebe, Nevanlinna, Grötzsch, Grunsky and Teichmüller\] 

L.V. Ahlfors, [*Variational methods in function theory*]{}. Lectures at Harvard University, 1953 transcribed by E.C. Schlesinger. \[$\spadesuit$ cited in Read 1958 [@Read_1958_Acta]. Does this contains another (more pedestrian) treatment of Ahlfors 1950 [@Ahlfors_1950]?\]$\bigstar$$\bigstar$ 

L.V. Ahlfors, [*On quasiconformal mappings*]{}, J. Anal. Math. (Jerusalem) 3 (1953–54), 1–58. ($+$Erratum) \[$\spadesuit$\] 

L.V. Ahlfors, [*Extremalprobleme in der Funktionentheorie*]{}, Ann. Acad. Sci. Fenn., A.I., 249/1 (1958), 9 pp. \[$\spadesuit$ survey like, but pleasant philosophy\] 

L.V. Ahlfors, [*The complex analytic structure of the space of closed Riemann surfaces*]{}, In: Analytic functions, Princeton Univ. Press, 1960, 45–60. \[$\spadesuit$\] 

L.V. Ahlfors, L. Sario, [*Riemann Surfaces*]{}, Princeton Univ. Press, 1960. 

L.V. Ahlfors, [*Classical and contemporary analysis*]{}, SIAM Review 3 (1961), 1–9. 

L.V. Ahlfors, [*Complex Analysis*]{}, McGraw-Hill Book Co. (2nd ed.), 1966. \[$\spadesuit$ p.243–253 proof of the PSM (and other radial/circular avatars) via the Dirichlet principle\] 

L.V. Ahlfors, [*Lectures on Quasiconformal mappings*]{}, Van Nostrand, Princeton, NJ, 1966, 146pp. \[$\spadesuit$\] 

L.V. Ahlfors, [*Collected Papers, Vol. 1, 1929–1955*]{}, Birkhäuser, 1982. \[$\spadesuit$ p.438 is worth quoting in extenso: “The point of departure in \[30\](=Ahlfors 1947 [@Ahlfors_1947]) is Painlevé’s problem: Given a compact set $E\subset {\Bbb C}$, when does there exist a nonconstant bounded analytic function $f(z)$ on ${\Bbb C}\setminus E$? I was really interested in the function with the smallest upper bound of $\vert f(z)\vert$ when normalized so that $f(z)\sim
1/z$ at $\infty$. This smallest maximum is now called the [*analytic capacity*]{}[^112] of $E$, and the Russians[^113] used to refer to the extremal function, if it exists[^114], as the “Ahlfors function”, an unexpected and probably unearned distinction. In this form Painlevé’s problem is closely related to the precise form of Schwarz’s lemma[^115] for an arbitrary region, and that is what the paper is actually about. To be specific: if $\Omega\subset {\Bbb C}$ is a region and if $\vert f(z)\vert \le 1$ in $\Omega$ while $f(z_0)=0$ for a given $z_0\in \Omega$, exactly how large can $\vert
f'(z_0)\vert$ be?—The difficulty lies in the fact that while $u=\log\vert f(z)\vert$ is a harmonic function with a logarithmic pole at $z_0$, the single-valuedness of $f$ translates into diophantine conditions on the conjugate harmonic function $\nu$. Quite obviously this makes the problem much harder than if only the absolute value $\vert
f(z)\vert$ were required to be single-valued.—In my paper I restrict myself to a region $\Omega$ of finite connectivity $n$, and my aim is to describe the extremal function $f(z)$. I show that $\vert f(z)\vert=1$ on the boundary and that $f$ has exactly $n-1$ zeros[^116]. In other words, $f$ maps $\Omega$ on an $n-1$ times covered disk[^117]. In addition there are conditions on the location of the zeros[^118]. When writing the paper I overlooked a minor difficulty in the proof. This was corrected in \[36\](=Ahlfors 1950 [@Ahlfors_1950]).—The purpose of \[36\](=Ahlfors 1950 [@Ahlfors_1950]) was to study open Riemann surfaces by solving extremal problems on compact subregions and passing to the limit as the subregions expand. The paper emphasizes the use of harmonic and analytic differentials in the language of differential forms. It is closely related to \[35\](=Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950]), but differs in two respects: (1) It deals with Riemann surfaces rather than plane regions and (2) the differentials play a greater role than the functions.—I regard \[36\] as one of my major papers. It was partly inspired by R. Nevanlinna, who together with P.J. Myrberg (1954[^119]) had initiated the classification theory of open Riemann surfaces, and partly by M. Schiffer (1943 [@Schiffer_1943]) and S. Bergman (1950 [@Bergman_1950]), with whose work I had become acquainted shortly after the war. The paper also paved the way for my book on Riemann surfaces with L. Sario (1960 [@Ahlfors-Sario_1960]), but it is probably more readable because of its more restricted contents.—I would also like to acknowledge that when writing this paper I made important use of an observation of P. Garabedian to the effect that the relevant extremal problems occur in pairs connected by a sort of duality. This is of course a classical phenomenon[^120], but in the present connections it was sometimes not obvious how to formulate the dual problem.”\] 

L.V. Ahlfors, [*The Joy of function theory*]{}, ca. 1984. \[$\spadesuit$ p.443: “It has been customary to write about the joy of everything, from the joy of cooking to the joy of sex, so why not the joy of function theory?” $\spadesuit$ p.444: “I remember vividly how he \[=Lindelöf\] encouraged me to read the collected papers of Schwarz and also of Cantor, but he warned me not to become a logician, for which I am still grateful. Riemann was considered too difficult, and Lindelöf never quite approved of the Lebesgue integral.” $\spadesuit$ p.445: “It is impossible to change an analytic function at or near a single point without changing it everywhere. This crystallized structure is a thing of great beauty, and it plays a great role in much of nineteenth-century mathematics, such as elliptic functions, modular functions, etc. On the other hand, it was also an obstacle, perhaps most strongly felt in what somewhat contemptuously was known as “Abschätzungsmathematik”. Consciously or subconsciously there was a need to embed function theory in a more flexible medium. For instance, Perron used the larger class of subharmonic functions to study harmonic functions, and it had also been recognized, especially by Nevanlinna and Carleman, that harmonic functions are more malleable that analytic functions, and therefore a more useful tool.”\] 

V.B. Alekseev, [*Abel’s Theorem in Problems and Solutions*]{}, Nauka, Moscow, 1976. (Russian).$\bigstar$ 

Ju.E. Alenicyn, [*On some estimates for functions regular in a region of finite connectivity*]{}, (Russ.) Mat. Sb. N. S. 49 (1959), 181–190. 78 $\bigstar$ 

Ju.E. Alenicyn, [*An extension of the principle of subordination to multiply connected regions*]{}, (Russ.) Trudy Mat. Inst. Steklov 60 (1961), 5–21; Amer. Math. Soc. Transl. (2) 43, 281–297. 78 $\bigstar$ 

Ju.E. Alenicyn, [*Conformal mappings of a multiply connected domain onto many-sheeted canonical surfaces*]{}, (Russ.) Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 607–644. 78 

Ju.E. Alenicyn, [*An estimate of the derivative in certain classes of function, analytic in a multiply connected domain*]{}, Zap. Nauch. Sem. Leningrad Otdel. Mat. Inst. Steklov 24 (1972), 6–15; English transl. (1974), 565–571. 78 \[$\spadesuit$ Ahlfors extremal problem is recalled and extended to a more general setting, where instead of considering functions bounded-by-one in modulus, there is some continuous positive function $\lambda(z)$ defined on the contour which acts as the upper-bound over the permissible modulus via $\lim\sup_{z\to
z_0\in \partial D} \vert f(z) \vert \le \lambda(z_0)$\] 

Ju.E. Alenicyn, [*Inequalities for generalized areas for multivalent conformal mappings of domains with circular cuts*]{}, Translated from Matematicheskie Zametki 29 (1981) 387–395; \[$\spadesuit$ extension of the result of Vo Dang Thao 1976 [@Vo-Dang-Thao_1976] and Gaier 1977 [@Gaier_1977-Roumaine] (which the latter ascribes to Grötzsch 1931 [@Groetzsch_1931]) $\spadesuit$ this is close to (but not exactly) the desideratum that Bieberbach’s minimum problem (1914 [@Bieberbach_1914]) yields another interpretation of the Ahlfors circle map when extended to multiply-connected domains $\spadesuit$ p.202 a cross-reference to Nehari 1952 [@Nehari_1952-BOOK] is given, but this does not really answer our question whether the minimal map (of least area) is a circle map (it is just observed that in higher-connectivity it is [*not*]{} schlicht)\] 

Ju.E. Alenicyn, [*Least area of the image of a multiconnected domain of $p$-sheeted conformal mappings*]{}, Translated from Matematicheskie Zametki 30 (1981) 807–812; \[$\spadesuit$ extension of the result of Vo Dang Thao 1976 [@Vo-Dang-Thao_1976] and Gaier 1977 [@Gaier_1977-Roumaine] (which Gaier 1978 [@Gaier_1978-JDMV] ascribes to Grötzsch 1931) $\spadesuit$ this is close to (but not exactly) the desideratum that Bieberbach’s minimum problem (1914 [@Bieberbach_1914]) yields another interpretation of the Ahlfors circle map when extended to multiply-connected domains\] 

J.W. Alexander, [*Functions which map the interior of the unit circle upon simple regions*]{}, Ann. of Math. (2) 17 (1915), 12–22. \[$\clubsuit$\] 

N.L. Alling, [*A proof of the corona conjecture for finite open Riemann surfaces*]{}, Bull. Amer. Math. Soc. 70 (1964), 110–112. \[$\clubsuit$ applies Ahlfors 1950 [@Ahlfors_1950] to the corona as a lifting procedure of the disc-case established in Carleson 1962 [@Carleson_1962] $\spadesuit$ for an alternative proof of the same result avoiding the Ahlfors map but using uniformization instead, cf. Forelli 1966 [@Forelli_1966]\] 

N.L. Alling, [*Extensions of meromorphic function rings over non-compact Riemann surfaces. I*]{}, Math. Z. 89 (1965), 273–299. \[$\clubsuit$ idem as Alling 1964 [@Alling_1964] with full details\] 

N.L. Alling, [*Extensions of meromorphic function rings over non-compact Riemann surfaces. II*]{}, Math. Z. 93 (1966), 345–394. 50 \[$\clubsuit$ p.346: “Finally, I am indebted to Professor Royden for his excellent paper, [*The boundary values of analytic and harmonic functions*]{}, \[24\](=Royden 1962 [@Royden_1962]), which not only gave a new proof of the existence of the Ahlfors’ map, but also gave generalizations of the classical boundary value theorems over the disc. …” $\spadesuit$ p.345: “As in Alling 1965, theorems are frequently proved for $\overline{X}$ \[=a finite open Riemann surface\] by lifting the corresponding classical result for the disc, using the Ahlfors map in conjunction with various algebraic facts. For example, Fatou’s theorem and Nevanlinna’s theorem (about functions of bounded characteristics) are easily proved in this way.”\] 

N.L. Alling, in MathReviews, Review of Stout 1965, Bounded holomorphic functions on finite Riemann surfaces. \[$\clubsuit$ quoting an extract of the text: “It is now clear that a great many of the results for the disc $U$, which can be found, for example in K. Hoffman’s book (=1962 [@Hoffman_1962]), also hold for $R$ \[=the interior of a compact bordered surface\]. The choice of technique to extend such results depends then on the ease of proof, the intuition generated by the setting, and the predisposition of the investigator. Uniformization and the algebraic approach \[based upon Royden’s idea (1958=[@Royden_1958]) of a lifting procedure along an Ahlfors map\] seem to have an advantage over annular analysis in that they treat the whole space and the whole ring simultaneously. Still, special advantages in using uniformization and in using the algebraic approach persist. For example, the theory of the closed ideals in the standard algebra on $R$, ${\cal A} (R)$, and the theory of invariant subspaces have been worked out by M. Voichick (=1964 [@Voichick_1964]), using uniformization, but has not been achieved yet using the algebraic approach. (See also Voichick 1966 [@Voichick_1966], and a paper by Hasumi now in preprint \[=Hasumi 1966 [@Hasumi_1966]\], all of which deal with the invariant subspace problem.)” $\spadesuit$ \[13.10.12\] for an upgrade giving full answer to Alling’s desideratum of an Ahlfors-map proof of the closed ideals, see Stanton 1971 [@Stanton_1971] $\spadesuit$ the review is concluded with the following: “Finally, concerning the corona problem,as far as the reviewer knows, no one has given a new proof of Carleson’s theorem or re-proved it on $R$; everyone, to generalize it to $R$, has merely lifted the result to $R$ \[Or “descended” in the case of the uniformizing approach.\]. A substantially simpler and more lucid proof of Carleson’s theorem still remains the most challenging question in this subject.” $\spadesuit$ possible upgrades the new proofs à la Hörmander/Wolff (cf. e.g. Gamelin 1980 [@Gamelin_1980-Wolff's-proof]), and also the localization of the corona done by Gamelin 1970 [@Gamelin_1970-Localization-of] should be satisfactory answers. Yet our impression is that eventually any sharper understanding of the geometry of Ahlfors map (e.g. Gabard’s improved bound (2006 [@Gabard_2006]) on the degree of the Ahlfors circle maps) could implies modest quantitative refinements in the corona with bounds (cf. Hara-Nakai 1985 [@Hara-Nakai_1985] and Oh 2008 [@Oh_2008])\] 

N.L. Alling, N. Greenleaf, [*Klein surfaces and real algebraic function fields*]{}, Bull. Amer. Math. Soc. 75 (1969), 869–872. \[$\clubsuit$ the first paper (to the best of my knowledge) which makes explicit the link between Ahlfors 1950 [@Ahlfors_1950] and the much older Kleinian theory (1876–82) of orthosymmetric (=dividing) real algebraic curves, see Klein 1876 [@Klein_1876] and Klein 1882 [@Klein_1882]\] 

N.L. Alling, N. Greenleaf, [*Foundations of the Theory of Klein Surfaces*]{}, Lecture Notes in Math. 219, Springer-Verlag, Berlin, 1971. \[$\clubsuit$ repeat the same Klein-Ahlfors connection (cf. comments to the previous entry Alling-Greenleaf 1969 [@Alling-Greenleaf_1969]), and develop a systematic theory of Klein surfaces, a new jargon derived from Berzolari 1906 [@Berzolari_1906] $\clubsuit$ Ahlfors’ theorem (compare p.16, Theorem 1.3.6) is stated as follows: “Theorem 1.3.6 (Ahlfors \[${\rm A}_1$\]). Let $\frak
X$ be \[a\] compact, connected, orientable Klein surface with non-void boundary. There exists $\underline{f}\in E(\frak X)$ such that $\partial X=\Gamma_{\underline f}$. $\spadesuit$ if I do not mistake Ahlfors’ result is only stated but not reproved in the text (perhaps quite contrary to the hope borne out by the cross-citation in Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000]) $\spadesuit$ personal reminiscence \[03.09.12\]: I can remind clearly that I knew this famous Alling-Greenleaf text quite early (ca. Spring 1999), but did not appreciated directly the significance of Ahlfors result, and had to rediscover it later (ca. 2001) after some intense own work (ca. 2 years of efforts) $\spadesuit$ this is a bit ironical for showing how one can severely miss a crucial information through quick reading, but permitted me to develop an independent approach which ultimately turned out to give a sharper result than Ahlfors’ $\spadesuit$ so this is probably a perfect illustration of how a poor knowledge of the literature is sometimes beneficial for the creativity in “young men games” (if we can borrow Hardy’s bitter joke)\] 

N.L. Alling, B.V. Limaye, [*Ideal theory on non-orientable Klein Surfaces*]{}, Ark. Mat. ?? (1972), 277–292. \[$\clubsuit$ extension of the Beurling-Rudin result for the disc to non-orientable bordered surfaces, hence cannot employ the Ahlfors map (whose existence is confined to the orientable case), for which case see Stanton 1972 [@Stanton_1971] who uses the Ahlfors map for the same purpose (extension of Beurling-Rudin)\] 

N.L. Alling, [*Analytic geometry on real algebraic curves*]{}, Math. Ann. 207 (1974), 23–46. \[$\spadesuit$\] 

D. Alpay, V. Vinnikov, [*Indefinite Hardy spaces on finite bordered Riemann surfaces*]{}, J. Funct. Anal. 172 (2000), 221–248. 50 \[$\clubsuit$ p.240 Ahlfors 1950 [@Ahlfors_1950] is cited and other references are given namely Alling-Greenleaf 1971 [@Alling-Greenleaf_1971] (where however no existence-proof is given), Fay 1973 [@Fay_1973] (where perhaps only the schlicht case is treated?), and finally Fedorov 1991 [@Fedorov_1991] (where probably only the planar case is treated) $\spadesuit$ still on p.240 it is asserted that $g+1$ is the minimal possible degree for expressing a compact bordered Riemann surface as ramified covering of the unit-disc ($g$ being as usual the genus of the double, cf. p.230) $\spadesuit$ if correct this assertion would (blatantly) corrupt the main result of Gabard 2006 [@Gabard_2006] (which by virtue of the incertitude principle could be false) $\spadesuit$ however the sharpness of $g+1$ in general is easily corrupted on the basis of simple concrete example of Klein’s Gürtelkurve (real quartic with two nested ovals) projected from a real point situated in the inner oval (cf. Figure 6 in Gabard 2006 [@Gabard_2006 p.955]), and therefore Alpay-Vinnikov’s assertion looks slightly erroneous. NB: this little misconception about the sharpness of Ahlfors bound seems to originate in Fay’s book, cf. Fay 1973 [@Fay_1973] $\spadesuit$ of course this sloppy detail does not entail at all the intrinsic beauty of this paper namely the study of Hardy spaces on finite bordered Riemann surface: “Furthermore, each holomorphic mapping of the finite bordered Riemann surface onto the unit disk (which maps boundary to boundary)determines an explicit isometric isomorphism between this space \[a certain Kreĭn space\] and a usual vector-valued Hardy space on the unit disk with an indefinite inner product defined by an appropriate Hermitian matrix.”\] 

E. van Andel, [*Extending Riemann mapping capabilities for the sage mathematics package*]{}, Calvin College, 2011. \[$\spadesuit$ p.1: “computation and visualization tools for the Riemann mapping”, “Ahlfors spiderweb”; p.3: “the Ahlfors map conformally maps multiply-connected regions to the unit circle. \[Of course in this case it is more traditional (at least correct) to speak of the Bieberbach-Grunsky map.\] This map is such that for a region with $n$ holes, $n+1$ points in the original region will map to $1$ point in the unit circle.”\] 

C. Andreian Cazacu, [*On the morphisms of Klein surfaces*]{}, Rev. Roumaine Math. Pures Appl. 31 (1986), 461–470. \[$\clubsuit$ inspired by Alling-Greenleaf 1971 [@Alling-Greenleaf_1971] and Stïlow 1938 [@Stoilow_1938-Lecons] $\spadesuit$ \[17.10.12\] for another (more elementary) proof of this result, cf. a paper by Cirre 1997\] 

C. Andreian Cazacu, [*Interior transformations between Klein surfaces*]{}, Rev. Roumaine Math. Pures Appl. 33 (1988), 21–26. \[$\clubsuit$ from the Introd.: “The interior transformations were introduced by Simion Stoilow in order to solve Brouwer’s problem: the topological characterization of analytic functions. By means of these transformations he founded a vast topological theory of analytic functions with essential implications in the study of Riemann surfaces \[8\](=Soilow 1938 [@Stoilow_1938-Lecons]). In this paper we show that interior transformations are a powerful tool in Klein surfaces theory \[2\](=Alling-Greenleaf 1971 [@Alling-Greenleaf_1971]) too. \[$\dots$\]”\] 

C. Andreian Cazacu, [*Complete Klein coverings*]{}, Bull. Soc. Sci. Lett. [Ł]{}ódź Sér. Rech. Déform. 37 (2002), 7–14. \[$\clubsuit$ the notion of the title is introduced as a generalization of the Ahlfors-Sario notion of complete Riemann coverings (1960 [@Ahlfors-Sario_1960 p.42, §21A]), i.e. any point in the range has a neighborhood whose inverse image consists only of compact components. For the case of coverings with a finite number of sheets, it is shown that a Klein covering is complete iff it is total, in the sense of Stoïlow (1938 [@Stoilow_1938-Lecons]), that is any sequence tending to the boundary has an image tending to the boundary. $\spadesuit$ \[13.10.12\] such purely topological conceptions are mentioned for they subsume the topological behaviour of Ahlfors circle maps (i.e. full covering of the circle, alias disc)\] 

A. Andreotti, [*Un’applicazione di un teorema di Cecioni ad un problema di rappresentazione conforme*]{}, Ann. Sc. Norm. Super. Pisa (3) (1950), 99–103. 60, but not in 78 \[$\clubsuit$ seems to extend the result of Matildi 1948 [@Matildi_1945/48] to the case of several contours, hence could be an (independent) proof of the existence of a circle map (than that of Ahlfors 1950 [@Ahlfors_1950]) $\clubsuit$ in fact the writer (Gabard) was not able to follow all the details of Andreotti’s proof but I have no specific objection to make (it would be a good challenge if somebody is convinced by the argument to translate it in English to make the argument more generally accessible, ask maybe Coppens or Huisman) $\clubsuit$ it would be interesting to see which degree is obtained by this method (presumably the genus of the double plus one, i.e. $p+1$ cf. p.101, where $k>p$ \[by Riemann-Roch\]) $\clubsuit$ maybe a last comment is that in Andreotti’s result it is not perfectly clear if the circumference can be arranged to coincide, so has to get an Ahlfors circle map\] 

P. Appell, E. Goursat, [*Théorie des fonctions algébriques*]{}, Paris, 1895. \[$\spadesuit$\] 

P. Appell, E. Goursat, [*Théorie des fonctions algébriques d’une variable et des transcendantes qui s’y rattachent*]{}, Deuxième édition revue et augmentée, Tome II, Fonction automorphes, par Pierre Fatou, Paris, Gauthier-Villars, 1930. 60 \[$\spadesuit$ discusses Klein’s orthosymmetry\] 

E. Arbarello, M. Cornalba, [*Footnotes to a paper of Beniamino Segre. The number of $g^1_d$’s on a general $d$-gonal curve, and the unirationality of the Hurwitz spaces of $4$-gonal and $5$-gonal curves*]{}, Math. Ann. 256 (1981), 341–362. 

E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris, [*Geometry of algebraic curves, Volume I*]{}, Grundlehren der math. Wiss. 267, Springer-Verlag, 1985. \[$\spadesuit$ p.217: “The existence theorem for $g_1^d$’s was first proved by Meis \[1\](=Meis 1960 [@Meis_1960]). Later, at a time when the techniques of enumerative geometry were better understood, the first fundamental theorem of the theory was established with a completely modern approach. In fact (partly under the influence of unpublished work of Mumford) simultaneously Kempf and Kleiman–Laksov gave the first rigorous proof of the Existence Theorem, and of Theorem (1.3). (See Kempf \[1\](=1971/72 [@Kempf_1971]), Kleiman–Laksov \[1,2\](=1972 [@Kleiman-Laksov_1972], 1974 [@Kleiman-Laksov_1974]))” $\spadesuit$ \[09.10.12\] again one may wonder if this enumerative geometry technology is susceptible to adapt to the context of the Ahlfors map, which amounts to real curves of the orthosymmetric(=dividing) type (ideally the goal would be to adapt the Kempf/Kleiman–Laksov proof to recover the bound of Gabard 2006 [@Gabard_2006] interpreted as a bordered avatar of Meis 1960 [@Meis_1960]) $\spadesuit$ another reason for quoting this book in connection with the Ahlfors map is the issue about generalized Ahlfors maps taking values not in the disc but in another finite bordered Riemann surface. Then there is a certain evidence that such Ahlfors maps generally fail to be full covering surface, for the doubled map relates two closed Riemann surfaces. But the latter are severely restricted and generally not existing. This can be either argued via a moduli count as in Griffiths-Harris 1980 [@Griffiths-Harris_1980] or via Exercise C-6. given on p.367 (of the book under review): “From the preceding exercise and the theorem on global monodromy proved in Chapter X conclude that a general curve of genus $g\ge 2$ does not admit a nonconstant map to a curve of positive genus.” $\spadesuit$ of course the statement is a bit sloppy for there is always the identity map as a trivial counterexample, but probably maps of non-unity degree are excluded tacitly. The proof given seems to use the fact that given a branched covering of curves the fundamental class of the inverse image of the Jacobian variety of the image curve is not a rational multiple of $\theta^{g-h}$, where $\theta$ is the theta divisor and $g,h$ are the resp. genuses of the curves\] 

R. Arens, [*The closed maximal ideals of algebras of functions holomorphic on a Riemann surface*]{}, Rend. Circ. Mat. Palermo 7 (1958), 245–260.\[$\spadesuit$\] 

V.I. Arnold, [*Distribution of ovals of the real plane algebraic curves, the involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms*]{}, Funkt. Anal. Prilozen 5 (1971), 1–9; English transl., Funct. Anal Appl. 5 (1972), 169–175. \[$\spadesuit$ some revolutionary ideas preparing the terrain for Rohlin’s breakthrough\] 

V.I. Arnold, [*Index of a singular point of a vector field, the Petrovskii-Oleinik inequalities, and mixed Hodge structures*]{}, Funkt. Anal. Prilozen 12 (1978), 1–14; English transl., Funct. Anal. Appl. 12 (1978), 1–12. \[$\spadesuit$\] 

V.I. Arnold, O.A. Oleinik, [*The topology of real algebraic varieties*]{}, Vestnik Moscov.Gos. Univ. Ser. 1 (1979), 7–17; English transl., Moscow Univ. Math. Bull. 34 (1979), 5–17. \[$\spadesuit$ a survey (?) oft cited, e.g. in Viro 1986/86 [@Viro_1986/86-Progress]\] 

V.I. Arnold, [*The branched covering ${\Bbb C}P^2\to S^4$, hyperbolicity and projective topology*]{}, Sibirsk. Mat. Zh. 29 (1988), 36–47; English transl., Sib. Math. J. 29 (1989), 717–726. \[$\spadesuit$ compare also Anosov’s “obituary of Pontrjagin” where this famous homeomorphism ${\Bbb C}P^2/ {\rm
conj}\approx S^4$ (Kuiper-Massey-Marin) is ascribed back to Pontrjagin $\spadesuit$ Rohlin expressed (orally) the same opinion, cf. e.g. Finashin 1995/98 [@Finashin_1995/98] $\spadesuit$ yet according to Arnold this result can be traced back to Maxwell\] 

V.I. Arnold, [*Topological content of the Maxwell theorem on multipole representation of spherical functions*]{}, Topol. Methods Nonlinear Anal. 7 (1996), 205–217. \[$\spadesuit$ cited in Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.757]: “as explained in \[5\](=Arnold 1996=this entry [@Arnold_1996]), this beautiful explicit proof was essentially known to Maxwell; \[…\]”\] 

V.I. Arnold, [*Symplectization, complexification and mathematical trinities*]{}, Fields Inst. Communications ?? (20??), ?–?. \[$\spadesuit$ p.7: “Near 1970 Petrovsky asked me to help in evaluating a thesis of a mathematician Gudkov from Nizhni Novgorod (it was Gorky at that time). He was studying the Hilbert problem 16, the question on the plane algebraic curves of degree 6: what are the possible shapes of the set $f(x,y)=0$, if $\deg
f=6$?—The classical answers for degree $2$ were extended to degrees $3$ and $4$ by Newton and Descartes. But then the difficulties starts. Hilbert was unable to solve the case of degree $6$, and this problem was explicitly formulated in his list. One may also consider the affine version but it is more complicated and instead we may consider the projective one, dealing with \[the\] curves in ${\Bbb R}P^2$. Even to this, easier question no answer was known at Hilbert’s time.—The only known thing was the celebrated theorem of Harnack \[…\] Gudkov claimed to obtain the complete possible configurations list of ovals of degree $6$ curves but Petrovsky was doubtful of his result. Let us describe it. The list contains three $M$-curves. \[…\] And this relation $\chi\equiv k^2 \pmod 8$ was observable in all examples of $M$-curves of degree $2k$ which Gudkov was able to construct for higher degrees. But there were no explanations for this behavior.—I was aware that congruences modulo $8$ were standard in $4$-dimensional topology. So my idea was that there existed somewhere a $4$-dimensional manifold which governed the topology of the real plane curve. But how to construct it? This was the place where the complexification came into the game and became very helpful. \[…\] p.14 [**Question.**]{} [Did Gudkov get the recommendation for his thesis?]{}—[**Answer**]{} The thesis was of course defended even though I was never able to read it. But as a result I invented all the matter I have explained to you: I was working hard for a month and after this I proved his conjecture modulo 4. The most difficult thing was some lemma which I was able to guess but not to prove. I always had very good undergraduate students and at that time I asked Varchenko to help me. \[…\] Unfortunately Varchenko had declined to sign the final paper as a coauthor.—D.A. Gudkov became the leader of a strong team in real algebraic geometry at Nizhni Novgorod (Utkin, Polotovskii, Shustin, …). Some of the results of Gudkov and his student were recently rediscovered by C.T.C. Wall. \[…\]”\] 

V.I. Arnold, [*I.G. Petrovskii, Hilbert’s topological problems and modern mathematics*]{}, Uspekhi Math. Nauk 57 (2002), 197–207; English transl., Russain Math. Surveys 57 (2002), 833–845. \[$\spadesuit$ p.197: “The content of Hilbert’s problem is [*to give a topological classification of real algebraic curves and manifolds (of fixed degree)*]{}. It is one of the principal and eternal problem of mathematics which is also important for many of its application (where these curves and manifolds describe laws of nature). [*What algebraic curves look like*]{}; even today this is unnown, even for plane curves of degree $8$ consisting of 22 connected ovals \[…\].” $\spadesuit$ p.834: “There are 1812 [*topologically possible*]{} arrangements of $11$ ovals in the plane. Hilbert’s result stated that of all these arrangement only two are realised by [*algebraic*]{} curves of degree $6$.—This result of Hilbert is wrong, as was shown 70 years later by D.A. Gudkov, who was a student of both Petrovskii and the physicist A.A. Andronov. Gudkov showed that there are three, not two, realizable arrangements.”\] 

V.I. Arnold, [*From Hilbert’s superposition problem to dynamical systems*]{}, Amer. Math. Monthly 111 (2004), 608–624. \[$\spadesuit$ p.608: “Some people, even though they study, do so without enough zeal, and therefore live long.”—Archbishop Genady of Novgorod, ca. 1500. $\spadesuit$ p.608: philosophy of the mushroom $\spadesuit$ p.622: “[**Question**]{} (J. Milnor) You often told us about important mathematical work in Russia that we did not know about and you gave another example today. I wonder if you can explain to us how to locate something interesting in the literature starting with zero information.”—[**Answer**]{}. \[…\] I usually start with the German [*Encyclop[æ]{}dia …*]{}. In Klein’s [*Vorlesungen über die Entwicklung der Math. im 19. Jahrhundert*]{} there is a lot of information on whatever happened in the nineteenth century and before. \[…\]”\] 

(On) V.I. Arnold, by A.A. Davydov, S.M Gusein-Zade, Yu.S. Ilyashenko, M.E. Kazaryan, A.G. Khovanskii, A.G. Kushnirenko, S.K. Lando, A.N. Varchenko, V.A. Vassiliev, and V.M. Zakayukin, [*Vladimir Igorevich Arnold in the eyes of his students*]{}, Proc. Steklov Inst. Math. 259 (2007), 1–5. \[$\spadesuit$ p.3: “Arnold’s seminar covered everything, for example real algebraic geometry. Hilbert spent a lot of effort constructing real plane algebraic curves of a given degree that have the maximum possible number of ovals. Unsuccessful attempts to construct such curves with an a priori possible topology of arrangement on the projective plane convinced him that not all possibilities are feasible. Hilbert collected open problems of real algebraic geometry in his 16th problem. D.A. Gudkov solved one of these problems for curves of degree $6$; however, the general picture remained unclear. Arnold general surprisingly fine topological obstacles showing that many a priori possible arrangements of curves with the maximal number of ovals cannot be realized. Arnold’s studies were picked up by V.A. Rokhlin, D.A. Gudkov, and their students. As a result real algebraic geometry has reached a completely new modern level.”\] 

V.I. Arnold, [*Topological properties of eigenoscillations in mathematical physics*]{}, Proc. Steklov Inst. Math. 273 (2011), 25–34. \[$\spadesuit$ discussion of Courant’s theorem on the number of residual component of the nodal hypersurface of an oscillating manifold (vibrating membrane) and its relationship with Hilbert’s 16th problem $\spadesuit$ precisely, Abstract: “Courant proved that the zeros of the $n$th eigenfunction of the Laplace operator on a compact manifold $M$ divide this manifold into at most $n$ parts. He conjectured that a similar statement is also valid for any linear combination of the first $n$ eigenfunctions. However, later it was found out that some corollaries to this generalized statement contradict the results of quantum field theory. Later explicit counterexamples were constructed by O. Viro. \[…\] ”\] 

D.S. Arnon, S. McCallum, [*A polynomial time algorithm for the topological type of a real algebraic curve*]{}, J. Symb. Comput. 5 (1988), 213–236. \[$\spadesuit$ cited in Kalla-Klein 2012 [@Kalla-Klein_2012-Computation-cite-Gabard]\] 

N. Aronszajn, [*Theory of reproducing kernels*]{}, Trans. Amer. Math. Soc. 68 (1950), 337–404. \[$\spadesuit$ abstract unified view on the theory of the reproducing kernel containing the special cases of Bergman and Szegö, etc.\] 

C. Arzelà, [*Sul Principo di Dirichlet*]{}, Nota letta alla R. Accademia delle Scienze dell’Instituto di Bologna nell’Adunza del 24 Gennaio 1897. \[$\spadesuit$ cited in Zaremba 1910 [@Zaremba_1910] as a precursor of Hilbert’s resurrection of the Dirichlet principle\]$\bigstar$$\bigstar$$\bigstar$ 

M.F. Atiyah, [*Riemann surfaces and spin structures*]{}, Ann. Sci. École Norm. Sup. (4) 4 (1971), 47–62. 

G. Aumann, C. Carathéodory, [*Ein Satz über die konforme Abbildung mehrfach zusammenhängender ebener Gebiete*]{}, Math. Ann. 109 (1934), 756–763. 78, but not in 60 

H.F. Baker, [*Abel’s theorem and the allied theory including the theory of the theta function*]{}, Cambridge Univ. Press, Cambridge, 1897. \[$\spadesuit$\] 

J.A. Ball, [*Operators of class $C_{00}$ over multiply-connected domains*]{}, Michigan Math. J. 25 (1978), 183–196. 47 \[$\spadesuit$ p.187, Ahlfors 1947 [@Ahlfors_1947] is cited for the following result (in fact due to Bieberbach 1925 [@Bieberbach_1925] in this formulation): “If $R$ is a domain in the complex-plane bounded by $n+1$ nonintersecting analytic Jordan curves, there exists a complex-valued inner function on $R$, which is analytic on a neighborhood of $\bar R$, has precisely $n+1$ zeros in $R$, and wraps each component of the boundary of $R$ once around the unit disk \[sic, but “disk” should rather be “circle”\] $\spadesuit$ this theorem (of Bieberbach-Grunsky-Ahlfors) is then applied to a problem in operator theory\] 

J.A. Ball, [*A lifting theorem for operator models of finite rank on multiply-connected domains*]{}, J. Operator Theory 1 (1979), 3–25. 47 \[$\spadesuit$ Ahlfors 1947 [@Ahlfors_1947] is cited on p.11 (in a context where perhaps Bieberbach 1925 [@Bieberbach_1925] would have been logically sufficient) $\spadesuit$ again the philosophy of the paper seems to transplant via the Ahlfors function a certain lifting theorem for operator models on the disc (due to Sz.-Nagy-Foiaş) to the more general case of a multi-connected domain $\spadesuit$ one can of course wonder about extension on bordered Riemann surfaces, probably established meanwhile (?)\] 

J.A. Ball, K. Clancey, [*Reproducing kernels for Hardy classes on multiply-connected domains*]{}, Integral Equations Operator Theory 25 (1996), 35–57. \[$\spadesuit$ extension to finite bordered Riemann surface, try Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000]\] 

E. Ballico, [*Real algebraic curves and real spanned bundles*]{}, Ricerche di Matematica 50 (2001), 223–241. \[$\spadesuit$\] 

E. Ballico, G. Martens, [*Real line bundles on $k$-gonal real curves*]{}, Abh. Math. Sem. Univ. Hamburg 71 (2001), 251–255. \[$\spadesuit$ real curves, “symmetric” Riemann surfaces, diasymmetric/orthosmmetric, p.251: “One cannot expect irreducible moduli for real curves of genus $g$. But as already indicated by Klein (in permitting shrinkage of components of $X({\Bbb R})$ to isolated double points, \[9\]=(Klein 1892 [@Klein_1892_Realitaet])), the set of (real or complex, whatsoever) isomorphism classes of real stable curves of arithmetic genus $g\ge 1$ can be equipped with a topology making it a connected space (\[14\]=Seppälä 1991 [@Seppala_1991-Moduli]). $\spadesuit$ p.252: “In particular the classical bound $k_{{\Bbb C}}\le (g+3)/2$ may be false for the real gonality $k$ (\[5\](=Chaudary 1995 [@Chaudary_1995-Thesis]), \[10\](=Martens 1978 [@Martens_1978])). If $n(X)>0$ one knows (\[3\]=Ballico 2003 [@Ballico_2003]) that $k\le (g+3)/2+3$; it seems not known if this is sharp for some $g$.”\] 

E. Ballico, [*Gonality and Clifford index for real algebraic curves*]{}, Collectanea Math. 53 (2002). \[$\spadesuit$\] 

E. Ballico, [*Codimension 1 subvarieties of ${\cal M}_g$ and real gonality of real curves*]{}, Czechoslovak Math. J. 53 (2003), 917–924. \[$\spadesuit$ some results seem to be reanalyzed in Coppens-Huisman 2010...\] 

E. Ballico, [*Real curves with fixed gonality and empty real locus*]{}, Le Matematiche 60 (2005), 129–131. \[$\spadesuit$\] 

E. Ballico, [*Real ramifications points and real Weierstrass points of real projective curves*]{}, Glasnik Mat. 41 (2006), 233–238. \[$\spadesuit$\] 

S. Banach, [*Théorie des opérations linéaires*]{}, Warsaw, 1932. \[$\spadesuit$ cited in Read 1958 [@Read_1958_Acta], where the Hahn-Banach theorem is put in connection to the Ahlfors map\] 

C. Bandle, M. Flucher, [*Harmonic radius and concentration of energy; hyperbolic radius and Liouville’s equations $\Delta
U= e^U$ and $\Delta U=U^{\frac{n+2}{n-2}}$*]{}, SIAM Review 38 (1996), 191–238. 47 \[$\spadesuit$ Ahlfors 1947 [@Ahlfors_1947] is cited on p.200 as follows: “Corollary 4 extends Liouville’s formula to multiply connected planar domains and so does the following formula from Mityuk’s monograph \[79\](=1985). Denote by $f\colon \Omega
\to B$ an [*Ahlfors map*]{} of $\Omega$ (cf. Ahlfors \[1\](=Ahlfors 1947 [@Ahlfors_1947]), obtained as a solution of the same extremal problem that we used for the definition of the Riemann map (§1). Then the inner radius of $\Omega$ is given by $r(x)=\frac{1-\vert f(x)\vert^2}{\vert f'(x)\vert}
\exp\bigl(-2\pi \sum_{\{y\neq x: f(y)=x\}} G_x(y)\bigr)$, provided $f'(x)\neq 0$. Note that on a $k$-connected domain the Ahlfors map is a $k$-sheeted branched covering. A modified formula involving higher derivative of $f$ holds at the branch points. The proof is similar to that of Corollary 4.” $\spadesuit$ perhaps instead of the mentioned monograph, the original articles of Mitjuk already contain this (multi-connected) extended formula, cf. Mitjuk 1965 [@Mitjuk_1965-inner-radius] (and also Mitjuk 1968 [@Mitjuk_1968] for the statement (in English), yet without the proof)\] 

V. Bangert, C. Croke, S. Ivanov, M. Katz [*Filling area conjecture and ovalless real hyperelliptic surfaces*]{}, Geom. Funct. Anal. ?? (2005), ?–?. \[$\spadesuit$ solve the hyperelliptic case of the filling area conjecture due to Gromov, hence in particular the genus-one case $p=1$ $\spadesuit$ the hearth of the argument seems to be an old result of Hersch\] 

W.H. Barker II, [*Kernel functions on domains with hyperelliptic double*]{}, Trans. Amer. Math. Soc. 231 (1977), 339–347. (Diss. under M.M. Schiffer) \[$\spadesuit$ p.345, the Ahlfors (extremal) function of a domain is discussed by referring to Bergman 1950 [@Bergman_1950], Heins 1950 [@Heins_1950], and also the original treatment due to Ahlfors 1947 [@Ahlfors_1947] and that of Garabedian 1949 [@Garabedian_1949]\] 

E. Bedford, [*Proper holomorphic mappings*]{}, Bull. Amer. Math. Soc. (N.S.) 10 (1984), 157–19?. 50 \[$\spadesuit$ p.159, Ahlfors 1950 is quoted as follows: “The existence of many proper mappings is given by a result of Grunsky \[55\](=[@Grunsky_1937]) and Ahlfors \[1\](=1950 [@Ahlfors_1950]). [Theorem]{}. [*If $M$ is a finite Riemann surface with nondegenerate boundary components, then there exists a proper mapping $f\colon M \to \Delta$.*]{} In general, however, given two Riemann surfaces $M$ and $N$, it does not seem easy to say whether there exists a proper mapping $f\colon M \to N$.”\] 

H. Behnke, H. Zumbusch, [*Konforme Abbildung von Bereichen auf in ihnen liegenden Bereiche*]{}, Semester-Ber. math. Sem. Münster 8 (1936), 100–121. \[$\spadesuit$ quoted in Grunsky 1940 [@Grunsky_1940 p.233], in connection with the definition of the Carathéodory metric (first appearance in Carathéodory 1926 [@Caratheodory_1926]) for multi-connected domains\] 

H. Behnke, K. Stein, [*Entwicklungen analytischer Funktionen auf Riemannschen Flächen*]{}, Math. Ann. 120 (1947/49), 430–461. \[$\spadesuit$ proves that any open Riemann surface carries a nonconstant analytic function $\spadesuit$ in the case where the Riemann surface is the interior of a compact Riemann surface this also follows form Ahlfors 1950 [@Ahlfors_1950] in the much sharper form of a branched covering of the disc $\spadesuit$ naive question \[01.10.12\]: by using an exhaustion of the open Riemann surface by finite bordered ones what sort of functions can be constructed on the whole surface? Is it in particular possible to subsume the Behnke-Stein theorem to that of Ahlfors? (looks a bit naive I confess)\] 

H. Behnke, F. Sommer, [*Theorie der analytischen Funktionen einer komplexen Veränderlichen*]{}, Die Grundlehren der math. Wiss. in Einzeldarstellungen, Bd.77, Springer-Verlag, Berlin, 1955; Third Edition, Springer-Verlag, New York, 1965. \[$\spadesuit$ pp.581–2 is quoted in Černe-Forstnerič 2002 [@Cerne-Forstneric_2002] for the “(Schottky) double” $\spadesuit$ other sources for this purposes are Klein 1882 [@Klein_1882] (in romantic pre-axiomatic style), else Koebe 1928 [@Koebe_1928-Acta], or Teichmüller 1939 [@Teichmueller_1939] and of course also Springer’s book 1957 [@Springer_1957-BOOK-Introd-to-RS] or Schiffer-Spencer 1954 [@Schiffer-Spencer_1954]\] 

S.R. Bell, [*Numerical computation of the Ahlfors map of a multiply connected planar domain*]{}, J. Math. Anal. Appl. 120 (1986), 211–217. \[$\spadesuit$ from the Introd.: “N. Kerzman and E.M. Stein discovered in \[6\](=1978 [@Kerzman-Stein_1978]) a method for computing the Szegö kernel of a bounded domain $D$ in the complex plane with $C^{\infty}$ smooth boundary. In case $D$ is also simply-connected, the Kerzman-Stein method yields a powerful technique for computing the Riemann mapping function associated to a point $a\in D$ (see \[6\](=Kerzman-Stein 1978 [@Kerzman-Stein_1978]), \[7\](=Kerzman-Trummer 1984)). In this note, we show how the Kerzman-Stein method can be generalized to yield a method for computing the Ahlfors map associated to a point in a finitely connected, bounded domain in the plane with $C^2$ smooth boundary. The Ahlfors map is a proper holomorphic mapping of $D$ onto the unit disc which maps each boundary component of $D$ one-to-one onto the boundary of the unit disc.—The Ahlfors map might prove to be useful in certain problems arising in fluid mechanics. For example, in the problem of computing the transonic flow past an obstacle in the plane, a conformal map of the outside of the obstacle onto the unit disc is used to set up a grid which is most convenient for making numerical computations (see \[5\](=Jameson 1974, “Iterative solution of transonic flows over airfoils and wings, including flows at Mach $1$”)). The Ahlfors map could be used in a similar way in problems of this sort in which more than one obstacle is involved. \[…\]”\] 

S.R. Bell, [*The Szegö projection and the classical objects of potential theory in the plane*]{}, Duke Math. J. 64 (1991), ?–?. \[$\spadesuit$ quoted in McCullough 1996 [@McCullough_1996] for the result that the Ahlfors function acquires distinct (simple) zeros when the center $a$ (the place where the derivative is maximized) is chosen near enough the boundary of the domain $\spadesuit$ a probably related result is to be found in Ovchintsev 1996/96 [@Ovchintsev_1996/96] $\spadesuit$ question \[20.09.12\]: does this result extend to bordered surfaces\] 

S.R. Bell, [*The Cauchy transform, potential theory, and conformal mapping*]{}, CRC Press, Boca Raton, Florida, 1992. 50 \[$\spadesuit$\]$\bigstar$ 

S.R. Bell, [*Complexity of the classical kernel functions of potential theory*]{}, Indiana Univ. Math. J. 44 (1995), 1337–1369. \[$\spadesuit$\] 

S.R. Bell, [*Ahlfors maps, the double of a domain, and complexity in potential theory and conformal mapping*]{}, J. d’Anal. Math. 78 (1999), 329–344. \[$\spadesuit$ proof that generically two Ahlfors maps suffice to generate the field of meromorphic function of the double of the domain (so-called primitive pairs)\] 

S.R. Bell, [*Complexity in complex analysis*]{}, Adv. Math. 172 (2002), 15–52. 50 

S.R. Bell, [*Möbius transformations, the Carathéodory metric, and the objects of complex analysis and potential theory in multiply connected domains*]{}, Michigan Math. J. 51 (2003), 352–361. \[$\spadesuit$ p.361: “It is also a safe bet that many of the results in this paper extend to the case of Riemann surfaces. I leave these investigations for the future.”\] 

S.R. Bell, [*Quadrature domains and kernel function zipping*]{}, Ark. Math. 43 (2005), 271–287. \[$\spadesuit$ p.271 (Abstract): “It is proved that quadrature domains are ubiquitous in a very strong sense in the realm of smoothly bounded multiply connected domains in the plane. In fact they are so dense that one might as well assume that any given smooth domain one is dealing with is a quadrature domain, and this allows access to a host of strong conditions on the classical kernel functions associated to the domain.”\] 

S.R. Bell, [*The Green’s function and the Ahlfors map*]{}, Indiana Univ. Math. J. 57 (2008), 3049–3063. \[$\spadesuit$ yet another fascinating paper among the myriad produced by the author, where now a striking expression is given for the Green’s function of a finitely connected domain in the plane in terms of a single Ahlfors mapping answering thereby (see third page of the introd.) a subconscious desideratum of Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949]\] 

S.R. Bell, [*The structure of the semigroup of proper holomorphic mappings of a planar domain to the unit disc*]{}, Complex Methods Function Theory 8 (2008), 225–242. \[$\spadesuit$ a description of all circle maps is given by returning to the original papers of Bieberbach and Grunsky\] 

S.R. Bell, [*The Szegö kernel and proper holomorphic mappings to a half plane*]{}, Comput. Methods Funct. Theory 11 (2011), 179–191. 50 \[$\spadesuit$ construction (for domains) of proper holomorphic maps of arbitrary mapping degree, reminiscent of Heins’ argument 1950 [@Heins_1950] about positive harmonic functions\] $\bigstar$$\bigstar$ 

E. Beltrami, [*Saggio di interpretazione della geometria non-euclidea*]{}, Giornale di Matematiche 6 (1868), 262–280. \[$\spadesuit$\] 

E. Beltrami, [*Teoria fundamentale degli spazii di curvatura costante*]{}, Annali di Matematica (2) 2 (1868), 232–255. \[$\spadesuit$\] 

R. Benedetti, J.-J. Risler, [*Real Algebraic and Semi-algebraic Sets*]{}, Hermann, Paris, 1990. \[$\spadesuit$ contains much material (and where from I initially learned the Brusotti theorem in “un lavoro di Diploma” under the guidance of Felice Ronga) $\spadesuit$ p.260: elementary properties of separating curves $\spadesuit$ exposition (not always with complete proofs) of some results of the Germano-Russian school: Harnack, Hilbert, Gudkov, Arnold, Rohlin, etc. $\spadesuit$ p.288: “…it can easily by\[=be\] (sic!) proved that any configuration under the broken line of figure 5.24, can be realized by a smooth curve of degree 6.” This is a bit sloppy, for Gudkov’s skill is required!\] 

M. Berger, [*Riemannian geometry during the second half of the twentieth century*]{}, Jber. d. Dt. Math.-Verein. 100 (1998), 45–208. \[$\spadesuit$ p.147: “The simplest filling volume, namely that for the circle $S^1$, was only obtained in (\[N.\] Katz, 1998).”), where the reference is (cf. p.196) “Katz, N. (1998). Filling volume of the circle.” $\spadesuit$ This work, presaging a complete solution to Gromov’s filling conjecture, has apparently never been published and probably turned out to contain a gap.\] 

M. Berger, [*A Panoramic View of Riemannian geometry*]{}, Springer, 2002. \[$\spadesuit$\] 

S. Bergman\[n\], [*Über die Entwicklung der harmonischen Funktionen der Ebene und des Raumes nach Orthogonalfunktionen*]{}, Math. Ann. 86 (1922), 238–271. (Thesis, Berlin, 1921.) \[$\spadesuit$ formulates—like Bieberbach 1914 [@Bieberbach_1914]—the desideratum that the function minimizing the area integral $\int \int \vert
f'(z)\vert^2 d\omega$ is the Kreisabbildung (alias Riemann mapping) $\clubsuit$ this desideratum will be only accomplished in the late 1940’s, i.e. Garabedian/Lehto’s era\] 

S. Bergman\[n\], [*Über eine Darstellung der Abbildungsfunktion eines Sternbereiches*]{}, Math. Z. 29 (1929), 481–486. \[$\spadesuit$ Minimalbereich in a special case\] 

S. Bergman\[n\], [*Über unendliche Hermitesche Formen, die zu einem Bereiche gehören, nebst Anwendunden auf Fragen der Abbildung durch Funktionen von zwei komplexen veränderlichen*]{}, Math. Z. 29 (1929), 641–677. \[$\spadesuit$ p.641 formulates—inspired by Bieberbach 1914 [@Bieberbach_1914]—the concept of a Minimalbereich, by referring to 3 of his previous work (alas no precise cross-references)\] 

S. Bergman\[n\], [*Eine Bemerkung über schlichte Minimalabbildungen*]{}, Sitzgsber. Berliner Math. Ges. 30 (1932) \[$\spadesuit$ quoted in Lehto 1949 [@Lehto_1949] for yet another formulation—like Bieberbach 1914 [@Bieberbach_1914]—of the desideratum that the function minimizing the area integral $\int \int \vert f'(z)\vert^2
d\omega$ is the Kreisabbildung (alias Riemann mapping) $\clubsuit$ this desideratum will be only accomplished in the late 1940’s, i.e. Garabedian/Lehto’s era, cf. Lehto 1949 [@Lehto_1949 p.46]\] 

S. Bergman\[n\], [*Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande*]{}, J. Reine Angew. Math. 169 (1933), 1–42; and 172 (1934), 89–128. \[$\spadesuit$ p.3 footnote 2 contains some brief indication on the case of multi-connected domains (in one complex variable) and a cross-ref. to Zarankiewicz 1934 [@Zarankiewicz_1934-numerisches]\] 

S. Bergman, [*Partial differential equations, Advanced topics*]{} (Conformal mapping of multiply connected domains), Publ. of Brown Univ., Providence, R.I., 1941. \[$\spadesuit$ probably completely incorporated in Bergman 1950 [@Bergman_1950]\] $\bigstar$$\bigstar$$\bigstar$ 

S. Bergman, [*A remark on the mapping of multiply-connected domains*]{}, Amer. J. Math. 68 (1946), 20–28. \[$\spadesuit$ uniformize via the Bergman kernel domains of finite connectivity, and via Koebe (1914/15) can be used for the Kreisnormierung.\] 78 

S. Bergman\[n\], [*Sur les fonctions orthogonales de plusieurs variables complexes avec les applications à la théorie des fonctions analytiques*]{}, Mémorial des Sci. Math. 106 (1947), 1–63. \[$\spadesuit$ p.32 points out that the old desideratum of Bieberbach-Bergman 1922 [@Bergman_1922] of reproving RMT via the problem of least area was still not achieved until this date of 1947, except for the special case of starlike domains (Bergman 1932 [@Bergman_1932] and Schiffer 1938 [@Schiffer_1938-CRAS-domaines-minima]). The breakthrough may have occurred only by Garabedian and Lehto’s Thesis, cf. e.g. [@Lehto_1949]\] 

S. Bergman\[n\], [*Sur la fonction-noyau d’un domaine et ses applications dans la théorie des transformations pseudo-conforme*]{}, Mémorial des Sci. Math. 108 (1948). \[$\spadesuit$ quoted in Maschler 1956 [@Maschler_1956] for the theory of minimal domains $\spadesuit$ p.41, Kufarev [@Kufareff_1935/37] is credited for the issue that for a doubly-connected domain the least area map is not univalent(=schlicht) $\spadesuit$ of course it looks evident that univalence fails as well in higher connectivity, cf. e.g. Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949] $\spadesuit$ yet nobody seems to claim that the range is a circle\] 

S. Bergman, [*The kernel function and conformal mapping*]{}, Mathematical Surveys 5, Amer. Math. Soc., New York, 1950. 50, 60, 78 \[$\spadesuit$ p.87 existence of a circle map for domains via an explicit formula (p.86) as a ratio of two kernel functions $\spadesuit$ a second revised edition was published in 1970\] 

S. Bergman, M. Schiffer [*Kernel functions and conformal mapping*]{}, Compositio Math. 8 (1951), 205–249. 60, 78 

A. Bernard, J.B. Garnett, D.E. Marshall, [*Algebras generated by inner functions*]{}, J. Funct. Anal. 25 (1977), 275–285. \[$\spadesuit$ p.282, the Ahlfors function is briefly mentioned as follows: “To show the inner functions separates the points of $X$ we modify the well-known construction of the Ahlfors function for a Denjoy domain.” $\spadesuit$ the bulk of the paper is devoted to the question of knowing when the unit ball of an uniform algebra (typically $H^{\infty}(\Omega)$ for $\Omega$ a finitely connected domain) is the closed convex hull of the inner functions $\spadesuit$ Corollary 5.2 (p.285) gives this conclusion provided the inner functions separate the points of the Shilov boundary, but the authors seem to confess that they do not know whether this proviso is automatically fulfilled (note: of course the simple argument of Stout 1966 [@Stout_1966/67 p.375] saying that inner functions separates points on the Riemann surface (just because the Ahlfors function based at the two given points do separate them) does not apply here, as we are truly looking at mystical points of the Shilov boundary) $\spadesuit$ p.276, one reads: “Minor modifications of the proof for this case \[i.e. finitely connected plane domain\] will give the result when $\Omega$ is a finite open Riemann surface, but we leave those details to the interested reader.” $\spadesuit$ conclusion: since the whole paper actually seeks for an extension of a disc result of Marshall (asserting precisely that the unit ball of the disc algebra $H^{\infty}(\Delta)$ is the closed convex hull of the inner functions), one could wonder if there is not a more naive strategy exploiting more systematically the Ahlfors function\] 

L. Bers, [*Quasiconformal mappings and Techmüller’s theorem*]{}, in: Analytic functions, Princeton Univ. Press, 1960, 89–119. \[$\spadesuit$ modernized account of Teichmüller theory\] 

L. Bers, [*Uniformization by Beltrami equation*]{}, Comm. Pure Appl. Math. 14 (1961), 215–228. \[$\spadesuit$ contain striking results on the Kreisnormierung\] 

L. Bers, [*Automorphic forms for Schottky groups*]{}, Adv. Math. 16 (1975), 332–371. \[$\spadesuit$ modernized proof via quasiconformal deformation of the retrosection theorem (alias Rückkehrschnitttheorem=RST) going back to Klein 1882 [@Klein_1882_Ruckkehrschnitt] and first seriously proved in Koebe 1910 UAK2 [@Koebe_1910_UAK2] $\spadesuit$ a question of some interest is whether the bordered avatar of this RST implies the Ahlfors circle mapping\]$\bigstar$ 

L. Berzolari, [*Allgemeine Theorie der höheren ebenen algebraischen Kurven*]{}, in: Enzyklopädie der math. Wissenschaften, Bd. III, 2, 1, 313–455, Leipzig, 1906. \[$\spadesuit$ a short survey of Klein’s theory of symmetric surfaces while coining first the designation “[*Klein’s surfaces*]{}” made popular much later by Alling-Greenleaf 1971 [@Alling-Greenleaf_1971].\] 

L. Berzolari, [*Algebraische Transformationen und Korrespondenzen*]{}, in: Enzyklopädie der math. Wissenschaften, Bd. III, C, 1, 1, 1781–2218, Leipzig, 1933. \[$\spadesuit$\] 

E. Betti, [*Sopra gli spazi di un numero qualunque di dimensioni*]{}, Annali di Matematica (2) 4 (1871), 140–158. \[$\spadesuit$ inspired from Riemann 1852/53 Fragment aus der Analysis situs [@Riemann_1852/53-Fragment-aus-der-Analysis-Situs], and will in turn inspire (modulo being misspelled as Brioschi) Poincaré 1895 [@Poincare_1895-Analysis-Situs] $\spadesuit$ this well-known line of thoughts leads to “homology theory” and is inasmuch relevant to the present article for the issue that several peoples (starting with Riemann, Klein, Poincaré, Brouwer, etc.) used topological methods in function theory, and in the specialized context of Ahlfors circle maps (similar inferences were used by Garabedian 1949 [@Garabedian_1949], Mizumoto 1960 [@Mizumoto_1960], and Gabard 2004/06 [@Gabard_2006])\] 

A. Beurling, [*Sur un problème de majoration*]{}, Thèse, Upsala, 1935, 109 pp. 

A. Beurling, [*On two problems concerning linear transformation in Hilbert space*]{}, Acta Math. 81 (1949), 239–255. \[$\spadesuit$ the so-called Beurling’s invariant subspaces theorem $\spadesuit$ for an extension to finite bordered Riemann surface see M. Hasumi 1966 [@Hasumi_1966] (and also related work by Voichick 1964 [@Voichick_1964]), yet without using the Ahlfors map, but cite Royden 1962 [@Royden_1962] which is closely allied $\spadesuit$ \[03.10.12\] one can wonder if like for the corona problem/theorem there is a direct inference of the Ahlfors map upon Beurling’s invariant subspaces (as Alling 1964 [@Alling_1964] managed to do for the corona)\] 

G.V. Beylĭ, [*On Galois extensions of the maximal cyclotomic field*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 269–276; English transl., Math. USSR Izv. 14 (1980), 247–256. \[$\spadesuit$ proof that a closed surface is defined over $\Qbar$ iff it ramifies only above 3 points of the sphere $\clubsuit$ can this be extended to bordered surfaces in the context of Ahlfors maps?\] 

G.V. Beyli, [*A new proof of the three points theorem*]{}, Sb. Math. 193 (2002), 329–332. \[$\spadesuit$\] 

L. Bieberbach, [*Über ein Satz des Herrn Carathéodorys*]{}, Gött. Nachr. (1913), 552–560. 

L. Bieberbach, [*Zur Theorie und Praxis der konformen Abbildung*]{}, Rend. del Circolo mat. di Palermo 38 (1914), 98–112. \[$\spadesuit$ this had some influence over Bergman’s Thesis 1921/22 [@Bergman_1922], and is in turn inspired by W. Ritz ca. 1908–09 $\spadesuit$ p.100, first formulation of the principle that the function minimizing the area integral $\int \int \vert f'(z)\vert^2 d\omega$ is the Kreisabbildung (alias Riemann mapping), and the hope is expressed of getting an independent proof of its existence through this least area problem $\clubsuit$ this desideratum (vividly sustained in Bergman’s Thesis 1921/22 [@Bergman_1922] and Bochner’s 1922 [@Bochner_1922]) will be only achieved in the late 1940’s, i.e. Garabedian/Lehto’s era (see Lehto 1949 [@Lehto_1949]) $\clubsuit$ another desideratum (Gabard 16-$\varepsilon$ June 2012, but perhaps already known) would be that such an extremal problem (closely allied to the theory of the Bergman kernel) yields an alternative proof of the Ahlfors mapping $\clubsuit$ even more since it is eminently geometric can we crack—via this Bieberbach-Bergman philosophy—the Gromov filling area conjecture? (Recompense 50 Euros)\] 

L. Bieberbach, [*Einführung in die konforme Abbildung*]{}, Sammlung Göschen, Berlin, 1915. \[$\spadesuit$ pp.94–108 deal specifically with Bieberbach’s minimizing principle (cf. Bieberbach 1914 [@Bieberbach_1914])\]

L. Bieberbach, [*$\Delta u= e^u$ und die automorphen Funktionen*]{}, Math. Ann. 77 (1916), 173–212. \[$\clubsuit$ p.175 speaks of (Klein’s) orthosymmetry, and write a sentence (which when read ouside of its context) bears strange resemblance with the Ahlfors circle map: “[*Wir nehmen die Fläche orthosymmetrisch an, d.h. sie möge durch diese Symmetrielinien in zwei symmetrische Hälften zerlegt werden, so da[ß]{} es sich also im Hauptkreisfalle um die konforme Abbildung eines berandeten Flächenstückes—der einen Flächenhälfte—auf das Innere des Einheitskreises handelt.*]{}” $\spadesuit$ the real issue in this paper is to implement Schwarz’s desideratum (Göttinger Preisaufgabe 1889) (primarily followed by Picard and Poincaré) of uniformizing (compact) Riemann surfaces via the Liouville equation whose geometric interpretation amounts searching a conformal metric of constant Gaussian curvature $\spadesuit$ \[03.10.12\] probably the above should not be interpreted as an Ahlfors map but rather as the fact that the interior of any compact bordered Riemann surface is uniformized by the unit disc (i.e. the universal covering of the interior is the unit disc)\] 

L. Bieberbach, [*Über einige Extremalprobleme im Gebiete der konformen Abbildung*]{}, Math. Ann. 77 (1916), 153–172. \[$\spadesuit$ includes (among other nice geometrical things) the first proof of Koebe’s Viertelsatz with the sharp constant $1/4$ upon the radius of a disc contained in the range of a schlicht map of the unit disc normed by $\vert f'(0) \vert
=1$\] 

L. Bieberbach, [*Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln*]{}, S.-B. Preuss. Akad. Wiss. Berlin (1916), 940–955. \[$\spadesuit$ where the Bieberbach/coefficient conjecture is first formulated. Solution: de Branges 1984/85.\] 

L. Bieberbach, [*Über einen Riemannschen Satz aus der Lehre von der konformen Abbildung*]{}, Sitz.-Ber. Berliner Math. Ges. 24 (1925), 6–9. 60, 78 (also cited in Courant 1939 [@Courant_1939]) \[$\clubsuit$ the schlicht(artig) case of Ahlfors 1950 [@Ahlfors_1950] is proved, and earlier work by Riemann 1857/58 [@Riemann_1857_Nachlass] and Schottky 1877 [@Schottky_1877] is put in perspective\] 

L. Bieberbach, [*Lehrbuch der Funktionentheorie*]{}, vols. 1 and 2, Berlin, New York, 1945. (Photographic reprint of the 4th edition of Band I (1934) and the 2nd edition of Band II (1931)) \[$\spadesuit$ cited by Bergman 1950 [@Bergman_1950 p.24] for the proof that the minimum function for the problem $\int \int_{B} \vert f'(z)\vert^2
d\omega$ has circle range; of course the original source is Bieberbach 1914 [@Bieberbach_1914]\] $\bigstar$$\bigstar$$\bigstar$ 

L. Bieberbach, [*Conformal mapping*]{}, Chelsea, New York, 1953. \[$\spadesuit$\]$\bigstar$ 

L. Bieberbach, [*Eine Bemerkung zur konformen Abbildung zweifach zusammenhängender Gebiete*]{}, Math. Z. 67 (1957), 99–102. 78 

L. Bieberbach, [*Einführung in die konforme Abbildung*]{}, Sammlung Göschen Bd. 768/768a, Walter de Gruyter and Co. (6th ed., 1967). 78 \[$\spadesuit$ includes a proof of the Hilbert-Koebe PSM (in infinite connectivity) $\spadesuit$ presumably an earlier edition (as the one cited in Burckel 1979 [@Burckel_1979]) do the job as well\] 

L. Bieberbach, [*Das Werk Paul Koebes*]{}, Jahresber. Deutsche Math.-Verein. 70 (1968), 148–158. 78 \[$\spadesuit$ contains a complete tabulation of Koebe’s work\] 

E. Bishop, [*Subalgebras of functions on a Riemann surface*]{}, Pacific J. Math. 9 (1959), 629–642. \[$\spadesuit$\] 

E. Bishop, [*Abstract dual extremal problems*]{}, Notices Amer. Math. Soc. 12 No.1 (1965), 123. \[$\spadesuit$  cited in O’Neill-Wermer 1968 [@O'Neill-Wermer_1968] for an abstract version of Ahlfors’ extremal problem pertaining to a function algebra over a compact space $X$\] 

E. Biswas, [*On line bundles over real algebraic curves*]{}, Bull. Sci. Math. 134 (2010), 447–449. \[$\spadesuit$\] 

A. Bloch, [*La conception actuelle de la théorie des fonctions entières et méromorphes*]{}, L’Enseign. Math. 25 (1926), 83–103. \[$\spadesuit$ great French prose and finitistic philosophy à la Kronecker, culminating to the slogan “Nihil est infinito…”\] 

A. Bloch, [*Les fonctions holomorphes et méromorphes dans le cercle-unité*]{}, Mémorial des Sci. Math. 20 (1926), 1–61. 

, Dokl. Akad. Nauk SSSR 295 (1987), 268–272; English transl., Soviet Math. Dokl. 36 (1988), 38–42. \[$\spadesuit$\] 

Some propositions concerning the geometric representation of imaginaries. [*Ann. of Math. 7*]{} (1892/93), 70–76. \[$\spadesuit$\] 

J. Bochnack, W. Kucharz, [*A characterization of dividing real algebraic curves*]{}, Topology (1996), 451–455. \[$\spadesuit$ definition of dividing curves à la Klein $\spadesuit$ albeit the title could be perfectly adequate to reflect the Ahlfors circle map existence theorem, the paper treat another characterization of dividing curves in terms of “regular mapping” (in the sense of real algebraic geometry) and their Brouwer’s topological degree, plus the Hopf’s theorem (classification of sphere valued mappings up to homotopy by the Brouwer degree)\] 

J. Bochnack, W. Kucharz, R. Silhol, [*Morphisms, line bundles and moduli spaces in real algebraic geometry*]{}, Publ. Math. IHES (1997 ca.), 5–65. \[$\spadesuit$ p.12: “dividing curves” appear, and occur in some problems about approximation of the smooth category by the algebro-geometric one\] 

S. Bochner, [*Über orthogonale Systeme analytischer Funktionen*]{}, Math. Z. 14 (1922), 180–207. (Thesis, Berlin, 1921.) \[$\spadesuit$ p.184: like Bieberbach 1914 [@Bieberbach_1914] and Bergman 1922 [@Bergman_1922] the author confesses to be not able to reprove the RMT via Bieberbach’s minimum problem (least area map) $\spadesuit$ this problem will be cracked (independently) in Garabedian and Lehto’s Thesis (cf. Garabedian 1949 [@Garabedian_1949] and Lehto 1949 [@Lehto_1949])\] 

S. Bochner, [*Fortsetzung Riemannscher Flächen*]{}, Math. Ann. 98 (1927), 406–421. \[$\spadesuit$ any Riemann surface of finite genus embeds conformally into a closed Riemann surface of the same genus $\spadesuit$ any Riemann surface embeds into a non-prolongeable Riemann surface\] 

C.F. Bödigheimer, [*Configuration models for moduli spaces of Riemann surfaces with boundary*]{}, Abh. Math. Seminar Hamburg (2006). \[$\spadesuit$\] 

M.D. Bolt, S. Snoeyink, E. van Andel, [*Visual representation of the Riemann and Ahlfors maps via the Kerzman-Stein equation*]{}, Involve 3 (2010), 405–420. \[$\spadesuit$ from MR: “The paper provides an elementary description of the Riemann and Ahlfors maps using the Szegö kernel. It further describes a numerical implementation of the maps.”\]$\bigstar$$\bigstar$ 

E. Borel, [*Leçons sur la théorie des fonctions*]{}, Gauthier-Villars, Paris, 1898. \[$\spadesuit$ complex function theory, but also the starting point of modern measure theory (influence over Lebesgue)\] 

J.B. Bost, [*Introduction to compact Riemann surfaces, Jacobians and Abelian varieties*]{}, in: From Number Theory to Physics, Springer-Verlag, 1992, Second Corrected Printing 1995. \[$\spadesuit$ p.99–104 contains an account of the Belyi-Grothendieck theorem as well as its geometric traduction in terms of equilateral triangulations\] 

M. Brandt, [*Ein Abbildungssatz für endlich-vielfach zusammenhängende Gebiete*]{}, Bull. de la Soc. des Sciences et des Lettres de Łódź XXX, 3 (1980). \[$\spadesuit$ extension of Koebe’s KNP to shapes with arbitrary contours; similar result in Harrington 1982 [@Harrington_1982] $\spadesuit$ variant of proof in Schramm 1996 [@Schramm_1996]\] $\bigstar$$\bigstar$$\bigstar$ 

M. Brelot, G. Choquet, [*Espaces et lignes de Green*]{}, Ann. Inst. Fourier 3 (1951), 199–263. 60 \[$\spadesuit$ the paper is started with a result of Evans (1927) that the streamlines of the Green’s function $G(z,t)$ \[with pole at $t$\] in a simply-connected plane domain have almost all (in the angular sense about $t$) finite length and therefore converge to a frontier-point $\spadesuit$ this is adapted to domains of arbitrary connectivity (as well as to “superior spaces”) $\spadesuit$ presumably as well as to bordered surfaces: \[11.08.12\] incidentally one could dream of a proof of Gromov’s FAC just via the Green’s function, while using the corresponding isothermic coordinates to calculate the area\] 

M. Brelot, [*La théorie moderne du potentiel*]{}, Ann. Inst. Fourier 4 (1952), 113–140. \[$\spadesuit$ p.114 “Mais si l’on peut dire que tout est dans l’[œ]{}uvre de Gauss, il apparut bientôt que la [*rigueur*]{} était insuffisante” $\spadesuit$ a historical survey starting from Poisson, then Gauss 1840 (who considers as evident that the minimum energy is attained, electrostatic influence, problème du balayage), and the culmination of Frostman’s Thesis (1935); meanwhile Dirichlet, Riemann and Hilbert; and also Neumann, Schwarz, Harnack and Poincaré’s balayage; next Fredholm’s theory (1900) and its application to Dirichlet and Neumann; Zaremba’s works; Lebesgue’s integral found an application in Fatou’s study of the Poisson integral and Evans introduced the Radon integral in potential theory; Perron and Wiener renewed the Dirichlet problem; F. Riesz introduced subharmonic functions (precursors like Poincaré and Hartogs are signaled on p.134); ca. 1930 de la Vallée Poussin took up again the méthode du balayage to study “les masses balayées”, etc.\] 

E. Brettler, [*Absolute Galois groups of real function fields in one variable*]{}, Diss. McGill, Univ. Montréal 1972. \[$\spadesuit$ quoted in Geyer-Martens 1977 [@Geyer-Martens_1977]\]

E. Brieskorn, H. Knörrer, [*Ebene algebraische Kurven*]{}, Birkäuser Boston, 1981; English translation: [*Plane Algebraic Curves*]{}, Trans. from the German by John Stillwell, Birkäuser Basel, 1986. \[$\spadesuit$ p.ii: “Es ist die Freude an der Gestalt in einem höheren Sinne, die den Geometer ausmacht. (Clebsch, in memory of Julius Plücker, Gött. Abh. Bd. 15).”\]

A. Brill, M. Nöther, [*Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie*]{}, Math. Ann. 7 (1874), 269–310. \[$\spadesuit$\]

A. Brill, M. Noether, [*Bericht über die Entwicklung der Theorie der algebraischen Functionen in älterer und neuerer Zeit*]{}, Jahresber. Deutsche Math.-Verein. 3 (1894), 107–566. \[$\spadesuit$ a monumental historiography spreading over more than 400 pages, from Descartes to Riemann and much more\] 

A. Brill, [*Über dden Weierstra[ß]{}chen Vorbereitungssatz*]{}, Math. Ann. 64 (1910), 538–549. \[$\spadesuit$\] 

L.E.J. Brouwer, [*Über die topologischen Schwierigkeiten des Kontinuitätsbeweises der Existenztheoreme eindeutig umkehrbarer polymorpher Funktionen auf Riemannschen Flächen*]{}, Gött. Nachr. (1912), 603–606. 60 \[$\spadesuit$ topological methods as applied to uniformization\] 

L.E.J. Brouwer, [*Über die Singularitätenfreiheit der Modulmannigfaltigkeit*]{}, Gött. Nachr. (1912), 803–806. 60 \[$\spadesuit$ idem\] 

L.E.J. Brouwer, [*Ueber eineindeutige, stetige Transformationen von Flächen in sich (6. Mitt.)*]{}, KNAW Proceedings 21 (1919), 707–710. \[$\spadesuit$ Brouwer seems to vindicate his priority over Koebe for a topological proof of uniformization via the continuity method\] 

L. Brusotti, [*Sulla generazzione di curve piane algebriche reali mediante “piccola variazione” di una curva spezzata*]{}, Annali di Mat. (3) 22 (1913), 117–169. \[$\spadesuit$ systematic small perturbation method for the independent smoothings of nodal plane curves (based upon an extrinsic version of Riemann-Roch (??), worked out over ${\Bbb C}$ by Severi $\spadesuit$ forerunners Plücker 1839 [@Plücker_1839], Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn], Harnack 1876 [@Harnack_1876], etc\] 

L. Brusotti, [*Curve generatrici e curve aggregate nella costruzione di curve piane d’ordine assegnato dotate del massimo numero di circuiti*]{}, Rend. Circ. Mat. Palermo 42 (1917), 138–144. \[$\spadesuit$ cited in Mikhalkin 2000 [@Mikhalkin_2000]\] 

L. Brusotti, [*Sulla “piccola variazione” di una curva piana algebrica reale*]{}, Rend. Rom. Acc. Lincei (5) 30 (1921), 375–379. \[$\spadesuit$ systematic small perturbation method for the independent smoothings of nodal plane curves (based upon an extrinsic version of Riemann-Roch, worked out over ${\Bbb C}$ by Severi $\spadesuit$ forerunners Plücker 1839 [@Plücker_1839], Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn], Harnack 1876 [@Harnack_1876], etc. $\spadesuit$ immediate follower Gudkov 1962 [@Gudkov_1962] (extension to cusps), Gudkov 1980/80 [@Gudkov_1980/80-Brusotti] (extension to other surfaces), etc.\] 

L. Brusotti, [*Su talune questioni di realita nei loro metodi, resultati e problemi*]{}, in: Colloque sur les questions de réalité en géométrie, (Liège 1955), Georges Johne, Liège, et Masson, Paris, 1956, 105–129. \[$\spadesuit$ briefly discussed in Viro’s survey 1989/90 [@Viro_1989/90-Construction] and also cited in Viro 1986/86 [@Viro_1986/86-Progress]\] 

E. Bujalance, J.J. Etayo, J.M. Gamboa, G. Gromadzki, [*Automorphisms groups of compact bordered Klein surfaces*]{}, Lecture Notes in Math. 1439, Springer-Verlag, Berlin, 1990. \[$\spadesuit$\] 

E. Bujalance, A. Costa, S. Natanzon, D. Singerman, [*Involutions of compact Klein surfaces*]{}, Math. Z. 211 (1992), 461–478. \[$\spadesuit$\] 

E. Bujalance, G. Gromadzki, D. Singerman, [*On the number of real curves associated to a complex curve*]{}, Proc. Amer. Math. Soc. 120 (1994), 507–513. \[$\spadesuit$\] 

J. Burbea, [*The Carathéodory metric in plane domains*]{}, Kodai. Math. Sem. Rep. 29 (1977), 159–166. \[$\spadesuit$ application of the Ahlfors function to a curvature estimate of the Carathéodory metric (defined via the analytic capacity) along the line of Suita’s works $\spadesuit$ Abstract: “Let $D\notin O_{AB}$ be a plane domain \[i.e., supporting non-constant bounded analytic functions\] and let $C_D(z)$ be its analytic capacity at $z\in D$ \[that is the maximum distortion of a circle-map centered at $z$\]. Let ${\cal K}_{D}(z)$ be the curvature of the Carathéodory metric $C_D(z) \vert dz \vert$. We show that ${\cal K}_D(z)<-4$ if the Ahlfors function of $D$ w.r.t. $z$ has a zero other than $z$. For finite \[domains\] $D$, ${\cal
K}_D(z) \le -4$ and equality holds iff $D$ is simply-connected. As a corollary we obtain a result proved first by Suita, namely, that ${\cal K}_D(z)\le -4$ if $D\notin O_{AB}$. Several other properties related to the Carathéodory metric are proven.” $\spadesuit$ a little anachronism is noteworthy, here logically the Ahlfors function and the allied analytic capacity (1947 [@Ahlfors_1947]) precedes the Carathéodory metric (1926 [@Caratheodory_1926] and 1927 [@Caratheodory_1927]), but of course in view of the real history, especially Carathéodory 1928 [@Caratheodory_1928] the definitional aspect is essentially compatible with the historical flow\] 

J. Burbea, [*The curvatures of the analytic capacity*]{}, J. Math. Soc. Japan 29 (1977), 755–761. \[$\spadesuit$ p.755: Ahlfors function à la Havinson 1961/64 [@Havinson_1961/64], i.e. for domains $D\notin O_{AB}$, analytic capacity, method of the minimum integral w.r.t. the Szegö kernel\] 

J. Burbea, [*Capacities and spans on Riemann surfaces*]{}, Proc. Amer. Math. Soc. 72 (1978), 327–332. \[$\spadesuit$ p.329: “Ahlfors function” is mentioned (in connection with the analytic capacity, yet it is not clear to me \[03.10.12\] if this definition is meaningful not for a domain but also on a finite Riemann surface)\] 

J. Burbea, [*The Schwarzian derivative and the Poincaré metric*]{}, Pacific J. Math. 85 (1979), 345–354. \[$\spadesuit$\] 

J. Burbea, [*The Cauchy and the Szegö kernels on multiply connected regions*]{}, Rend. Circ. Mat. Palermo (2) 31 (1982), 105–118. \[$\spadesuit$ Ahlfors function mentioned on p.106 an p.116\] 

R.B. Burckel, [*An Introduction to Classical Complex Analysis*]{}, Vol.1, Mathematische Reihe 64, Birkhäuser, 1979. \[$\spadesuit$ p.357 some nice comments upon the literature about PSM\] 

W. Burnside, [*On functions determined from their discontinuities, and a certain form of boundary condition*]{}, Proc. London Math. Soc. 22 (1891), 346–358. \[$\spadesuit$ detected \[30.07.12\] via W. Seidel’s bibliogr. (1950/52), who summarize the paper as: a method is given for mapping a region bounded by $m$ simple closed curves $C_i$ on an $n$-sheeted Riemann surface over the $w$-plane, where the curves $C_i$ correspond to rectilinear slits $\spadesuit$ surprisingly this paper is not quoted in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] nor in Grunsky 1978 [@Grunsky_1978] $\spadesuit$ the topic addressed bears some vague resemblance with the Bieberbach-Grunsky-Ahlfors paradigm of the circle map\] 

W. Burnside, [*On a class of automorphic functions*]{}, Proc. London Math. Soc. (3) 23 (1892), 49–88. \[$\spadesuit$\] 

P. Buser, M. Seppälä, R. Silhol, [*Triangulations and moduli spaces of Riemann surfaces with group actions*]{}, Manuscr. Math. 88 (1995), 209–224. \[$\spadesuit$ connectedness of the moduli space of real curves when projected down in the complex moduli\] 

A.P. Calderón, [*Cauchy integrals on Lipschitz curves and related operators*]{}, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1324–1327. \[$\spadesuit$ implies a resolution of the so-called Denjoy conjecture, according to which a subset of a rectifiable curve is removable for the class of bounded analytic functions (alias Painlevé null-sets) iff it has zero length $\spadesuit$ the explicit link from Calderón-to-Denjoy is made explicit in Marshall [@Marshall_1978?], upon combining a long string of previous works (Garabedian, Havinson, Davie 1972 [@Davie_1972])\] 

A.P. Calderón, [*Commutators, singular integrals on Lipschitz curves and applications*]{}, ICM Helsinki 1978, 85–96. \[$\spadesuit$ the Denjoy’s conjecture is mentioned as an application of Calderón 1977 [@Calderon_1977], as follows (p.90): “Now let us turn to applications. Let $\Gamma$ be a simple rectifiable arc in the complex plane. Then the function $G(z)=\frac{1}{2\pi i}\int_\Gamma
\frac{f(w)}{w-z} dw$, where $f(w)$ is a function on $\Gamma$ which is integrable w.r.t. arc length, has a limit almost everywhere in $\Gamma$ as $z$ approaches nontangentially a point of $\Gamma$. \[…\] Another application is the following result due to D.E. Marshall (personal communication) which confirms an old conjecture of A. Denjoy (1909 [@Denjoy_1909-Painleve/Sur-les-fct-anal-unif-a-sing-discontinues]): the analytic capacity $\gamma(E)$ of a compact subset $E$ of a rectifiable arc in the complex plane is zero if and only if its one-dimensional Hausdorff measure vanishes.” $\spadesuit$ for the detailed proof see Marshall [@Marshall_1978?] (and maybe also Melnikov 1995 [@Melnikov_1995])\] 

A.P. Calderón, [*Acceptance speech for the Bôcher Price*]{}, Notices A.M.S. 26 (1979), 97–99. \[$\spadesuit$ the solution to the Denjoy’s conjecture is mentioned as one of the most significant application of the article Calderón 1977 [@Calderon_1977]\]$\bigstar$$\bigstar$ 

A. Candel, [*Uniformization of surface laminations*]{}, Ann. Sci. Ecole Norm. Sup. (1993). \[$\spadesuit$ ... To have the same relation between Riemann surface laminations and oriented surface laminations with riemannian metric we then need a regularity theorem for the Beltrami equation depending on parameters. This is precisely what Ahlfors and Bers proved in their classical ... \] 

C. Carathéodory, [*Sur quelques applications du théorème de Landau-Picard*]{}, C.R. Acad. Sci. Paris 144 (1907), 1203–1204; also in: Ges. Math. Schriften, Band 3, 6–9. \[$\spadesuit$ first modern proof of the Schwarz lemma, acknowledging E. Schmidt, compare footnote 2: “Je dois cette démonstration si élégante d’un théorème connu de M. Schwarz (Ges.Abh.,t.2,p.108) à une communication orale de M. E. Schmidt.”\] 

C. Carathéodory, [*Über die Variabilitätsbereich der Koeffizienten von Potenzreihen die gebebene Werte nicht annehmen*]{}, Math. Ann. 64 (1907), 95–115. \[$\spadesuit$ this and the next entry where the first work bringing together Minkowski’s theory of convex sets with complex function theory $\spadesuit$ for an extension of this Carathéodory theory to finite Riemann surface, cf. Heins 1976 [@Heins_1976]\] 

C. Carathéodory, [*Über die Variabilitätsbereich der der Fourierschenkonstanten von positiven harmonischen Funktionen*]{}, Rend. Circ. Mat. Palermo 32 (1911), 193–217. \[$\spadesuit$\] 

C. Carathéodory, L. Fejér, [*Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landauschen Satz*]{}, Rend. Circ. Mat. Palermo 32 (1911), 218–239. \[$\spadesuit$ for a (vague?) interconnection of this article with the Ahlfors map, cf. Jenkins-Suita 1979 [@Jenkins-Suita_1979]\] 

C. Carathéodory, [*Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten*]{}, Math. Ann. 72 (1912), 107–144. 78 

C. Carathéodory, [*Über die gegenseitige Beziehung der Ränder bei der konformen Abbildung des Inneren einer Jordanschen Kurve auf einen Kreis*]{}, Math. Ann. 73 (1913), 305–320. 

C. Carathéodory, [*Über die Begrenzung einfach zusammenhängender Bereiche*]{}, Math. Ann. 73 (1913), 323–370. 

C. Carathéodory, [*Elementarer Beweis für den Fundamentalsatz der konformen Abbildung*]{}. In: Mathematische Abhandlungen, Hermann Amandus Schwarz zu seinem fünfzigjähren Doktorjubiläum am 6. August 1914 gewidmet von seinen Freunden und Schülern, 19–41; also in: Ges. Math. Schriften, Band 3, 273–299. \[$\spadesuit$ p.294 perhaps the first usage of the jargon “[*quasikonform*]{}”, compare Ahlfors’ memory failure reported in Kühnau 1997 [@Kuehnau_1997] $\spadesuit$ more importantly the classical square-root procedure is developed in detail\] 

C. Carathéodory, [*Über das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Veränderlichen*]{}, Math. Ann. 97 (1926), 76–98. \[$\spadesuit$ quoted in Grunsky 1940 [@Grunsky_1940 p.233], who discusses the connection between the Carathéodory metric and the “Ahlfors” function (which in the present connection should be definitively better called the “Grunsky-Ahlfors function”)\] 

C. Carathéodory, [*Über eine spezialle Metrik, die in der Theorie der analytischen Funktionen auftritt*]{}, Atti Pontifica Acad. Sc., Nuovi Lincei 80 (1927), 135–141. \[$\spadesuit$ where the so-called Carathéodry metric is first defined (but see also the previous entry Carathéodory 1926 [@Caratheodory_1926]), which in turn turned out to be closely related to the Ahlfors function, cf. e.g. Grunsky 1940 [@Grunsky_1940], Simha 1975 [@Simha_1975], Burbea 1977 [@Burbea_1977-Caratheodory], Krantz 2008 [@Krantz_2008]\] 

C. Carathéodory, [*Bermerkungen zu den Existenztheoremen der konformen Abbildung*]{}, Bull. Calcutta Math. Soc. 20 (1928), 125–134; also in: Ges. Math. Schriften, Band 3, 300–310. 60 \[$\clubsuit$ along lines initiated by Fejér-Riesz (published by Radó 1922/23 [@Rado_1922-3]) a new proof of RMT is given via an extremal problem, which is a simply-connected prelude to Ahlfors 1950 [@Ahlfors_1950] $\clubsuit$ as pointed out by Remmert 1991 [@Remmert_1991] Carathéodory’s elegant proof appears rarely in book form (exception Narasimhan’s book), and is somewhat less popular than the variant of Fejér-Riesz-Bieberbach-Ostrowski $\spadesuit$ the article involves (cf. p.303) the extremal problem $\max \vert f(z_1) \vert$ of maximizing the modulus of the function at an auxiliary point $z_1$, whereas the other method (Fejér-Riesz, etc.) maximizes the derivative at the basepoint $z_0$ $\spadesuit$ it is precisely Carathéodory’s version which is extended in Ahlfors 1950 [@Ahlfors_1950], but of course the other formulation lead likewise to a circle map\] 

C. Carathéodory, [*Conformal representation*]{}, Cambridge Tracts in Math. and Math. Physics 28, London 1932. (2nd edition 1958) 60, 78 \[$\spadesuit$ an introduction to problem of conformal mapping\] 

C. Carathéodory, [*On Dirichlet’s problem*]{}, Amer. J. Math. 59 (1937), 709–731. \[$\spadesuit$ surprisingly this item is not quoted in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] $\spadesuit$ p.710: “In the foregoing chapter, I have tried to give a very elementary treatment of the principal properties of harmonic functions culminating in the existence proof for Dirichlet’s problem devised by O. Perron \[=1923 [@Perron_1923]\] and very much simplified by T. Radó and F. Riesz \[=1925 [@Rado-Riesz_1925]\]. I have done this in order to show how the whole theory can be condensed if one puts systematically from the outset Poisson’s Integral in the limelight.”\] 

C. Carathéodory, [*A proof of the first principal theorem on conformal representation*]{}, Studies and Essays presented to R. Courant on his 60th birthday, Jan. 8, 1948, Interscience Publ., 1948, 75–83; also in: Ges. Math. Schriften, Band 3, 354–361. 60, 78 \[$\spadesuit$ another proof of RMT through an iterative method, using square-roots operations, Schwarz’s lemma and Montel $\spadesuit$ naive question, although this might be more in line with the earlier approach ca. 1910 of Koebe-Carathéodory, this approach looks more involved than the extremum problem in the previous item [@Caratheodory_1928], and perhaps less susceptible of extension to Riemann surfaces\] 

C. Carathéodory, [*Funktionentheorie, I, II*]{}, Birkhäuser, Basel, 1950. 60, 78 

C. Carathéodory, [*Bemerkung über die Definition der Riemannschen Flächen*]{}, Math. Z. 52 (1950), 703–708. 60 \[$\spadesuit$ purist approach to uniformization via extremal problems, similar ideas in several papers by Grunsky not listed here, cf. his Coll. Papers $\spadesuit$ the (Grenzkreis) uniformization appears also in Carathéodory 1928 [@Caratheodory_1928]\] 

A.L. Carey, K.C. Hannabuss, [*Infinite dimensional groups and Riemann surfaces field theories*]{}, Comm. Math. Phys. 176 (1996), 321–351. \[$\spadesuit$\] 

T. Carleman, [*Über ein minimal Problem der mathematischen Physik*]{}, Math. Z. 1 (1918), 208–212. 78 \[$\spadesuit$ used in Gaier 1978 [@Gaier_1978-JDMV] and Alenycin 1981/82 [@Alenicyn_1981/82] in connection with an extension of the (Bieberbach 1914) minimum area problem to multiply-connected regions\] 

T. Carleman, [*Sur la représentation conforme des domaines multiplement connexes*]{}, C.R. Acad. Sci. Paris 168 (1919), 843–845. 78 \[$\spadesuit$ another proof of KNP=Kreisnormierungsprinzip, originally due to Koebe 1907/1920, if not (implicit in) Schottky 1877 [@Schottky_1877]\] 

T. Carleman, [*Über die Approximation analytischer Funktionen durch linear Aggregate von vorgegebenen Potenzen*]{}, Arkiv för mat., astron. o. fys. 17 (1922). \[$\spadesuit$ credited in Lehto 1949 [@Lehto_1949 p.8] for some work (independent of Bergman 1922 [@Bergman_1922] and Bochner’s 1922 [@Bochner_1922]) inaugurating the usage of orthogonal systems in the theory of conformal mappings\] 

L. Carleson, [*On bounded analytic functions and closure problems*]{}, Ark. Mat. 2 (1952), 283–291. \[$\spadesuit$\]$\bigstar$$\bigstar$ 

L. Carleson, [*Interpolations by bounded analytic functions and the Corona problem*]{}, Ann. of Math. (2) 76 (1962), 547–559. \[$\spadesuit$ one of the super-famous problem solved by Carleson, and which received (thanks Alling 1964 [@Alling_1964] and others) an extension from the disc to any compact bordered Riemann surface via the Ahlfors circle map\] 

L. Carleson, [*Selected problems on exceptional sets*]{}, Van Nostrand, Princeton, 1967. \[$\spadesuit$ p.73–82 uniqueness of the Ahlfors extremal function \[the one maximizing the derivative at a fixed point amongst functions bounded-by-one\] for domains of infinite connectivity; similar work in Havinson 1961/64 [@Havinson_1961/64] and simplifications in Fisher 1969 [@Fisher_1969]\] 

L. Carleson, [*Lars Ahlfors and the Painlevé problem*]{}. In: [*In the tradition of Ahlfors and Bers*]{} (Stony Brook, NY, 1998), 5–10. Contemp. Math. 256, Amer. Math. Soc., Providence, RI, 2000. Nostrand, Princeton, 1967. \[$\spadesuit$ survey of the theory of removable sets for bounded analytic functions (a.k.a. Painlevé null-sets) from Painlevé, Denjoy to G. David, via Ahlfors (analytic capacity), Garabedian and Melnikov (Menger curvature). Future research is suggested along the 3 axes: (i) develop a theory of periods for the conjugate of positive harmonic functions (ii) sharper study of the extremal function (and induced measure) that appear in Garabedian 1949 [@Garabedian_1949] (iii) to continue the study of the Cauchy integrals in relation with Menger curvature and rectifiability\] 

F. Carlson, [*Sur le module maximum d’une fonction analytique uniforme. I*]{}, Ark. Mat. Astron. Fys. 26 (1938), 13pp. 78 \[$\spadesuit$ quoted (joint with Teichmüller 1939 [@Teichmueller_1939] and Heins 1940 [@Heins_1940-Extremal]) in Grunsky 1940 [@Grunsky_1940] as one of the precursors of the extremal problem for bounded analytic functions\] $\bigstar$$\bigstar$$\bigstar$ 

H. Cartan, [*Théorie él’ementaire des fonctions analytiques d’une ou plusieurs variables complexes*]{}, Hermann, Paris, 1961. \[$\spadesuit$\] 

E. Casas-Alvero, Roots of complex polynomials and foci of real algebraic curves, L’Enseign. Math. 2013. \[$\spadesuit$\] 

A.L. Cauchy, [*Mémoire sur la théorie des intégrales définies*]{}, communicated to the Paris Academy in 1814, and first published in 1827. \[$\spadesuit$ first occurrence of the Cauchy-Riemann equations, as criterion for the analyticity (holomorphy) of functions of a complex variable, modulo earlier occurrences in the work of Euler on hydrodynamics, and even earlier in the work of Jean le Rond d’Alembert (1717–1783) (on behalf of p.12 of Monastyrsky 1987/99 [@Monastyrsky_1987/99/08-even-1979], who do not give the exact sources)\] 

A.L. Cauchy, [*Mémoire sur les intégrales définies prises entre des limites imaginaires*]{}, De Bure Frères, Paris, 1825, posthumous papers not published until 1874, in: [Œ]{}uvres de Cauchy, 1876, Série II, tome XV, 41–89. \[$\spadesuit$ definition of the integral of a function in the complex domain, including the case of singularity in which case the integral may depend on the path $\spadesuit$ first appearance of the notion of residue\] 

A.L. Cauchy, [*???*]{}, work completed in 1831, published in 1836. \[$\spadesuit$ power series expansion of an analytic function and the integral representation of $f(z)$ inside a circle (Cauchy formula)\] 

A.L. Cauchy, [*Considérations nouvelles sur les intégrales définies qui s’étendent à tous les points d’une courbe fermée, et sur celles qui sont prises entre des limites imaginaires*]{}, C.R. Acad. Sci. Paris 23 (1846), 689. \[$\spadesuit$ Cauchy’s residue theorem\] 

F. Cecioni, [*Sulla rappresentazione conforme delle aree piane pluriconnesse su un piano in cui siano eseguiti dei tagli paralleli*]{}, Rend. Circ. Mat. Palermo 25 (1908), 1–19. 78 \[$\spadesuit$ another derivation of the parallel-slit map of Schottky 1877 [@Schottky_1877], via several citation to Picard’s book for the foundational aspects $\spadesuit$ as Schottky’s proof depends on a heuristic moduli count, this paper of Cecioni may well be regarded as the first rigorous existence proof of PSM (cf., e.g., Grunsky 1978 [@Grunsky_1978 p.185])\] 

F. Cecioni, [*Sulla rappresentazione conforme delle aree pluriconnesse appartenenti a superficie di Riemann*]{}, Annali delle Università Toscane 12, nuova serie (1928), 27–88. \[$\spadesuit$ cited via Matildi 1948 [@Matildi_1945/48]; WARNING: this entry looks much like the next item, yet differs in the pagination\] $\bigstar$$\bigstar$ 

F. Cecioni, [*Sulla rappresentazione conforme delle aree pluri-connesse appartenenti a superficie di Riemann*]{}, Rend. Accad. d. L. Roma (6) 9 (1929), 149–153. 60 \[$\spadesuit$ cited via Ahlfors-Sario 1960 [@Ahlfors-Sario_1960]\] $\bigstar$$\bigstar$ 

F. Cecioni, [*Osservazioni sopra alcuni tipi aree e sulle loro curve caratteristiche nella teoria della rappresentazione conforme*]{}, Rend. Palermo 57 (1933), 101–122. \[$\spadesuit$ la parole “curve catteristiche” means the Schottky(-Klein) double $\spadesuit$ contains several nice remarks about the Klein correspondence when particularized to orthosymmetric curve tolerating a direct-conformal involution which is sense reversing on the ovals\] 

F. Cecioni, [*Un teorema su alcune funzioni analitiche, relative ai campi piani pluriconnessi, usate nella teoria della rappresentazione conforme*]{}, Ann. Pisa (2) 4 (1935), 1–14. 78 

M. Černe, J. Globevnik, [*On holomorphic embedding of planar domains into ${\Bbb C}^2$*]{}, J. Anal. Math. 81 (2000), 269–282. 50 \[$\spadesuit$ Koebe’s Kreisnormierungsprinzip is combined with the Ahlfors function to show that every bounded, finitely connected domain of ${\Bbb C}$ without isolated boundary points embeds properly holomorphically into ${\Bbb C}^2$ $\spadesuit$ of course, those are not the sole ingredients for otherwise the method would probably extend to positive genus surfaces in view of Ahlfors 1950 [@Ahlfors_1950], and positive genus extensions of KNP due to Haas 1984 [@Haas_1984]/Maskit 1989 [@Maskit_1989]\] 

M. Černe, F. Forstnerič, [*Embedding some bordered Riemann surfaces in the affine plane*]{}, Math. Research Lett. 9 (2002), 683–696. 50 \[$\spadesuit$ Ahlfors 1950 is cited at several places $\spadesuit$ p.684: “On each smoothly bounded domain $\Omega\Subset {\Bbb C}$ with $m$ boundary components there exists an inner function $f$ with $\deg(f)=m$ \[Ahl\](=Ahlfors 1950 [@Ahlfors_1950])[^121]. The map $F(x)=(f(x),x)\in{\Bbb C}^2$ for $x\in \overline\Omega$ satisfies the hypothesis of Theorem 1.2 and hence $\Omega$ embeds in ${\Bbb C}^2$. This is the theorem of Globevnik and Stens[ø]{}nes \[GS\](=1995).” $\spadesuit$ p.684: “We shall call a bordered Riemann surface $\cal R$ hyperelliptic if its double is hyperelliptic. Such \[an\] $\cal R$ has either one or two boundary components[^122] and it admits a pair of inner functions $(f,g)$ which embed ${\rm int}{\cal R}$ in the polydisc $U^2$ such that $b{\cal R}$ is mapped to the torus $(bU)^2$; moreover, one of the two functions has degree $2g_{\cal
R}+m$ and the other one has degree $2$ (see \[Ru1\](=Rudin 1969 [@Rudin_1969]) and sect.2 in \[Gou\](=Gouma 1998 [@Gouma_1998])). Thus $\cal R$ is of class ${\cal F}$ and we get:—[**Corollary 1.3**]{} [*If ${\cal R}$ is a hyperelliptic bordered Riemann surface then ${\rm int} {\cal R}$ admits a proper holomorphic embedding in ${\Bbb C}^2$. In particular, each torus with one hole embeds properly holomorphically into ${\Bbb C}^2$.*]{} $\spadesuit$ p.686: “[**Comments regarding class ${\cal F}$.**]{} It is proved in \[Ahl,pp.124–126\](=Ahlfors 1950 [@Ahlfors_1950]) that on every bordered Riemann surface $\cal
R$ of genus $g_{\cal R}$ with $m$ boundary components there is an inner function $f$ with multiplicity $2 g_{\cal R}+m$ (although the so-called Ahlfors functions may have smaller multiplicity). A generic choice of $g\in A^1({\cal R})$ gives an immersion $F=(f,g)\colon {\cal R}\to \overline U \times {\Bbb C}$ with at most finitely many double points (normal crossings). The main difficulty is to find $g$ such that $F=(f,g)$ is injective on ${\cal R}$. We do not know whether such $g$ always exists as Oka’s principle does not apply in this situation (Proposition 2.2).” $\spadesuit$ Ahlfors 1950 is cited once more on p.687 during the proof of Theorem 1.1 stating that there is no topological obstruction to holomorphic embeddability in ${\Bbb C}^2$, in the following sense (p.683) “[**Theorem 1.1**]{} [*On each bordered surface $\cal R$ there exists a complex structure such that the interior ${\rm int} {\cal R}={\cal R}\setminus \partial
{\cal R}$ admits a proper holomorphic embedding in ${\Bbb C}^2$.*]{} $\spadesuit$ p.693: “[**Remark.**]{} As already mentioned, Ahlfors \[Ahl\](=1950) constructed inner functions of multiplicity $2g_{\cal R}+m$ on any bordered Riemann surface. Proposition 4.1 shows that such functions are stable under small perturbations of the complex structure. On the other hand this need not be true for the Ahlfors function $f_p$ which maximizes the derivative at a given point $p\in {\cal R}$ since the degree of $f_p$ may depend on $p$.” $\diamondsuit$ \[28.09.12\] maybe there is a somewhat more elementary approach to the main result (no topological obstruction) by looking at some real algebraic models in ${\Bbb
P}^2$ or ${\Bbb P}^1\times {\Bbb P}^1$, for instance taking a saturated pencil on the Gürtelkurve (cf. Gabard 2006 [@Gabard_2006]) and removing an imaginary line of this pencil one gets an embedding of the bordered surface (half of the real quartic $C_4$) into ${\Bbb C}^2$\] 

M. Černe, [*Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces*]{}, Amer. J. Math. 126 (2004), 65–87. 50 \[$\spadesuit$\] 

M. Černe, M. Flores, [*Generalized Ahlfors functions*]{}, Trans. Amer. Math. Soc. 359 (2007), 671–686. \[$\spadesuit$ a promising generalization of the Ahlfors function is given where the (static) unit-circle is replaced by a dynamical family $\{\gamma_z \}_{z\in \partial F}$ of Jordan curves enclosing the origin parametrized by the boundary of the bordered surface $F$\] 

M. Černe, F. Forstnerič, [*Embedding some bordered Riemann surfaces in the affine plane*]{}, Math. Res. Lett. 9 (2002), 683–696. \[$\spadesuit$ it is shown that there is no topological obstruction to embed membranes in ${\Bbb C}^2$ $\spadesuit$ this is a good starting toward the difficult conjecture about proper embedding of open Riemann surfaces in the same recipient\] 

S. Chaudary, [*The Brill-Noether theorem for real algebraic curve*]{}, Ph.D. Thesis. Duke University, 1995. \[$\spadesuit$\]$\bigstar$ 

?. Chebotarev, [*?*]{}, ? , 1948. \[$\spadesuit$ a textbook on analytic functions and Riemann surfaces cited in Gudkov 1974/74 [@Gudkov_1974/74] for the intrinsic proof of Harnack’s inequality $\spadesuit$ this entry is possibly the source of some historical confusion crediting Hurwitz instead of Klein for this result (compare comments after Gudkov 1974/74 ) $\spadesuit$ Hurwitz extended to the case of singular curves, but modulo the normalization (desingularization) everything reduces to the smooth case\] 

A.L. Cheponkus, [*On nests of real plane algebraic curves*]{}, Litovsk. Mat. Sb. 16 (1976), 239–243, 257; English transl., Lithuanian Math. J. 16 (1976), 634–637. \[$\spadesuit$ cited in Rohlin 1978 [@Rohlin_1978], cf. also a paper by Marin 1988 [@Marin_1988] for another proof $\spadesuit$ in fact Cheponkus’ proof turned out to be incorrect (cf. e.g. Viro 1986/86 [@Viro_1986/86-Progress p.68] and especially Marin 1988 [@Marin_1988] for a specific objection)\] 

I.V. Cherednik, [*Reality condition in “finite-zone integration”*]{}, Dokl. Akad. Nauk SSSR 252 (1980), 1104–1108; English transl., Soviet Phys. Dokl. 25 (1980), 450–452. \[$\spadesuit$ cited in Dubrovin 1983/85 [@Dubrovin_1983/85]\] 

S.S. Chern, P. Hartman, A. Wintner, [*On isothermic coordinates*]{}, Comment. Math. Helv. 28 (1954), 301–309. \[$\spadesuit$\] 

S.S. Chern, [*Complex Manifolds without Potential Theory*]{}, Second Edition. Springer-Verlag, Berlin, 1979. \[$\spadesuit$\] 

C. Chevalley, [*Introduction to the Theory of Algebraic Functions of One Variable*]{}, Math. Surveys 6, Amer. Math. Soc., New York, 1951, 188 pp. \[$\spadesuit$\] 

Yu.S. Chislenko, [*Pencils of real algebraic curves*]{}, Leningrad Topology Conference, 1982, 28. \[$\spadesuit$ correct proof of a special case of Cheponkus’ theorem, namely through 13 points in general position in the real projective plane passes a connected quartic $\spadesuit$ for a generalization based upon Klein’s remark (1876), cf. Marin 1988 [@Marin_1988]\] 

Yu.S. Chislenko, [*$M$-curves of degree $10$*]{}, Zap. Nauch. Sem. Leningrad 122 (1982), 142–161; English transl., 1984. \[$\spadesuit$\] 

M. Christ, [*A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral*]{}, Colloq. Math 60/61 (1990), 1367–1381. \[$\spadesuit$\] 

E. Christoffel, [*Ueber die Abbildung einer $n$-blättrigen einfach zusammenhängender ebenen Fläche auf einen Kreise*]{}, Gött. Nachr. (1870), 359–369. \[$\spadesuit$ the so-called Schwarz-Christoffel formula effecting the (one-to-one) conformal representation of a polygon upon the disc $\spadesuit$ more precisely Schwarz 1869 [@Schwarz_1869-Ueber-einige-Abbildungsaufgaben] stated that the formula is easily generalized to the case of a multi-sheeted domain bounded by straight lines and containing branch points, and Christoffel considers here this generalization in some detail $\spadesuit$ \[07.10.12\] can we connect the Schwarz-Christoffel theory with that of the Ahlfors map? try perhaps Kühnau 1967 [@Kuehnau_1967]\] 

Y.-B. Chung, [*The Ahlfors mapping function and an extremal problem in the plane*]{}, Houston J. Math. 263 (1993), 263–273. \[$\spadesuit$\]$\bigstar$ 

Y.-B. Chung, [*The Bergman kernel function and the Ahlfors mapping in the plane*]{}, Indiana Univ. Math. J. 42 (1993), 1339–1348. \[$\spadesuit$ Ahlfors mapping, Bergman kernel, etc.\] 

Y.-B. Chung, [*Higher order extremal problem and proper holomorphic mapping*]{}, Houston Math. J. 27 (2001), 707–718. 50 \[$\spadesuit$ Ahlfors extremal problem (in the domain-case) with multiplicity (i.e. some first derivatives are imposed to be $0$ at some base-point $a$)\]$\bigstar$$\bigstar$$\bigstar$\[MR-OK\] 

Y.-B. Chung, [*The Bergman kernel function and the Szegö kernel function*]{}, J. Korean Math. Soc. 43 (2006), 199–213. \[$\spadesuit$ the “Ahlfors map” of a smoothly bounded domain in the plane occurs several times through the paper\] 

C. Ciliberto, C. Pedrini, [*Annibale Comessatti and real algebraic geometry*]{}, Rend. Cont. Circ. Mat. Palermo (2) Suppl. 36 (1994), 71–102. \[$\spadesuit$\] 

C. Ciliberto, C. Pedrini, [*Real abelian varieties and real algebraic curves*]{}. In: Lectures in Real Geometry, F. Broglia (ed.), de Gruyter Exp. in Math. 23 (1996), 167–256. \[$\spadesuit$ a modernized (neoclassical) account of the theories of Klein, Weichold and Comessatti\] 

K. Clancey, [*Representing measures on multiply connected planar domains*]{}, Illinois J. Math. 35 (1991), 286–311. \[$\spadesuit$ what about Riemann surface? try Alpay-Vinnikov 200 [@Alpay-Vinnikov_2000], and also Nash 1974 [@Nash_1974]\] 

A. Clebsch, [*Ueber die Anwendung der Abelschen Functionen in der Geometrie*]{}, J. Reine Angew. Math. 63 (1863), 189–243. \[$\spadesuit$\] 

A. Clebsch, [*Ueber diejenigen ebenen Curven, deren Coordinaten rationale Functionen eines Parameters sind*]{}, J. Reine Angew. Math. 64 (1865), 43–65. \[$\spadesuit$ coins the nomenclature genus, conceptually put in the limelight by Riemann (plus maybe Abel in some algebraic disguise)\] 

A. Clebsch, P. Gordan, [*Theorie der Abelschen Functionen*]{}, Teubner, Leipzig, 1866. \[$\spadesuit$\] 

A. Clebsch, [*Zur Theorie der Riemann’schen Flächen*]{}, Math. Ann. 6 (1872), 216–230. 60 \[CHECK date for Ahlfors-Sario 1960, it is 1873?\] 

C.H. Clemens, [*A Scrapbook of Complex Curve Theory*]{}, Plenum Press, New York, 1980, 186 pp. \[$\spadesuit$\] 

W.K. Clifford, [*On the space-theory of matter*]{}, Cambridge Philos. Society’s Proc. 2 (1876), 157–158. \[$\spadesuit$\] 

W.K. Clifford, [*On the canonical form and dissection of a Riemann’s surface*]{}, Proc. London Math. Soc. 8 (1877), 292–304. \[$\spadesuit$ not cited in Ahlfors-Sario 1960!\] 

R. Coifman, G. Weiss, [*A kernel associated with certain multiply connected domains and its application to factorization theorems*]{}, Studia Math. 28 (1966), 31–68. 78 \[$\spadesuit$ p.31: “Our main result is a generalization of the classical factorization theorem for function in the Nevanlinna class of the unit disc.”\] 

R.R. Coifman, A. McIntosh, Y. Meyer, [*L’opérateur de Cauch définit un opératuer borné sur $L^2$ pour les courbes lipschitziennes*]{}, Ann. of Math. (2) 116 (1982), 361–387. \[$\spadesuit$\] 

Y. Colin de Verdière, A. Marin, [*Triangulations presques équilatérales des surfaces*]{}, J. Differential Geom. 32 (1990), 199–207. \[$\spadesuit$\] 

H. Comessatti, [*Fondamenti per la geometria sopra le superficie razionali dal punto di vista reale*]{}, Math. Ann. 43 (1912), 1–72. \[$\spadesuit$\] 

H. Comessatti, [*Sulla connessione delle superficie razionali reale*]{}, Math. Ann. ?? (1914), 215–283. \[$\spadesuit$\] 

H. Comessatti, [*Sulle varietà abeliane reali, I, II*]{}, Ann. Mat. Pura Appl. (4) 2 (1924), 67–106; (4) 4 (1926), 27–71. 

H. Comessatti, [*Sulla connessione delle superficie algebriche reali*]{}, Verhandl. Internat. Math. Kongress Zürich, vol.2, p.129. \[$\spadesuit$\] 

H. Comessatti, [*Reelle Fragen in der algebraischen Geometrie*]{}, Jahresb. d. Deutsch. Math. Verein. 41 (1932), 107–134. \[$\spadesuit$\] 

H. Comessatti, [*Problemi di realtà per le superficie e varietà algebriche*]{}, Reale Accad. Ital. (Fondaz. A. Volta), Atti dei Convegni, vol.9 (1939), Rome 1943, 15–41. \[$\spadesuit$ cited in Nikulin 1983/84\] 

. Oxford University Press, London, 1931.

M. Coppens, [*One-dimensional linear systems of type II on smooth curves*]{}, Ph.D. Thesis, Utrecht, 1983. \[$\spadesuit$\] 

M. Coppens, G. Martens, [*Linear series on general $k$-gonal curves*]{}, Abh. Math. Sem. Univ. Hamburg 69 (1999), 347–371. \[$\spadesuit$\] 

M. Coppens, [*Totally non-real divisors in linear ssystems on smooth real curves*]{}, Adv. Geometry 8 (2008), 551–555. \[$\spadesuit$\] 

M. Coppens, G. Martens, [*Linear pencils on real algebraic curves*]{}, J. Pure Appl. Algebra 214 (2010), 841–849. 50 \[$\clubsuit$ Ahlfors 1950 [@Ahlfors_1950] is cited in the following fashion (p.843): “Let $X$ be a real curve of genus $g$ with $s\ge 1$ real components and $g^1_d$ be a basepoint free pencil on $X$. Since $X({\Bbb R})\neq \varnothing$ the image curve $X'$ of the morphism $\varphi$ induced by the pencil is the rational real curve ${\Bbb P}^1_{\Bbb R}$. Assume that the fibre of $\varphi$ at every real point of $X'$ consists entirely of real points of $X$ (or, what is the same, that $\varphi$ separates conjugate points of $X_{\Bbb C}$: $\varphi(\sigma P)\neq
\varphi(P)$ for any non-real point $P\in X_{\Bbb C}$); we call such a $g^1_d$ [*totally real*]{}. Then $\varphi$ is a ramified covering of bordered real surfaces (in the topological sense, cf. \[7, part3\](=Geyer-Martens 1977 [@Geyer-Martens_1977])), and the induced covering $X({\Bbb R})\to X'({\Bbb R})\cong S^1$ is unramified. In particular, $s\le d$. Since $X'=({\Bbb P}^1_{\Bbb
C} {\rm\; mod\; conjugation})$, a half-sphere with boundary, is an orientable real surface it follows that also the Klein surface \[of\][^123] $X$ must be orientable which implies $s\not\equiv g
\mod 2$ (cf. \[7, part.2\](=Geyer-Martens 1977 [@Geyer-Martens_1977]))[^124]. Hence the assumed property that every divisor of $X$ in the $g^1_d$ is entirely made up by real points puts severe restrictions on $X$. So we cannot expect to find such a pencil on every real curve. More precisely, by a result of Ahlfors \[10\](=1950 [@Ahlfors_1950]) there is a totally real pencil of degree $g+1$ on $X$ iff the Klein surface $X$ is orientable thus giving an interesting algebraic characterization of a topological property.”\] 

M. Coppens, J. Huisman, [*Pencils on real curves*]{}, arXiv (2011). \[$\clubsuit$\] 

M. Coppens, [*The separating gonality of a separating real curve*]{}, arXiv (2011); or Monatsh. Math. 2012. \[$\clubsuit$ the spectacular result is proven that all intermediate gonalities compatible with Gabard’s bound ($\le r+p$) are realized by some compact bordered Riemann surface $\spadesuit$ the work is written in the language of real algebraic geometry, especially dividing (or separating) curve and is a tour de force involving several techniques: Kodaira-Spencer deformation theory, Meis’ bound and its phagocytose into modernized Brill-Noether theory, stable curves à la Deligne-Mumford (1969), geometric Riemann-Roch, Hilbert scheme, etc.\] 

M. Coppens, [*Pencils on separating $(M-2)$-curves*]{}, arXiv (2012). \[$\clubsuit$\] 

A.F. Costa, [*On anticonformal automorphisms of Riemann surfaces with nonembeddable square*]{}, Proc. Amer. Math. Soc. 124 (1996), 601–605. \[$\spadesuit$\] 

A.F. Costa, [*Embeddable anticonformal automorphisms of Riemann surfaces*]{}, Comment. Math. Helv. 72 (1997), 203–215. \[$\spadesuit$ outgrowth of the work by Garsia/Rüedy on conformal embeddings, in particular Prop.1.1. [*Let $f$ be an anticonformal involution of a Riemann surface $S$ then $f$ is embeddable iff either $S/f$ is orientable or $S/f$ is non-orientable without boundary.*]{} $\spadesuit$ Since the problem of conformal embeddings of Reimann surface as classical surface was first posed by Klein, it is easy to imagine how this result is a double Kleinian synthesis, which would have much pleased the “magister ludis”\] 

A.F. Costa, M. Izquierdo, [*On the connectedness of the locus of real Riemann surfaces*]{}, Ann. Acad. Sci. Fenn. Math. 27 (2002), 341–356. \[$\spadesuit$ a new proof is offered of a result due Buser-Seppälä-Silhol 1995 [@Buser-Seppala-Silhol_1995], stating the connectedness of the projection of the real moduli down to the complex one (upon forgetting the anti-holomorphic involution) $\spadesuit$ intuitively this means that any symmetric Riemann surface can be deformed so as to create a new symmetry and one can explore the full real moduli space (doing some jump when one switch the symmetry)\] 

A.F. Costa, M. Izquierdo, [*On real trigonal Riemann surfaces* ]{}, Math. Scand. (2006). \[$\spadesuit$ A closed Riemann surface X which can be realized as a $3$-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. A trigonal Riemann surface $X$ is called real trigonal if there is an anticonformal involution (symmetry) ...\] 

R. Courant, [*Über die Anwendung des Dirichletschen Prinzipes auf die Probleme der konformen Abbildung*]{}, Math. Ann. 71 (1912), 141–183. 78 \[$\spadesuit$ Diese Arbeit ist bis auf einige redaktionelle Änderungen ein Abdruck meiner Inauguraldissertation, Göttingen 1910.\] 

R. Courant, [*Über eine Eigenschaft der Abbildungsfunktion\[en\] \[sic!?\] bei konformer Abbildung*]{}, Gött. Nachr. (1914), 101–109. 78 \[$\spadesuit$ this work is regarded by Gaier 1978 [@Gaier_1978-JDMV p.43] (and probably many others) as the first apparition of the length-area principle, which will be largely exploited by Grötzsch (Flächenstreifenmethode) and Ahlfors-Beurling (extremal length), etc., and which in the long run should obviously constitutes one of the key to the resolution of the Gromov filling conjecture $\spadesuit$ uses also the area integral $\int \int \vert f'(z)\vert^2 dx dy$ like Bieberbach 1914 [@Bieberbach_1914]\] 

R. Courant, [*Über konforme Abbildung von Bereichen, welche nicht durch alle Rückkehrschnitte zerstückelt werden, auf schlichte Normalbereiche*]{}, Math. Z. 3 (1919), 114–122. 60 

R. Courant, D. Hilbert, [*Methoden der mathematischen Physik. I*]{},Springer-Verlag, Berlin, 1931. (Reedited 1968) \[$\spadesuit$ cited e.g. in Simha 1975 [@Simha_1975] for an explicit formula for Jacobi theta function, the latter being involved in an explicit description of the Ahlfors map and the Carathéodory metric\] 

R. Courant, [*Plateau’s problem and Dirichlet’s principle*]{}, Ann. of Math. 38 (1937), 679–725. 

R. Courant, [*Remarks on Plateau’s and Douglas’ problem*]{}, Proc. Nat. Acad. Sci. U.S.A. 24 (1938), 519–522. \[$\clubsuit$ this is first place where the theorem of Bieberbach-Grunsky is reproved via Plateau, yet without citing them $\spadesuit$ a more detailed proof is given in the next entry (Courant 1939 [@Courant_1939])\] 

R. Courant, [*Conformal mapping of multiply-connected domains*]{}, Duke Math. J. 5 (1939), 814–823. 60, 78 \[$\clubsuit$ the Bieberbach-Grunsky theorem is re-proved à la Plateau; now Bieberbach 1925 [@Bieberbach_1925] and Grunsky 1937 [@Grunsky_1937] are cited as well as Riemann (as an oral tradition)\] 

R. Courant, [*The existence of minimal surfaces of given topological structure under prescribed boundary condition*]{}, Acta Math. 72 (1940), 51–98. \[$\spadesuit$ specializing to the case of ambient dimension 2 might perhaps reprove a theorem like the Ahlfors circle map $\spadesuit$ recall however that Tromba 1983 [@Tromba_1983-PREPRINT] seems to express doubts about the validity of Courant’s proof, compare also Jost 1985 [@Jost_1985]\] 

R. Courant, M. Manel, M. Shiffman, [*A general theorem on conformal mapping of multiply connected domains*]{}, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 503–507. 78 \[¶Result generalized in Schramm’s Thesis ca. 1990, cf. arXiv\] 

R. Courant, [*The conformal mapping of Riemann surfaces not of genus zero*]{}, Univ. Nac. Tucumán Revista A. 2 (1941), 141–149. 60 \[$\spadesuit$ detected only the 13.06.2012, via Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] $\spadesuit$ alas Gabard could not a find a copy of this article, and it seems unlikely that the article contains material not overlapping with previous and subsequent work by Courant, especially it is unlikely that the paper contains an existence of circle maps à la Ahlfors\] $\bigstar$ 

R. Courant, [*Dirichlet’s principle, Conformal Mapping, and Minimal Surfaces*]{}, with an appendix by M. Schiffer. Pure and appl. math. 3, New York, Interscience Publishers, 1950. 60, 78 \[$\clubsuit$ overlap much with the previous ref., but somehow hard to read due to its large content and mutatis mutandis type proof, in particular it is not clear if p.183 contains another proof of the circle map of Ahlfors $\spadesuit$ p.169 contains a proof of the Kreisnormierung in finite connectivity\] 

R. Courant, [*Flow patterns and conformal mapping of domains of higher topological structure*]{}. In: [*Construction and Applications of Conformal Maps*]{}, Proc. of a Sympos. held on June 22–25 1949, Applied Math. Series [*18*]{}, 1952, 7–14. 

D. Crowdy, J. Marshall, [*Green’s functions for Laplace’s equation in multiply connected domains*]{}, IMA J. Appl. Math. (2007), 1–24. \[$\spadesuit$ p.13–14, contains beautiful pictures of the levels of the Green’s function on some circular domains\] 

D. Crowdy, [*Conformal mappings from annuli to canonical doubly connected Bell representations*]{}, J. Math. Anal. Appl. 340 (2008), 669–674. \[$\spadesuit$ p.670, the Ahlfors map is briefly mentioned in connection with the work of Jeong-Oh-Taniguchi 2007 [@Jeong-Oh-Taniguchi_2007] on deciding when Bell’s doubly-connected domain $A(r)=\{
z\in{\Bbb C} : \vert z+z^{-1} \vert<r \}$ is conformally equivalent to the Kreisring $\Omega(\rho^2)=\{ \zeta \in {\Bbb
C}: \rho^2<\vert \zeta \vert <1 \}$\] 

G. David, [*Unrectifiable $1$-sets have vanishing analytic capacity*]{}, Rev. Mat. Iberoam. 14 (1998), 369–479. 47 \[$\spadesuit$ p.369: “[**Abstract.**]{} We complete the proof of a conjecture of Vitushkin that says that if $E$ is a compact set in the plane with finite $1$-dimensional Hausdorff measure, then $E$ has vanishing analytic capacity iff $E$ is purely unrectifiable (i.e., the intersection of $E$ with any curve of finite length has zero $1$-dimensional Hausdorff measure). \[…\]” $\spadesuit$ \[29.09.12\] this is quite close to a solution of Painlevé’s problem, but just not so due to the proviso $H^1(E)<\infty$, which cannot be relaxed for p.370: “Actually Vitushkin’s conjecture also said something about the case when $H^1(K)=+\infty$[^125], but this part turned out to be false (\[Ma1\]=(Mattila 1986 [@Mattila_1986]))”\] 

G. David, [*Analytic capacity, Calderón-Zygmund operators, and rectifiability*]{}, Publ. Mat. 43 (1999), 3–25. 47 \[$\spadesuit$\] 

A.M. Davie, [*Analytic capacity and approximation problems*]{}, Trans. Amer. Math. Soc. 171 (1972), 409–414. \[$\spadesuit$ a reduction is effected of the Denjoy conjecture (on removable sets lying on rectifiable curves) to the case where the supporting curve is $C^1$, giving one of the ingredient toward the ultimate solution of Denjoy’s conjecture (compare Marshall [@Marshall_1978?]), where the last piece of the puzzle is the contribution of Calderón 1977 [@Calderon_1977]\] 

A.M. Davie, B. [Ø]{}ksendal, [*Analytic capacity and differentiability properties of finely harmonic functions*]{}, Acta Math. 140 (1982), 127–152. \[$\spadesuit$\] 

P. Davis, H. Pollak, [*A theorem for kernel functions*]{}, Proc. Amer. Math. Soc. 2 (1951), 686–690. \[$\spadesuit$ parallel-slit mapping via Bergman kernel\] 78

B. Deconinck, M. van Hoeij, [*Computing Riemann matrices of algebraic curves*]{}, Physica D 152/153 (2001), 28–46. \[$\spadesuit$\]$\bigstar$ 

R. Dedekind, [*??*]{}, Abhandlungen der Königl. Gesellchaft der Wiss. zu Göttingen 13 (1868). \[$\spadesuit$ first published report of Riemann’s Habilitationsvortrag (1854 [@Riemann_1854-Habilitation/Ueber-die-Hypothesen]) analyzing primarily the mathematical side of Riemannian geometry $\spadesuit$ in particular the so-called Laplace-Beltrami operator ought to be discussed here; there is somewhere a commented version in French\] 

R. Dedekind, H. Weber, [*Theorie der algebraischen Funktionen einer Veränderlichen*]{}, Crelles J. 92 (1879). \[$\spadesuit$ “arithmetized” account of algebraic function and the allied Riemann surfaces, cf. also in the same spirit Hensel-Landsberg 1902 [@Hensel-Landsberg_1902] and H. Weber 1908 [@Weber_1908]\] 

A. Degtyarev, V. Kharlamov, [*Topological properties of real algebraic varieties: Rokhlin’s way*]{}, Uspekhi Mat. Nauk 55 (2000) 129–212; English transl., Russian Math. Surveys 55 (2000), 735–814. \[$\spadesuit$ p.736: “Another fundamental result difficult to overestimate is Rokhlin’s formula for complex orientations. The notion of complex orientation of a dividing real curve (see below), as well as Rokhlin’s formula and its proof, seem incredibly transparent at first sight. The formula settles, for example, two of Hilbert’s conjectures on 11 ovals of plane sextics, which Hilbert himself tried to prove in a very sophisticated way and then included in his famous problem list (as the sixteenth problem).” $\spadesuit$ p.739: “Note that the topological properties of abstract, not embedded, real curves are simple and have been understood completely since Klein’s time; see for example, \[87\](=Natanzon 1990 [@Natanzon_1990/90]) and \[99\](=Rohlin 1978 [@Rohlin_1978]).” $\spadesuit$ p.757: “According to Arnol’d, the following result is due to Maxwell.—[**3.2.1 Theorem**]{}. The orbit space ${\Bbb P}^2/{\rm
conj}$ is diffeomorphic to $S^4$.” $\spadesuit$ p.785: “[**4.6.8. Klein’s statement.**]{} If a real curve of type I undergoes a Morse surgery through a non-degenerate double point, then the number of connected components of this curve cannot increase. \[…\]” (For related literature see Klein 1876 [@Klein_1876]), Rohlin 1978 [@Rohlin_1978], and Marin?, etc. $\spadesuit$ p.788: “[**Digression: real rational curves**]{}. As far as we know, the following problem is still open: is it possible to draw an irreducible real rational curve (or more precisely a connected component of it) of degree $q$ through any set of $3q-1$ real points in general position? In \[99\](=Rohlin 1978 [@Rohlin_1978]) the question is answered in the affirmative; however, the proof has never been published; possibly it contained a gap. \[…\] The first non-trivial case (and the only case in which the complete answer is known) is $q=3$, namely through 8 generic points one can draw 12 rational cubics; depending on the arrangement of the points, the number of real cubics among them can be $8,10$ or $12$. (All three va,ues occur; the 12 rational cubics in a pencil are real iff the pencil contains 2 cubics with a solitary real double point.)”\] 

K. de Leeuw, W. Rudin, [*Extreme points and extremum problems in $H_1$*]{}, Pacific J. Math. 8 (1958), 467–485. \[$\spadesuit$ gives a characterization of the extreme points of the unit ball of the disc-algebra $H^1(\Delta)$, an analogue of which for the same algebra attached to a finite bordered Riemann surface will be given in Gamelin-Voichick 1968 [@Gamelin-Voichick_1968] upon making use of the Ahlfors map or at least techniques closely allied to its existence-proof (as given by Ahlfors 1950 [@Ahlfors_1950])\] 

P. Deligne, D. Mumford, [*Irreducibility of the space of curves of given genus*]{}, Publ. Math. Inst. Hautes. Études Sci. 36 (1969), 75–109. \[$\spadesuit$\] 

A. Denjoy, [*???*]{}, C.R. Acad. Sci. Paris 14? (1907), 258–260. 60 \[$\spadesuit$ yet another Denjoy conjecture (not to be confounded with that of the next entry) on the number of asymptotic values of entire functions of finite order $\spadesuit$ formulated by Denjoy at age 21, it was solved by Ahlfors in 1928 (at age 21), 21 years after its formulation (arithmetical curiosity noticed by Denjoy)\]$\bigstar$

A. Denjoy, [*Sur les fonctions analytiques uniformes à singularités discontinues*]{}, C.R. Acad. Sci. Paris 149 (1909), 258–260. 60 \[$\spadesuit$ the following theorem is proved (or rather asserted since a gap was later located in proof) but Denjoy’s assertion turned out to be ultimately correct via Calderón 1977 [@Calderon_1977] and Marshall [@Marshall_1978?]: “a closed set of positive length lying on a rectifiable arc is unremovable in the class of bounded analytic functions” $\spadesuit$ this became the famous “Denjoy conjecture” $\spadesuit$ partial positive results on it where obtained by Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950] in the case where the supporting arc is a segment (for this case they credit Denjoy himself) and then they extend the result to an analytic curve via conformal mapping $\spadesuit$ Ivanov treated the case of curves slightly smoother than $C^1$ $\spadesuit$ Davie 1972 [@Davie_1972] proved that it sufficed to assume the curve $C^1$ (i.e. the rectifiable case of Denjoy can be reduced to the $C^1$ case) $\spadesuit$ then, Caderón 1977 [@Calderon_1977] proved that the Cauchy integral operator, for $C^1$ curves, is bounded on $L^p$, $1<p<\infty$ $\spadesuit$ at this stage, Marshall [@Marshall_1978?] put the “touche finale” by writing a note explaining how Calderón implies Denjoy via classical results of Garabedian, Havinson and finishing the proof with Davie’s reduction to the $C^1$-case, validating thereby Denjoy’s assertion announced ca. 7 decades earlier $\spadesuit$ Calderón himself was first not aware of the relevance of his work to Denjoy’s (as one learns e.g. from Verdera 2004 [@Verdera_2004 p.29]), but in the acceptance speech for the Bôcher price (see Calderón 1979 [@Calderon_1979]), Calderón mentions the solution to the Denjoy conjecture as one of the most significative application of his article (see also Calderón 1978 [@Calderon_1978-ICM], ICM lecture)\]$\bigstar$

H. Denneberg, [*Konforme Abbildung einer Klasse unendlich-vielfach zusammenhängender schlichter Bereiche auf Kreisbereiche*]{}, Ber. Verhd. Sächs. Akad. Wiss. Leipzig 84 (1932), 331–352. 60, 78 \[$\spadesuit$ a contribution to KNP\]$\bigstar$

J. Dieudonné, [*Cours de géométrie algébrique*]{}, Presses universitaires de France, Paris, 1974. \[$\spadesuit$ as mixture of Bourbakist pesanteur mixed with the usual charming touch of the gifted “God-given” writer\]

J. Diller, [*Green’s functions, electric networks, and the geometry of hyperbolic Riemann surfaces*]{}, Illinois J. Math. 45 (2001), 453–485. \[$\spadesuit$ p.456, Swiss cheese description of Hardt-Sullivan’s work (1989 [@Hardt-Sullivan_1989]) on the Green’s function for a bordered Riemann surface given as a branched cover of the unit-disc $\spadesuit$ so this Hardt-Sullivan work may possibly interact with the Ahlfors function\]

P.G. Lejeune Dirichlet, [*??*]{}, Crelle’s Journal 1829. \[$\spadesuit$ first convergence proof of the Fourier series toward the given (continuous) function\] 

P.G. Le Jeune Dirichlet, [*Vorlesungen über die im umgekehrten Verhältniss des Quadrats der Entfernung wirkenden Kräfte*]{}, herausgegeben von Dr. F. Grube, Leipzig, 1876. \[$\spadesuit$ first version of DP available in print under (essentially) Dirichlet’s own pen, as Grube reproduced a Dirichlet Göttingen lecture (ca. 1856) $\spadesuit$ alas Dirichlet’s formulation was a bit ill-posed, as came very apparent through the example of Prym 1871 [@Prym_1871] (compare also Elstrodt-Ullrich 1999 [@Elstrodt-Ullrich_1999])\]$\bigstar$ 

S. Donaldson, [*Yang-Mills invariants of smooth four-manifolds*]{}, in: Geometry of low-dimensional manifolds, vol. 1, Cambridge Univ. Press, 1990, 5–40. \[$\spadesuit$ contains some trick to transmute to holomorphic the antiholomorphic involution induce by Galois on a real quartic surface (or more general K3 surfaces=Kummer-Kähler-Kodaira in Weil’s designation)\] 

S. Donaldson, [*Complex curves and surgery*]{}, Publ. Math. Inst. Hautes Ét. Sci. 1989, 91–97. \[$\spadesuit$ a discussion of Thom’s conjecture, p.91: “An entrancing problem in Geometric Topology, usually ascribed to R. Thom, asks whether $C$ minimises the genus among all $C^\infty$ representatives for the homology class.” p.92 Lee Rudolph (1984 [@Rudolph_1984]) counterexamples for “topologically locally flat” surfaces are mentioned $\spadesuit$ Kirby’s 1970 list [@Kirby_1970--95] of problems already contains (conjecturally) an extension of Thom to any complex projective surface\] 

J. Douglas, [*Solution of the problem of Plateau*]{}, Trans. Amer. Math. Soc. 33 (1931), 263–321. \[$\spadesuit$ a new proof of RMT is given via Plateau, including the Osgood-Carathéodory refinement about the boundary behaviour of the Riemann map $\spadesuit$ reduces the mapping problem to that of minimizing a functional (named after Douglas by now)\] 

J. Douglas, [*Some new results in the problem of Plateau*]{}, Amer. J. Math. 61 (1939), 590–608. 

J. Douglas, [*Minimal surfaces of higher topological structure*]{}, Ann. of Math. (2) 40 (1939), 205–298. 78 

J. Douglas, [*The most general form of the problem of Plateau*]{}, Amer. J. Math. 61 (1939), 590–608. 60 

R.G. Douglas, W. Rudin, [*Approximation by inner functions*]{}, Pacific J. Math. 31 (1969), 313–320. \[p.314 the Ahlfors function (in the very trivial case of an annulus $D=\{z \colon r_1<\vert z\vert < r_2\}$) is involved in the proof of the following theorem: the set of all quotients of inner functions is norm-dense in the set of unimodular functions\] 

B. Drinovec Drnovšek, [*Proper discs in Stein manifolds avoiding complete pluripolar sets*]{}, Math. Res. Lett. 11 (2004), 575–581. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited\] 

B. Drinovec Drnovšek, F. Forstnerič, [*Holomorphic curves in complex spaces*]{}, Duke Math. J. 139 (2007), 203–252. \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited in the bibliography, but apparently not within the text $\spadesuit$ yet the connection with Ahlfors is evident in view of the following extract of the review of the paper: “Since the early 1990’s a series of papers, motivated mainly by works J. Globevnik and of Forstnerič, has been devoted to constructing holomorphic discs $f\colon \Delta \to M$ in complex manifolds that are proper. The article under review offers a culmination of the subject, lowering as much as possible the convexity assumptions, working on complex spaces with singularities, and “properizing” not only discs, but general open Riemann surfaces whose boundary consists of a finite number of closed Jordan curve. \[…\]”\] 

V.N. Dubinin, S.I. Kalmykov, [*A majoration principle for meromorphic functions*]{}, Sbornik Math. 198 (2007), 1737–1745. \[p.1740 a majoration principle is specialized to the Ahlfors function upon using the formula expressing the logarithm of the modulus of the Ahlfors function as a superposition of Green’s functions with poles at the zeros of the Ahlfors function\] 

B.A. Dubrovin, I.M. Krichever, S.P. Novikov, [*The Schrödinger equation in a periodic field and Riemann surfaces*]{}, Dokl. Akad. Nauk SSSR 229 (1976), 15–18; English transl., ?? ? (197?), ?–?. \[$\spadesuit$ cited in Dubrovin 1983/85 where some connection with Klein’s orthosymmetry/separating type I is given\] 

B.A. Dubrovin, S.M. Natanzon, [*Real two-zone solutions of the sine-Gordon equation*]{}, Funkt. Anal. Prilozhen. 16 (1982),27–43; English transl., Funct. Anal. Appl. 16 (1982), 21–33. \[cited in Vinnikov 1993 [@Vinnikov_1993] who claims a simplified proof\] 

B.A. Dubrovin, [*Matrix finite-zone operators*]{}, (Itogi Nauki i Tekhniki) 23 (1983), 33–78; English transl., Contemporary problems in math. ? (1985), 20–50. \[$\spadesuit$ cited in Vinnikov 1993 [@Vinnikov_1993 p.478] for a proof of the rigid-isotopy of any two smooth plane real curves having a deep nest (a result first established by Nuij 1968 [@Nuij_1968]) $\spadesuit$ p.48: “We shall now list themost important properties of real Riemann surfaces. A Riemann surface is called real if on it there is given an antiholomorphic involution \[…\] There are two possible cases: I) the union of real ovals decomposes $\Gamma$ into two components \[…\]; or II) the union of ovals does not decompose $\Gamma$. Surfaces of type I we call surfaces of separating type, while those of type II we call surfaces of nonseparating type.” $\spadesuit$ p.43: “The Riemann surface $\Gamma$ with antiinvolution $\tau$ belongs to \[the\] separating type. (the proof given on p.43–44 seems to use a sort of total reality?) $\spadesuit$ p.41: “separating type” $\spadesuit$ p.42: Fay 1973 [@Fay_1973] is cited, yet not clear to Gabard \[12.01.13\] if Dubrovin’s paper has any dep connection with Ahlfors 1950 [@Ahlfors_1950] $\spadesuit$ p.43: Rohlin 1978 [@Rohlin_1978] is cited for the simple fact that a plane curve with a deep nest is separating\] 

B.A. Dubrovin, [*Theory of operators and real algebraic geometry*]{}, in: Global Analysis and Math. Physics, III, Voronezh State Univ., 1987; English transl., Lecture Notes in Math. 1334, 1988, 42–59. \[$\spadesuit$\]$\bigstar$ 

B.A. Dubrovin, S.M. Natanzon, [*Real theta-function solutions of the Kadomtsev-Petviashvili equation*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 267–286; English transl., Math. USSR Izv. 32 (1989), 269–288. \[$\spadesuit$\]$\bigstar$ 

B.A. Dubrovin, S.P. Novikov, A.T. Fomenko, [*Modern Geometry: Methods and Applications*]{}, 3rd ed., Nauka, Moscow, 1986; English transl., Part I, II, III, Springer, 1984, 1985, 1990. \[$\spadesuit$\]$\bigstar$ 

C.J. Earle, A. Marden, [*On Poincaré series with application to $H^{p}$ spaces on bordered Riemann surfaces*]{}, Illinois J. Math. 13 (1969), 202–219. \[$\spadesuit$ cited in Forelli 1979 [@Forelli_1979], where the automorphic uniformization is employed to construct the Poisson kernel of a finite bordered Riemann surface, which in turn is involved in a new derivation of Ahlfors circle maps of controlled degree $\le
r+2p$\] 

C.J. Earle, A. Schatz, [*Teichmüller theory for surfaces with boundary*]{}, J. Differential Geom. 4 (1970), 169–185. \[$\spadesuit$\] 

C.J. Earle, [*On the moduli of closed Riemann surfaces with symmetries*]{}, In: Advances in the Theory of Riemann Surfaces, Annals of Math. Studies 66, Princeton Univ. Press and Univ. of Tokyo Press, Princeton, N.J., 1971, 119–130. \[$\spadesuit$ modernized account of Klein 1882 [@Klein_1882] and Teichmüller 1939 [@Teichmueller_1939], cf. also related works by Natanzon and Seppälla 1978 [@Seppala_1978-Teich-spaces-of-Klein-surfaces] $\spadesuit$ [*Warning*]{}. According to Natanzon 1999 [@Natanzon_1999-Moduli-real-alg-surf.superanal-differ-spinors p.1101], Earle’s description of the topological structure of the components of the moduli space of real algebraic curves (as being each diffeomorphic to ${\Bbb R}^{3g-3}/ {\rm Mod}_{g,r,\varepsilon}$ for a suitable discrete modular group) while being correct, its proof (using the theory of quasiconformal maps) is not, since it relies on a Kravetz (1959 [@Kravetz_1959]) theorem “which turned out latter to be wrong”. Still according to Natanzon () “A correct proof based on the theory of quasiconformal maps was obtained in Seppälä 1978 [@Seppala_1978-Teich-spaces-of-Klein-surfaces].”\] 

T. Ekedahl, S. Lando, M. Shapiro, A. Vainshtein, [*Hurwitz numbers and intersections on moduli spaces*]{}, Invent. Math. 146 (2001), 297–327. \[$\spadesuit$ a new derivation of Hurwitz’s count of the number of branched coverings of the sphere having prescribed ramification\] 

A. El Soufi, S. Ilias, [*Le volume conforme et ses applications d’après Li et Yau*]{}, Sém. Théo. Spectrale Géom. (1983/84), 15pp. \[$\spadesuit$ exploits the optimum $[\frac{g+3}{2}]$ gonality (Riemann-Brill-Noether-Meis) in the realm of spectral theory\] 

J. Elstrodt, P. Ullrich, [*A real sheet of complex Riemannian function theory: a recently discovered sketch in Riemann’s own hand*]{}, Historia Math. 26 (1999), 268–288. 

, edited by Kiyosi Itô, Vol. II, Second Edition, English transl. (1987) of the third (Japanese) edition (1968) \[sic!\]. 50 \[$\spadesuit$ on p.1367 the result of Ahlfors 1950 [@Ahlfors_1950] is quoted as follows (with exact source omitted but given on the next page p.1368): “L. Ahlfors proved that a Riemann surface of genus $g$ bounded by $m$ contours can be mapped conformally to an at most $(2g+m)$-sheeted unbounded covering surface of the unit disk.”\] 

F. Enriques, [*Sul gruppo di monodromia delle funzioni algebriche, appartenti ad una data superficie di Riemann*]{}, Rom. Acc. L. Rend. 13 (1904), 382–384. 60 \[$\spadesuit$ just quoted to ponder a bit the severe diagnostic to be found in the next entry (i.e. Ahlfors was of course by no mean ignorant about the Italian algebro-geometric community)\] 

F. Enriques, O. Chisini, [*Lezioni sulla Teoria Geometrica delle Equazioni e delle Funzioni Algebriche*]{}, Zanichelli, Bologna, 1915–1918–1924. \[$\spadesuit$ appears to the writer as a clear-cut forerunner of both Bieberbach 1925 [@Bieberbach_1915] and Wirtinger 1942 [@Wirtinger_1942], as argued in Gabard 2006 [@Gabard_2006 p.949] (cf. also Huisman 2001 [@Huisman_2001] for a similar proof) $\spadesuit$ actually Enriques-Chisini give another derivation of Harnack’s bound (on the number of components of a real curve) via Riemann-Roch, but their argument supplies an immediate proof of the so-called [*Bieberbach-Grunsky theorem*]{} (cf. Bieberbach 1925 [@Bieberbach_1925], Grunsky 1937 [@Grunsky_1937] and for instance A. Mori 1951 [@Mori_1951]), that is, the planar version of the Ahlfors map $\spadesuit$ as far as I know this little anticipation of Enriques-Chisini over Bieberbach-Grunsky has never been noticed (or admitted?) by the function-theory community (say Bieberbach, Grunsky, Wirtinger, Ahlfors, A. Mori, Tsuji, …) showing an obvious instance of lack of communication between the analytic and geometric communities\] 

F. Enriques, [*Sulle curve canoniche di genere $p$ dello spazio a $p-1$ dimensioni*]{}, Rend. Accad. Sci. Ist. Bologna 23 (1919), 80–82. \[$\spadesuit$ so-called canonical curve termed “Normalkurve der $\varphi$” in Klein 1892 [@Klein_1892_Realitaet] and also studied by M. Noether\] 

B. Epstein, [*Some inequalities relating to conformal mapping upon canonical slit-domains*]{}, Bull. Amer. Math. Soc. ?? (1947), ??–??. \[$\spadesuit$\] 

B. Epstein, [*The kernel function and conformal invariants*]{}, J. Math. Mech. 7 (1958). \[$\spadesuit$ quoted in Gustafsson 2008\]  $\bigstar$$\bigstar$$\bigstar$

L. Euler, [*Instit. Calc. Integr. Petrop.*]{}, 1768–70, 2, 1169. \[$\spadesuit$ cited in Petrowsky 1938 [@Petrowsky_1938] as one of the tool used in the proof of the Petrovskii’s inequalities via the so-called Euler-Jacobi interpolation formula (Kronecker also involved) concerning solutions of systems of algebraic equations and yielding a highbrow extensions to curves of higher orders of the results of Hilbert-Rohn for sextics\] 

G. Faber, [*Neuer Beweis eines Koebe-Bieberbachschen Satzes über konforme Abbildung*]{}, Sitz.-Ber. math.-phys. Kl. Bayer. Akad. Wiss. (1916), 39–42. \[$\spadesuit$ related to the so-called area principle of Gronwall 1914/15 [@Gronwall_1914/15], Bieberbach 1916 [@Bieberbach_1916]\]  $\bigstar$$\bigstar$$\bigstar$

G. Faber, [*Über den Hauptsatz aus der Theorie der konformen Abbildung*]{}, Sitz.-Ber. math.-phys. Kl. Bayer. Akad. Wiss. (1922), 91–100. 78 \[$\spadesuit$ must be another proof of RMT $\spadesuit$ regarded in Schiffer 1950 [@Schiffer_1950-Appendix-Courant p.313] as one of the originator of the method of [*extremal length*]{} (jointly with Grötzsch (1928) and Rengel 1932/33 [@Rengel_1932-33]), cf. also the introductory remarks of Bieberbach 1957 [@Bieberbach_1957] $\spadesuit$ maybe another origin is Courant 1914 [@Courant_1914] (at least for the length-area principle), cf. e.g. Gaier 1978 [@Gaier_1978-JDMV]\] $\bigstar$$\bigstar$$\bigstar$ 

G. Faltings, [*Endlichkeitssätze für abel’sche Varietäten über Zahlkörpern*]{}, Invent. Math. 73 (1983), 349–366. \[$\spadesuit$ proof of the so-called Mordell conjecture that a curve defined over ${\Bbb Q}$ (or a more general number field, i.e. a finite extension of ${\Bbb Q}$) has only finitely many rational points provided the genus $g$ of the underlying complex curve has genus $g\ge 2$ $\spadesuit$ it would we interesting to detect if the finer Kleinian invariants allied to real curves also have some similar arithmetical repercussion (to my knowledge nothing is known along this way, even at the conjectural level)\] 

G. Faltings, [*Real projective structures on Riemann surfaces*]{}, Compos. Math. 48 (1983), 223–269. \[$\spadesuit$ p.231: “Any Riemann surface may be considered as an algebraic curve defined over ${\Bbb C}$. Sometimes this algebraic variety is already definable over the real numbers. This happens precisely if there exist an antiholomorphic involution on the surface, and these involutions correspond bijectively to the different real models of the curve.—The basic example here is the double of a Riemann surface with boundary, which has a canonical real structure. The real points of this real curve are the fixed-points of the involution, hence the points in the boundary of our original Riemann surface.—Not every real curve is of this form, since for example there exist curves $X$ over ${\Bbb R}$ for which $X({\Bbb
C})-X({\Bbb R})$ is connected. ($X({\Bbb C})$, $X({\Bbb R})$ denote the ${\Bbb C}$-respectively ${\Bbb R}$-valued points of a real algebraic curve $X$.) We shall see that all counterexample are of this form.” (Okay but all this is of course trivial since Felix Klein.) $\spadesuit$ \[20.12.12\] an evident “Jugendtraum” of mine (and probably of many others, Gross, Faltings, etc.?) since ca. 1999/2000 is whether the finer topological invariants of Klein of a real curve (as opposed to the sole Riemannian genus fixing the topology of the underlying complex curve) have any arithmetical repercussion, à la Mordell-Faltings, namely finiteness of the rational points $C({\Bbb Q})$ whenever the genus $g\ge 2$. To my knowledge not a single result of the sort is known and it is quite hard to speculate about any such topologico-arithmetical connection. Crudely speaking the implication could be of the format if the curve is dividing then the cardinality of $C({\Bbb Q})$ is even, but this is surely wrong\] 

H.M. Farkas, I. Kra, [*Riemann surfaces*]{}, Second Edition, Grad. Texts in Math. 71, Springer, 1992. (1st edition published in 1980) 

P. Fatou, [*Séries trigonométriques et séries de Taylor*]{}, Acta Math. 30 (1906), 335–400. \[$\spadesuit$ influenced by Lebesgue, and will in turn influence F. Riesz (so called Fischer-Riesz theorem)\] 

J. Fay, [*Theta functions on Riemann surfaces*]{}, Lecture Notes in Math. 352, Springer, 1973. 47, 50 \[$\spadesuit$ cite Ahlfors 1950 [@Ahlfors_1950] and write down explicit formulas for the Ahlfors function (at least in the planar case) in terms of theta-functions $\spadesuit$ gives perhaps another proof of Ahlfors 1950 (cf. Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000]) but this hope is probably not borne out (Fay probably only recovers the Ahlfors circle map in the planar case) $\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited thrice in this booklet $\spadesuit$ on p.108 (just for the double) $\spadesuit$ on p.116: “It has been proved in \[3, p.126\](=Ahlfors 1950 [@Ahlfors_1950]) that there are always unitary functions with exactly $g+1$ zeroes [*all*]{} in $R$; and when $R$ is a planar domain, it is shown in Prop.6.16 that $S_{0,\dots,0}\cap \Sigma_a$ is empty for $a\in R$ and that the unitary functions holomorphic on $R$ with $g+1$ zeroes are parametrized by the torus $S_0$.” \[Added by Gabard \[10.09.12\]: of course one can wonder how much of this is anticipated in Bieberbach 1925 [@Bieberbach_1925]\] $\spadesuit$ p.129: “Using this result, a solution can be given to an extremal problem for bounded analytic functions as formulated in \[3, p.123\](=Ahlfors 1950):” where the Ahlfors function is expressed in terms of the theta function and the prime form, yet it should be noted that unfortunately at some stage Fay’s exposition is confined to the case of planar domains $\spadesuit$ somewhat earlier in the text (in a portion not yet confined to the planar case) we read on p.114: “The spaces $S_{\mu}$ parametrize the generic unitary functions on $C$ with the minimal $(g+1)$ number of zeroes:”, maybe this claim of minimality is erroneous as it could be incompatible with Gabard 2006 [@Gabard_2006], and even if the latter is incorrect there is basic experimental evidence violating this minimality claim on the bound $g+1$, compare our remarks after Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000]\] 

S.I. Fedorov, [*Harmonic analysis in a multiply connected domain, I*]{}, Math. USSR Sb. 70 (1991), 263–296. \[$\spadesuit$ credited by Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000 p.240] (and also Yakubovich 2006 [@Yakubovich_2006]) for another existence-proof of the Ahlfors map (at least for planar domains), cf. p.271–275 $\spadesuit$ on p.272 it is remarked that one cannot prescribe arbitrarily the $n$ zeroes of a circle-map on an $n$-connected domain of minimum degree $n$ as follows: “Unfortunately we cannot prescribe $n$ points on $\Omega_{+}$ arbitrarily in such a way that their union will be the set of zeros of an $n$-sheeted inner function $\theta$ of the form (3), since the zeros of an $n$-sheeted function $\theta$ must satisfy the rather opaque condition $\sum_{k=1}^n
\omega_s(z_k), \; s=1, \dots, n-1,$ where $\omega_s$ is the harmonic measure of the boundary component $\Gamma_s$.” $\spadesuit$ \[26.09.12\] it seems to the writer (Gabard) that this condition already occurs (at least) in A. Mori 1951 [@Mori_1951] $\clubsuit$ it would be interesting to analyze carefully Fedorov’s argument (or Mori’s) to see if it can be extended to the positive genus case (this is perhaps already done in Mitzumoto 1960 [@Mizumoto_1960]) $\spadesuit$ p.272 desideratum of a constructive procedure for building all $n$-sheeted inner functions on an $n$-connected domain, which is answered on p.274 via “Theorem 1. Let $z_1, \dots, z_n$ be arbitrary points with $z_k\in \Gamma_k$, $k=1,\dots, n$. Then there exist positive numbers $\lambda_1,\dots, \lambda_n$ such that the function $w=\int_{z_\Gamma}^z \sum_{j=1}^n \lambda_j \nu_{z_j}$, $z_\Gamma\in \Gamma$, $z_\Gamma\neq z_j$, $j=1,\dots,n$, is a single-valued $n$-sheeted function on $\hat \Omega$, real-valued on $\Gamma$, with positive imaginary part on $\Omega_+$. The function $\theta=\frac{w-i}{w+i}$ is an $n$-sheeted inner function.” $\spadesuit$ of course in substance (or essence) this is nothing but what Japaneses calls the Bieberbach-Grunsky theorem (cf. Mori 1951 [@Mori_1951] or Tsuji 1956 [@Tsuji_1956])\] 

J.L. Fernandez, [*On the existence of Green’s function in Riemannian manifolds*]{}, Proc. Amer. Math. Soc. 96 (1986), 284–286. \[$\spadesuit$\] 

$\bigstar$ Fiedler, student of Rohlin, ca. 1978.

T. Fiedler, [*Eine Beschränkung für die Lage von reellen ebenen algebraischen Kurven*]{}, Beiträge Algebra Geom. 11 (1981), 7–19. \[$\spadesuit$ the eminent DDR student of Rohlin, who seems to have been the first to notice the simple fact that orientation-preserving smoothings conserve the dividing character of curves, compare also Rohlin 1978 [@Rohlin_1978] where the contribution of Fiedler is already mentioned\] 

T. Fiedler, [*Geraden Büschel und die Topologie der reellen algebraischen Kurven*]{}, Dissertation, 1981. \[$\spadesuit$ (in part) reproduced in the next entry Fiedler 1982/83 [@Fiedler_1982/83-Pencil]\] 

T. Fiedler, [*Pencils of lines and the topology of real algebraic curves*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 853–863; English transl., Math. USSR Izv. 21 (1983), 161–170. \[$\spadesuit$ p.161 (Abstract): “\[…\] a new invariant of the strict isotopy type of the curve is given, which in particular distinguishes some seventh degree $M$-curves with the same complex scheme.”\] 

T. Fiedler, [*New congruences in the topology of real plane algebraic curves*]{}, Dokl. Akad. Nauk SSSR 270 (1983), 56–58; English transl., Sov. Math. Dokl. 27 (1983), 566–568. \[$\spadesuit$\] 

T. Fiedler, [*New congruences in the topology of singular real plane algebraic curves*]{}, Dokl. Akad. Nauk SSSR 286 (1986), 1075–1079; English transl., Sov. Math. Dokl. 33 (1986), 262–266. \[$\spadesuit$\] 

T. Fiedler, [*Additional inequalities in the topology of real plane algebraic curves*]{}, Izv. Akad. Nauk SSSR Ser. Mat. ?? (1986), ?–?; English transl., Math. USSR Izv. 27 (1986), 183–191. \[$\spadesuit$\] 

T. Fiedler, [*Real points on complex plane curves*]{}, Math. Ann. 284 (1989), 267–284. \[$\spadesuit$\] 

S. Fiedler-Le Touzé, [*Orientations complexes des courbes algébriques réelles*]{}, Thèse doctorale 2000. \[$\spadesuit$ cited in the entry Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]\] 

S. Fiedler-Le Touzé, [*Cubics as tools to study the topology of $M$-curves of degree $9$ in ${\Bbb R}P^2$*]{}, J. London Math. Soc. (2) 66 (2002), 86–100. \[$\spadesuit$ p. dividing curves\] 

S. Fiedler-Le Touzé, [*Pencils of cubics with eight base points lying in convex position in $\RR P^2$*]{}, arXiv, v2, 53 pages, Sept. 2012. \[$\spadesuit$ contains foundations required in the next entry Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]\] 

S. Fiedler-Le Touzé, [*Totally real pencils of cubics with respect to sextics*]{}, a marvellous preprint received the 1 March 2013 (v.1), and a second version (v.2) the 3 March 2013 (where the basepoints are assigned on the ovals instead of in their insides like in v.1). Final version on arXiv 18–19 March 2013. \[$\spadesuit$ a seminal work containing proofs of Rohlin’s 1978 (unproven) total reality assertion for certain $(M-2)$-sextics totally swept out by suitable pencil of cubics $\spadesuit$ this is the first non-trivial (i.e. not involving pencil of lines or conics) extrinsic manifestation of Ahlfors theorem $\spadesuit$ another but much more modest phenomenon of total reality occurs for $M$-curves (as slowly discovered by Gabard, cf. Theorem \[total-reality-of-plane-M-curves:thm\]) but this is merely at the level of the Bieberbach-Grunsky theorem, i.e. the genus zero case of Ahlfors theorem $\spadesuit$ \[20.03.13\] as brilliantly explained in the paper in question (p.3), Le Touzé proves actually a slightly weaker statement that Rohlin’s original claim, namely the dividing character is not deduced a priori from total reality (as Rohlin claimed being able to do), but rather the dividing character is taken as granted via the Rohlin-Kharlamov-Marin congruence while total reality of the pencil of cubics is built upon this preliminary knowledge. Hence it could still be of some interest to reconstruct a proof [*purely a priori*]{} assuming of course that there is a such. This looks quite likely, yet apparently quite elusive to implement.\] 

S.M. Finashin, [*The topology of the complement of a real algebraic curve in ${\Bbb C}P^2$*]{}, Zap. Nauch. Sem. LOMI 122 (1982), 137–145; English transl., J. Soviet Math. 26 (1984), 1684–1689. \[$\spadesuit$ briefly discussed in Viro 1986/86 [@Viro_1986/86-Progress]\] 

S.M. Finashin, [*Differential topology of quotients of complex surfaces by complex conjugation*]{}, Zap. Nauch. Sem 231 (1995), 215–221; English transl., J. Math. Sciences 91 (1998), 3472–3475. \[$\spadesuit$ p.3472: “A well-known example is $X={\Bbb C} P^2$, for which \[the quotient by conj is\] $Y \cong
S^4$. According to V.A. Rohlin, the last equality was quite widely known in the mathematical folklore, in any case to those who reflected on the four-dimensional Poincaré conjecture, for example, to Pontryagin. However, the author knows no mention of this account before Arnold’s paper \[2\](=1971 [@Arnold_1971/72]) and no published proofs before the papers of Kuiper \[9\](=1974) and Massey \[10\](=1973).” $\spadesuit$ according to some subsequent publication by Arnold, the result goes back to Maxwell!\] 

S.M. Finashin, [*Rokhlin conjecture and quotients of complex surfaces by complex conjugation*]{}, J. reine angew. Math. 481 (1996), 55–71. \[$\spadesuit$ p.68: some remarks on sextics, e.g. Fig.10 gives the $(M-1)$-scheme $10$ via a perturbation of a line arrangement $\spadesuit$ p.68: “It is well known and not difficult to see directly from the Hilbert and Gudkov constructions of nonsingular real sextics (cf. \[V\](=Viro 1986/86 [@Viro_1986/86-Progress])), that the ones with schemes $\langle \alpha \sqcup 1 \langle \beta \rangle$ can be deformed to the both schemes \[having resp. one less outer oval or inner oval\] by passing through a cross-like real node which connects the ambient oval with an exterior oval resp. with an interior one. The only exceptio is the scheme $\langle 9 \sqcup 1 \langle 1 \rangle$ \[ of Harnack\] which can be reduced to $\langle 10 \rangle$ only by contracting the inner oval.”\] 

S.M. Finashin, [*On the topology of real plane algebraic curves with nondegenerate quadratic singularities*]{}, Algebra i Analiz 8 (1996), 186–204; English transl., St. Petersburg Math. J. 8 (1997), 1039–1051. \[$\spadesuit$\] 

A.E. Fischer, A.J. Tromba, [*On a purely “Riemannian” proof of the structure and dimension of the unramified moduli space of a compact Riemann surface*]{}, Math. Ann. (1983). \[$\spadesuit$ ... Ahlfors \[2\]. The space of extremal quasi-conformal maps between two Riemann surfaces (the so-called Teichmüller space) is in fact a ramified covering of the space of conformal classes of Riemann surfaces of prescribed genus (the real moduli space). In \[4\] Ahlfors shows that ...\] 

S.D. Fisher, [*Exposed points in spaces of bounded analytic functions*]{}, Duke Math. J. 36 (1969), 479–484. 50 \[$\spadesuit$ cite Ahlfors 1950 [@Ahlfors_1950] and the following result is obtained: the exposed points of the algebras $A(\overline R)$ (resp. $H^{\infty}(R)$) are uniformly dense in the unit sphere of the respective space\]$\bigstar$$\bigstar$$\bigstar$ 

S.D. Fisher, [*Another theorem on convex combination of unimodular functions*]{}, Bull. Amer. Math. Soc. ?? (1969), 1037–1039. \[$\spadesuit$ finite Riemann surfaces, inner functions and it is proved that the closed convex-hull of the inner functions is the unit ball (for the sup norm) of the algebra $A(R)$ of analytic functions continuous up to the border $\spadesuit$ this is proved via an interpolation lemma due to Heins 1950 [@Heins_1950], which is closely allied to the Ahlfors function (plus maybe some Garabedian) $\spadesuit$ this is stated as: “Lemma 1: Let $z_1,\dots,z_N$ be distinct points of $R$ (=a finite Riemann surface) and let $h$ be an analytic function on $R$ bounded by $1$. Then there is an inner function $f$ (i.e. of modulus one on the boundary $\partial R$) in $A(R)$ with $f(z_j)=h(z_j)$, $j=1,\dots,
N$.” $\spadesuit$ one can take $h\equiv 1$ then $f$ looks strange for it maps inner points to the boundary point 1, yet still $f=1$ works $\spadesuit$ the question is of course whether this reproves Ahlfors 1950, but this looks unlikely especially as no control is supplied on the degree, but see Heins 1950 [@Heins_1950], which suitably modified should recover Ahlfors result by controlling appropriately the bound involved\] 

S.D. Fisher, [*On Schwarz’s lemma and inner functions*]{}, Trans. Amer. Math. Soc. 138 (1969), 229–240. 47, 78 \[$\spadesuit$ after Havinson 1961/64 [@Havinson_1961/64] and Carleson 1967 [@Carleson_1967-book], study the Ahlfors map for domains of infinite connectivity $\spadesuit$ subsequent ramifications in Röding 1977 [@Roeding_1977_Ahlfors], Minda 1981 [@Minda_1981-image-Ahlfors-fct], Yamada 1983–92 [@Yamada_1983-rmk-image-Ahlfors-fct; @Yamada_1992-Ahlfors-fct-on-Denjoy]\] 

S.D. Fisher, [*The moduli of extremal functions*]{}, Michigan Math. J. 19 (1972), 179–183. 47 \[$\spadesuit$ the Ahlfors function of a domain (supporting nonconstant bounded analytic functions) is shown to be of unit modulus on the Šilov boundary of $H^{\infty}$\] 

S.D. Fisher, [*Non-linear extremal problems in $H^{\infty}$*]{}, Indiana Univ. Math. J. 22 (1973), 1183–1190. 50 \[$\spadesuit$ p.1183/7 speaks of the “Ahlfors-Royden extremal problem” $\spadesuit$ the author explains that in Ahlfors extremal problem the class of competing functions is convex, explaining uniqueness of the soution and studies a variant of the problem with a side-condition amounting to require “no other zeros” which leads to a non-convex problem lacking uniqueness $\spadesuit$ p.1187/88, grasp of the geometric quintessence of Ahlfors’ argument: “By a theorem of Ahlfors \[A1; §4.2\] there is a set of $r+1$ points $p_j$ in $\Gamma$ such that if $v_i$ is the period vector of a unit mass at $p_j$, then $v_0,\dots, v_r$ form the vertices of a simplex in ${\Bbb R}^{r}$ which contains the origin as an interior point.”\] 

S.D. Fisher, [*Function theory on planar domains*]{}. A second course in complex analysis. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1983. 

S.D. Fisher, D. Khavinson, [*Extreme Pick-Nevanlinna interpolants*]{}, Canad. J. Math. 51 (1999), 977–995. \[$\spadesuit$ Ahlfors function (in the domain case only), its connection with Blaschke products and the Green’s functions, Pick bodies (jargon of Cole, Lewis, Wermer) and interpolation\] 

H. Florack, [*Reguläre und meromorphe Funktionen auf nicht geschlossenen Riemannschen Flächen*]{}, Schr. Math. Inst. Univ. Münster no. 1 (1948), 34pp. 60 \[$\spadesuit$ cited also in Royden 1962 [@Royden_1962] (yet not within the text?) and briefly summarized in a ICM talk ca. 1954 of Behnke\]$\bigstar$$\bigstar$$\bigstar$ 

F. Forelli, [*Bounded holomorphic functions and projections*]{}, Illinois J. Math. 10 (1966), 367–380. \[$\spadesuit$ the universal covering method is employed to derive another proof of the corona theorem for interiors of compact bordered Riemann surfaces, relativizing thereby the ubiquitousness of the Ahlfors function given in Alling 1964 [@Alling_1964] $\spadesuit$ Forelli’s proof uses the following tools: $\bullet$ (p.368) “measure and Hilbert space theory, and the harmonic analysis that goes with the Hilbert space $H^2$” $\bullet$ (p.373,374) existence of analytic differentials with prescribed periods on the Schottky double (via Pfluger 1957 [@Pfluger_1957]) $\bullet$ Beurling’s invariant subspace theorem (p.366), but this can be dispensed in the compact bordered case by appealing to a holomorphic function continuous up to the border “whose zeros are the critical point of the Green’s function with pole at $t(0)$” (p.377)\] 

F. Forelli, [*Extreme points in $H^1(R)$*]{}, Canad. J. Math. 19 (1967), 312–320. \[$\spadesuit$\]$\bigstar$$\bigstar$ 

F. Forelli, [*The extreme points of some classes of holomorphic functions*]{}, Duke Math. J. 46 (1979), 763–772. \[$\spadesuit$ study of the extreme points of the family of analytic functions with positive real part on a given finite Riemann surface normalized to take the value $1$ at a given point $\spadesuit$ the paper Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF] supplements the results of Forelli by precise characterizing results for the case where the genus of $S$ is positive $\clubsuit$ \[11.10.12\] in fact this Forelli paper is a jewel (that I was only able to read today=\[11.10.12\], shame on me!) $\clubsuit$ despite presenting itself too humbly as a modest appendix to Heins 1950 [@Heins_1950], its main result (Theorem 3.2, p.766) gives the chain of inclusions $N_q(W,
\zeta)\subset
\partial N(W, \zeta) \subset
\bigcup_{q}^{2p+q} N_k(W, \zeta)$, which readily implies a new proof of circle maps of degree $\le 2p+q$ (like Ahlfors 1950 [@Ahlfors_1950]). To understand this point, first recall Forelli’s notation: $\overline{W}$ is a compact bordered Riemann surface of genus $p$ with $q$ contours, $W$ is of course its interior; $N(W, \zeta)$ is the class of holomorphic functions $f$ on $W$ with positive[^126] real part (${\rm Re} f>0$) normalized by $f(\zeta)=1$ at some fixed $\zeta \in W$ (it is easily verified that $N(W, \zeta)$ is convex and compact in the compact-open topology) \[notion due to Arens/Fox, if I remember well???\]; the symbol $\partial$ used above refers [*not*]{} to the boundary but to the set of all extreme points of a convex body, i.e. those points of the body not expressible as convex (=barycentric) combination $t x + (1-t)y$ ($t\in
[0,1]$) of two (distinct) points $x,y$ of the body. This is also the smallest subset of the body permitting its complete reconstruction via the convex-hull operation; finally $N_k(W,
\zeta)$, for $k>0$ a positive integer, is the subclass of $N(W,\zeta)$ consisting of functions that cover the right half-plane $k$ times. $\clubsuit$ having explained notation, it is plain to deduce Ahlfors’ result. Indeed from the cited properties of convexity and compactness for $N(W,\zeta)$ one deduces (via Krein-Milman) existence of extreme points, i.e. $\partial N(W,\zeta)\neq \varnothing$ (this issue is not explicit in Forelli’s paper, but so evident that it is tacit, cf. e.g., Heins’ commentary in 1985 [@Heins_1985-Extreme-Pick-Nevanlinna p.758]: “My paper \[7\](=Heins 1950 [@Heins_1950]) showed the existence of minimal positive harmonic functions on Riemann surfaces using elementary standard normal family results without the intervention of the Krein-Milman theorem[^127] and gave applications to qualitative aspects of Pick-Nevanlinna interpolation on Riemann surfaces with finite topological characteristics and nonpointlike boundary components.” $\clubsuit$ Now Forelli’s second inclusion implies immediately the desideratum (existence of circle maps of degree $d$ such that $q\le d\le
2p+q$) $\spadesuit$ note of course that the first set of the string, that is $N_q(W, \zeta)$, can frequently be empty. Consider e.g. $\overline W$ be one-half of [*Klein’s Gürtelkurve*]{}[^128], that is any real plane smooth quartic, $C_4 \subset
{\Bbb P}^2$, with two nested ovals, then $q=2$ but quartics and more generally smooth plane curves of order $m$ are known to be $(m-1)$-gonal). For an even simpler example, consider any bordered surface $W$ with only one contour ($q=1$) and of positive genus $p>0$, then there cannot be a circle-map of degree $d=q=1$ for a such would be an isomorphism (by the evident branched covering features of analytic maps), violating the topological complexity prompted by $p>0$ $\spadesuit$ several questions arise naturally form Forelli’s work. A first one is the perpetual question about knowing if the method can recover the sharper bound $p+q$ ($\approx r+p$) of Gabard 2006 [@Gabard_2006]. (Here and below $\approx $ refers to notational conversion from Forelli’s notation to the one used in the present paper). Again it is our belief that the ultimate convex geometry reduction of the problem (already explicit in Ahlfors) could be slightly improved so as to do this (compare below for more details). Another problem is to understand the distribution of degrees corresponding to extreme points of Forelli’s convex body $\partial N(W, \zeta)$ (maybe call it the Carathéodory-Heins-Forelli body to reflect better the historical roots of the technique, brilliantly discussed in Heins 1985 [@Heins_1985-Extreme-Pick-Nevanlinna]). For instance is the least degree half-plane map (equivalently circle map) always an extreme point, as the nebulous principle of economy ($\approx$ least effort) could suggest? (Nature always tries to relax itself along an equilibrium position necessitating the minimum existential stress-tensor!??) Finally one would like to see the connection between Ahlfors extremals and the extreme points of Heins-Forelli. Of course there is a little tormenting routine to switch from the one to the others via a Möbius-Cayley transformation from the disc to the half-plane. Yet loosely it seems that Ahlfors functions are a subclass of the extreme points, for they former depend on less parameters. For instance as noted by Forelli in the special planar case $p=0$, the above chain of inclusions collapses to give the clear-cut equation $\partial N(W,
\zeta)=N_q(W,\zeta)$ characterizing the set of extreme points in, essentially, purely topological terms. Yet the Bieberbach-Grunsky theorem (1925 [@Bieberbach_1925], or A. Mori [@Mori_1951]) tell us that circle maps are in this case ($p=0$) fairly flexible insofar that we can preassign one point on each contour and find a circle map (of degree $q$) taking those points over the same boundary point[^129]. Hence for large values of $q$ such minimal degree circle maps depends on essentially $q$ real parameters, whereas for Ahlfors maps we can only specify the basepoint undergoing maximum distortion (hence just 2 real free parameters). $\clubsuit$ Finally some words about Forelli’s method of proof: It uses some “functional analysis” in the form of measure theory. Specifically Radon measures are mentioned, and a proposition permitting to express extreme points of a body $B$ specified by $n$ linear integral conditions as combination of $(n+1)$ extreme probability measures (cf. Prop.2.1 for the exact statement identified as dating back to Rosenbloom 1952 [@Rosenbloom_1952], \[but in geometric substance a similar lemma is already employed in Heins 1950 [@Heins_1950], as well as in Ahlfors 1950 [@Ahlfors_1950])\]. This is then specialized to the case where the space $X$ is the boundary of the bordered surface $\partial W$[^130], and the $n$ conditions amounts essentially to ask the vanishing of the periods along representatives of a homology basis of $\overline W$, consisting of $n:=2p+(r-1)$ cycles. The crucial potential theory is done via the Poisson integral inducing a bijective map $\#\colon P(\partial W) \to
h_+(W, \zeta)$ between probability measures on the boundary and positive harmonic functions normalized by taking $\zeta$ to $1$. It is defined by $\mu^{\#}(w)=\int_{\partial W} Q(w,y)
d\mu (y)$, where $Q(w,y)$ is the Poisson kernel of $W$ ($w\in
W, y\in
\partial W$). Now to find and describe (extreme) half-plane maps in $\partial N(W, \zeta)$, we are reduced via the above correspondence to a special set $B$ of measure verifying $n$ integral equations. On applying (Rosenbloom’s) proposition, the measure $\mu$ defined by $\mu^{\#}={\rm Re} f$ where $f\in
N(W, \zeta)$ is decomposed as a convex sum (i.e. with positive coefficient $t_k$) of Dirac measures $\mu=\sum_1^m t_k
\delta_k$ concentrated at some boundary points $y_k\in
\partial W$, where $m\le n+1$. It follows by calculation (Poisson+Dirac’s trick) that ${\rm Re}
f(z)=\sum_1^m t_k Q(w, y_k)$ (because integrating a function against the Dirac measure concentrated at some point just amounts evaluating the function at that point). Of course notice at this stage that the Poisson function $Q(w,y)$ is nothing else than the Green function with pole pushed to the boundary (so the object that we manipulated during our attempt to decipher Ahlfors’ proof). At this stage the proof is essentially finished. $\spadesuit$ as a matter of details Forelli further discuss the construction of the Poisson kernel taking inspiration from techniques of Earle-Marden 1969 [@Earle-Marden_1969-On-Poincare], using primarily the uniformization of Poincaré-Koebe. To sum up Forelli’s is able to reprove existence of circle maps but needs uniformization, admittedly in a simple finitistic context. Of course Ahlfors proof seems to avoid this dependance, which is anyway perhaps not so dramatic. $\spadesuit$ The latter issue should of course not detract us from the geometrical main aspect of the proof. First Forelli’s proof uses heavily a little yoga between measures and harmonic functions converting the one to the others via the Poisson integral. This technique involves so Poisson, then Stieltjes and finally the so-called Herglotz-Riesz (1911 [@Herglotz_1911-U-Potenzreihen]) (representation) theorem, a special incarnation of Fischer-Riesz (1907). Of course the yoga in question boils down to the Dirichlet principle when the measure has continuous density so that Herglotz-Riesz is just the Dirichlet problem enhanced by Lebesgue integration. Of course all this is beautiful, yet probably not fully intrinsic to the problematic of half-plane (or the allied circle) maps, which can probably be arrived upon via more classical integration theories (and in particular the classical Dirichlet problem, plus the allied potential functions, Green’s, Poisson’s or whatever you like to call them). I personally used the term Red’s function (somewhere in this text) as colorful contrast to evergreens tree, honoring George Green, but of course Poisson’s function might be historically more accurate. (After all, human beings descend from fishes rather than vegetables, and Green himself quotes of course Poisson, and Dirichlet was a Poisson student). $\clubsuit$ but now the key issue would be to penetrate even deeper in the geometry of Forelli’s proof. Again the hearth of the problem is the possibility of expressing a certain point as convex combination of [*at most*]{} $(n+1)$ points; in Forelli’s treatment cf. Prop.2.1, where however the “at most” proviso is not explicit but implicitly used later in the proof of Theorem 3.2. Like in our attempt to push Ahlfors proof down to recover Gabard’s bound, we believe that a better inspection of this convex geometry could corroborate the possibility of locating half-plane maps of lower degree. The situation we have in mind is the following (to which we were reduced by reading carefully Ahlfors 1950 [@Ahlfors_1950]): suppose we are given in ${\Bbb R}^(n\approx g)$ a collection of $q\approx r$ curves forming a balanced configuration (all $\approx$ signs just amounts to conversion from Forelli’s notation to the one used in the present text), in the sense that the convex hull encloses the origin, then it is of course possible to express the origin as convex sum of $\le
n+1\approx g+1$ point (recovering thereby Ahlfors’ result). However it must be also possible to be more economical by using a more special, lower-dimensional simplex, able to cover the origin with a smaller quantity of points. We hope that this is a problem of pure (Euclid/convex/Minkowski) geometry (perhaps involving some topological tricks like in the Borsuk-Ulam (ham-sandwich) theorem, which can concomitantly be proved via more simple center of masses considerations, cf. e.g. Fulton’s book on “topology”). Alas I can only try to convince the reader by looking at the (very special) case where $n\approx g=2$ coming (via $g=2p+(r-1)$) from the values $p=1, r=1$. Then we have one balanced circle in the plane ${\Bbb R}^2$. If we follow Ahlfors, we just have the plain remark that there is $g+1=r+2p=1+2\cdot 1=3$ points spanning a simplex covering the origin (which is trivial for dimensional reason), however it is evident that a more special and lucky constellation (Stonehenge alinement) of two points situated on the topological circle (Jordan curve) corresponding to the contour of the bordered surface, suffice to cover the origin with a $1$-simplex, giving existence of a circle map of degree $2$, like the $r+p$ bound predicted in Gabard 2006 [@Gabard_2006] $\spadesuit$ of course all we are saying does not detract the possibility that the extreme points studied by Forelli always contain an element landing in the highest possible degree $2p+q \approx 2p+r=g+1$\] 

J.E. Fornaess, N. Sibony, [*Some open problems in higher dimensional complex analysis and complex dynamics*]{}, Publ. Mat. 45 (2001), 529–547. \[$\spadesuit$ p.539: “Question 3.16. [*Can one embed all Stein Riemann surfaces as closed complex submanifolds of ${\Bbb C}^2$? (See [\[GS\]=Globevnik-Stensones 1995 [@Globevnik-Stensones_1995]]{})*]{}”\] 

O. Forster, [*Riemannsche Flächen*]{}, Springer, Berlin, 1977, 223 pp; English trans., available. \[$\spadesuit$ sheaf-theoretic approach\] 

F. Forstnerič, E.F. Wold, [*Bordered Riemann surfaces in ${\Bbb C}^2$*]{}, J. Math. Pures Appl. 91 (2009), 100–114. \[$\spadesuit$ reduction of the big problem of embedding open Riemann surfaces in the affine plane to that of embedding compact bordered surfaces, which looks more tractable due to its finitary nature, yet apparently completely out of reach\] 

F. Forstnerič, E.F. Wold, [*Embeddings of infinitely connected planar domains in ${\Bbb C}^2$*]{}, arXiv (2012). \[$\spadesuit$ “Abstract. We prove that every circled(=circular) domain (=Koebe’s Kreisbereich) in the Riemann sphere admits a proper holomorphic embedding (=PHE) in ${\Bbb C}^2$.” This is yet another spectacular advance on the proper embedding problem, giving insights on how to crack the general problem. (This may be restated as, p.1: “[*Is every open Riemann surface biholomorphic to a smoothly embedded, topologically closed complex curve in ${\Bbb C}^2$.*]{}”) Of course, when combined the He-Schramm 1993 [@He-Schramm_1993] uniformization result this gives the “Theorem 1.1.—[*Every domain in the Riemann sphere with at most countably many boundary components, none of which are points, admits a PHE in ${\Bbb C}^2$*]{}.” p.2: This result gives a wide extension of the similar statement in finite connectivity due to Globevnik-Stens[ø]{}nes 1995 [@Globevnik-Stensones_1995]. p.17: “There exists a Cantor set in ${\Bbb P}^1$ whose complement embeds PH into ${\Bbb C}^2$ (Orevkov 2008 [@Orevkov_2008]), but it is an open problem whether this holds for each Cantor set.”\] 

J. Fourier, [*Théorie analytique de la chaleur*]{}, 1822. \[$\spadesuit$ trigonometric series expansion of an arbitrary function (so-called Fourier series), despite some earlier appearance of them in works by by Clairaut and Euler $\spadesuit$ Fourier’s first work on the topic was presented to Paris Academy in 1807, yet rejected by Lagrange, Laplace and Legendre\] 

W.F. Fox, [*Harmonic functions with arbitrary singularity*]{}, Pacific J. Math. (1961), 153–164. \[$\spadesuit$ discusses and rederives old results of Schwarz 1870, Koebe while pointing out to the developments made by Sario $\spadesuit$ p.153 probably corroborates the intuition that the solvability of the Dirichlet principle on a compact bordered Riemann surface was first treated by Schwarz 1870\] 

A. Fraser, R. Schoen, [*The first Steklov eigenvalue, conformal geometry, and minimal surfaces*]{}, Adv. in Math. 226 (2011), 4011–4030. 50 \[$\clubsuit$ applies Ahlfors 1950 [@Ahlfors_1950] (and even Gabard 2006 [@Gabard_2006]) to spectral theory, especially first Steklov eigenvalue. For higher eigenvalues, cf. Girouard-Polterovich 2012 [@Girouard-Polterovich_2012], and for Dirichlet-Neumann eingenvalues, cf. Gabard 2011 [@Gabard_2011].\] 

I. Fredholm, [*Sur une classe d’équations fonctionnelles*]{}, Acta Math. 27 (1903), 365–390. \[$\spadesuit$ early influence of Abel (1823), then Neumann’s approach to the Dirichlet problem and Volterra (1896) where Neumann’s method was successfully applied to an integral equation\] 

R. Fricke, F. Klein, [*Vorlesungen über die Theorie der automorphen Functionen*]{}, Two volumes, Teubner, Leipzig, 1897, 1912, 634 pp., 668 pp.; Reprinted by Johnson Reprint Corp., New York and Teubner, Stuttgart, 1965. \[$\spadesuit$ contains versions of RST (=Rückkehrschnitttheorem), while the completion of the second volume seem to have received some helping hand from Paul Koebe $\spadesuit$ p.180ff. contains anothe account of the classification of Klein’s syymmetric Riemann surfaces\] 

G. Fubini, [*Il Principio di minmo i teoremi di esistenza per i problemi al contorno relativi alle equazioni alle derivate parziale di ordini pari*]{}, Rend. Circ. Mat. Palermo (1907). \[$\spadesuit$ cited in Zaremba 1910 [@Zaremba_1910] as another extension (beside Beppo Levi 1906 [@Beppo-Levi_1906] and Lebesgue 1907 [@Lebesgue_1907]) of Hilbert’s resurrection of the Dirichlet principle\] 

B. Fuchs, [*Sur la fonction minimale d’un domaine, I, II*]{}, Mat. Sbornik N.S. 16 (58) (1945); 18 (60) (1946). \[$\spadesuit$ quoted in Lehto 1949 [@Lehto_1949] and consider the problem of least momentum, i.e. minimizing $\int\!\!\int_B \vert f(z)\vert^2 d\omega$ under the side-condition $f(t)=1$ at some interior point\] $\bigstar$$\bigstar$$\bigstar$\[part I OK, part II still not found\] 

A. Gabard, [*Topologie des courbes algébriques réelles: une question de Felix Klein*]{}, L’Enseign. Math. 46 (2000), 139–161. \[$\spadesuit$ furnish a complete answer to a question raised by Klein as a footnote to his Coll. Papers, using an inequality due to Rohlin 1978 [@Rohlin_1978]. Previous (unpublished) work on the same question due to Kharlamov-Viro in the Leningrad seminar of topology supervised by V.A. Rohlin. Confirms incidentally a desideratum of Gross-Harris 1981 [@Gross-Harris_1981].\] 

A. Gabard, [*Sur la topologie et la géométrie des courbes algébriques réelles*]{}, Thèse, Genève, 2004. 50 \[$\clubsuit$ includes the improved bound $r+p$ upon the degree of a circle map of a membrane of genus $p$ with $r$ contours. Up to minor redactional change this is the same as the next entry Gabard 2006 [@Gabard_2006]\] 

A. Gabard, [*Sur la représentation conforme des surfaces de Riemann à bord et une caractérisation des courbes séparantes*]{}, Comment. Math. Helv. 81 (2006), 945–964. 50 (This result also appeared previously in the Ph.D. Thesis of the author published in 2004, cf. the previous item.) \[$\clubsuit$ proposes an improved bound upon Ahlfors 1950 [@Ahlfors_1950], as discussed in the previous item $\spadesuit$ for an update regarding the question about the sharpness of the bound so obtained see Coppens 2011 [@Coppens_2011] $\spadesuit$ \[03.10.12\] all this is fairly good yet a certain discrepancy with Ahlfors viewpoint is annoying and much remains to be clarified $\spadesuit$ \[03.10.12\] further one can wonder if there is not a Teichmüller-theoretic proof of the existence of such circle maps, parallelling that of Meis 1960 [@Meis_1960] in the case of closed surfaces, and conversely one can of course wonder if Meis cannot be proved via the topological method used in the present entry (Gabard 2006 [@Gabard_2006])\] 

A. Gabard, [*A separable manifold failing to have the homotopy type of a $CW$-complex*]{}, arXiv 2006, and another (simpler?) proof suggested by the referee in, Archiv der Math. (2008). \[$\spadesuit$ this little note was primarily intended to give a counterexample to an assertion made by Milnor in 1959, to the effect that all separable manifolds have the homotopy type of a CW-complex. Alas, this is completely wrong (as soon as one familiar with the Prüfer surface 1922–25). Notwithstanding, more mature knowledge of mine (ca. 2009) I realized that Milnor was not wrong at all, except that for him separable meant at that time second countable or metrizable (compare for instance sone of his preprint ca. 1958–59 available on the net). So the explanation is simply that the term “separable” had a different meaning in the first half of the 20th century (up to some residues moving as high as Milnor’s 1959 article [@Milnor_1959])\] 

A. Gabard, D. Gauld, [*Jordan and Schoenflies in non-metrical analysis situs*]{}, arXiv 2010. \[$\spadesuit$\] 

A. Gabard, D. Gauld, [*Dynamics of non-metric manifolds*]{}, arXiv 2011. \[$\spadesuit$ this is just cited for the proof of the implication: simply-connected $\Rightarrow$ schlichtartig $\Rightarrow$ orientable, which may be reduced to the five lemma\] 

A. Gabard, [*Compact bordered Riemannian surfaces as vibrating membranes: an estimate à la Hersch-Yang-Yau-Fraser-Schoen*]{}, arXiv 2011. 50 \[$\clubsuit$ inspired by Fraser-Schoen 2011 [@Fraser-Schoen_2011], this adapts Hersch 1970 [@Hersch_1970] (isoperimetric property of spherical vibrating membranes) to configurations of higher topological structure using the Ahlfors circle map with the bound of Gabard 2006 [@Gabard_2006] $\spadesuit$ notice an obvious (but superficial) connection with Gromov’s filling area conjecture (FAC) (1983 [@Gromov_1983]) positing the minimality of the hemisphere among non-shortening membranes, hence it would be fine that conformal geometry/transplantation—enhanced perhaps by Weyl’s asymptotic law for the high vibratory modes (out of which we can ‘hear’ the area of the drum)—affords a proof, either geometric or acoustic, of FAC. This would maybe be a spectacular application of the Ahlfors map, or maybe some allied conformal maps, e.g. that of Witt-Martens [@Witt_1934], [@Martens_1978], for non-orientable membranes. Recall indeed Gromov’s trick of cross-capping (à la von Dyck) the boundary contour of the membrane reduces the filling area problem (in genus zero) to Pu’s systolic inequality for the projective plane\] 

A. Gabard, [*Ebullition in foliated surfaces vs. gravitational clumping*]{}, arXiv 2011. \[$\clubsuit$ not relevant to the present topic, but just cited for a Jordan separation argument via covering theory that can be ascribed to Riemann with some imagination\] 

A. Gabard, [*Euler-Poincaré obstruction for pretzels with long tentacles à la Cantor-Nyikos*]{}, arXiv, Dec. 2011. \[$\clubsuit$ not relevant to the present topic, but just cited for some rudiment about Poincaré’s index formula for foliations\] 

A. Gabard, [*Ahlfors circle maps: historical ramblings*]{}, arXiv 2012. \[$\clubsuit$ this is the present article available on the arXiv, but which soon afterward (2013) was expanded so has to reinforce the connection with Rohlin’s work on Hilbert’s 16th problem. The pivotal motivation for this junction between Ahlfors-Rohlin is Rohlin’s cryptical claim of total reality for certain $(M-2)$-sextics having a real scheme forcing the type I. Also instrumental for this expansion (of material) is the Rohlin conjecture (or at least the vestige thereof post Shustin) that a scheme of type I is maximal. This problem (still open) bears some connection with earlier speculations of Klein (1876) which however could not resist Shustin’s work 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin] (much based upon the Viro revolution as well as deep Bézout-style obstructions coming from Fiedler-Viro)\] 

D. Gaier, [*Konforme Abbildung mehrfach zusammenhängender Gebiete durch direkte Lösung von Extremalproblemen*]{}, Math. Z. 82 (1963), 413–419. 78 \[$\spadesuit$ what sort of maps via which (extremal) method? Essentially the PSM via the Ritz-Ansatz (ca. 1908), à la Bieberbach-Bergman (1914/22), plus Nehari’s 1949 integral representation of such slit mappings\] 

D. Gaier, [*Konstruktive Methoden der konformen Abbildung*]{}, Ergebnisse d. Angew. Math. 3, Springer, Berlin, 1964. 78 \[$\spadesuit$ Chap.III discusses in details the extremal properties of the Riemann mapping for a plane simply-connected region (distinct of ${\Bbb C}$), namely that the range of the map normalized by $f'(z_0)=1$ has minimal area (first in Bieberbach 1914 [@Bieberbach_1914]) or that the boundary of the range has minimal length (probably first in Szegö 1921 [@Szego_1921]) $\spadesuit$ this material was also presented (in book format) by Julia 1931 [@Julia_1931]\]$\bigstar$ 

D. Gaier, [*Über ein Flächeninhaltsproblem und konforme Selbstabbildungen*]{}, Rev. Roumaine Math. Pures Appl. 22 (1977), 1101–1105. \[$\spadesuit$ cited for the same reasons as the next item and complement some details of it (especially in the sharpness of cross-references)\] 

D. Gaier, [*Konforme Abbildung mehrfach zusammenhängender Gebiete*]{}, Jber. d. Dt. Math.-Verein. 81 (1978), 25–44. \[$\spadesuit$ p.34–35, §C, brilliant proof (of a fact discovered and briefly handled by Grötzsch 1931 \[alas no precise cross-ref.\]) via his [*Flächenstreifenmethode*]{} that “the” (non-unique!) map minimizing the area integral $\int\int \vert f'(z) \vert^2 d \omega$ (à la Bieberbach 1914 [@Bieberbach_1914]–Bergman\[n\] 1922 [@Bergman_1922], but extended to the multiply-connected setting) under the schlichtness proviso (and the normalizations $f(z_0)=0, f'(z_0)=1$) maps the domain upon a circular slitted disc (with concentric circular slits centered about the origin) $\spadesuit$ Gaier’s proof is based upon a Carleman isoperimetric property of rings relating the modulus to the area enclosed by the inner contour, plus Bieberbach 1914 [@Bieberbach_1914] (first area theorem) to the effect that a schlicht normed map ($f'(a)=1$) from the disc inflates area, unless it is the identity $\clubsuit$ a natural (naive?) question of the writer (\[13.07.12\]) is what happens if we relax schlichtness of the map? Do we recover an Ahlfors circle map? Try maybe to get the answer from the entry Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949]\] 

T.W. Gamelin, M. Voichick, [*Extreme points in spaces of analytic functions*]{}, Canad. J. Math. 20 (1968), 919–928. 50 \[$\spadesuit$ Ahflors 1950 [@Ahlfors_1950] is quoted several times through the paper, the most relevant being $\clubsuit$ p.926: “According to \[1, §4.2\], there exist $r+1$ ($r=g$ in our notation) points $w_1, \dots, w_{r+1}$ on $bR$ such that if $B_j$ is the period vector of the singular function $T_j$ corresponding to a unit point mass at $w_j$, then $B_1, \dots, B_{r+1}$ are the vertices of a simplex in ${\Bbb R}^r$ which contains $0$ as an interior point.” $\clubsuit$ This is indeed the geometric heart of Ahlfors’ existence proof of a circle map of degree $\le g+1=r+2p$ $\spadesuit$ \[28.09.12\] the obvious game is whether one can lower the number of $w_j$ to recover the degree predicted in Gabard 2006 [@Gabard_2006] $\spadesuit$ as to the content of this entry, it is involved with an extension of the de Leeuw-Rudin (1958 [@deLeeuw-Rudin_1958]) characterization of the extreme points of the unit ball of the disc-algebra $H^1(\Delta)$ as the outer functions of norm $1$, and as usual this is obtained upon appealing to the Ahlfors map, or techniques closely allied to its existence-proof\] 

T.W. Gamelin, [*Embedding Riemann surfaces in maximal ideal spaces*]{}, J. Funct. Anal. 2 (1968), 123–146. \[$\spadesuit$ p.130: “Let $R$ be a finite bordered Riemann surface with boundary $\Gamma$. Let $A$ be the algebra of functions continuous on $R\cup \Gamma$ and analytic on $R$. Let $\varphi$ be the evaluation at some point $z_0$ of $R$. Then the harmonic measure for $z_0$ on $\Gamma$ is a unique Arens-Singer measure for $\phi$ on $\Gamma$. The spaces $N_c$ consists of the boundary values along $\Gamma$ of the analytic differentials on the doubled surface of $R$, the so-called Schottky differentials of $R$. The space $N_c$ is finite-dimensional.” $\spadesuit$ p.133: “Since $P$ admits a finite-sheeted covering map over $\{\vert \lambda
\vert<1\}$, $P$ must be one-dimensional.” $\spadesuit$ it is not clear (to Gabard) if this Gamelin argument makes tacit use of the Ahlfors map\] 

T.W. Gamelin, [*Uniform algebras*]{}, Prentice Hall, 1969. \[$\spadesuit$ p.195–200, analytic capacity as the first coefficient in the Laurent expansion of the Ahlfors function $\spadesuit$ p.197, existence and uniqueness of the Ahlfors function for a general open set in the plane $\spadesuit$ p.198, proof of the following convergence property of the Ahlfors function $f_E$ of a compact plane set $E$ (meaning the one, centered at $\infty$, of the outer component of $E$, i.e. the component of the complement of $E$ containing $\infty$): if $E_n$ is decreasing sequence of compacta with intersection $E$, and $f_n$ be the Ahlfors functions of $E_n$, then $f_n$ converge to $f$ uniformly on compact subsets of the outer component of $E$, and the corresponding analytic capacities converge $\gamma(E_n) \to
\gamma(E)$ $\spadesuit$ \[21.09.12\] this reminds perhaps one the famous conjecture (e.g. of Bing) about knowing if a descending sequence of plane (topological) discs must necessarily converge to a compactum satisfying the fixed-point property, even when the latter has the ugliest possible ‘dendrite’ shape $\spadesuit$ one may wonder if function theory, especially boosted version of RMT, could crack the problem (this is of course just a naive challenge)\]$\bigstar$ 

T.W. Gamelin, [*Localization of the corona problem*]{}, Pacific J. Math. 34 (1970), 73–81. 78 

T.W. Gamelin, J. Garnett, [*Distinguished homomorphisms and fiber algebras*]{}, Amer. J. Math. ?? (1970), 455–474. \[$\spadesuit$ p.474 Ahlfors function mentioned as follows: “It is more difficult to relate the Shilov boundary of $H^{\infty}(D)$ to the Shilov boundaries of the fiber algebras. The problem is to decide whether the distinguished homomorphisms $\phi_\lambda$ lie in the Shilov boundary of $H^{\infty}(D)$. This question was resolved negatively by Zalcman \[11\](=1969 [@Zalcman_1969-TAMS]) for the domains he considered, because in this case the Ahlfors function of $D$ could be seen to have unit modulus on the Shilov boundary of $H^{\infty}(D)$.”\] 

T.W. Gamelin, [*The algebra of bounded analytic functions*]{}, Bull. Amer. Math. Soc. 79 (1973), 1095–1108. 47, 50, 78 \[$\spadesuit$ p.1104: “The Ahlfors function tries hard to be unimodular on the boundary of an arbitrary domain.” The following result of Fisher is quoted (and reproved) “The Ahlfors function for a bounded domain $D$ in ${\Bbb C}$ has unit modulus on the Šilov boundary of $H^{\infty}(D)$.” $\spadesuit$ circa 12 occurrences of “Ahlfors function” throughout the paper $\spadesuit$ p.1104: “Incidentally, the preceding proof \[via the Šilov boundary\] also establishes the uniqueness of the Ahlfors function.” $\spadesuit$ p.1104: “Combined with cluster value theory, Fisher’s theorem yields information on the Ahlfors function which is already sharper than that which had been obtained by classical means.” p.1106–07: “if the harmonic measure for $D$ is carried by the union of an at most countable number of boundary components of $D$, then the Ahlfors function $G$ for $D$ is inner; that is, the composition $G\circ \pi$ with the universal covering map $\pi\colon \Delta \to D$ has radial boundary values of unit modulus a.e. ($d\theta$). Without the hypothesis on the harmonic measure, the Ahlfors function needs not be inner, and an example is given in \[17\](=Gamelin, to appear) of a domain $D$ with Ahlfors function $G$ satisfying $\vert G \circ \pi
\vert<1$ a.e. ($d\theta$) on $\partial \Delta$.” $\clubsuit$ the paper Ahlfors 1950 [@Ahlfors_1950] is quoted in the following brief connection: “For dual extremal problems on Riemann surfaces, see \[2\](=Ahlfors 1950) and \[36\](=Royden 1962).”\]  

T.W. Gamelin, [*Extremal problems in arbitrary domains*]{}, Michigan Math. J. 20 (1973), 3–11. 50, 78 \[$\spadesuit$ quoted in Hayashi 1987 [@Hayashi_1987] for the issue that the following property: “the natural map of a Riemann surface $R$ into its maximal ideal space $\frak M (R)$ (this is an embedding if we assume that the algebra $H^{\infty}(R)$ of bounded analytic functions separates points) is a homeomorphism onto an open subset of $\frak M
(R)$” has some application to the uniqueness of the Ahlfors function, as well as to its existence via Hayashi 1987 [@Hayashi_1987] $\spadesuit$ Royden 1962 [@Royden_1962] is cited instead of the original work Ahlfors 1950 [@Ahlfors_1950] for the treatment of extremal problems on finite bordered Riemann surfaces\] 

T.W. Gamelin, [*Extremal problems in arbitrary domains, II*]{}, Michigan Math. J. 21 (1974), 297–307. 78 \[$\spadesuit$ p.297, Ahlfors function is quoted as follows: “Hejhal proof’s depends on the methods developed by Havinson 1961/64 [@Havinson_1961/64], who proved the uniqueness of the Ahlfors function of arbitrary domains. Now there is in \[4\](=Gamelin 1973 [@Gamelin_1973-Extremal-I]) an economical proof of Havinson’s theorem that depends on function-algebraic techniques (see also \[3\](=Gamelin 1972, La Plata Notas) and \[5\](=Gamelin 1973 [@Gamelin_1973-BAMS])\] 

T.W. Gamelin, [*The Shilov boundary of $H^{\infty}(U)$*]{}, Amer. J. Math. 96 (1974), 79–103. \[$\spadesuit$ p.79, the Ahlfors function is cited and the author finds a bounded domain in the plane whose Ahlfors function fails to be inner (violating thereby a guess formulated, e.g. in Rubel 1971 [@Rubel_1971]) $\spadesuit$ let us quote the text (p.79): “Let $U$ be a bounded domain in the plane, and let $H^{\infty}(U)$ be the algebra of bounded analytic functions on $U$, and $\frak M (U)$ be its maximal ideal space. Our object here is to study the Shilov boundary $S(U)$ of $H^{\infty}(U)$. It will be shown that $S(U)$ is extremely disconnected, and that every positive continuous function on $S(U)$ is the modulus of a function in $H^{\infty}(U)$. Fisher \[7\](=1972 [@Fisher_1972-The-moduli-of-extremal-fctions]) has shown that there exist nonconstant functions in $H^{\infty}(U)$ with unit modulus on $S(U)$. In fact, he proves that the Ahlfors function for $U$ is unimodular. We will show that there is an abundant supply of unimodular functions in $H^{\infty}(U)$, sufficiently many to separate $S(U)$ from the points of ${\frak M} (U )\setminus S(U)$ which are adherent to $U$. In the negative direction, we show that the property of having unit modulus on the Shilov boundary of $H^{\infty}(U)$ does not yield a great deal of information concerning the classical boundary values of functions in $H^{\infty}(U)$. In fact, an example is given of a reasonably well-behaved domain $U$ with the following property: If $f$ is any nonconstant function in $H^{\infty}(U)$ such that $\|f\|\le 1$, then the lift of $f$ to the open unit disc via the universal covering map has radial boundary values of modulus $<1$ a.e. ($d\theta$).” $\spadesuit$ the latter assertion specialized to an Ahlfors function (at some center) shows that the latter can fail to be inner (indeed not even hypo-inner in the sense of Rubel)\] 

T.W. Gamelin, J.B. Garnett, L.A. Rubel, A.L. Shields, [*On badly approximable functions*]{}, J. Approx. Theory 17 (1976), 280–296. \[$\spadesuit$ if $F$ is a finite bordered Riemann surface, let $A(F)$ be the algebra of functions, analytic in the interior with continuous extension to the boundary $\Gamma:=\partial F$. The boundary value map $A(F)\to
C(\Gamma)$ is injective (upon splitting into real/imaginary parts and applying the uniqueness of the Dirichlet problem). The algebra $C(\Gamma)$ (complex-valued functions on the boundary $\Gamma$) is endowed with the sup-norm $\|\varphi\|=\sup_{z\in \Gamma} \vert \varphi(z) \vert$. Now given any $\varphi \in C(\Gamma)$ there must be a best analytic approximant $f\in A(F)$, that is minimizing $\|
\varphi - f\|$. The authors (following Poreda 1972) call $\varphi\in C(\Gamma)$ [*badly approximable*]{} if its distance $d(\varphi, A(D))$ to the space $A(D)$ is equal to the norm $\|\varphi\|$. This amounts saying that the best analytic approximant of $\varphi$ is $0$ (zero function). $\spadesuit$ \[01.10.12\] of course such badly approximable function are the opposite extreme of the boundary-values of an Ahlfors function (or of a circle map), since the latter coincide with their best analytic approximant. Despite this contrast, badly approximable functions are shown to have constant modulus along the boundary (Theorem 1.2, p.281) sharing a distinctive feature of circle maps, but deviates from them by having a small index (=winding number), namely ${\rm ind} (\varphi)<2p+(r-1)$, where $p$ is the genus and $r$ the contour number of $F$. Precisely Theorem 8.1 (p.294) states: “[*If $\varphi\in C(\Gamma)$ is badly approximable, then $\varphi$ has nonzero constant modulus, and ${\rm ind}(\varphi)<2p+(r-1)$.*]{}” The proof involves the theory of Toeplitz operators and reduces ultimately to the theory of Schottky differentials (forming a real vector space of dimension equal to the genus $g$ of the double which is precisely the upper bound involved above). Hence the connection with Ahlfors 1950 [@Ahlfors_1950] is evident (at least at some subconscious level), and accentuated by the numerous citations to the allied paper Royden 1962 [@Royden_1962]. $\spadesuit$ finally, let us maybe observe that the converse of the above statement (Theorem 8.1) can be foiled as follows: via Gabard 2006 [@Gabard_2006] there is always a circle map $f$ of degree $d\le r+p$. Its boundary restriction $\partial f=:\varphi$ has index equal to this degree ${\rm ind } (\varphi) =d\le
r+p\buildrel{!}\over{<}2p+(r-1)$, provided $p>1$. Yet the map $\varphi$ is not badly approximable, for by construction it admits a perfect analytic approximant.\] 

T.W. Gamelin, [*Cluster values of bounded analytic functions*]{}, Trans. Amer. Math. Soc. 225 (1977), 295–306. \[$\spadesuit$ several aspects of the Ahlfors function are discussed, and some new property (extending a result of Havinson) is given. To be more precise, we quote some extracts $\spadesuit$ p.296: Recall that the Ahlfors function $G$ of $D$, depending on the point $z_0 \in D$, is the extremal function for the problem of maximizing $\vert f'(z_0)\vert$ among all $f\in H^{\infty}(D)$ satisfying $\vert f \vert \le
1$; $G$ is normalized so that $G'(z_0)>0$, and then $G$ is unique. If $\zeta$ is an essential boundary point of $D$, then $\vert G \vert =1$ on $\amalg\!\!\!\,\amalg_{\zeta}$ (Šilov boundary). Furthermore, either $\lim_{D\ni z \to
\zeta} \vert G(z)\vert=1$ or ${\rm Cl}(G, \zeta)=
\overline{\Delta}$(=closed unit disc). S.Ya. Havinson \[7, Theorem 28\] has proved that $G$ assumes all values in $\Delta$, with the possible exception of a subset of $\Delta$ of zero analytic capacity. $\spadesuit$ p.297: we conclude the following sharper version of Havinson’s Theorem. [**1.2 Corollary.**]{} Let $G$ be the Ahlfors function of $D$, and let $\zeta$ be an essential boundary point of $D$ such that ${\rm
Cl}(G, \zeta)=\overline{\Delta}$. Then values in $\Delta$ are assumed infinitely often by $G$ in every neighborhood of $\zeta$, with the exception of those lying in a set of zero analytic capacity.\] 

T.W. Gamelin, [*Wolff’s proof of the corona theorem*]{}, Israel J. Math. ?? (1980), ??–??. \[$\spadesuit$ “Abstract. An expository account is given of T. Wolff’s recent elementary proof of Carleson’s Corona Theorem (1962). The Corona Theorem answers affirmatively a question raised by S. Kakutani (1957) as to whether the open unit disc in the complex plane is dense in the …”\] 

T.W. Gamelin, M. Hayashi, [*The algebra of the bounded analytic functions on a Riemann surface*]{}, J. Reine Angew. Math. 382 (1987), 49–73. \[$\spadesuit$ p.72 some sophisticated (but lucid) questions about the Grunsky-Ahlfors (abridged Grahl=Graal=Sangreal) extremal problem of maximizing the derivative $f'(p)$ among functions bounded-by-one $\vert f
\vert \le 1$ (where $p$ is a given point and the derivative is taken w.r.t. a fixed local coordinates). The following questions are posed under the proviso that $H^{\infty}(R)$ separates points. [**Problem 1.**]{} For a fixed $p\in R$, is there an $f\in H^{\infty}(R)$ such that $f'(p)\neq 0$? If such an $f$ exists, then any extremal function for the Grahl-problem normalized so that $f'(p)>0$ is termed an [*Ahlfors function*]{} corresponding to $p$. [**Problem 2.**]{} For fixed $p\in R$, assume the Grahl-extremal problem is non-trivial. Is the Ahlfors function unique? Does it have unit modulus on the Shilov boundary of $H^\infty(R)$? $\spadesuit$ the writer (Gabard) is not aware of any update on those questions, yet it may be emphasized that partial answers are sketched in Hayashi 1987 [@Hayashi_1987], namely that under the assumption that the natural map of $R$ to its maximal ideal space $\frak M (R)$ takes $R$ homeomorphically onto an open set of $\frak M (R)$, then existence and uniqueness of the Ahlfors function is ensured\] 

M. Gander, G. Wanner, [*From Euler, Ritz and Galerkin to modern computing*]{}, (2012), 49–73. \[$\spadesuit$ a historical survey about Galileo, Bernoulli, Euler, Lagrange, Chladni,…, Ritz, Galerkin and their influence upon modern computing\] 

P.R. Garabedian, M.M. Schiffer, [*Identities in the theory of conformal mapping*]{}, Trans. Amer. Math. Soc. 65 (1949), 187–238. 60, 78 \[$\spadesuit$ p.201, the problem of least area is considered (i.e. minimization of $\int\!\!\int \vert f'(z)
\vert^2 d\omega$) among [*all*]{} (not necessarily schlicht) mappings $f$ normed by $f(a)=0, f'(a)=1$ defined on an $n$-connected domain $\spadesuit$ it should be emphasized that the solution of this problem was stated (without proof) by Grunsky 1932 [@Grunsky_1932 p.140]; Grunsky’s influence is recognized in the introduction (p.188), yet not made explicit at the relevant passage (p.201, Problem I.) for the specific result of the least area map $\spadesuit$ assert (without detailed proof) that the solution is at most $n$-valent $\spadesuit$ alas it is not asserted that those least-area maps are circle maps (which looks a natural conjecture)\] 

P.R. Garabedian, [*Schwarz’s lemma and the Szegö kernel function*]{}, Trans. Amer. Math. Soc. 67 (1949), 1–35. 60, 78 \[$\spadesuit$ includes the formula $f'(t)=2\pi k(t,t)$ for the derivative of the Ahlfors function in terms of Szegö’s kernel function, other expositions of the same result in Bergman 1950 [@Bergman_1950], Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950] and Nehari 1952 [@Nehari_1952-BOOK] $\spadesuit$ at several crucial stage this paper makes use of topological arguments (hence a possible connection with Gabaredian 2006 [@Gabard_2006] remains to be elucidated)\] 

P.R. Garabedian, [*The sharp form of the principle of hyperbolic measure*]{}, Ann. of Math. 51 (1950), 360–379. 60, 78 \[$\clubsuit$ claims to recover the full Ahlfors (1950 [@Ahlfors_1950]) theorem on existence of circle maps (by deploying a large array of techniques blending from Teichmüller 1939 [@Teichmueller_1939-Dreikreisesatzes], Grunsky 1940–42 [@Grunsky_1940], [@Grunsky_1942], Ahlfors 1947 [@Ahlfors_1947] and the variational method of Schiffer/Hadamard), but the detailed execution is limited to the planar case, and only the same bound as Ahlfors 1950 [@Ahlfors_1950] is obtained\] 

P.R. Garabedian, [*The class $L_p$ and conformal mapping*]{}, Trans. Amer Math. Soc. 69 (1950), 392–415. \[$\clubsuit$\] 

P.R. Garabedian, M.M. Schiffer, [*On existence theorems of potential theory and conformal mapping*]{}, Ann. of Math. (2) 52 (1950), 164–187. 78 \[$\spadesuit$ reprove RMT via the Bergman kernel (for smooth boundary, p.164), but the general case follows by topological approximation (exhaustion) $\spadesuit$ p.182 points out that circle maps lye somewhat deeper than slit mappings $\spadesuit$ p.181 recover the circle map for domains (of finite-connectivity) $\spadesuit$ recover also the parallel-slit mappings and cite Lehto 1949 [@Lehto_1949] for equivalent work $\spadesuit$ p.182 coins the designation “circle mapping”, to which we adhere in this survey.\] 

P.R. Garabedian, [*A new proof of the Riemann mapping theorem*]{}. In: [*Construction and Applications of Conformal Maps*]{}, Proc. of a Sympos. held on June 22–25 1949, Applied Math. Series [*18*]{}, 1952, 207–213. \[$\spadesuit$ consider a (strange) least area problem yet without making very explicit the range of the geometry of the extremal function\] 

P.R. Garabedian, [*Univalent functions and the Riemann mapping theorem*]{}, Proc. Amer. Math. Soc. 61 (1976), 242–244. \[$\spadesuit$ yet another new proof of RMT via an extremal problem and normal families $\spadesuit$ also cited for the reasons annotated after de Possel 1939 [@de-Possel_1939], namely the issue of avoiding the use of RMT in the extremal proof of PSM\] 

L. Garding, [*The Dirichlet problem*]{}, Math. Intelligencer 2 (1979/80), 43–53. \[$\spadesuit$ historical survey of the Dirichlet problem with Poisson, Gauss 1839 [@Gauss_1839], its influence upon Thomson 1847 [@Thomson_1847], Stokes (credited for the maximum principle!?), Dirichlet, Riemann, Weierstrass, Schwarz, Neumann, Poincaré (balayage) and its modern ramification by Perron [@Perron_1923] and Radó-Riesz 1925 [@Rado-Riesz_1925], up to Frostman, Beurling-Deny\] 

J. Garnett, [*Positive length but zero analytic capacity*]{}, Proc. Amer. Math. Soc. 24 (1970), 696–699. \[$\spadesuit$ simplifies the example of Vitushkin 1957 [@Vitushkin_1959] by taking advantage of the homogeneity of the compactum which is a simple planar Cantor set obtained by keeping only the 4 corner squares of a subdivision of the unit-square in $4\times
4$ congruent subsquares, and iterating ad infinitum $\spadesuit$ compare Murai 1987 [@Murai_1987] for another direct strategy (via Garabedian instead of Ahlfors) which is supposed to give more insight about the general problem\] 

J. Garnett, [*Analytic capacity and measures*]{}, Lecture Notes in Math. 297, Springer, Berlin, 1972, 138 pp. \[$\spadesuit$ p.18, Ahlfors function $\spadesuit$ p.36, Denjoy conjecture (cf. for its resolution Marshall [@Marshall_1978?] via Calderón mostly)\] 78 

J.B. Garnett, [*Bounded analytic functions*]{}, Pure and Appl. Math. 96, Academic Press, New York, 1981. \[$\spadesuit$ includes proofs of the corona theorem\] 

J. Garnett, J. Verdera, [*Analytic capacity, bilipschitz mappings and Cantor sets*]{}, Math. Res. Lett. 10 (2003), 515–522. \[$\spadesuit$\] 

A.M. Garsia, [*Calculation of conformal parameters for some imbedded Riemann surfaces*]{}, Pacific J. Math. 10 (1960), 121–165. \[$\spadesuit$\] 

A.M. Garsia, [*Imbeddeding of Riemann surfaces by canal surfaces*]{}, Rend. Circ. Math. Palermo (2) 9 (1960), 313–333. \[$\spadesuit$\] 

A.M. Garsia, E. Rodemich, [*An imbeddeding of Riemann surfaces of genus one*]{}, Pacific J. Math. 11 (1961), 193–204. \[$\spadesuit$ p.193: “Theorem. [*Any compact Riemann surface of genus one can be $C^\infty$ embedded in $3$-space.*]{}” $\spadesuit$ inspiration=Teichmüller 1944 [@Teichmueller_1944-Beweis-der-analytischen-Abhaengigkeit] and Ahlfors 1953/54 [@Ahlfors_1953/54-On-quasiconformal-mappings] $\spadesuit$ extension of the result in the next entry Garsia 1961 [@Garsia_1961]\] 

A.M. Garsia, [*An imbedding of closed Riemann surfaces in Euclidean space*]{}, Comment. Math. Helv. 35 (1961), 93–110. \[$\spadesuit$ yet another brilliant student of Loewner; it is shown that any closed Riemann surface admits a conformal model in Euclid’s $3$-space $E^3$. \[10.12.12\] Upon taking taking the Schottky double, the same assertion holds true for bordered Riemann surfaces, and this is perhaps enough when $E^3$ is replaced by the more generous $E^4$ to settle the Forstnerič-Wold 2009 [@Forstneric-Wold_2009] desideratum that any compact bordered Riemann surface embeds holomorphically in ${\Bbb C}^2$. \[11.12.12\] Warning: not at all enough for the image is merely a smooth surface, and not a complex analytic curve $\spadesuit$ p.94: “The main result of the present paper is a proof that there exists in Euclidean space a conformally equivalent $C^{\infty}$ model for every compact Riemann surface of genus $g\ge 2$.” Compare also Rüedy 1968 [@Ruedy_1968] $\spadesuit$ “The methods that we have followed are essentially an extension of those in \[9\]. However, here certain devices introduced by J. Nash in \[13\], together with some results of L. Ahlfors \[2\] and L. Bers \[3\] on spaces of Riemann surfaces are quite crucial…” $\spadesuit$ to be fair the main technique permitting the breakthrough on this almost centennial problem conjectured by Klein (realizability of all Riemann surfaces as classical surface in $E^3$) is primarily Teichmüller theory, especially the 1944 paper [@Teichmueller_1944-Beweis-der-analytischen-Abhaengigkeit]. $\spadesuit$ \[19.12.12\] Garsia’s result can be given the following metaphoric interpretation (for single people having the Riemann($\approx$ woman) surface) as sole sentimental partner during their whole life, e.g. Koebe who never married): in the vicinity of any surface embedded in Euclid’s $3$-space $E^3$ one can realize any conformal structure via small variations confined to the normal bundle of the initial surface. This holds true for arbitrarily small thicknesses $\varepsilon$ of the tubular neighborhood. Metaphorically, this amounts to say that if the Riemann surface becomes a woman surface (materialized by the skin of some naked woman) then a minim variation of the skin permits to explore all other (women) surfaces by epidermic bubbling, alas generically akin to a cellulite formation. $\spadesuit$ This metaphor seems again to say something on the Forstnerič-Wold 2009 [@Forstneric-Wold_2009] desideratum. First we know (from Černe-Forstnerič 2002 [@Cerne-Forstneric_2002]) that any topological type of bordered surface contains a representative holomorphically embedded in ${\Bbb C}^2$. Applying the high-dimensional version of Garsia (due to Ko 1989 [@Ko_1989-compact], plus subsequent articles) we can realize all Riemann surfaces within a normal tubular neighborhood via an (infimal) normal variation. This is akin to a cellulite bubbling, alas destroying a priori the holomorphic character of the initial model. However it is not to be excluded that better controlled vibrations of the pudding[^131] permit to explore the full moduli space.\] 

A.M. Garsia, [*On the conformal type of algebraic surfaces in euclidean space*]{}, Comment. Math. Helv. 37 (1962–63), 49–60. \[$\spadesuit$ “It has been an open question for some time whether or not the classical (i.e. $C^2$ surfaces) of Euclidean space (3-dimensional) exhaust all possible conformal types. In the non compact case the question is still open. \[UPDATE: Rüedy 1971 [@Ruedy_1971]\] In the compact case it can be shown (see \[1\]=Garsia-Rodemich 1961 [@Garsia-Rodemich_1961] and \[2\]=Garsia 1961 [@Garsia_1961]) that among the $C^\infty$ surfaces of Euclidean space there are surfaces conformally equivalent to any given compact Riemann surface.—In this paper we are to improve the results in \[1\] and \[2\]. It will be shown that for any given compact Riemann surface conformally equivalent models can be found among the algebraic surfaces of ordinary space. Here by “algebraic surface” we mean a surface satisfying an equation of the type $F(x,y,z)=0$, where $F(x,y,z)$ is a real polynomial in its argument.—\[...\] it is not known whether or not the affine images of the tori of revolution contain all conformal types of surfaces of genus one.—Perhaps it should be noted that the ease with which the results in \[2\] and specially those of this paper are obtained illustrate once more the power of the Teichmüller results on quasiconformal mappings and the usefulness of the concept of Teichmüller space for the study of families of compact Riemann surfaces. $\spadesuit$ of course in view of Garsia’s result one would like to bound the degree of the representing algebraic surface..., for more on this cf. Pinkall 1985 [@Pinkall_1985]\] 

C.F. Gauss, 1811 (unpublished) correspondence with F.W. Bessel. \[$\spadesuit$ complex integration and the formula $\frac{1}{2\pi i} \int_l \frac{dz}{z}=1$, where $l$ is a circle enclosing the origin\] 

C.F. Gauss, Allgemeine Auflösung der Aufgabe: die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden, dass die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird. Als Beantwortung der von der königlichen Societät der Wissenschaften in Copenhagen für 1822 aufgegebnen Preisfrage, in: [*Schumacher’s Astronomische Abhandlungen*]{}, Drittes Heft, pp. 1–30, Altona 1825. (Also in: Werke, Bd.4, 189–216.) \[$\spadesuit$ This is probably the only record in print which may be regarded as a weak (very local) forerunner of the RMT. This text was of course known to Riemann, while adumbrating the conformal plasticity of 2D-mappings $\spadesuit$ in fact this Gauss text 1822/25 is the pre-big-bang, for it is the only reference cited in Riemann’s Thesis 1851 [@Riemann_1851], who however had several other inspirators like Dirichlet, Cauchy, etc. $\spadesuit$ some antecedents of this Gauss work is that by Lagrange 1779 [@Lagrange_1779] involved with a cartography problem, yet failing to prove, as Gauss do (in op.cit.), that locally any surface is conformally flat\] 

C.F. Gauss, Disquisitiones generales circa superficies curvas, 1827. \[$\spadesuit$ concept of Gaussian (total) curvature, theorema egregium (the curvature $K$ is isometry-invariant, e.g. under bending), etc.\] 

C.F. Gauss, [*Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrates der Entfernung wirkenden Anziehungs- und Abstossungs-Kräfte*]{}, Magnetischer Verein (1839). Werke vol. 5, 195–242. \[$\spadesuit$ a forerunner of the Dirichlet principle. This text was known to Riemann $\spadesuit$ this Gauss work is supposed to lack in rigor, yet encompass the substance of the all potential theory (compare Brelot 1952 [@Brelot_1952] for a modern appreciation)\] 

P.M. Gauthier, M. Goldstein, [*From local to global properties of subharmonic functions on Green spaces*]{}, J. London Math. Soc. (2) 16 (1977), 458–466. \[$\spadesuit$ p.465, includes the following application of the Ahlfors function. Let $\overline{\Omega}$ be a compact bordered Riemann surface with interior $\Omega$ and contour $C=\partial \overline{\Omega}$. Given $f\colon C \to
\overline{\Bbb R}=[-\infty,+\infty]$ an extended real-valued continuous function, one says that $f$ is Dirichlet soluble if it continuously extends to $\overline{\Omega}$ so that its restriction to the interior $\Omega$ is harmonic. In this case, $f^{-1}(+\infty)$ is a closed set of HMZ(=harmonic measure zero). Now the authors shows the converse statement. Indeed, given $E\subset C$ closed and of HMZ, its image under an Ahlfors function (cf. Ahlfors 1950 [@Ahlfors_1950]) $F\colon \overline{\Omega} \to \overline{\Delta}$ is a subset of the circumference $S^1=\partial\overline{\Delta}$ of measure zero (Sard required?). According to Fatou 1906 [@Fatou_1906] any null-set of the circle occurs as $u_1^{-1}(\infty)$ for a continuous function $u_1\colon
\overline{\Delta}\to \overline{{\Bbb R}}$ harmonic in the interior. The composed map $u_1\circ F$ has the desired properties $\spadesuit$ note however that the trick of the Ahlfors function seems not well suited for reducing the Dirichlet problem (even with non-extended boundary values) on a compact bordered Riemann surface to the case of the disc where it is soluble via the Poisson integral (albeit this may have been a partial intention in Bieberbach 1925 [@Bieberbach_1915])\] 

W.-D. Geyer, [*Ein algebraischer Beweis des Satzes von Weichold über reelle algebraische Funktionenkörper*]{}, In: [*Algebraische Zahlentheorie*]{} (Ber. Tagung Math. Forschungsinst. Oberwolfach, 1964), 83–98. \[$\spadesuit$ includes a new proof of the theorem of Witt 1934 [@Witt_1934]\] 

W.-D. Geyer, G. Martens, [*Überlagerungen berandeter Kleinscher Flächen*]{}, Math. Ann. 228 (1977), 101–111. 50 \[$\clubsuit$ after Alling-Greenleaf 1969 [@Alling-Greenleaf_1969], Ahlfors 1950 [@Ahlfors_1950] is also interpreted in terms of Klein’s orthosymmetric real curves, specifically p.106: “Gewisserma[ß]{}en als Umkehrung von a) ist das resultat von Ahlfors (\[1\], §4) anzusehen, wonach jede Kleinsche Fäche vom Typ $+(g,r)$ mit $r>0$ eine $(g+1)$-blättrige verzweigte Überlagerung der zur reellen projectiven Geraden ${\Bbb P}_1$ gehörenden Kleinschen Fläche $\overline{\Bbb C}/\sigma$ (=Riemannsche Zahlenhalbkugel) ist.” $\spadesuit$ p.101: “Seit Klein \[6, 12\] zieht man zum Studium reeller algebraischer Funktionenkörper $F$ einer Variablen mit Erfolg die zur Komplexifizierung von $F$ gehörige Riemannsche Fläche, versehen mit einer antiholomorphen Involution $a$, heran, oder auch die Kleinsche Fläche \[…\].—Unter den algebraischen Körpererweiterungen $E\vert F$ gibt es gewisse, durch ihr Realitätsverhalten ausgezeichnete Typen, die zuerst von Knight in \[7\](=1969 [@Knight_1969]) betrachtet wurden. Wir nennen $E\vert F$ total reell, wenn über reellen Stellen von $F$ nur reelle Stellen von $E$ liegen. Da die reellen Stellen von $F$ auf $\frak R$ eine disjunkte Vereinigung endlich vieler Kreise $Z1, \dots, Z_r$ bilden, induziert eine total reelle Erweiterung $E\vert F$ (unverzweigte) Überlagerungen der $Z_i$.” $\clubsuit$ p.102: “Die Kleinsche Fläche $\frak K=\frak R /
\sigma$ is ebenfalls kompakt und zusammenhängend; sie ist genau dann orientierbar wenn $\frak R-\cal C ({\Bbb R})$ unzusammenhängend ist \[3,6\](=Alling-Greenleaf 1971 [@Alling-Greenleaf_1971], Klein 1882 [@Klein_1882]). Eine algebraische Kennzeichnung der Orientierbarkeit gab Ahlfors in \[1\](=1950 [@Ahlfors_1950]): $\frak K$ ist genau dann orientierbar, wenn es eine Funktion $f\in F$ gibt, die nur auf $\cal C ({\Bbb R})$ reelle Werte annimt.” $\spadesuit$ p.103: “Ein Morphismus $\varphi\colon \cal B\to
\cal C$ reeller Kurven hei[ß]{}t [*total reell*]{}, wenn $\varphi^{-1} {\cal C} ({\Bbb R})={\cal B} ({\Bbb R})$ ist. Dann ist also eine Erweiterung $E\vert F$ reeller Funktionenkörper total reell, wenn jede reelle Stelle von $F$ nur reelle Fortsetzungen hat, und ein Morphismus Kleinscher Flächen is total reell, wenn der Rand respektiert wird, d.h. nur Randpunkte auf Randpunkte abgebildet werden.”\] 

W.-D. Geyer, [*Reelle algebraische Funktionen mit vorgegeben Null- und Polstellen*]{}, Manuscripta Math. 22 (1977), 87–103. \[$\spadesuit$ p.91: “Ein Morphismus $\varphi\colon Y\to X$ reeller Kurven hei[ß]{}e total reell, wenn $\varphi^{-1}
X(K)=Y(K)$ ist, d.h. wenn alle reellen Stellen von $F=K(X)$ nur relle Forsetzungen in $E=K(Y)$ haben.”\] 

P. Gilmer, [*Algebraic curves in $RP^1 \times RP^1$*]{}, Proc. Amer. Math. Soc. 113 (1991), 47–52. \[$\spadesuit$ switching the Hilbert problem to ${\Bbb P}^1\times {\Bbb P}^1$\] 

P. Gilmer, [*Real algebraic curves and link cobordism*]{}, Pacific J. Math. 153 (1992), 31–69. \[$\spadesuit$ promise (in the next entry Gilmer 1996 [@Gilmer_1996-Link-cobordismII]) a new derivation of the Gudkov-Rohlin congruence for $M$-curves, as well as the related congruence for $(M-1)$-curves\] 

P. Gilmer, [*Real algebraic curves and link cobordism, II*]{}, in: Topology of Real Algebraic Varieties and Related Topics, ed. by V. Kharlamov et al., Amer. Math. Soc. Transl. Ser. 2 173, Amer. Math. Soc.., Providence, Ri, 1996, 73–84. \[$\spadesuit$ new derivation of the Gudkov-Rohlin congruence for $M$-curves, as well as the related congruence for $(M-1)$-curves\] 

A. Girouard, I. Polterovich, [*Steklov eigenvalues*]{}, arXiv (2012). 50 \[$\clubsuit$ extension of Fraser-Schoen 2011 [@Fraser-Schoen_2011] to higher eigenvalues\] 

A.M. Gleason, [*Function algebras*]{}, Seminar on analytic functions, Institute for Advanced Study, Princeton, N.J., 1957. \[$\spadesuit$ where the Gleason parts are defined as the equivalence classes of the following relation $\spadesuit$ for an arbitrary function algebra $A$ on a compact metrizable space $X$, let $M$ be its maximal ideal space and $S$ its Shilov boundary. Realizing $A$ as a function algebra on $M$, two points $m_1, m_2 \in M $ are (Gleason) equivalent if $\sup \{ \vert f(m_1)\vert: f(m_2)=0, \|
f\| \le 1  \}<1$. $\spadesuit$ for a connection with the Ahlfors map cf. e.g. O’Neill-Wermer 1968 [@O'Neill-Wermer_1968]\] 

J. Globevnik, B. Stens[ø]{}nes, [*Holomorphic embeddings of planar domains in ${\Bbb C}^2$*]{}, Math. Ann. 303 (1995), 579–597. \[$\spadesuit$ it is show that any plane domain of finite connectivity without point-like (punktförmig) boundaries has a proper holomorphic embedding in the affine complex plane $\spadesuit$ for a wide extension to infinite connectivity, cf. Forstnerič-Wold 2012 [@Forstneric-Wold_2012]\] 

G.M. Golusin, [*Auflösung einiger ebener Grundaufgaben der mathematischen Physik im Fall der Laplaceschen Gleichung und mehrfach zusammenhängender Gebiete, die durch Kreise begrenzt sind*]{}, (Russian, German Summary) Mat. Sb. 41 (1934), 246–276. 78 \[$\clubsuit$ Seidel’s summary: a harmonic function $U$ of two real variables is sought exterior to the circles $C_1, \dots, C_n$, with $U(\infty)$ finite, which on $C_k$ assumes preassigned continuous values $f_k$. The problem is reduced to the solution of a finite system of functional equations which are solved by successive approximations. The method is applied to solve Neumann’s problem and other similar problems for Laplace’s equation and for regions of the above type. The Green’s functions of such regions and the functions which map them on slit planes are determined\]$\bigstar$ 

G.M. Golusin, [*Sur la représentation conforme*]{}, (French, Russian Summary) Mat. Sb.=Rec. Math. 1 (43) (1936), 273–282. 78 \[$\clubsuit$ p.273, Lemme 1 gives another proof of a basic lemma about areas of rings under conformal maps $\spadesuit$ Pólya-Szegö 1925 are cited, but it should go back to Carleman 1918 [@Carleman_1918] $\spadesuit$ for the relevance of this lemma to the least area problem of multi-connected under schlicht maps see Gaier 1977 [@Gaier_1977-Roumaine] where a dissection process shows that a solution (non-unique!) to this problem effects a representation upon a circular slit disc $\spadesuit$ incidentally the proof of Thm 1, p.274 looks very akin to Gaier’s argument of 1977 [@Gaier_1977-Roumaine]\] 

G.M. Golusin, [*Iterationsprozesse für konforme Abbildungen mehrfach zusammenhängender Bereiche*]{}, (Russian, German Summary) Mat. Sb. N.S. 6 (48) (1939), 377–382. 78 \[$\clubsuit$ Iterative methods are established by means of which a schlicht conformal map of regions of finite connectivity on some canonical domains is reduced to a sequence of conformal maps of simply-connected regions\]$\bigstar$ 

G.M. Golusin, [*Geometrische Funktionentheorie*]{}, Übersetzung aus dem Russischen. Hochschulbücher f. Math. Bd. 31, Berlin, VEB Deutscher Verlag d. Wiss., 1957. English transl.: Geometric theory of functions of a complex variable, 1969. (Russian original published in 1952.) 60, 78 \[$\spadesuit$ p.240–4, proof of a circle map in the schlicht(artig) case following Grunsky 1937–41 (potential-theoretic) $\spadesuit$ p.412–8, the extremal approach is presented (Ahlfors 1947 [@Ahlfors_1947] is cited, and ref. to Grunsky 1940–42 [@Grunsky_1940; @Grunsky_1942] where added by the German editors (probably Grunsky himself) $\spadesuit$ p.200–217 present a proof of Koebe’s KNP via the continuity method (approached via Brouwer’s invariance of the domain)\] 

T. Gouma, [*Ahlfors functions on non-planar Riemann surfaces whose double are hyperelliptic*]{}, J. Math. Soc. Japan 50 (1998), 685–695. 50 \[$\clubsuit$ detailed study of the degrees of the Ahlfors map in the hyperelliptic case $\clubsuit$ a complement (tour de force) is to be found in Yamada 2001 [@Yamada_2001] $\spadesuit$ for an application to proper holomorphic embeddings in ${\Bbb C}^2$, cf. Černe-Forstnerič 2002 [@Cerne-Forstneric_2002] $\spadesuit$ Köditz’s summary (MathReviews): “Let $R$ be a finite bounded \[=bordered\] Riemann surface with genus $p$ and $q$ contours and let $P$ be a point in $R$. The author studies the set of Ahlfors functions on $R$. These functions are the extremal functions obtained by maximizing the derivative $\vert f'(P) \vert$ (in some local parameter at $P$) in the class of holomorphic functions on $R$ bounded by one. Each Ahlfors function has modulus $1$ on the boundary of $R$ and gives a complete[^132] covering of the unit disk. It is known that the degree $N$ of any Ahlfors function satisfies $q\le N \le 2p+q$ (Ahlfors, 1950=[@Ahlfors_1950]). The set of degrees $N(R)$ of Ahlfors functions on a given Riemann surface $R$ is not well known. In this paper, the author deals with Ahlfors functions on non-planar Riemann surfaces whose doubles are hyperelliptic. Among others, examples for such Riemann surfaces with $N(R)=\{2, 2p+q\}$ are constructed.”\] 

W.H. Gottschalk, [*Conformal mapping of abstract Riemann surfaces*]{}, Published by the author, Univ. of Pennsylvania, Philadelphia, 1949, 77p. $\bigstar$ 60 

L.B. Graĭfer, S.Ja. Gusman, V.V. Dumkin, [*An extremal problem for forms with singularities on Riemannian manifolds*]{}, Perm. Gos. Univ. Učen. Zap. 218 (1969), 47–52. \[$\spadesuit$ from MathReview (by Kiremidjian): “In 1950, Ahlfors showed that a number of extremal problems on compact subregions of open Riemann surfaces could be solved by studying the class of Schottky differentials \[Ahlfors 1950 [@Ahlfors_1950]; errata, MR 13, p.1138\]. In recent years, certain aspects of Ahlfors’ work were investigated in the case of $n$-dimensional orientable differentiable manifolds \[the second author, 1966\]. I the present paper, the authors study the class of Schottky-Ahlfors forms with singularities.” $\spadesuit$ so those cited works constitute a rare but foolhardy attempt to extend Ahlfors’ theory to higher dimensions $\spadesuit$ perhaps one is prompted by the (naive!) question if one could formulate a theory able to (re)prove the famous 3D-conjecture of Poincaré-Perelman in its bordered incarnation: any compact bordered $3$-manifold is topologically equivalent to the $3$-ball, provided it is contractible or simply-connected and bounded by the $2$-sphere (of course the modest antecedent being the fact that one can prove the Schoenflies theorem via RMT thanks to Osgood/Carathéodory) $\spadesuit$ the Ahlfors function $W^3\to \Delta^3$ has then perhaps to be a harmonic map with maximal distortion at some basepoint, and if the contours are surfaces distinct from the sphere then there is no chance to have a covering along the boundary, but otherwise e.g. for $W^3$ the interspace of two concentric spheres it is not difficult to visualize a 3D-avatar of the Ahlfors map (just by taking the revolution of a map from a annulus to the disc, cf. our Fig.\[Annulus:fig\])\] 

J. Gray, [*On the history of the Riemann mapping theorem*]{}, Rend. Circ. Mat. Palermo (2) 34 (1994), 47–94. \[$\spadesuit$ from Riemann to Koebe’s area, through Osgood, etc.\] 

J. Gray, M. Micallef, [*The work of Jesse Douglas on minimal surfaces*]{}, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 293–302. \[$\spadesuit$ contains several critiques (mostly raised by Tromba 1983 [@Tromba_1983-PREPRINT]) about the rigor of the work of Douglas/Courant on the Plateau problem, especially when it comes to higher topological structure\] 

H. Grassmann, [*Die lineare Ausdehnungslehre*]{}, 1844. \[$\spadesuit$ summarized in Briekorn-Knörrer 1981/86 [@Brieskorn-Knörrer_1981/1986 p.122]: “A very important contribution to the development of the manifold concept was made by H. Grassmann in \[t\]his work, in which he spoke of $n$-tuply extended manifolds for the first time and developped, among other things, modern $n$-dimensional analytic geometry anfd linear algebra, in which the mathematical theory structure is worked out in a coordinated-free way, allowing the simplest treatment of problems in $n$-dimensional geometry, and in other fields as well.”\] 

G. Green, [*An essay on the application of mathematical analysis to the theories of electricity and magnetism*]{}. Printed for the Author by Whellhouse T. Nottingham, 1828, 72 pp. Also in: Mathematical Papers of George Green, Chelsea Publishing Co., 1970, 1–115; and reprinted in three parts in J. Reine Angew. Math. 39 (1850), 73–89; 44 (1852), 356–374; 47 (1854), 161–221. \[$\spadesuit$ this Crelle reprint was organized by W. Thomson $\spadesuit$ contains a form of the Dirichlet principle, presumably the first ever put in print $\spadesuit$ as to the connection with our problem of the Ahlfors map, the connection is evident and implicit in Ahlfors 1950 paper [@Ahlfors_1950], albeit the latter employs a variant of the Green’s function with “dipole” singularity placed at a boundary point\] 

P. Griffiths, J. Harris, [*Principles of Algebraic Geometry*]{}. Wiley, New York, 1978, 813 pp.; Wiley Classic Library edition, 1994; Russian transl., Vol.1, Mir, Moscow, 1982. \[$\spadesuit$ contains both an heuristic and formal proof of the $[\frac{g+3}{2}]$ gonality of closed Riemann surfaces of genus $g$, a result predicted since Riemann 1857 [@Riemann_1857] but only firmly validated in the modern era through the work of Meis 1960 [@Meis_1960] $\spadesuit$ see especially p.358, (special linear systems) and the proof presented is presumably quite close (??) to that of Kempf 1971 [@Kempf_1971]\] 

P. Griffiths, J. Harris, [*On the variety of special linear systems on a general algebraic curve*]{}, Duke Math. J. 47 (1980), 233–272. \[$\spadesuit$ p.236/7 gives a parameter count argument (via Riemann-Hurwitz and Riemann’s $3g-3$ moduli) to show that “a general curve $C$ of genus $g\ge 2$ cannot be expressed as a multiple cover of any curve $C'$ of genus $g'\ge 1$.” $\spadesuit$ this can be employed to show that the avatar of the Ahlfors map with range not a disc but a membrane of higher topological complexity fails generally to share the property of the usual circle-valued Ahlfors map of taking the boundary to the boundary\] 

M. Gromov, V.A. Rohlin, [*Embeddings and immersions in Riemannian geometry*]{}, Russian Math. Surv. 25 (1969), 1–57. \[$\spadesuit$ p.14: “In Appendix 4 we show that the real projective plane with a metric of positive curvature, in particular, the elliptic plane, cannot be isometrically $C^2$-embedded in ${\Bbb R}^4$.”\] 

M. Gromov, [*Filling Riemannian manifolds*]{}, J. Differential Geom. 18 (1983), 1–147. \[$\spadesuit$ present a modernized proof of the Loewner-Pu isosystolic inequality, by quoting Jenkins, hence indirectly Grötzsch, so back to Koebe–Poincaré, genealogically. Of course the uniformization required for Loewner (torus) and Pu (projective plane) are of a simpler nature, (Abel and Riemann, Schwarz resp.).\] 

M. Gromov, [*Spaces and questions*]{}, Preprint (1999).

T.H. Gronwall, [*Some remarks on conformal representation*]{}, Ann. of Math. (2) (1914/15), 72–76. \[$\spadesuit$ probably one of the first usage of the area-principle, cf. also Bieberbach 1914 [@Bieberbach_1914], Bieberbach 1916 [@Bieberbach_1916] and Faber 1916 [@Faber_1916]\] 

B. Gross, [*Real algebraic curves and their Jacobians*]{}, preprint, 1979. \[$\spadesuit$ this is cited in Jaffee 1980[@Jaffee_1980], yet probably never appeared as a such but might have been phagocytosed in the next entry Gross-Harris 1981 [@Gross-Harris_1981]\] 

B.H. Gross, J. Harris, [*Real algebraic curves*]{}, Ann. Sci. École Norm. Sup. (4) 14 (1981), 157–182. \[$\spadesuit$ modern account of Klein’s theory of real curves with many innovative ideas and viewpoints $\spadesuit$ the question posed on p.177 about the number of ovals for dividing plane smooth curves easily follows from the ideas of Rohlin[^133] 1974/75 [@Rohlin_1974/75], 1978 [@Rohlin_1978], compare Gabard 2000 [@Gabard_2000] for a detailed discussion\] 

A. Grothendieck, [*Techniques de construction en géométrie analytique*]{}, Sém. H. Cartan 1960/61, Exp.7, 9–17, Paris, 1962. \[$\spadesuit$ Teichmüller theory à la Grothendieck\] 

A. Grothendieck, [*Techniques de construction et théorèmes d’existence en géométrie algébrique IV. Les schéms de Hilbert.*]{} Sém. Bourbaki 221, 1960/61. \[$\spadesuit$\] 

A. Grothendieck, [*Techniques de construction et théorèmes d’existence en géométrie algébrique V. Les schéms de Picard.*]{} Sém. Bourbaki 232, 1960/61. \[$\spadesuit$\] 

A. Grothendieck, [*Esquisse d’un programme*]{}, 1984; reproduced in: L. Schneps and P. Lochak (eds), Geometric Galois Actions I. Around Grothendieck’s Esquisse d’un programme, London Math. Soc. Lecture Note Ser. 242, Cambridge Univ. Press, 1997, 5–48. \[$\spadesuit$ Teichmüller, Thurston, legos, etc. plus the Belyi-Grothendieck theorem that a closed Riemann surface is defined over $\Qbar$ iff it has only 3 ramifications over the sphere\] 

H. Grötzsch, [*Über einige Extremalprobleme der konformen Abbildung* ]{}, Ber. Verh. Sächs. Akad. Wiss. Leipzig 80 (1928), 497–502. 60, 78 \[$\spadesuit$ credited by Nehari 1953 [@Nehari_1953-Inequalities] for the solution of maximizing the derivative (distortion) at a given point of a multi-connected domain among schlicht functions bounded-by-one (extremals mapping upon a circular slit disc)\] 

H. Grötzsch, [*Über konforme Abbildung unendlich vielfach zusammenhängender schlichter Bereiche mit endlich vielen Häufungsrandkomponenten*]{}, Ber. Verh. Sächs. Akad. Wiss. Leipzig (1929), 51–86. 60, 78 \[$\spadesuit$ first proof of the circular slit disc mapping in infinite connectivity, see also Reich-Warschawski 1960 [@Reich-Warschawski_1960] for more subsequent references\] 

H. Grötzsch, [*Das Kreisbogenschlitztheorem der konformen Abbildung schlichter Bereiche*]{}, Ber. Verh. Sächs. Akad. Wiss. Leipzig (1931), 238–253. 60, 78 \[$\spadesuit$ another proof of the circular slit disc mapping in infinite connectivity, compare Grötzsch 1929 [@Groetzsch_1929]\] 

H. Grunsky, [*Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Bereiche*]{}, (Diss.) Schriften Math. Semin., Inst. angew. Math. Univ. Berlin 1 (1932), 95–140. \[$\spadesuit$ \[26.07.12\] p.140, Grunsky announces (without proofs) the result that a suitable combination $c(\frak x (\zeta;z) - \frak y (\zeta;z))$ of the horizontal $\frak x$ (resp. vertical $\frak y$) \[those being the fraktur letters for $x$ resp. $y$!\] slit-maps affords the solution to the problem of least area among all analytic functions normed by $f'(z)=1$ $\spadesuit$ on reading the rest of the paper it seems that the image might fail to be a disc, compare esp. p.135 where a similar least area problem is handled $\spadesuit$ this topic is addressed again in Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949] and Nehari 1952 [@Nehari_1952-BOOK]\] 60, 78 

H. Grunsky, [*Über die konforme Abbildung mehrfach zusammenhängender Bereiche auf mehrblättrige Kreise*]{}, Sitzungsber. Preu[ß]{}. Akad. (1937), 40–46. 60, 78 \[$\clubsuit$ new potential-theoretic proof of the circle map for domains\] 

H. Grunsky, [*Über die konforme Abbildung mehrfach zusammenhängender Bereiche aud mehrblättrige Kreise, II*]{}, Abh. Preu[ß]{}. Akad. Wiss. Math.-nat. Kl. 11 (1941), 1–8. 60, 78 \[$\clubsuit$ idem as the previous item\] 

H. Grunsky, [*Eindeutige beschränkte Funktionen in mehrfach zusammenhängenden Gebieten I*]{}, Jahresb. d. Deutsch. Math.-ver. 50 (1940), 230–255. 78 \[$\clubsuit$ extremal-problem description of circle maps for domains\] 

H. Grunsky, [*Eindeutige beschränkte Funktionen in mehrfach zusammenhängenden Gebieten II*]{}, Jahresb. d. Deutsch. Math.-ver. 52 (1942), 118–132. 78 \[$\clubsuit$ sequel of the previous item\] 

H. Grunsky, [*Zur Funktionentheorie in mehrfach zusammenhängenden Gebieten*]{}, Ber. Mathematikertagung Tübingen (1946), 68–69; in Coll. Papers, 245–6. 78 

H. Grunsky, [*Nachtrag zu meinen Arbeiten über “Eindeutige beschränkte Funktionen in mehrfach zusammenhängenden Gebieten”*]{}, Math. Z. 52 (1950), 852. 78 

H. Grunsky, [*Über die Fortsetzung eines auf einer berandeten Riemannschen Fläche erklärten meromorphen Differentials*]{}, Math. Nachr. 39 (1969), 87–96. \[$\clubsuit$ one of the rare work by Grunsky concerned with bordered surfaces, yet it does not seem to reprove the existence of a circle map à la Ahlfors\] 

H. Grunsky, [*Lectures on Theory of Functions in Multiply Connected Domains*]{}, Studia Mathematika, Skript 4, Vandenhoeck and Ruprecht in Göttingen, 1978. \[$\clubsuit$ all inclusive account but focusing to the case of domains (no Riemann surfaces)\] 

$\bigstar$ Gudkov (1918–1992), student of Andronov and Petrovskii. Famous for his sextic solution to Hilbert’s 16th problem (1 st part thereof on the mutual arrangements of ovals), and in particular for its refutation of Hilbert’s conjecture that there is only two possible configurations in degree $6$ (his own and the earlier one of Harnack). Influential over Arnold 1971 [@Arnold_1971/72], hence also over Rohlin, etc. Gudkov’s students includes Utkin, Polotovskii, Shustin, Korchagin.

D.A. Gudkov, [*Establishing all existing types of non-singular plane algebraic curves of the sixth order with real coefficients*]{}, Ph.D. Dissertation, Gorki, 1952. \[$\spadesuit$ the title sounds slightly overambitious, compare comment in the next entry Gudkov 1954 [@Gudkov_1954]\] 

D.A. Gudkov, [*The complete topological classification of non-singular real algebraic curves of the sixth order in the projective plane*]{}, Dokl. Akad. Nauk SSSR 98 (1954), 521–524. \[$\spadesuit$ albeit this paper contains some mistakes (too prohibitive), it must contain the first serious prohibitions via the Hilbert-Rohn method consolidated by Andronov-Pontrjagin theory of roughness (structural stability) $\spadesuit$ thus the method is still of interest, and its charming power is perhaps only eclipsed by more topological variants of the Arnold-Rohlin era, e.g. Arnold 1971 [@Arnold_1971/72] or Rohlin 1972/72 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov]\] 

D.A. Gudkov, [*On certain questions in the topology of plane algebraic curves*]{}, Mat. Sb. (N.S.) 58 (1962), 95–127. \[$\spadesuit$ some overlap with Brusotti 1921 [@Brusotti_1921], yet an extension thereof to curves having “turning points” (=cusps)\] 

D.A. Gudkov, [*On the topology of plane algebraic curves*]{}, Doctor’s Thesis, Gor’kii, 1969, 1–351. \[$\spadesuit$ first disproof of Hilbert’s conjecture about the nonexistence of the scheme $\frac{5}{1}5$. According to Polotovskii 1996 [@Polotovskii_1996-D-A-Gudkov]’s overview: “It is interesting to remark that the first proof of this fact in \[18\](=this entry) was extraordinarily complicated. It takes up $28$ pages of text, is a “pure existence proof”, and was obtained by means of a combination of the Hilbert-Rohn method with quadratic transformations. Shortly after D.A. Gudkov suggested significantly simpler [*constructions*]{} of curves having this scheme, see \[19\](=1971 [@Gudkov_1971-const-new-ser-M-curv]), \[21\], \[23\]. ”\] 

D.A. Gudkov, [*Complete topological classification of the disposition of ovals of a sixth order curve in the projective plane*]{}, Gor’kov. Gos. Univ. Ucen. Zap. Vyp. 87 (1969), 118–153. \[$\spadesuit$ where Hilbert got corrected?!\] 

D.A. Gudkov, G.A. Utkin, [*The topology of sixth-order curves and fourth-order surfaces*]{}, Gor’kov. Gos. Univ. Učen. Zap. Vyp. 87 (1969), 154–211; English transl., in: Nine papers on Hilbert’s 16th problem, Amer. Math Soc. Transl. (2) 112 (1978). \[$\spadesuit$ where Hilbert got corrected by Gudkov: disproof of one of one of Hilbert’s conjectures on the arrangement of ovals of plane $M$-sextic (that is Harnack-maximal). Gudkov corrected Hilbert’s conjecture by including the newly discovered so-called Gudkov curve and showed that only the trinity Harnack 1876 [@Harnack_1876], Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege] and Gudkov’s 1969 can exist, yielding a complete classification up to isotopy of $M$-sextic (when combined with the prohibition à la Hilbert-Rohn which Gudkov was the first to implement correctly in 1954 [@Gudkov_1954] upon combining the classical method with “roughness” à la Andronov-Pontrjagin) $\spadesuit$ more generally this work contains a complete classification of all 56 isotopy classes realized by plane sextic (hence a complete solution to Hilbert’s 16th problem), but the story does not end here (cf. e.g. Rohlin’s 1978 enhancement by complex characteristics à la Klein, hence concomitant with Ahlfors, cf. our discussion in Sec.\[Klein-Rohlin-conj:sec\]) $\spadesuit$ another subsequent step is Nikulin 1979/80 [@Nikulin_1979/80] stronger classification of $C_6$ up to rigid-isotopy showing the completeness of the Klein-Rohlin invariants (the proof uses the whole apparatus of the complex geometry of K3 surfaces)\] 

D.A. Gudkov, [*Construction of a new series of $M$-curves*]{}, Dokl. Akad. Nauk SSSR 200 (1971), 1269–1272; English transl., Soviet Math. Dokl. 12 (1971), 1559–1563. \[$\spadesuit$ Gudkov construction is reproduced (at least in abridged form) in Gudkov 1974/74 [@Gudkov_1974/74], in A’Campo 1979 [@A'Campo_1979]\]$\bigstar$$\bigstar$$\bigstar$ 

D.A. Gudkov, [*Construction of a curve of degree $6$ of type $\frac{5}{1}5$*]{}, Izv. Vyssh. Uchebn. Zaved. Mat. 3 (1973), 28–36; English transl., Soviet Math. (Iz. VUZ) (1973). \[$\spadesuit$ simplification in the disproof of Hilbert’s conjecture on the arrangement of ovals of plane $M$-sextics (that is Harnack-maximal). Gudkov corrected Hilbert’s conjecture by including the newly discovered so-called Gudkov curve (this contribution appears first in Gudkov’s Doctor Thesis (1969 [@Gudkov_1969-Doctor's-Thesis]), yet in a much more sophisticated way\] 

D.A. Gudkov, A.D. Krakhnov, [*On the periodicity of the Euler characteristic of real algebraic $(M-1)$-manifolds*]{}, Funkt. Anal. Prilozhen. 7 (1973), 15–19; English transl., Funct. Anal. Appl. 7 (1973), 98–102 \[$\spadesuit$ same result as in Kharlamov 1973 [@Kharlamov_1973/73], i.e. an obstruction on $(M-1)$-schemes, e.g. for plane curves of degree $2k$, which is independent of the Hilbert-Rohn-Gudkov geometric method\] 

D.A. Gudkov, [*The topology of real projective algebraic varieties*]{}, Uspekhi Mat. Nauk 29 (1974), 3–79; English transl., Russian Math. Surveys 29 (1974), 1–79. \[$\spadesuit$ a masterpiece survey full of historical details and mathematical tricks $\spadesuit$ contains an extensive bibliography (157 entries) of early real algebraic geometry (in Germany, Italy and Russia), mostly in the spirit of Hilbert (by contrast to Klein’s more Riemannian approach) $\spadesuit$ p.2 and p.17 contain in my opinion a historical inaccuracy which imbued alas some of the subsequent literature (e.g. A’Campo 1979 [@A'Campo_1979 p.01], Jaffee 1980 [@Jaffee_1980 p.82]), namely Hurwitz 1891–92 is jointly credited for the intrinsic proof of Harnack’s inequality ($r\le
g+1$), while it goes back of course to Klein 1876 [@Klein_1876] (and not only Klein 1892 lectures as cited by Gudkov)\] 

D.A. Gudkov, [*On the topology of algebraic curves on a hyperboloid*]{}, Uspekhi Mat. Nauk 34 (1979), 26–32; English transl., Russian Math. Surveys 34 (1979), 27–35. \[$\spadesuit$ p.27: “Algebraic curves on a hyperboloid of one sheet have been studied for a long time. A start was made by Plücker and Chasles in the mid-nineteenth century (see \[1\], Ch.IV)=(Klein 1926 [@Klein_1926-Vorlesungen-über-die-Entwicklung]). In a fundamental article \[2\](=Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege]) Hilbert proved \[…\]”\] 

D.A. Gudkov, [*Generalization of a theorem of Brusotti for curves on a surface of second order*]{}, Funkt. Anal. Prilozhen. 14 (1980), 20–24, 96; English transl., Funct. Anal. Appl. 14 (1980), 15–18. \[$\spadesuit$ p.15: “Brusotti \[1\](=1921 [@Brusotti_1921]) proved the following assertion in 1921.—[*Brusotti’s theorem.*]{} If all the singular points of a curve $F$ in the complex projective plane ${\Bbb C}P^2
(x_0:x_1:x_2)$ are simple double points, then the simplifications of these singular points are independent.—This means that by adding an arbitrarily small term of degree $m$ (with real coefficients) to the polynomial $F$, it is possible to get a curve $\Phi$ such that each real singular point of the curve $F$ either remains or else simplifies in one of the two possible ways (depending on our choice, cf. \[5\]), and each pair of imaginary conjugate singular points either remains or vanishes (depending on our choice).”\] 

D.A. Gudkov, G.M. Polotovskii, [*Stratification of a space of fourth-order curves. Contiguity of strata*]{}, Uspekhi Mat. Nauk 42 (1987), 152. \[$\spadesuit$\] 

D.A. Gudkov, [*Plane real projective quartic curves*]{}, in: Lecture Notes in Math. 1346, Springer, Berlin, 1988, 341–347. \[$\spadesuit$ cited in Shustin 1990/91 [@Shustin_1990/91-Geom-of-discr-alg-curve] for a complete description of the discriminant of quartics $\spadesuit$ presumably some overlap with the previous entry $\spadesuit$ is this description of Gudkov compatible with our crazy disconnection result in Sec.\[Disconnection-of-the-empty-locus:sec\]\] 

D.A. Gudkov, [*N.I. Lobachevskii. Biographical enigmas*]{}, Nizhnii Novgorod, 1992 (monograph in print). \[$\spadesuit$\] 

L. Guillou, A. Marin, [*Une extension d’un théorème de Rohlin sur la signature*]{}, C.R. Acad. Sci Paris Ser. A 285 (1977), 95–98. \[$\spadesuit$ useful in correcting Rohlin’s proof of the Gudkov hypothesis $p-n\equiv k^2 \pmod 8$ (compare Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000]) $\spadesuit$ this paper is an announcement and more details are to be found in the book Guillou-Marin 1986 [@Guillou-Marin_1986 p.97–118]\] 

L. Guillou, A. Marin, [*A la recherche de la topologie perdue, I Du côté de chez Rohlin, II Le côté de Casson*]{}, Progress in Math. 62, Birkäuser, Boston, Basel, Stuttgart, 1986. \[$\spadesuit$ contains French translation of Rohlin’s ground-breaking works in low-dimensional differential topology, plus some of its applications to real algebraic geometry\] 

R.C. Gunning, [*Lectures on Riemann surfaces*]{}. Princeton Acad. Press, Princeton, 1966. \[$\spadesuit$\] $\bigstar$ 

R.C. Gunning, R. Narasimhan, [*Immersion of open Riemann surfaces*]{}, Math. Ann. 174 (1967), 103–108. \[$\spadesuit$ no directly visible connection with Ahlfors 1950, but there must be some link in the long run\] 

R.C. Gunning, [*Lectures on Riemann surfaces: Jacobi varieties*]{}, Princeton Univ. Press, Princeton, N.J., 1972, 189 pp. \[$\spadesuit$ new (essentially topological?) proof of Meis’ result upon the gonality of complex curves (=closed Riemann surfaces) $\spadesuit$ \[21.06.12\] the following extract of H.H. Martens’s review in MathReviews is capital for it brings the hope to gain a Teichmüller theoretic approach to the existence of circle maps with the best possible bounds (hence hinting how to recover Ahlfors and even Gabard 2006 [@Gabard_2006] by an analytic (or rather geometric!) approach competing seriously with the naive topological proof of the writer): “A pièce de résistance is served in the appendix in the form of a proof of the existence of functions of order $\le [\frac{1}{2}(g+3)]$ on any closed Riemann surface. This result was previously obtained by T. Meis 1960 [@Meis_1960] using Teichmüller space techniques, and it is a special case of the more general results of Kleiman-Laksov 1972 [@Kleiman-Laksov_1972] and Kempf 1971 [@Kempf_1971].”\] $\bigstar$$\bigstar$$\bigstar$ 

B. Gustafsson, [*Quadrature identities and the Schottky double*]{}, Acta Appl. Math. 1 (1983), 209–240. \[$\spadesuit$ \[13.10.12\] can the theory be extended to non-planar domains?\] 

B. Gustafsson, [*Applications of half-order differentials on Riemann surfaces to quadrature identities for arc-length*]{}, J. Anal. Math. 49 (1987), 54–89. \[$\spadesuit$\] 

A. Haas, [*Linearization and mappings onto pseudocircle domains*]{}, Trans. Amer. Math. Soc. 282 (1984), 415–429. \[$\spadesuit$ Koebe’s Kreisnormierungsprinzip for positive genus, uniqueness complement in Maskit 1989 [@Maskit_1989]\] 

J. Hadamard, [*Sur le principe de Dirichlet*]{}, Bull. Soc Math. France (1906). \[$\spadesuit$ p.135 an example is given of a continuous function on the boundary of a domain such that none functions satisfying the boundary prescription has finite Dirichlet integral $\spadesuit$ a similar example was given in Prym 1871 [@Prym_1871], where a continuous function is given on the circle such that the harmonic function matching this boundary data (whose existence is derived by another procedure, e.g. the Poisson integral) has infinite Dirichlet integral $\spadesuit$ of course, heuristically any Prym’s boundary data must be of the Hadamard type (precisely by virtue of the just corrupted Dirichlet principle!): if the harmonic solution explodes any vulgar solution (hence less economical) must explode as well\] 

J. Hadamard, [*Mémoire sur le problème d’analyse relatif à l’équilibre de plaques élastiques encastrées*]{}, Mémoires présentés par divers savants à l’Académie des Sciences 33 (1908), 128pp. \[$\spadesuit$ Discussion of the famous method, named after Hadamard, of variation of domains $\spadesuit$ further developed by Schiffer especially\] 

G. Halphen, [*Mémoire sur la classification des courbes gauche algébriques*]{}, J. École Polytech. 52 (1882), 1–200. \[$\spadesuit$ sharing the price with M. Noether\] 

R.S. Hamilton, [*The Ricci flow on surfaces*]{}, In: Mathematics and General Relativity (Santa Cruz, CA, 1986). Contemporary Mathematics 71, Amer. Math. Soc., Providence, 1988, 237–262. \[$\spadesuit$ uniformization of surfaces via the 2D-Ricci flow (at least in the compact case)\] $\bigstar$$\bigstar$$\bigstar$

M. Hara, M. Nakai, [*Corona theorem with bounds for finitely sheeted disks*]{}, Tôhoku Math. J. 37 (1985), 225–240. 50 \[$\clubsuit$ applies Ahlfors mapping in a quantitative fashion (making use of its degree in contrast to Alling 1964 [@Alling_1964]) $\clubsuit$ naive question (ca. Sept. 2011) can we improve the bounds by appealing instead to Gabard 2006 [@Gabard_2006]\] 

A. Harnack, [*Ueber die Vieltheiligkeit der ebenen algebraischen Curven*]{}, Math. Ann. 10 (1876), 189–198. \[$\spadesuit$ a proof is given (via Bézout’s theorem) that a smooth plane real curve of order $m$ possesses at most $g+1=\frac{(m-1)(m-1)}{2}+1$ components (reellen Züge) and such Harnack-maximal curves are constructed for each degree via a method of small perturbation $\spadesuit$ as everybody knows a more intrinsic proof was given by Klein 1876 [@Klein_1876] by simply appealing to Riemann’s definition of the genus as the maximum number of retrosections not morcellating the surface $\spadesuit$ a more exotic derivation of the Harnack bound (using Riemann-Roch) is to be found in Enriques-Chisini 1915 [@Enriques-Chisini_1915-1918], whose argument actually supplies a proof of the so-called Bieberbach-Grunsky theorem (cf. Bieberbach 1925 [@Bieberbach_1925], Grunsky 1937 [@Grunsky_1937] and for instance A. Mori 1951 [@Mori_1951]) which is the planar version of the Ahlfors map\] 

A. Harnack, [*Die Grundlagen der Theorie des logarithmischen Potentiales, und der eindeutigen Potentialfunktionen in der Ebene*]{}, Teubner, Leipzig, 1887. 

R. Hardt, D. Sullivan, [*Variation of the Green function on Riemann surfaces and Whitney’s holomorphic stratification conjecture*]{}, Publ. Math. I.H.E.S. (1989), 115–138. \[$\spadesuit$ \[10.08.12\] the starting point of the paper (p.115) is a representation of a Riemann surface as a $k$-sheeted branched covering of the unit disc (denoted $B$) with branch point $a_1,
\dots, a_l$ in $B_{1/2}$ (ball of radius one-half) $\spadesuit$ this situation resembles sufficiently to Ahlfors 1950 [@Ahlfors_1950] to ask if a precise connection can be made $\spadesuit$ of course one may notice that a map of the type required (by Hardt-Sullivan) exists for any interior of a compact bordered Riemann surface: indeed take a Ahlfors map or just a circle map (existence ensured by Ahlfors 1950 [@Ahlfors_1950], or other sources, e.g. Gabard 2006 [@Gabard_2006]) and then upon post-composing by a power-map $z\mapsto z^n$ we may contract the modulus of the branch points to make them as small as we please upon choosing $k$ large enough $\spadesuit$ perhaps the dual game of looking at largest possible winding points should relate to the problem of finding the circle maps of lowest possible degrees $\spadesuit$; at least one should be able to define a conformal invariant of a bordered surface $F$ by looking at the largest possible modulus of a branch point of a circle map (of course composing with a disc-automorphism, the branch point can be made very close to $1$, so one requires a normalization, e.g. mapping a base-point of $F$ to $0$) $\spadesuit$ this defines a $[0,1)$-valued numerical invariant of a marked compact bordered Riemann surface $(F,b)$; how does it depends on $b$ when the latter is dragged through the (fixed) surface and does this invariant takes the value $0$ only for when $F$ is the disc $\spadesuit$ as another variant without marking, we may always assume that $0$ is nor ramified, and we may look for the largest radius free of ramification, this defines another numerical invariant taking values in $]0,1]$; obviously it takes the value one only when $F$ is topologically a disc (Riemann mapping theorem maybe in the variant firmly established by Schwarz) $\spadesuit$ maybe in the spirit of Bloch there is an absolute (strictly) positive lower bound on this “schlicht radius” at least for prescribed topological characteristic (i.e. the invariant $p$ and $r$ counting the genus and the contours) $\spadesuit$ call this constant $B_{p,r}$: how does it depend on $p,r$ asymptotically (maybe convergence to $0$ if $p,r \to \infty$); further is the infimum achieved by some surfaces, if so can we describe the extremal surfaces (naive guess the ramification is then cyclotomic); compare maybe work of Minda ca. 1983 for related questions\] 

G.H. Hardy, [*On the mean modulus of an analytic function*]{}, Proc. London Math. Soc. 14 (1915), 269–277. \[$\spadesuit$\] 

A.N. Harrington, [*Conformal mappings onto domains with arbitrarily specified boundary shapes*]{}, J. d’Anal. Math. 41 (1982), 39–53. \[$\spadesuit$ extension of Koebe’s KNP; similar result in Brandt 1980 [@Brandt_1980] $\spadesuit$ method: potential theory and (algorithmic) Brouwer’s fixed point $\spadesuit$ variant of proof in Schramm 1996 [@Schramm_1996]\] 

J. Harris, [*On the Severi problem*]{}, Invent. Math. 84 (1986), 445–461. \[$\spadesuit$ based on virtually the same idea as Severi 1921, and Brusotti 1921, cf. e.g. Shustin 1990/91 [@Shustin_1990/91-Geom-of-discr-alg-curve]\] 

R. Hartshorne, [*Algebraic Geometry*]{}, Grad. Texts in Math. 49, Springer-Verlag, 1977. \[$\spadesuit$ some elementary aspects of curves and surfaces via the sheaf theoretic approach (Leray, etc.)\] 

M. Hasumi, [*Invariant subspaces for finite Riemann surfaces*]{}, Canad. J. Math. 18 (1966), 240–255. \[$\spadesuit$ extension of Beurling’s theorem (1949 [@Beurling_1949]) for the disc to the case of finite bordered Riemann surface, yet without using the Ahlfors map, but cite Royden 1962 [@Royden_1962] which is closely allied\] 

O. Haupt, [*Ein Satz über die Abelschen Integrale 1. Gattung*]{}, Math. Z. 6 (1920), 219–237. \[$\spadesuit$ only cited for the Riemann parallelogram method, which bears (perhaps?) some resemblances with Gabard 2006 [@Gabard_2006] $\spadesuit$ work is influenced by Prym, tries to answer a question of Klein, further influence of Wirtinger, etc. $\spadesuit$ for modern ramification cf. Gerstenhaber 1953 [@Gerstenhaber_1953]\] 

S.Ya. Havinson, [*On an extremal problem in the theory of analytic functions*]{}, (Russian) Uspekhi Mat. Nauk. 4 (1949), 158–159. \[$\spadesuit$\]$\bigstar$ 

S.Ya. Havinson, [*On extremal properties of functions mapping a region on a multi-sheeted circle*]{}, Doklady Akad. Nauk. SSSR (N.S.) 88 (1953), 957–959. (Russian.) 60 $\bigstar$ 

S.Ya. Havinson, [*Extremal problems for certain classes of analytic functions in finitely connected regions*]{}, Mat. Sb. (N.S.) 36 (78) (1955), 445–478; Amer. Math. Soc. Transl. 5 (1957), 1–33. 78 \[$\spadesuit$ generalized linear extremal problems (finite connectivity), i.e. maximization of the modulus of the derivative replaced by an arbitrary linear functional\]$\bigstar$ 

S.Ya. Havinson, G.C. Tumarkin, [*Existence of a single-valued function in a given class with a given modulus of its boundary values in multiply connected domains*]{}, (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 543–562. \[$\spadesuit$ quoted in Khavinson 1984 [@Khavinson-Dimitri_1984 p.378], and the same problem of prescribing the boundary modulus had been already treated by Grunsky 1942 [@Grunsky_1942]\]$\bigstar$ 

S.Ya. Havinson, [*Analytic capacity of sets, joint nontriviality of various classes of analytic functions and the Schwarz lemma in arbitrary domains*]{}, Mat. Sb. 54 (96) (1961), 3–50; English transl., Amer. Math. Soc. Transl. (2) 43 (1964), 215–266. 47, 50, 78 \[$\spadesuit$ uniqueness of the (Ahlfors) extremal function for domains of infinite connectivity (similar work in Carleson 1967 [@Carleson_1967-book]); but Khavinson’s work goes deeper (according to Hejhal 1972 [@Hejhal_1972]) into the study of the behavior of the extremal function\] 

S.Ya. Havinson, [*Factorization theory for single-valued analytic functions on compact Riemann surfaces with boundary*]{}, Uspekhi Math. Nauk. 44 (1989), 155–189; English transl., Russian Math. Surveys 44 (1989), 113–156. \[$\spadesuit$ p.117 explains the usual trick of annihilating the $2p+(h-1)$ periods along essential cycles on a finite Riemann surface, for which we may take any $r-1$ of the boundary contours, as well as meridians and parallels taken along each handle $\spadesuit$ so this is quite close to our naive attempt to reprove Ahlfors’ theorem, compare Sec.\[Ahlfors-proof:sec\]\] 

S.Ya. Havinson, [*Duality relations in the theory of analytic capacity*]{}, St. Petersburg Math. J. 15 (2004), 1–40. (Russian version published in 2003.) \[$\spadesuit$ Ahlfors function appears on pp.2,11,13,20 $\spadesuit$ the terminology “analytic capacity” (or “Ahlfors capacity”) is credited to V.D. Erokhin’s: “In accordance with V.D. Erokhin’s proposal (1958), the quantity $\gamma(F)$ has been called the [*analytic capacity*]{} or the [*Ahlfors capacity*]{} since that time.”\] 

N.S. Hawley, M. Schiffer, [*Half-order differentials on Riemann surfaces*]{}, Acta Math. 115 (1966), 199–236. \[$\spadesuit$\] 

N.S. Hawley, M. Schiffer, [*Riemann surfaces which are double of plane domains*]{}, Pacific J. Math. 20 (1967), 217–222. 78 \[$\spadesuit$\] 

N.S. Hawley, [*Weierstrass points of plane domains*]{}, Pacific J. Math. 22 (1967), 251–256. 78 \[$\spadesuit$ addresses the question of the distribution of Weierstrass points upon the Schottky double of a plane domain. Precisely, for a planar membrane with hyperelliptic double, all W-points are located on the boundary. The author gives an example, derived from a real quartic with 4 ovals, whose W-points are not confined to the boundary. Such questions make good sense over positive genus membranes and are perhaps worth investigating further. Probably updates are already known, and one would like to explicit any possible relation between W-points and the degree of the Ahlfors function. Compare for this issue, Yamada 1978 [@Yamada_1978]\] 

M. Hayashi, [*The maximal ideal space of the bounded analytic functions on a Riemann surface*]{}, J. Math. Soc. Japan 39 (1987), 337–344. \[$\spadesuit$ the following property: “the natural map of a Riemann surface $R$ into its maximal ideal space $\frak M (R)$ (this is an embedding if we assume that the algebra $H^{\infty}(R)$ of bounded analytic functions separates points) is a homeomorphism onto an open subset of $\frak M (R)$” has some application to the uniqueness of the Ahlfors function (cf. Gamelin 1973 [@Gamelin_1973-Extremal-I]), as well as to its existence $\spadesuit$ the bulk of this paper consists in giving examples where this property fails answering thereby a question of Gamelin 1973\] 

Z.-X. He, [*Solving Beltrami equation by circle packing*]{}, Trans. Amer. Math. Soc. 322 (1990), 657–670. \[$\spadesuit$ includes another proof of GKN (generalized Kreisnormierung) where a compact bordered Riemann surface is conformally mapped upon a circular domain in a space-form (=constant curvature) \[of the same genus?\] $\spadesuit$ similar statement obtained by Haas 1984 [@Haas_1984] and Maskit 1989 [@Maskit_1989] (curiously non-cited here)—maybe also Jost 1985 [@Jost_1985] $\spadesuit$ perhaps the “syntax” of the main result (Thm 5.1, p.669) must be slightly corrected, probably by assuming the contours of $\partial{\overline{\Omega}}$ to bounds discs in the surface $M$ (equivalently to be null-homotopic)\] 

Z.-X. He, O. Schramm, [*Fixed points, Koebe uniformization and circle packings*]{}, Ann. of Math. (2) 137 (1993), 369–406. \[$\spadesuit$ the deepest advance upon the KNP=Kreisnormierungsprinzip (raised by Koebe 1908 UbaK3 [@Koebe_1908_UbaK3]), which is established for countably many boundary components $\spadesuit$ The general case is still unsettled today (2012), and maybe undecidable within ZFC? (just a joke, of course)\] 

Z.-X. He, O. Schramm, [*On the convergence of circle packings to the Riemann map*]{}, Invent. Math. 125 (1996), 285–305. \[$\spadesuit$ improvement and generalization of the Rodin-Sullivan proof (1987 [@Rodin-Sullivan_1987]), making it logically independent of RMT (thus reproving it via the technology of circle packings) $\spadesuit$ \[08.10.12\] what about the same game for the Ahlfors map?\] 

E. Heine, [*Ueber trigonometrische Reihen*]{}, J. Reine Angew. Math. 71 (1870), 353–365. \[$\spadesuit$ the rôle of uniform convergence is emphasized (i.e. Weierstrass’ notion, yet first only familiar to his direct circle of students)\] 

M.H. Heins, [*Extremal problems for functions analytic and single-valued in a doubly connected region*]{}, Amer. J. Math. 62 (1940), 91–106. 78 \[$\spadesuit$ quoted (joint with Carlson 1938 [@Carlson_1938] and Teichmüller 1939 [@Teichmueller_1939-Dreikreisesatzes]) in Grunsky 1940 [@Grunsky_1940] as one of the forerunners of the extremal problem for bounded analytic functions (alias Ahlfors map, subsequently)\] 

M. Heins, [*On the iteration of functions which are analytic and single valued in a given multiply connected region*]{}, Amer. J. Math. 63 (1941), 461–480. 78 \[$\spadesuit$ regarded by Minda 1979 [@Minda_1979 p.421] as the proper originator of the [*annulus theorem*]{} (i.e., an analytic self-map of an annulus can take the generator of the fundamental group only upon a $0$ or $\pm 1$ multiple of itself, and the $\pm 1$ case forces the map to be a conformal automorphism)\] 

M. Heins, [*A lemma on positive harmonic functions*]{}, Ann. of Math. (2) 52 (1950), 568–573. 60, 78 \[$\spadesuit$ may contain another proof of the existence of the Ahlfors function (at least a circle map), yet not very clear which degree Heins’ argument supplies $\clubsuit$ in fact since the quantity $m$ appearing on p.571 for a generating system of the fundamental group is easily found to be $m=2p+(r-1)$ (where $p$ is the genus and $r$ the number of contours) it is quite likely (albeit the writer has no certitude!) that Heins’ method may reproduce (by specialisation) exactly Ahlfors upper bound upon the degree of a circle map $\spadesuit$ \[06.10.12\] for a possible corroboration of this intuition, check also the subsequent paper Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF] which truly seems to get again the $r+2p$ bound of Ahlfors (1950) $\spadesuit$ treats Pick-Nevanlinna interpolation for a bordered surface (extending the work of Garabedian 1949 [@Garabedian_1949])\] 

M. Heins, [*Symmetric Riemann surfaces and boundary problems*]{}, Proc. London Math. Soc. (3) 14A (1965), 129–143. \[$\spadesuit$ looks closely allied to Ahlfors 1950 [@Ahlfors_1950], which is not cited, but so are some direct descendants, Read 1958 [@Read_1958_Acta] and Royden 1962 [@Royden_1962] $\spadesuit$ enters into the category of “transplanting papers” where some result for the disc is lifted to a compact bordered surface (=membrane) $\spadesuit$ in the present case M. Riesz’s theorem on the conjugate Fourier series, and the unique decomposition of $f\in L^p$ into interior/exterior Fatou boundary functions of functions in $H_p$\] 

M. Heins, [*Hardy classes on Riemann surfaces*]{}, Lecture Notes in Mathematics 98, Springer, 1969. \[$\spadesuit$ p.59–65 contains a re-exposition of Heins 1950 [@Heins_1950], yielding probably an alternate proof of the Ahlfors circle maps\]$\bigstar$$\bigstar$$\bigstar$

M. Heins, [*Nonpersistence of the Grenzkreis phenomenon for Pick-Nevanlinna interpolation on annuli*]{}, Ann. Acad. Sci. Fenn. Ser. A. 596 (1975), 1–21. 50, 78 \[$\spadesuit$ cited in Jenkins-Suita 1979 [@Jenkins-Suita_1979] $\spadesuit$ from MathReviews: “Let $A$ be a subset of the open unit disc $\Delta$. Consider the family of functions $f$ regular in $\Delta$ such that $\vert f\vert \le 1$ on $\Delta$ and at each point of $A$, a specified initial Taylor section is assigned. For $b\in \Delta$, let $W(b)$ denote the set of values assumed by the functions of the family at $b$. \[As Heins explains in the original article “W(b) is termed the [*Wertevorrat*]{} of the family at $b$.”\] The Pick-Nevanlinna-Grenzkreis phenomenon asserts that if there is more than one function in the family and $b\in \Delta-A$, then the set $W(b)$ is a closed circular disc of positive radius. The author constructs a counter example to show that this is no longer true for multiply connected domains. Let $\Omega$ be the annulus $r<\vert z\vert< r^{-1}$ and let $B(c)$ denote the set of functions $f$, analytic in $\Omega$, such that $\vert f\vert \le
1$, $f(-1)=0$ and $f'(-1)=c$, where $c$ is small and positive. The author shows that in this case $W(b)$, $b\neq -1$, is a set with nonempty interior but is not a circle.—A result of Garabedian 1949 [@Garabedian_1949] asserts that if $\Omega$ is a domain of finite connectivity such that no boundary component reduces to a point and if the values of the function are assigned at a finite number of points, then the unique extremal function which takes at $b$ a given value on ${\rm Fr} W(b)$ maps $\Omega$ onto $\Delta$ with constant valency. The author shows that this remains true for his example although the initial Taylor section assigned is of order one at $z=-1$. There is also a general discussion of the problem in the general setting of Riemann surfaces with finite topological characteristics.” $\spadesuit$ \[07.10.12\] as a modest task one may wonder if Heins’ paper reproves Ahlfors’ existence of circle maps of degree $\le r+2p$. As a pessimistic remark it seems that there is a wide variety of extremal problems, somehow reflecting our mankind capitalistic/competitive aberration, making it unclear what the God given problem is, especially the one capturing circle maps of lowest possible degree $\spadesuit$ more optimistically it is clear that there is a fascinating body of knowledge among such problems (interpolation by prescribed Taylor section). Given a finite Riemann surface $\overline F$ (bordered), choose a finite set $A$ each point being decorated by a Taylor section (w.r.t. a local uniformizer), look at all functions bounded-by-one matching the Taylor data. For any $b\in F-A$, define $W(b)\subset \Delta$ as the set of values assumed at $b$ by functions of the family. $\spadesuit$ as above we look at the function $f_{b,w}$ taking at $b$ a given value $w$ of the frontier of $W(b)$. Q1. Is then Garabedian’s result on the constant valency of $f_{b,w}\colon F \to \Delta$ true in this non-planar setting? If yes what is the degree of the corresponding circle map (Q2). Of course the case where $A=\{a\}$ is a singleton with Taylor section $f(a)=0$ ($b\neq a$) and $w$ chosen so as to maximize the modulus in the set $W(b)$ gives exactly the Ahlfors map $f_{a,b}$ studied in Ahlfors 1950 [@Ahlfors_1950]. This induces (via the assignment $\overline{F}\mapsto \vert f_{a,b}(b)\vert$) a real-valued function ${\cal M}_{r,p}\to ]0,1[$ on the moduli space of surfaces with two marked points. One can dream about understanding the Morse theory of this function. $\spadesuit$ The answer to our two naive questions (Q1, Q2) is apparently already in Heins’ paper, for Jenkins-Suita 1979 [@Jenkins-Suita_1979 p.83] write: “Quite recently Heins \[10\](=1975 [@Heins_1975]) proved uniqueness of the extremal function $f_0$ which maximizes ${\rm Re}(e^{i\theta} f(z_0))$ among the class of analytic functions $f$ bounded by unity and with given Taylor sections \[…\] on a compact bordered Riemann surface $\Omega$. He also proved the extremal $f_0$ maps $\Omega$ onto a finite sheeted covering of the unit disc and gave a bound on the number of sheets called the [*Garabedian bound*]{}.” $\spadesuit$ \[07.10.12\] as a micro-objection the terming “Garabedian bound” is probably slightly unfair for Ahlfors as the latter probably knew it (in the case of a single interpolating point) without Garabedian’s helping hand (at least for circle maps, yet arguably not for the Ahlfors’ extremals) (cf. of course the acknowledgments to be found in Ahlfors 1950 [@Ahlfors_1950], but see also Nehari 1950 [@Nehari_1950] where the Ahlfors upper bound $r+2p$ is credited back to Ahlfors’ Harvard lectures in Spring 1948) $\clubsuit$ \[12.10.12\] Heins’ statement is as follows (p.18): “(3) [*The Garabedian bound*]{}. We consider a determinate Pick-Nevanlinna problem relative to $\Omega$ with a finite set of data and denote the solution by $f$. \[…\] For an interpolation point $b$ we let $\nu(b)$ denote the order of interpolation at $b$ augmented by one. We let $\nu$ denote the sum of the $\nu(b)$ taken over the interpolation points $b$. The Euler characteristic of $\Omega$ will be denoted by $\chi$. We shall show—[**Theorem 8.2**]{} [*f has at most $\nu+\chi$ zeros counted by multiplicity.*]{} $\clubsuit$ this statement subsumes the upper estimate of Garabedian, but also that of Ahlfors: indeed Ahlfors extremal problem is the case where there is a single interpolating point of order zero. So $\nu=0+1=1$. Now given a bordered surface $\Omega$ of genus $p$ with $r$ contours, we have $\chi(\Omega)=2-2p-r$ \[beware that Heins seems to work with the old convention about the sign of the Euler characteristic, hence just change his formula to $\nu-\chi$\]. So we get $\deg f \le \nu-\chi= \nu-2+2p+r\approx
r+2p$ (note a little arithmetical discrepancy from Ahlfors, surely easily explained) $\clubsuit$ Heins’ proof uses the following tools: $\bullet$ basic facts concerning Hardy classes on Riemann surfaces for which one is referred to Heins 1969 [@Heins_1969-LNM-Hardy] $\bullet$ a variational formula of F. Riesz 1920 [@Riesz_1920-Ueber-Potenzreihen] $\bullet$ the theorem of Cauchy-Read (cf. Read 1958 [@Read_1958_Acta]) $\bullet$ the Fatou boundary function, $\bullet$ the Green’s function $\bullet$ the qualitative Harnack inequality $\spadesuit$ a slightly different proof of a much related result (on “Garabedian bound”) is given as Theorem 3 of Jenkins-Suita 1979 [@Jenkins-Suita_1979], which uses maybe less machinery (?), an instead of Read the closely allied paper Royden 1962 [@Royden_1962]. Yet Jenkins-Suita’s proof depend on Heins’ proof when it comes to the “interpolation divisor”\] 

M. Heins, [*Carathéodory bodies*]{}, Comm. in honorem Rolf Nevanlinna LXXX annos nato, Ann. Acad. Sci. Fenn. Ser. A.I, Math. 2 (1976), 203–232. \[$\spadesuit$ extension to the setting of finite Riemann surfaces of Carathéodory’s theory on the “Variabilitätsbereich” (1907 [@Caratheodory_1907-Variabilitaetsbereich], 1911 [@Caratheodory_1911-Variabilitaetsbereich]) of coefficient of analytic functions with positive real part (bringing together Minkowski’s theory of convex sets with complex function theory), while encompassing interpolation problems subsuming those of Pick-Nevanlinna type\] $\bigstar$ 

M. Heins, [*Extreme normalized analytic functions with positive real parts*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 10 (1985), 239–245. 50 \[$\spadesuit$ localized via Bell 2009/11 [@Bell_2009_PREPRINT/2011] $\spadesuit$ also quoted in Khavinson 1984 [@Khavinson-Dimitri_1984 p.377] for another proof of the Bieberbach-Grunsky theorem $\spadesuit$ Heins handles the more general non-planar case recovering probably the Ahlfors circle maps of 1950, and so seems indeed to be the case according to MathReviews (translated from Hervé’s review in Zentralblatt): “Let $P$ be the family of holomorphic functions $f$ on a given Riemann surface $S$ satisfying ${\rm Re f}>0$ on $S$ and $f(a)=1$ for a given point $a\in S$. If $S$ is the unit circle, the extremal elements of $P$ are the functions $z\to
(\eta+z)/(\eta-z)$, $\vert \eta \vert=1$. If $S$ is a bounded open plane region whose boundary consists of $c$ analytic Jordan curve $\Gamma_1,\dots, \Gamma_c$, the author associates the extremal elements $f_\zeta$ of $P$ with the system $\zeta=(\zeta_1, \dots,
\zeta_c)\in \Gamma_1\times \dots \Gamma_c$; ${\rm Re} f_{\zeta}$ is an appropriate linear combination of minimal harmonic functions $>0$ on $S$ with poles $\zeta_k$, $k=1,\dots, c$. This results extends to the case in which $S$ is an open region of a compact Riemann surface of genus $g$, but here the real parts of the extremal element of $P$ are linear combination of \[AT MOST\][^134] $2g+c$ minimal positive harmonic functions on $S$.” $\spadesuit$ \[06.10.12\] so it seems that this new work of Heins, albeit quite close to Heins 1950 [@Heins_1950], may be a bit more explicit and truly include the existence of (Ahlfors) circle map with the bound $r+2p$ like Ahlfors 1950 [@Ahlfors_1950] $\spadesuit$ \[06.10.12\] it would be of course of primary importance to study if Heins’ methods is susceptible of recovering the sharper $r+p$ bound asserted in Gabard 2006 [@Gabard_2006] $\spadesuit$ \[12.10.12\] after reading the original text, it must alas recognize that Heins’ proof is not perfectly satisfactory, for when it comes to the case of positive genus, he writes simply (p.243): “the corresponding developments of Section 3 \[=planar case\] may be paraphrased.” $\spadesuit$ hence the pedestrian reader will not find it easy to recover even Ahlfors basic (but deep) result from Heins’ account. So let me try once to degage the substance of the argument, while trying to locate “en passant” those critical steps which in our opinion is not made explicit in Heins’ exposition. (I shall use my notation hopefully for convenience of the reader.) We start as usual with $\overline{F}$ a compact bordered Riemann surface of genus $p$ and with $r$ contours. Let $a\in F$ be some fixed interior point. Heins considers $P$ the set of analytic functions $f$ on $F$ with ${\rm Re} f>0$ and $f(a)=1$. (The family $P$ is convex and compact, hence admits extreme points by Krein-Milman. Actually we shall probably not need this, albeit being an interesting viewpoint.) Let $g:=2p+(r-1)$ and $\gamma_1, \dots,
\gamma_g$ be representatives of the homology group $H_1(F)$. For $u$ harmonic on $F$, let $\pi(u)$ be the period vector given by $\pi(u)=(\int_{\gamma_1} \delta u, \dots, \int_{\gamma_g} \delta
u)$, where $\delta u$ is a certain abelian differential given by some local recipe. In fact it is perhaps more natural (and equivalent?) to define $\delta u$ as the conjugate differential $(du)^{\ast}$. For $\zeta \in \partial F$, Heins considers (p.241) $u_{\zeta}$ the minimal positive harmonic function on $F$ vanishing on $\partial F-\{\zeta\}$ and normalized by $u_\zeta(a)=1$. \[Maybe here Heins still relies subconsciously on Martin 1941 [@Martin_1941], yet arguably this is nothing else that the Green’s function with pole pushed to the boundary, what I called a Red’s function, but perhaps calls it a Poisson function, as may suggest the paper Forelli 1979 [@Forelli_1979].\] We seek to construct a half-plane map $f$ by taking a combination $u=\sum_{k=1}^d \mu_k u_{\zeta_k}$ of such elementary potentials, with $\mu_k>0$ while trying to arrange the free parameters (e.g. the $\zeta_k\in
\partial F$) so as to kill all periods of $(du)^{\ast}$. If this can be achieved for some $d$, then $f=u+iu^{\ast}$ (where $u^{\ast}$ is defined by integrating the differential $(du)^{\ast}$) supplies a half-plane map of degree $d$. (Recall indeed that $u$ vanishes continuously on the boundary $\partial F$, except at the $\zeta_k$ which are catapulted to $\infty$. Hence the map is boundary preserving and has therefore constant valency, here $d$.) To kill all periods, we may look at the map $\varphi\colon \partial F
\buildrel{u}\over{\to} h(F) \buildrel{\pi}\over{\to} {\Bbb
R}^g$, where $u(\zeta)=u_\zeta$ and $h(F)$ denotes the space of harmonic functions. At this stage it must be explained that the image $\varphi(\partial F)$ is “balanced”, i.e. not situated in a half space of ${\Bbb R}^g$. \[I am not sure that Heins explains this in details.\] If so then it is plain that there is a collection of $d\le g+1$ points (assume $d=g+1$ if you want) on $\varphi(\partial F)$ spanning a simplex containing $0$. This is just the principle that in Euclidean space of some dimension, a collection of one more points than the given dimension span a top-dimensional simplex with optimum occupation property of the territory (=Euclid space). Thus expressing the origin as convex combination of those $g+1$ points we find scalars $\mu_k>0$, which injected in the formula defining $u$, gives us an $u$ meeting the requirement. This reproves Ahlfors 1950, but alas I still do not have a simple explanation for the balancing condition. Next the challenge, is of course to improve the geometry by remarking that clever placements of points may span a lower dimensional simplex yet still covering the origin. Hopefully one may reprove the $r+p$ upper bound of Gabard 2006 [@Gabard_2006], along this path (which is essentially Ahlfors’ original approach).\] 

M. Heins, [*Extreme Pick-Nevanlinna interpolating function*]{}, J. Math. Kyoto Univ. 25–4 (1985), 757–766. \[$\spadesuit$ p.758: “It is appropriate to cite instances of convexity considerations related to the present paper. The pioneer work of Carathéodory \[2\](=1907 [@Caratheodory_1907-Variabilitaetsbereich]),\[3\](=1911 [@Caratheodory_1911-Variabilitaetsbereich]) on coefficient problems for analytic functions with positive real part is, as far I am aware, the first bringing together of the Minkowski theory of convex sets and complex function theory. Extreme points are present in the fundamental work of R.S. Martin \[12\](=1941 [@Martin_1941]) on the representation of positive harmonic functions as normalized minimal positive harmonic functions. My paper \[7\](=Heins 1950 [@Heins_1950]) showed the existence of minimal positive harmonic functions on Riemann surfaces using elementary standard normal families results without the intervention of the Krein-Milman theorem and gave application to qualitative aspects of Pick-Nevanlinna interpolation on Riemann surfaces with finite topological characteristics and nonpointlike boundary components. Such Riemann surfaces will be termed [*finite*]{} Riemann surfaces henceforth. In \[8\](=Heins 1976 [@Heins_1976]) the Carathéodory theory cited above was extended to the setting of finite Riemann surfaces for interpolation problems subsuming those of Pick-Nevanlinna type. Forelli \[5\](=1979 [@Forelli_1979]) has studied the extreme points of the family of analytic functions with positive real part on a given finite Riemann surface $S$ normalized to take the value $1$ at a given point of $S$. In my paper \[9\](=Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF]) the results of Forelli were supplemented by precise characterizing results for the case where the genus of $S$ is positive.”\] 

D.A. Hejhal, [*Linear extremal problems for analytic functions*]{}, Acta Math. 128 (1972), 91–122. 50, 78 \[$\spadesuit$ generalized extremal problem, existence as usual via normal families, and so uniqueness is given “a reasonably complete answer” (p.93) $\spadesuit$ p.119 Royden 1962 [@Royden_1962] is cited for another treatment of Ahlfors’ extremal problem\] 

D.A. Hejhal, [*Theta functions, kernel functions and Abelian integrals*]{}, Memoirs Amer. Math. Soc. 129, 1972. \[$\spadesuit$ quoted in Burbea 1978 [@Burbea_1978-Capacities], where like in Suita 1972 [@Suita_1972] an application of the Ahlfors function is given to an estimation of the analytic capacity\] $\bigstar$ 

D.A. Hejhal, [*Some remarks on kernel functions and Abelian differentials*]{}, Arch. Rat. Mech. Anal. 52 (1973), 199–204. 78 

D.A. Hejhal, [*Universal covering maps for variable regions*]{}, Math. Z. 137 (1974), 7–20. 78 \[$\spadesuit$ just quoted for the philosophical discussion on p.19, especially the issue that the (Koebe) Kreisnormierung (=circular mapping) \[not to be confused with our circle maps!\] is “[*somewhat more involved*]{} than the other canonical mappings, esp. the PSM\] 

D.A. Hejhal, [*Linear extremal problems for analytic functions with interior side conditions*]{}, Ann. Acad. Sci. Fenn. Ser.A 586 (1974), 1–36. 78 \[$\spadesuit$ cited in Jenkins-Suita 1979 [@Jenkins-Suita_1979]\]$\bigstar$ 

D.A. Hejhal, [*On Schottky and Teichmüller spaces*]{}, Adv. Math. 15 (1975), 133–156. \[$\spadesuit$\]$\bigstar$ 

B. Heltai, [*Symmetric Riemann surfaces, torsion subgroups and Schottky coverings*]{}, Proc. Amer. Math. Soc. 100 (1987), 675–682. \[$\spadesuit$\] 

?. Henoch, [*De Abelianarum Functionum Periodis*]{}, Inaugural-Dissertation, Berlin. \[$\spadesuit$ cited in both Hurwitz 1883 [@Hurwitz_1883] and Weichold 1883 [@Weichold_1883], who both relax hyperellipticity from Henoch results on the periods of Abelian integrals on real algebraic curve, while extending also Klein’s general version for $g=3$ (cf. Klein 1876 [@Klein_1876_Verlauf])\] 

K. Hensel, W. Landsberg, [*Theorie der algebraischen Funktionen einer Variabeln und ihre Anwendung auf algebraische Kurven und Abelsche Integrale*]{}, Leipzig, 1902. \[$\spadesuit$ contains the sharp estimate $[\frac{g+3}{2}]$ of Riemann-Brill-Noether upon the gonality of a closed surface, but the treatment is not considered as convincing (to contemporary scientists) until the work of Meis 1960 [@Meis_1960], compare e.g., H.H. Martens 1967 [@Martens_Henrik_1967] and Kleiman-Laksov 1972 [@Kleiman-Laksov_1972]\] 

G. Herglotz, [*Über Potenzreihen mit positivem reellen Teil im Einheitskreis*]{}, Ber. Verhandl. Sächs. Ges. Wiss. Leipzig 93 (1911), 501–511. \[$\spadesuit$ yet another theorem in the disc susceptible (???) of a transplantation to finite bordered Riemann surface, try e.g Agler-Harland-Raphael 2008 [@Agler-Harland-Raphael_2008] (multi-connected planar domains), or Heins 1985 [@Heins_1985-Extreme-Pick-Nevanlinna] $\spadesuit$ Herglotz’s representation theorem is concerned with the so-called Poisson-Stieltjes representation for analytic functions on the unit disc $\Delta$ with $\ge 0$ real part (simultaneous work by F. Riesz), and cf. the above cited work of Heins for an application (of Herglotz-Riesz 1911) to a description of extreme points of a class $I$ of analytic function arising from a Pick-Nevanlinna interpolation problem involving functions $\Delta \to \{{\rm Re} z>0 \}$: “Theorem 1. The extreme points of $I$ \[in the sense of convex geometry\] are precisely the members of $I$ having constant valence on $\{Re z>0\}$, the value $\nu$ of the valence satisfying $1+n<\nu<1+2n$.” \[$n$ being the number of interpolation points\] $\spadesuit$ such results are probably extensible to the situation where the source (=disc $\Delta$) is replaced by a finite bordered Riemann surface, and the resulting theory probably interacts with the Ahlfors map $\spadesuit$ \[13.10.12\] in fact the Poisson-Stieltjes-Herglotz-Riesz representation formula is rather involved in another proof of Ahlfors circle maps, see for this Forelli’s brilliant account (1979 [@Forelli_1979])\] 

J. Hersch, [*Quatres propriétés isopérimétriques de membranes sphériques homogènes*]{}, C. R. Acad. Sc. Paris 270 (1970), 1645–1648. \[$\spadesuit$ contains 4 spectral (eigenvalues) inequalities for disc-shaped membranes emphasizing the extremality of resp. the round sphere, the hemisphere, the quarter of sphere and of the octant of sphere $\spadesuit$ the first inequality has been extended via conformal transplantation to closed surfaces of higher topological type by Yang-Yau 1980 [@Yang-Yau_1980] (who failed to take advantage of the well-known sharp gonality bound of Riemann-Meis (1960 [@Meis_1960]), but see El Soufi-Ilias 1983/84 [@El-Soufi-Ilias_1983/84]) $\spadesuit$ the first inequality has been extended by Gabard 2011 [@Gabard_2011] upon using the Ahlfors map $\spadesuit$ \[08.10.12\] of course it would be also interesting trying to get extensions of the two remaining Hersch’s inequalities (involving the quarter of sphere and its octant resp.)\] 

J. Hersch, L.E. Payne, M.M. Schiffer [*Some inequalities for Steklov eigenvalues*]{}, Arch. Rat. Mech Anal. 57 (1973/74), 99–114. \[$\spadesuit$ contain estimates of Steklov eigenvalues along the method of Szegö 1954 [@Szego_1954], Weinstock 1954 [@Weinstock_1954], Weinberger 1956 [@Weinberger_1956], etc. (i.e. conformal transplantation) $\spadesuit$ in the light of Fraser-Schoen’s article (2011 [@Fraser-Schoen_2011]) one can remark two things: (1) it is strange that the present paper does not cite the planar avatar of the Ahlfors map (that is the Bieberbach-Grunsky theorem); this is perhaps done subconsciously in §5.2, p.106 (2) of course (at least since Fraser-Schoen’s paper 2011 [@Fraser-Schoen_2011]) it is obvious that the result can (via the Ahlfors map) be extended to bordered Riemann surfaces; for an exact implementation cf. Girouard-Polterovich 2012 [@Girouard-Polterovich_2012]\] 

M. Hervé, [*Quelques propriétés des transformations intérieures d’un domaine borné*]{}, Ann. sci. École norm. sup. (3) 68 (1951), 125–168. 78 \[$\spadesuit$ Grunsky’s works, as well as Ahlfors 1947 [@Ahlfors_1947] are cited, and it could be nice to look for extensions to bordered surfaces\] 

R.A. Hidalgo, A.F. Costa, [*Anticonformal automorphisms and Schottky coverings*]{}, Ann. Acad. Sci. Fenn 26 (2001), 489–508. \[$\spadesuit$\] 

R.A. Hidalgo, [*Schottky uniformization of stable symmetric Riemann surfaces*]{}, Notas de la Soc. Mat. de Chile (N.S.) 1 (2001), 82–91. \[$\spadesuit$\] 

R.A. Hidalgo, [*Real surfaces, Riemann matrices and algebraic curves*]{}, In: [*Complex Manifolds and Hyperbolic Geometry*]{}, Guanajuato 2001, Contemp. Math. 311, Amer. MAth. Soc., Providence, 2002. \[$\spadesuit$ a neoclassical account on the Rückkehrschnitttheorem of Klein 1882 $\spadesuit$ question: is it sufficient to reprove existence of Ahlfors circle maps?\] 

R.A. Hidalgo, B. Maskit, [*On Klein-Schottky groups*]{}, Pacific J. Math. 220 (2005), 313–328. \[$\spadesuit$\] 

R.A. Hidalgo, B. Maskit, [*On neoclassical Schottky groups*]{}, Trans. Amer. Math. Soc. 358 (2006), 4765–4792. \[$\spadesuit$\] 

R.A. Hidalgo, [*On the retrosection theorem*]{}, Proyecciones 27 (2008), 29–61. \[$\spadesuit$ a neoclassical account on the Rückkehrschnitttheorem of Klein 1882 $\spadesuit$ question: is it sufficient to reprove existence of Ahlfors circle maps?\] 

R.A. Hidalgo, [*On the inverse uniformization problem: real Schottky uniformization*]{}, Rev. Mat. Complut. 24 (2011), 391–420. \[$\spadesuit$ p.394: “The reciprocal is valid by the retrosection theorem \[13\](=Koebe 1910 UAK2 [@Koebe_1910_UAK2]) (see \[4\]=(Bers 1975 [@Bers_1975]) for a modern proof using quasiconformal deformation theory).”\] 

D. Hilbert, [*Über die reellen Züge algebraischen Kurven*]{}, Math. Ann. 38 (1891), 115–138; or Ges. Abhandl., Bd.II. \[$\spadesuit$ where Hilbert’s 16th problem (Paris 1900) starts taking shape, in the sense of asking for the isotopy classification of plane smooth real sextics in ${\Bbb R}P^2={\Bbb
P}^2 ({\Bbb R})$ $\spadesuit$ a method of oscillation is given permitting to exhibit a new scheme of $M$-sextic not available via Harnack’s method of 1876 (this is nowadays called Hilbert’s method) which is quite powerful (but not omnipotent) to analyze the topology of plane (real) sextics $\spadesuit$ in particular Hilbert develops the intuition that a sextic cannot have 11 unnested ovals, so must be nested yielding some noteworthy form of complexity of algebraic varieties $\spadesuit$ a complete proof of this assertion will have to wait for a longue durée series of attempt by his own students Kahn 1909 [@Kahn_1909] Löbenstein 1910 [@Löbenstein_1910] and especially Rohn 1911–13 [@Rohn_1913]. All these attempts where judged unconvincing, and from the Russian rating agency not judged as sufficiently rigorous until the intervention of Petrovskii 1933–38 [@Petrowsky_1938] and Gudkov 1948–1969, cf. e.g. Gudkov 1974/74 [@Gudkov_1974/74] $\spadesuit$ p.418 (in Ges. Abh., Bd.II): “Diesen Fall $n=6$ habe ich einer weiteren eingehenden Untersuchung unterworfen, wobei ich—freilich auf einem au[ß]{}erordentlich umständlichen Wege— fand, da[ß]{} die elf Züge einer Kurve 6-ter Ordnung keinesfalls sämtlich au[ß]{}erhalb un voneinander getrennt verlaufen können. Dieses Resultat erscheint mir deshalb von Interesse, weil er zeigt, da[ß]{} für Kurven mit der Maximalzahl von Zügen der topologisch einfachste Fall nicht immer möglich ist.” $\spadesuit$ for the next episode in Hilbert’s pen, cf. Hilbert 1909 [@Hilbert_1909] where Hilbert ascribes to his students a complete proof of the result (inexistence of the unnested scheme of 11 ovals)\] 

D. Hilbert, [*Mathematische Probleme*]{}, Arch. Math. Phys. (3) 1 (1901), 213–237; also in Ges. Abh., Bd.III. \[$\spadesuit$ includes Hilbert’s 16th problem on the mutual disposition of ovals of plane curves (especially sextics), completely solved by Gudkov ca. 1969, cf. Gudkov-Utkin 1969 [@Gudkov-Utkin_1969/78]\] 

D. Hilbert, [*Über das Dirichletsche Prinzip*]{}, Jahresb. d. Deutsch. Math.-ver. 8 (1900), 184–188. \[$\spadesuit$ \[08.10.12\] the technological breakthrough based upon the “direct method” in the calculus of variation, where one directly minimizes the integral (without transiting to its first variation, alias Euler-Lagrange equation) via the idea of minimizing sequences implying a topologization of the space of test functions while checking its compactness (=Fréchet’s jargon) of the resulting family $\spadesuit$ the method also differs from its predecessors Schwarz-Neumann-Poincaré where the problem was first solved for the disc and combinatorial tricks permitted proliferation to higher topological complexity\] 60 

D. Hilbert, [*Über das Dirichletsche Prinzip*]{}, Math. Ann. 59 (1904), 161–186. (Abdruck aus der Festschrift zur Feier des 150jährigen Bestehens der Königl. Gesellschaft der Wissenschaften zu Göttingen 1901.) 60  \[$\spadesuit$ p.161 (Introd.): “Unter dem Dirichletschen Prinzip verstehen wir diejenige Schlu[ß]{}weise auf die Existenz einer Minimalfunktion, welche Gauss (1839)\[=[@Gauss_1839]\], Thomson (1847)\[=[@Thomson_1847]\], Dirichlet (1856)\[=of course much earlier, at least as early as when Riemann studied in Berlin, ca. 1849–50!\] und andere Mathematiker zur Lösung sogennanter Randwertaufgaben angewandt haben und deren Unzulässigkeit zuerst von Weierstrass erkannt worden ist. \[…\] Durch das Dirichletsche Prinzip hat insbesondere Riemann die Exitenz der überall endlichen Integrale auf einer vorgelegten Riemannschen Fläche zu beweisen gesucht. Ich bediene mich im folgenden dieses klassischen Beispiels zur Darlegung meines strengen Beweisverfahrens.”\] 

D. Hilbert, [*Über das Dirichletsche Prinzip*]{}, J. Reine Angew. Math. 129 (1905), 63–67. (Abdruck eines Vortrages aus dem Jahresb. d. Deutsch. Math.-ver. 8 (1900), 184–188.) 60 

D. Hilbert, [*Zur Theorie der konformen Abbildung*]{}, Gött. Nachr. (1909), 314–323; Ges. Abh. 3, 73–80. 78 \[$\spadesuit$ parallel-slit mapping including positive genus (and infinite connectivity) $\spadesuit$ influenced much Courant, and also Koebe 1910 [@Koebe_1910_Hilbert]\] 

D. Hilbert, [*Über die Gestalt einer Fläche vierter Ordnung*]{}, Gött. Nachr. (1909), 308–313; Ges. Abh. 2, 449–453. \[$\spadesuit$ contains a good picture for the construction of Harnack-maximal sextic $\spadesuit$ p.453, Hilbert ascribes to his students G. Kahn 1909 [@Kahn_1909] and Löbenstein 1910 [@Löbenstein_1910] a complete proof that a real sextic cannot have 11 unnested ovals (but that was not judged solid enough by subsequent workers, e.g. Rohn, Petrovskii, and Gudkov 1974 [@Gudkov_1974/74]): “\[…\] eine ebene Kurve 6-ter Ordnung hervorgehen, die aus elf au[ß]{}erhalb voneinander getrennt verlaufenden Zügen bestände. Da[ß]{} aber eine solche Kurve nicht existiert, ist einer der tiefstliegenden Sätze aus der Topologie der ebenen algebraischen Kurven; derselbe ist kürzlich von G. Kahn und K. Loebenstein (Vgl. die Göttinger Dissertationen derselben Verfasserinnen.) auf einem von mir angegebenen Wege bewiesen worden.” $\spadesuit$ nowadays there is five-minute proof of what Hilbert called one of the deepest problem in the topology of plane curves, via Rohlin’s formula ca. 1974–78 (cf. e.g. our Sec.\[Rohlin-formula:sec\]), yet we believe that there is perhaps also a proof via the Ahlfors map (in the special case due to Riemann-Schottky-Bieberbach-Grunsky). This would be a fantastic project\] 

D. Hilbert, S. Cohn-Vossen, [*Anschauliche Geometrie*]{}, Springer, Berlin, 1932. (Translation: Geometry and the Imagination) \[$\spadesuit$\] 

S. Hildebrandt, H. von der Mosel, [*Conformal mapping of multiply connected Riemann domains by a variational approach*]{}, Adv. Calc. Var. 2 (2009), 137–183. \[$\spadesuit$ new proof of the Kreisnormierung for (planar) domains via Plateau-style method $\spadesuit$ question: can we apply the same method for the (Ahlfors) circle map? (cf. Courant 1939 [@Courant_1939] for the planar case \[$p=0$\]) $\spadesuit$ “Abstract. We show with a new variational approach that any Riemannian metric on a multiply connected schlicht domain in ${\Bbb R}^2$ can be represented by globally conformal parameters. This yields a “Riemannian version” of Koebe’s mapping theorem.”\] 

S. Hildebrandt, [*Plateau’s problem and Riemann’s mapping theorem*]{}, Milan J. Math. (2011), 67–79. \[$\spadesuit$ survey putting in perspective several recent developments, including the previous item\] 

F. Hirzebruch, [*Topological methods in algebraic geometry*]{}, Springer, 1978; translated from the Original German text ca. 1955. \[$\spadesuit$ how to put Pontrjagin, Thom, cobordism, the signature theorem, etc. into action to get the generalized Riemann-Roch theorem\] 

W. Hodge, [*The Theory and Applications of harmonic integrals*]{}, Cambridge, 1941. 

K. Hoffman, [*Banach spaces of analytic functions*]{}, Prentice-Hall (Englewood Cliffs), 1962; Dover Reprint, 1988. 

M. Homma, [*Separable gonality of a Gorenstein curve*]{}, Math. Contemp. ca. 2004. \[$\spadesuit$ cited in Ballico 2003 [@Ballico_2003]\] 

M. Horikawa, [*On deformations of holomorphic maps I*]{}, J. Math. Soc. Japan 25 (1973), 372–396. \[$\spadesuit$\] 

J. Huisman, [*Real quotient singularities and nonsingular real algebraic curves in the boundary of the moduli space*]{}, Compos. Math. 118 (1999), 42–60. \[$\spadesuit$\] 

J. Huisman, [*Real Teichmüller spaces and moduli of real algebraic curves*]{}, Contemp. Math. 253 (2000), 145–179. \[$\spadesuit$\] 

J. Huisman, [*On the geometry of algebraic curves having many real components*]{}, Rev. Mat. Complut. 14 (2001), 83–92. \[$\spadesuit$ p.87, Prop.3.2 contains an algebro-geometric proof of the so-called Bieberbach-Grunsky theorem (for antecedent along similar lines compare Enriques-Chisini 1915/18 [@Enriques-Chisini_1915-1918], Bieberbach 1925 [@Bieberbach_1925], and Wirtinger 1942 [@Wirtinger_1942]) $\spadesuit$ of course Huisman’s paper goes much deeper by exploring the properties of linear series on Harnack-maximal curves (alias $M$-curves)\] 

J. Huisman, [*Non-special divisors on real algebraic curves and embeddings into real projective spaces*]{}, Ann. di Mat. 182 (2003), 21–35. \[$\spadesuit$\] 

A. Hurwitz, [*Über die Perioden solcher eindeutiger, $2n$-fach periodischer Funktionen, welche im Endlichen überall den Charakter rationaler Funktionen besitzen und reell sind für reelle Werte ihrer $n$ Argumente*]{}, J. Reine Angew. Math. 94 (1883), 1–20. (Math. Werke, Bd. I) \[$\spadesuit$\] 

A. Hurwitz, [*Über Riemannsche Flächen mit gegebenen Verzweigungspunkten*]{}, Math. Ann. 39 (1891), 1–61. \[$\spadesuit$ \[13.10.12\] if we fix a ramification divisor in the sphere of degree $b$ and a mapping degree $d$ there is finite number of Riemann surfaces $F$ of Euler characteristic $\chi (F)=d
\chi(S^2)-b$ having the prescribed topological behaviour (Hurwitz is able to make a fine study, using of course the monodromy and so to get upper bounds on the number of admissible maps). It seems evident that the game should extend in the bordered setting in the context of Ahlfors circle maps, which are truly (upon doubling) real maps of a special kind (totally real, saturated or separating) from Klein’s orthosymmetric real curves to the real projective line. Then one can try to adventure into similar group theoretical (combinatorial) games as did Hurwitz in the complex case (in fact Hurwitz himself give close attention to reality questions) $\spadesuit$ a more modest question is whether a careful variation of branch points does not produce a quick “action-painting” or “sweeping out” proof of the existence of circle maps of lowest possible degree. $\spadesuit$ as yet I was never able to proceed along this way, which looks yet a reasonable strategy for in the complex case such argument yield at least the right prediction about the gonality of complex curves as divinized by Riemann 1857 (cf. e.g. the heuristic count in Griffiths-Harris 1978 [@Griffiths-Harris_1978/94]). I remind clearly that this idea was suggested by Natanzon (Rennes ca. 2001), and in Rennes 2001/02 (Winter) Johan Huisman also presented to me a simple moduli parameter count somehow comforting the bound $r+p$ (when I suggested him the possibility of the sharpened $r+p$ bound); for the details of Huisman’s count cf. \] 

A. Hurwitz, [*Über algebraische Gebilde mit eindeutigen Transformationen in sich*]{}, Math. Ann. 41 (1893), 403–442; or Math. Werke, Bd.I, Funktionentheorie. \[$\spadesuit$ it is proved that if a conformal self-map of a closed Riemann surface of genus $>1$ induces the identity on the first homology group then the self-map is the identity. Historically, one may wonder how this formulation borrowed from Accola ca. 1966 is reliable for the language of homology was not yet “invented” at least in this precise context (recall Poincaré 1895, but of course a myriad of people used the term “homology” in different contexts, e.g. Jordan) $\spadesuit$ despite this detail the assertion is correct\] 

A. Hurwitz, R. Courant, [*Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen*]{}, Grundl. der. Math. Wiss. 3, Springer-Verlag, Berlin, 1922. (Subsequent editions 1929, 1964, 706 pp.) \[$\spadesuit$ contains another proof of the KN(=Kreisnormierung) in finite connectivity, according to Schiffer-Hawley [@Schiffer-Hawley_1962], also quoted for this purpose in Stout 1965 [@Stout_1965] $\spadesuit$ Ahlfors once said (recover the source!!) that it this in this book that he learned the length-area principle so fruitful in the theory of quasi-conformal maps (roughly the pendant of Grötzsch’s Flächenstreifenmethode)\] 

Y. Imayoshi, [*Holomorphic families of Riemann surfaces and Teichmüller spaces*]{}, Ann. of Math. Stud, 1981. \[$\spadesuit$\] 

M.S. Ioffe, [*Extremal quasiconformal embeddings of Riemann surfaces*]{}, Sib. Math. J. (1975), 520–537; English transl. 1976. \[$\spadesuit$ Teichmüller theory for finite bordered surfaces (with optional punctures)\] 

I.V. Itenberg, [*Curves of degree 6 with one nondegenerate double point and groups of monodromy of nonsingular curves*]{}, in: Real Algebraic Geometry, Proceedings, Rennes 1991, Lecture Notes in Math. 1524, Springer-Verlag, Berlin, 1992. \[$\spadesuit$\] 

I.V. Itenberg, [*Contre-exemples à la conjecture de Ragsdale*]{}, C.R. Acad. Sci. Paris (Sér.I) 317 (1993), 277–282. \[$\spadesuit$ where Ragsdale conjecture is seriously destroyed, yet it remains open the case of $M$-curves, compare also Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale]\] 

I.V. Itenberg, [*Groups of monodromy of non-singular curves of degree 6*]{}, in: ???, 199?. \[$\spadesuit$ extending a private communication of Kharlamov, the monodromy groups (ovals permutation) of each chamber of the discriminant is described $\spadesuit$ relies on Nikulin’s classification via K3 surfaces and uses Coxeter, Vinberg, etc.\] 

I.V. Itenberg, [*Rigid isotopy classification of curves of degree 6 with one nondegenerate double point*]{}, in: Topology of Manifolds and Varieties, Advances in Soviet Math. 18, Amer. Math. Soc., 1994, 193–208. (English) \[$\spadesuit$ p.196: “Proposition 2.1. [*Each empty oval of a nonsingular curve of degree $6$ can be contracted and there is only one rigid-isotopy class of the result of such a degeneration.*]{}” $\spadesuit$ the method employed seems to depend upon “Nikulin’s approach for obtaining rigid-isotopy classification of nonsingular curves of degree $6$” (cf. p.193) $\spadesuit$ as noted by Viro (in the same volume, p.xiii): “I only want to formulate a conjecture, suggested by Itenberg’s Prop. 2.2: each empty oval of a nonsingular real algebraic plane projective curve can be contracted by a deformation of the curve in the class of curves of the same degree. According to Prop. 2.2, this is true for curves of degree $6$. It is easy to check for curves of degree $\le 5$. The first case for which it is unknown is the case of degree $7$.” $\spadesuit$ \[08.01.13\] In the same vein one can perhaps conjecture that any two ovals lying at the same depth can always coalesce to a single one.\] 

I.V. Itenberg, O. Viro, [*Patchworking algebraic curves disproves the Ragsdale conjecture*]{}, The Math. Intelligencer 18 (1996), 19–28. \[$\spadesuit$ self-explanatory title $\spadesuit$ contains (besides some fascinating historical sketches) in particular a formulation of the last vestige of Ragsdale’s conjecture which is still open for $M$-curves (as I learned personally from Th. Fiedler)\] 

C. Jacob, [*Sur le problème de Dirichlet dans un domaine plan multiplement connexe et ses applications à l’hydrodynamique*]{}, J. Math. Pures Appl. 18 (1939), 363–383. 78 \[$\spadesuit$ cf. also the next entry\] 

C. Jacob, [*Introduction mathématique à la mécanique des fluides*]{}, 1959. (ca. 1286 pp.) \[$\spadesuit$\] 

C.G.J. Jacobi, [*Considerationes generales de transcendentibus Abelianis*]{}, Crelle J. Reine Angew. Math. 9 (1832), 394–403. \[$\spadesuit$ Jacobi inversion problem, and first place where jargon like Abelian integrals are employed\]

C.G.J. Jacobi, [*Gesammelte Werke, III*]{}. \[$\spadesuit$ cited in Petrovskii 1938 [@Petrowsky_1938] for the so-called Euler-Jacobi formula, as being one of the tool towards Petrovskii’s proof of the extended Hilbert-Rohn theorem forcing the presence of nesting in $M$-curves (though Petrovskii’s inequalities have a universal validity)\]

S. Jacobson, [*Pointwise bounded approximation and analytic capacity of open sets*]{}, Trans. Amer. Math. Soc. 218 (1976), 261–283. \[$\spadesuit$ the Ahlfors function appears on p.261, 272, 274 in the context of analytic capacity, which is examined from the angle of the semi-additivity question (Vitushkin) $\spadesuit$ the latter aspect has meanwhile been settled in the seminal breakthrough of Tolsa 2003 [@Tolsa_2003] giving also a complete (geometric) solution to Painlevé’s problem\] 

A. Jaffe, S. Klimek, L. Lesniewski, [*Representation of the Heisenberg algebra on a Riemann surface*]{}, Comm. Math. Phys. 126 (1990), 421–433. \[$\spadesuit$\] 

H. Jaffee, [*Real algebraic curves*]{}, Topology 19 (1980), 81–87. \[$\spadesuit$ p.82: “We state Harnack’s Theorem in a slightly strengthened form which is probably due to Hurwitz:—[Theorem 2]{}. [*Let $(X, \rho, g,r)$ be as in §$1$. The number $c$ of components of the space $X-X^{\rho}$ is at most $2$. If $c=2$, then $r\le 1+g$ and $g-r$ is odd. If $c=1$, then $r\le g$.*]{}” $\spadesuit$ of course this is historical non-sense, read “Klein” in place of “Hurwitz” $\spadesuit$ \[05.01.13\] the explanation for this historical mistake (alas quite widespread in literature) seems to find its origin in Gudkov’s survey 1974/74 [@Gudkov_1974/74], where Klein’s priority is not sufficiently emphasized! $\spadesuit$ otherwise the paper is quite pleasant albeit quite elementary, especially it cites (p.86) a preprint of Gross 1979 [@Gross_1979-PREPRINT] which probably was phagocytosed in Gross-Harris 1981 [@Gross-Harris_1981]\] 

P. Järvi, [*On some function-theoretic extremal problems*]{}, Complex Variables Theory Appl. 24 (1994), 267–270. \[$\spadesuit$ related to the Ahlfors function\]$\bigstar$ 

J.A. Jenkins, [*On the existence of certain general extremal metrics*]{}, Ann. Math. (2) (1957). \[$\spadesuit$\] 

J.A. Jenkins, [*Some new canonical mappings for multiply-connected domains*]{}, Ann. Math. (2) 65 (1957), 179–196. 60, 78 \[$\spadesuit$ new derivation of the parallel-slit maps (and radial avatar) in the slightly extended context of rectangular multi-connected domains (resp. radioactive) domains bounded respectively by rectangles or by rectangles in polar coordinates $\spadesuit$ technique: the classical continuity method à la Brouwer-Koebe, but augmented by some quasi-conformal technology (à la Grötzsch, etc.)\] 

J.A. Jenkins, N. Suita, [*On the Pick-Nevanlinna problem*]{}, Kōdai Math. J. 2 (1979), 82–102. \[$\clubsuit$ includes probably an extension and thus also a new derivation of the Ahlfors circle map, compare also Heins 1975 [@Heins_1975] who probably already achieves this goal\] 

J.A. Jenkins, N. Suita, [*On analytic maps of plane domains*]{}, Kōdai Math. J. 11 (1988), 38–43. \[$\clubsuit$ for $D$ a plane bordered surface, an analytic map $f\colon D \to \Delta$ to another bordered surface is called [*boundary preserving*]{} it it takes boundary to boundary. “A boundary preserving map $f\colon D \to \Delta$ covers the image domain finitely many times. It can also be extended to the doubles as $\hat D\to \hat \Delta$. Now the Seveli-deFranchis’ theorem[^135] states finiteness for the number of nonconstant analytic maps between two closed Riemann surfaces of genuses both $>1$, hence we get as a dividend finiteness for the above boundary preserving maps, as soon as the genus of the doubles are $>1$. $\spadesuit$ p.40: “Since $f$ is boundary preserving, $f$ has no branch points on the boundary.”, this is completely akin to the Ahlfors map $\spadesuit$ of course the problematic addressed by Jenkins-Suita extends directly to bordered surface of positive genus, and it could be nice to work out corresponding bounds\] 

M. Jeong, [*The Szegö kernel and a special self-correspondence*]{}, J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 5 (1998), 101–108. \[$\spadesuit$ the Ahlfors map is briefly mentioned in the following connection: “Since the zeroes of the Szegö kernel are parts of the zeroes of the Ahlfors map and give rise to a particular basis for the Hardy space $H^2(b\Omega)$ (see \[5\]=Bell 1995 [@Bell_1995-Complexity-Indiana]), they can be the powerful tools for getting the properties of the mapping for planar domains.”\] 

M. Jeong, M. Taniguchi, [*Bell representations of finitely connected planar domains*]{}, Proc. Amer. Math. Soc. 131 (2002), 2325–2328. \[$\spadesuit$ a problem posed by Bell (1999/2000) is given a positive answer, even in the following sharper form: “Theorem 1.2. [*Every non-degenerate $n$-connected planar domain with $n>1$ is mapped biholomorphically onto a domain defined by $\{ \vert z+
\sum_{k=1}^{n-1}\frac{a_k}{z-b_k}\vert<1\}$ with suitable complex numbers $a_k$ and $b_k$.*]{}” $\spadesuit$ the philosophy of such Bell’s domain is a sort of reverse engineering: instead of constructing the Ahlfors function of a given domain one first gives the function $f$ and define the domain as $\vert f(z) \vert <1$ $\spadesuit$ p.2326, it is observed that Bell’s domains depend on $2n-2$ complex parameters (so $4n-4$ real parameters) exceeding the $3g-3=3(n-1)-3=3n-6$ real moduli predicted by Riemann-Schottky-Klein-Teichmüller $\spadesuit$ this discrepancy is explained by the fact “that every Bell domain is actually associated with an $n$-sheeted branched covering of the unit disk”, for knowing the $a_k,b_k$ we may construct the circle-map $f(z)=z+\sum_{k=1}^{n-1}\frac{a_k}{z-b_k}$ (call it the Bell representation) $\spadesuit$ of course the question arise of describing the “Ahlfors locus” (within the Hurwitz space) of those parameters $a_k,b_k$ such that the Bell representation is actually an Ahlfors extremal function $\spadesuit$ this problem is reposed again in Taniguchi 2004 [@Taniguchi_2004] $\spadesuit$ from the more traditional point view, starting form an $n$-connected domain (with contours $C_1, \dots, C_n$) one can construct a circle map (of minimal degree $n$) by prescribing boundary points $p_i\in
C_i$ mapping to $1\in S^1$ (Bieberbach-Grunsky), thus roughly speaking circle maps depends over $n$ parameters, whereas the Ahlfors functions $f_a$ (or $f_{a,b}$) depend only on 2 real parameters (resp. $4$) $\spadesuit$ compare maybe Agler-Harland-Raphael [@Agler-Harland-Raphael_2008] (and its MathReview summary) for a description of the Grunsky functions as the extreme points of the compact convex set of holomorphic functions with positive real parts normalized by $f(z_0)=1$ for some fixed interior point\] 

M. Jeong, [*The exact Bergman kernel and the extremal problem*]{}, Kangweon-Kyungki Math. J. 13 (2005), 183–191. \[$\spadesuit$ the Ahlfors map appears twice on p.185–6\] 

M. Jeong, J.-W. Oh, M. Taniguchi, [*Equivalence problem for annuli and Bell representations in the plane*]{}, J. Math. Anal. Appl. 325 (2007), 1295–1305. \[$\spadesuit$ the Ahlfors function is employed in the problem of determining the parameter for which a certain doubly connected domain of Bell, namely $\vert z+z^{-1}\vert < r$, is conformal to a circular (concentric) ring\] 

G. Jones, D. Singerman, [*Belyi functions, hypermaps and Galois groups*]{}, Bull. London Math. Soc. 28 (1996), 561–590. \[$\spadesuit$\] 

P.W. Jones, D.E. Marshall, [*Critical points of Green’s function, harmonic measure, and the corona problem*]{}, Ark. Mat. 23 (1985), 281–314. 47, 50 \[$\spadesuit$ p.293–4 the Ahlfors function enters into the dance as follows: “We mention one more method for solving the corona problem. The previous methods have the drawback that Green’s function does not ignore subsets of $\partial {\cal R}$ which have zero analytic capacity and positive logarithmic capacity. To avoid this we can use Ahlfors’ function, $A$, instead. Ahlfors’ function for a point $\zeta_0 \in{\cal R}$ is defined by $A'_{\zeta_0}(\zeta_0)=\sup \{{\rm Re} f'(z_0) : f\in
H^{\infty}({\cal R}), \|f\|_{\infty} \le 1\}$. \[…\] Ahlfors \[1\](=1947), \[2\](=1950) has shown that for our “nice” Riemann surfaces $\vert A_{\zeta_0}(\zeta)\vert =\exp \{
-\sum_{j=0}^{n-1} g(\zeta, \zeta_j)\}$ for some points $\zeta_1, \dots, \zeta_{n-1}\in{\cal R}$. \[…\] Then all of the results of this section hold for the critical points of Ahlfors’ function $\{w_{j,k}\}$ as well as for the critical points of $G$. One can easily construct Riemann surfaces where $\sum_k G(\zeta_k, \zeta')=\infty$, so that the methods using the critical points of $G$ will not work, yet this method using Ahlfors’ function gives solution to the corona problem. \[…\] We remark that we chose the Ahlfors function here because of its natural association with $H^{\infty}({\cal
R})$, but we could have chosen any function $F\in
H^{\infty}({\cal U})$ with $-\log \vert F(z)\vert=\sum_{j=1}^m
G(\pi(z), \alpha_j)$, $\alpha_j\in {\cal R}$.” $\spadesuit$ p.286: “If $\cal R$ is a planar domain, then it is a simple consequence of the argument principle that $G(\zeta, \pi(0))$ has $N-1$ critical points (counting multiplicity), where $N$ is the number of closed boundary curves. See e.g. \[33\](=Nehari 1952 [@Nehari_1952-BOOK])[^136] More generally, the number of critical points of $G$ isthe first Betti number, or the number of generators of the first singular homology group, of $\cal R$ \[46\](=Widom 1971 [@Widom_1971]), and hence is finite. See Walsh \[44, Chap.VII\](=Walsh 1950 [@Walsh_1950-The-location:AMS-Colloq.-Publ.]) for more information concerning the location of the critical points.”\] 

P.W. Jones, T. Murai, [*Positive analytic capacity but zero Buffon needle probability*]{}, Pacific J. Math. 133 (1988), 99–114. \[$\spadesuit$ self-explanatory, i.e. a counter-example to the Vitushkin conjecture (that a plane compactum is a Painlevé null-set iff it is invisible, i.e. a.e. projection of the set have zero length) $\spadesuit$ note: the Buffon needle problem was solved by Crofton in 1868: if $E$ is a compactum in the plane, let $\vert P_{\theta}(E)\vert$ be the Lebesgue measure of the orthogonal projection of $E_{\theta}$ on the line of angular slope $\theta$ and define the Crofton invariant as $CR(E)=\int_{0}^{2\pi}\vert P_{\theta}(E)\vert d\theta$. This quantity may be interpreted as the probability of the body $E$ falling over a system of parallel lines equidistantly separated by the diameter of $E$\] 

P.W. Jones, [*Square functions, Cauchy integrals, analytic capacity, and harmonic measure*]{}, in: Proc. Conf. on Harmonic Analysis and Partial Differential Equations, El Escorial 1987, Lecture Notes in Math. 1384, Springer-Verlag, 1989, 24–68. \[$\spadesuit$\] 

P.W. Jones, [*Rectifiable sets and the travelling salesman problem*]{}, Invent. Math. 102 (1990), 1–16. \[$\spadesuit$\] 

C. Jordan, [*Sur la déformation des surfaces*]{}, J. Math Pures Appl. (2) 11 (1866), 105–109. \[$\spadesuit$ after Möbius 1863 [@Moebius_1863] in the closed case, discuss a classification of compact orientable bordered surfaces, by the genus and the number of contours $\spadesuit$ quoted in Klein’s lectures 1892/93 [@Klein_1892_Vorlesung-Goettingen p.150], and in Weichold 1883 [@Weichold_1883 p.330], who need the non-orientable case as well\] 

J. Jost, [*Conformal mappings and the Plateau-Douglas problem in Riemannian manifolds*]{}, J. Reine Angew. Math. 359 (1985), 37–54. \[$\spadesuit$ reprove some results about conformal mapping (uniformization of real orthosymmetric curves) surely well-known since Koebe’s era (and conjectured by Klein) via the method of Plateau $\spadesuit$ then attack and solve a very general case of Plateau’s problem in a generality unifying the desire of Douglas (positive genus) and Morrey (curvy ambient Riemannian manifold instead of flat Euclid) $\spadesuit$ reports also some of Tromba’s critics over the solution of Courant to the Plateau-Douglas problem of higher genus $\clubsuit$ it is not clear to the writer if such critics (of Tromba) affects as well the whole content of Courant’s book 1950 [@Courant_1950] especially regarding the varied type of conformal maps $\spadesuit$ at any rate Jost propose a parade using techniques of Mumford and Schoen-Yau, but the “meandreousness” of the resulting proof is slightly criticized in Hildebrandt-von der Mosel 2009 [@Hildebrandt-von-der-Mosel_2009]\] 

J. Jost, [*Two-dimensional geometric variational problem*]{}, Wiley, New York, 1991. \[$\spadesuit$ from Sauvigny’s review in BAMS: “Chapter 3 deal with conformal representation of surfaces homeomorphic to the sphere $S^2$, circular domains, and closed surfaces of higher genus. The proof is given by direct variational methods and not as usual by uniformization, completing a fragmentary proof of Morrey.\] 

G. Julia, [*Sur la représentation conforme des aires simplement connexes*]{}, C.R. Acad. Sci. Paris 182 (1926), 1314–1316. \[$\spadesuit$ another characterization of the (Riemann) mapping function by a minimum principle\] 

G. Julia, [*Développement en série de polynômes ou de fonctions rationelles de la fonction qui fournit la représentation conforme d’une aire simplement connexe sur un cercle*]{}, Ann. Éc. Norm. Sup. 44 (1927), 289–316. \[$\spadesuit$ Seidel’s summary: a determination of a sequence of polynomials is given which converges to the properly normed (Riemann) mapping function of a simply-connected region\] 

G. Julia, [*Leçon sur la représentation conforme des aires simplement connexes*]{}, Gauthier-Villars, Paris, 1931. \[$\spadesuit$ one among the early book format exposition of the extremal properties of the Riemann mapping for a plane simply-connected region (distinct of ${\Bbb C}$), namely that the range of the map normalized by $f'(z_0)=1$ has minimal area (first in Bieberbach 1914 [@Bieberbach_1914]) or that the boundary of the range has minimal length (probably first in Szegö 1921 [@Szego_1921]) $\spadesuit$ for both those extremal principles see also the detailed treatment in the book Gaier 1964 [@Gaier_1964-BOOK-Konstruktive-methoden]\] 

G. Julia, [*Sur la représentation conforme des aires multiplement connexes*]{}, Ann. Sc. Norm. Sup. Pisa (2) 1 (1932), 113–138. 78 \[$\spadesuit$ still in great admiration for Schottky 1877 [@Schottky_1877] and use Klein’s jargon of orthosymmetry, yet confined to the case of domains $\spadesuit$ however the main purpose is the study of a new sort of mapping introduced by de la Vallée Poussin (and which will in turn fascinate Walsh and Grunsky)\] 

G. Julia, [*Reconstruction d’une surface de Riemann $\sigma$ correspondant à une aire multiplement connexe $\cal
A$*]{}, C.R. Acad. Sci. Paris 194 (1932), 423–425. 60 

G. Julia, [*Prolongement d’une surface de Riemann $\sigma$ correspondant à une aire multiplement connexe $\cal A$*]{}, C.R. Acad. Sci. Paris 194 (1932), 580–583. 60 

G. Julia, [*Leçon sur la représentation conforme des aires multiplement connexes*]{}, Gauthier-Villars, Paris, 1934, 94 pp. 60, 78 \[$\spadesuit$\] 

G. Julia, [*La représentation conforme des aires multiplement connexes*]{}, L’Enseign. Math. 33 (1935), 137–168. 78 \[$\spadesuit$ survey from Riemann, Schottky 1877 [@Schottky_1877] through Hilbert 1909 [@Hilbert_1909], Koebe, up to the extremal treatments by de Possel and Grötzsch (slit mappings in infinite connectivity)\] 

G. Julia, [*Quelques applications fonctionnelles de la topologie*]{}, Reale Accademia d’Italia Fondazione A. Volta, Att dei Convegni 9 (1939), Rome, 1943, 201–306. 60 \[$\spadesuit$ cited in Ahlfors-Sario 1960\]$\bigstar$$\bigstar$$\bigstar$ 

M. Juurchescu, [*A maximal Riemann surface*]{}, ???? ?? (1961?), 91–93. \[$\spadesuit$ p.91, a map between bordered Riemann surfaces taking boundary to boundary is termed [*distinguished*]{}\] 

G. Kahn, [*Eine allgemeine Methode zur Untersuchung der Gestalten algebraischer Kurven*]{}, Inaugural Dissertation, Göttingen, 1909. \[$\spadesuit$ Dissertation under Hilbert (cf. e.g. Hilbert 1909 [@Hilbert_1909-Ueber-die-Gestalt-sextic]), attempting to prohibit the real sextic scheme consisting of 11 unnested ovals $\spadesuit$ considered non-rigorous in Gudkov 1974 [@Gudkov_1974/74] $\spadesuit$ historical anecdote: Kahn’s work as well as the related Thesis by Löbenstein 1910 [@Löbenstein_1910] were instead considered as rigorous in Hilbert 1909 [@Hilbert_1909-Ueber-die-Gestalt-sextic])\] 

S. Kakutani, [*Rings of analytic functions*]{}, Lectures on functions of a complex variable, 71–83, Univ. of Michigan Press, Ann Arbor, 1955. \[$\spadesuit$\]$\bigstar$ 

C. Kalla, Ch. Klein, [*On the numerical evaluation of algebro-geometric solutions to integrable equations*]{}, Nonlinearity 25 (2012), 569–596. \[$\spadesuit$\] 

C. Kalla, Ch. Klein, [*Computation of the topological type of a real Riemann surface*]{}, arXiv (2012). \[$\spadesuit$\] 

L.V. Kantorovič, [*FOUR ARTICLES IN FRENCH in the period 33–34 including multi-connected and potentially based upon Bieberbach’s method*]{} $\bigstar$$\bigstar$$\bigstar$ 

L.V. Kantorovič, V.I. Krylov, [*Methods for the approximate solution of partial differential equations*]{}, Leningrad–Moscow, 1936, Russian. \[$\spadesuit$ Chap.V is devoted to conformal mapping. §1 is introductory. §2 takes up the method of Bieberbach (1914 [@Bieberbach_1914]) which reduces the problem of conformal mapping to a minimum principle (for the area). This is then solved by Ritz’s method. In §3 a second extremal property for mapping functions is discussed and Ritz’s method is again applied §4 takes up orthogonal polynomials of Szegö and Bochner-Bergman types and applies them to the above minimizing problems.\]$\bigstar$$\bigstar$$\bigstar$ 

V. Karimipour, A. Mostafazadeh, [*Lattice topological field theory on nonorientable surfaces*]{}, J. Math. Phys. 38 (1997), 49–66. \[$\spadesuit$\] 

M.G. Katz, S. Sabourau, [*Hyperellipticity and systoles on Klein surfaces*]{}, Geom. Dedicata 159 (2012), 277–293. \[$\spadesuit$\] 

S. Katz, C.-C.M. Liu, [*Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc*]{}, Geom. Topol. Monogr. 8 (2006), 1–47; reproduced from: Adv. in Theoret. Math. Phys. 5 (2002), 1–49. \[$\spadesuit$\] 

M.V. Keldysh, M.A. Lavrentief, [*Sur la représentation conforme des domaines limités par des courbes rectifiables*]{}, Ann. Sci. Éc. Norm. Sup. 54 (1937), 1–38. \[$\spadesuit$ only the case of simply-connected domains bounded by a rectifiable Jordan curve in the plane, but deep questions about the boundary behavior of the Riemann map $\varphi$ as well as the Smirnov problem of deciding when the harmonic function $\log \vert \varphi'(w)\vert$ is representable in the unit-disc by the Poisson integral of its (limiting) values on the circumference\] 

M.V. Keldysh, [*Sur la résolubilité et la stabilité du problème de Dirichlet*]{}, C.R. Acad. Sci. URSS 18 (1938), ???–??? (French). \[$\spadesuit$ quoted in Walsh-Sinclair 1965\]$\bigstar$$\bigstar$$\bigstar$ 

M.V. Keldysh, [*Sur l’approximation en moyenne quadratique des fonctions analytiques*]{}, Mat. Sb. (N.S.) 5 (1939), 391–401. \[$\spadesuit$ quoted in Walsh-Sinclair 1965\]$\bigstar$$\bigstar$$\bigstar$ 

M.V. Keldysh, [*Conformal mappings of multiply connected domains on canonical domains*]{}, (Russian) Uspehi Mat. Nauk 6 (1939), 90–119. 78 \[$\spadesuit$ a survey of the developments in the field, up to 1939\] 

O.D. Kellogg, [*Foundations of potential theory*]{}, Grundl. d. math. Wiss. 31, Springer, Berlin, 1929. \[$\spadesuit$ “Introduction to fundamentals of potential functions covers: the force of gravity, fields of force, potentials, harmonic functions, electric images and Green’s function, sequences of harmonic functions, fundamental existence theorems, the logarithmic potential, and much more.”\] 

G. Kempf, [*Schubert methods with an application to algebraic curves*]{}, Stichting mathematisch centrum, Amsterdam, 1971. \[$\spadesuit$ the first (simultaneous with Kleiman-Laksov 1972 [@Kleiman-Laksov_1972]) existence proof of special divisors in the general case, extending thereby the result of Meis 1960 [@Meis_1960]\] 

B. Kerékjártó, [*Vorlesungen über Topologie I, Flächen Topologie*]{}, Springer, Berlin, 1923. \[$\spadesuit$ a seminal work (Part II never occurred) with parcelled appreciation (disliked by Lefschetz but admired by Weyl) $\spadesuit$ cited in Natanzon 1993 [@Natanzon_1993 p.268] for the basic result that one may lift a complex structure under a branched covering $\spadesuit$ \[30.12.12\] boosting somewhat the method one could hope to reprove so the Ahlfors theorem\] 

M. Kervaire, J. Milnor, [*On $2$-spheres in $4$-manifolds*]{}, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1651–1657. \[$\spadesuit$ as noted in Kronheimer-Mrowka 1994 [@Kronheimer-Mrowka_1994]\] 

N. Kerzman, E.M. Stein, [*The Cauchy kernel, the Szegö kernel, and the Riemann mapping function*]{}, Math. Ann. 236 (1978), 85–93. \[$\spadesuit$ quite influential, especially over Bell\] 

G. Khajalia, [*Sur la représentation conforme des domaines doublement connexes*]{}, (French) Mat. Sb. N.S. 8 (1940), 97–106. 78 \[$\spadesuit$ Seidel’s summary: the problem of mapping a doubly connected finite region on a circular ring is reduced to minimizing an area integral for a certain class of functions. If the region is accessible from without, then a sequence of minimal rational fractions converges uniformly to the desired mapping function $\spadesuit$ in fact the condition in question seems to ensure the least area map (minimizing $\int\int_{B}\vert f'
(z)\vert^2 d\omega$) to be schlicht and maps it upon the concentric circular ring $1<\vert w-w_0\vert<R$, thus the problem is different from that à la Bieberbach-Bergman handled in Kufareff 1935/37 [@Kufareff_1935/37] where the least area map is not univalent $\spadesuit$ a naive question \[05.08.12\] is whether Khajalia’s method could perform the Kreisnormierung in higher connectivity\] 

V.M. Kharlamov, [*The maximum number of components of a surface of degree $4$ in ${\Bbb R}P^3$*]{}, Funkt. Anal. Prilozhen. 6 (1972), 101; English transl., Funct. Anal. Appl. ? (197?), ?–?. \[$\spadesuit$ besides Gudkov 1969, yet another solution to one part of Hilbert’s 16th problem (or even Rohn 1886 [@Rohn_1886]) asking for the maximum number of “sheets” of the 4th order in three dimensional space (Kharlamov’s answer $10$, already in his Master Thesis 1972)\] 

V.M. Kharlamov, [*New congruences for the Euler characteristic of real algebraic manifolds*]{}, Funkt. Anal. Prilozhen. 7 (1973), 74–78; English transl., Funct. Anal. Appl. 7 (1973), 147–150. \[$\spadesuit$ cited in Wilson 1978 [@Wilson_1978] for an $(M-1)$-avatar (i.e. $p-n=k^2\pm 1 \pmod
8$) of the Gudkov-Rohlin congruence modulo 8 (i.e. $p-n=k^2 \pmod
8$).\] 

V.M. Kharlamov, [*The topological type of nonsingular surfaces in ${\Bbb R}P^3$ of degree four*]{}, Funkt. Anal. Prilozhen. 10 (1976), 55–68; English transl., Funct. Anal. Appl. 10 (1976), 295–305. \[$\spadesuit$ topological classification of nonsingular quartics surfaces in 3-space resting on the theory of K3 surfaces (via Tyurina, but [*not*]{} via Torelli’s theorem of Pyatetsky-Shapiro-Shafarevich 1971/71 [@Pyatetsky-Shapiro-Shafarevich_1971/71])\] 

V.M. Kharlamov, [*Isotopic types of nonsingular surfaces of fourth degree in ${\Bbb R}P^3$*]{}, Funkt. Anal. Prilozhen. 12 (1978), 86–87; English transl., Funct. Anal. Appl. ? (197?), ?–?. \[$\spadesuit$\] 

V.M. Kharlamov, [*Real algebraic surfaces*]{}, Proc. Internat. Congr. of Mathematicians, Helsinki, 1978, 421–428. \[$\spadesuit$\] 

V.M. Kharlamov, O.Ya. Viro, [*Congruences for real algebraic curves with singularities*]{}, Uspekhi Mat. Nauk 35 (1980), 154–155; English transl., ?? (198?), ?–?. \[$\spadesuit$ cited in Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91]\] 

V.M. Kharlamov, [*Rigid classification up to isotopy of real plane curves of degree $5$*]{}, Funkt. Anal. Prilozhen. 15 (1981), 88–89; English transl., Funct. Anal. Appl. 15 (1981), 73–74. \[$\spadesuit$ as a historical curiosity the same result in degree $6$ was effected earlier in Nikulin 1979/80 [@Nikulin_1979/80]\] 

V.M. Kharlamov, [*On the classification of non-singular surfaces of degree $4$ in ${\Bbb R}P^3$ with respect to rigid*]{}, Funkt. Anal. Prilozhen. 18 (1984), 49–56; English transl., Funct. Anal. Appl. 18 (1984), 49–56. \[$\spadesuit$\] 

V.M. Kharlamov, O. Viro, [*Extensions of the Gudkov-Rohlin congruence*]{}, in: Topology and Geometry, Rohlin Seminar, edited by O.Ya. Viro, 1984–86, Lecture Notes in Math. 1346, Springer (1988 or 1991? CHECK DATE), 357–406. \[$\spadesuit$ p.359: “type I or dividing”\] 

V.M. Kharlamov, O. Viro, [*Easy reading on topology of real plane algebraic curve*]{}, UNDATED but (ca. 1978–2013), i.e. a shortened version of the book planned (but apparently never completed) by Rohlin-Kharlamov-Viro. \[$\spadesuit$ \[21.03.13\] p.15, contains valuable information on Ragsdale, yet overlapping with Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale]. It seems to me (Gabard) that one-half of the Ragsdale conjecture follows from Thom[^137], cf. Lemma \[Thom-implies-one-half-of-Ragsdale:lem\], and so answers one half of question 10.E posed on p.15 (where incidentally it seems to me that there is the same misprint as in Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale])\] 

D. Khavinson, [*On removal of periods of conjugate functions in multiply connected domains*]{}, Michigan Math. J. 31 (1984), 371–379. 50 \[$\spadesuit$ p.377 reproves the Bieberbach-Grunsky-Ahlfors theorem in the planar case while quoting Heins 1950 [@Heins_1950] and using the classical device of annihilating “the periods of the conjugate function”\] 

R. Kirby, [*Problems in low-dimensional topology*]{}, 1970, updated in 1995 (available on the net). \[$\spadesuit$ Thom’s conjecture is mentioned as Problem 4.36, where the proof of Kronheimer-Mrowka 1994 and Morgan-Szabó-Taubes 1995 are cited\] 

G. Kirchhoff, [*Über das Gleichgewicht und die Bewegung einer elastischen Scheibe*]{}, J. Reine Angew. Math. 40 (1850), 51–88. \[$\spadesuit$ Riemann was aware of this ref. in connection to the Dirichlet principle (cf. Neuenschwander 1981 [@Neuenschwander_1981]), yet never mentions it in print $\spadesuit$ the next big revolution is Ritz, see Gander-Wander 2012 [@Gander-Wanner_2012] for a thorough “mise en perspective”\] 

S. Kirsch, [*Transfinite diameter, Chebyshev constant and capacity*]{}, in: Handbook of Complex Analysis, Elsevier, 2005. 50 \[$\spadesuit$ extract from the web (whence no page): “Ahlfors generalized Garabedian’s result to regions on Riemann surfaces \[2\](=Ahlfors 1950 [@Ahlfors_1950]); see Royden’s paper \[159\](=1962 [@Royden_1962]) for another treatment as well as further references to the literature.” $\spadesuit$ compare (if you like) our (depressive) Sec.\[dissident:sec\] for a complete list of “dissident” authors having apparently (like me) some pain to digest Ahlfors proof, and therefore cross-citing often Royden $\spadesuit$ “Abstract. The aim of the present chapter is to survey alternate descriptions of the classical transfinite diameter due to Fekete and to review several generalizations of it. Here we lay emphasis mainly on the case of one complex variable. We shall generalize this notion…”\]$\bigstar$$\bigstar$ 

S.L. Kleiman, D. Laksov, [*On the existence of special divisors*]{}, Amer. J. Math. 94 (1972), 431–436. \[$\spadesuit$ cite Riemann 1857 [@Riemann_1857], Hensel-Landsberg 1902 [@Hensel-Landsberg_1902] for linear series of dimension $1$, and Brill-Noether 1874 [@Brill-Noether_1874], Severi 1921 [@Severi_1921-Vorlesungen-u-alg.-Geom-BUCH] in the general case $\spadesuit$ supplies an existence proof of its title via Schubert calculus, Poincaré’s formula, some EGA (=Grothendieck), and a bundle constructed in Kempf’s Thesis $\spadesuit$ compare Kempf 1971 [@Kempf_1971] for a simultaneous solution of the same fundamental problem $\spadesuit$ \[08.10.12\] since this Kempf-Kleiman-Laksov result includes as a special case the result of Meis 1960 [@Meis_1960], it enables one eradicating Teichmüller theory from the gonality problem (this is not so surprising for Poincaré’s formula is essentially “homology theory” (intersection theory) specialized to the Jacobian variety, and the theta-divisor, image the $(g-1)$-symmetric power $C^{(g-1)}$ of the curve into the Jacobian via the Abel map $\spadesuit$ thus roughly speaking (and with some imagination) we are back to the method used in Gabard 2006 [@Gabard_2006] $\spadesuit$ for less arrogant looseness it would be nice to adapt the methods of Kempf/Kleiman-Laksov to the problem of the Ahlfors mapping with sharp bounds (i.e. like in Gabard 2006 [@Gabard_2006] granting of course the latter to be correct, else)\] 

S.L. Kleiman, D. Laksov, [*Another proof of the existence of special divisors*]{}, Acta Math. 132 (1974), 163–176. \[$\spadesuit$ cite Gunning’s work of 1972 [@Gunning_1972] as an alternative to Meis’ (for linear series of dimension $1$) $\spadesuit$ novel proof via the theory of singularities of mappings (Thom polynomial, Porteous’ formula, plus influence of Mattuck) $\spadesuit$ \[08.10.12\] like in the previous entry, try again to specialize the Thom-Porteous technique to the context of real algebraic geometry (orthosymmetric curve à la Klein) so as to recover the circle maps of Ahlfors 1950 [@Ahlfors_1950], optionally with the bound of Gabard 2006 [@Gabard_2006] $\spadesuit$ of course the view point of special divisors (=essentially those moving in linear systems $g_d^r$ of dimensions higher than predicted by Riemann’s inequality $\dim \vert D \vert
\deg D -g$ (due to the $g$ constraints imposed by Abelian differentials) seems to indicate that the theory of the Ahlfors map is just the top of a much larger iceberg, probably already partially explored by experts (Coppens, Huisman, Ballico, Martens, Monnier, etc.)\] 

F. Klein, [*Über die sogenannte Nicht-Euklidische Geometrie*]{}, Math. Ann. 4 (1871), also in Ges. math. Abh. I, 244–253. \[$\spadesuit$\] 

F. Klein, [*Über Flächen dritter Ordnung*]{}, Math. Ann. 6 (1873), also in Ges. math. Abh. II, 11–62. \[merely cited for Plücker 1839 [@Plücker_1839] as being the oldest user (recorded) of the method of “small perturbation”, compare also Gudkov 1974/74 [@Gudkov_1974/74] whose first entry in his Refs. list is Plücker 1839 $\spadesuit$ “Wenn eine Kurve mit Doppelpunkten gezeichnet vorliegt, so kann man aus ihr Kurven derselben Ordnung ohne Doppelpunkt oder mit weniger Doppelpunkten schematisch ableiten, indem man die in den Doppelpunkten oder einigen derselben zusammensto[ß]{}enden Kurvennäste durch ähnlich verlaufende, sich nicht treffende ersetzt. Nach diesem ebenso einfachen als fruchtbaren Prinzip \[footnote=Wer diese Prinzip zuerst verwertet hat, lä[ß]{}t sich bei dessen gro[ß]{}er Selbstverständlichkeit wohl kaum festellen. Dem Verf. is dasselbe, sowie namentlich das Beispiel der Erzeugung einer Kurve $n$-ter Ordnung aus $n$ geraden Linien, von Plücker her bekannt: vgl. z.B. dessen Theorie der algebraischen Kurven (1839), in welcher fortwärend ähnliche Überlegungen angewandt werden.\] erhählt man z.B. ohne weiteres die beiden Grundformen der ebenen Kurven dritter Ordnung, wenn \[…\]”\] 

F. Klein, [*Bemerkungen über den Zusammenhang der Flächen*]{}, (zwei Aufsätze aus den Jahren 1874 und 1875/76), Math. Ann. 7, 9 (1874, 1875/76), also in Ges. math. Abh. II, 63–77. \[$\spadesuit$ some discussions with Ludwig Schläfli about the topology of surfaces (especially in the non-orientable case) $\spadesuit$ taken together with the earlier works of Riemann, Möbius 1860/63 [@Moebius_1863] and Jordan 1866 [@Jordan_1866] this constitutes a complete classification of finite(=compact) surfaces be they orientable or not, bordered or closed $\spadesuit$ this classification is of course instrumental to Klein’s classification of the topology of real algebraic curves (equivalently symmetric Riemann surfaces), as discussed in Klein 1876 [@Klein_1876], Klein 1882 [@Klein_1882] or Klein 1892 [@Klein_1892_Realitaet], as well as in Weichold 1883 [@Weichold_1883]\] 

F. Klein, [*Über eine neue Art der Riemannschen Flächen*]{} (Erste Mitteilung), Math. Ann. 7 (1874), also in Ges. math. Abh. II, 89–98. \[$\spadesuit$ first apparition of some “new” types of Riemann surface, which later will evolve to the concept of “Klein surfaces”, but at this stage this is merely a synthetic visualization of the complex locus of a plane curve defined over the reals upon the real projective plane via the map assigning the unique real point of an imaginary line. Also this is not yet “[*was ich den “echten” Riemann zu nennen pflege*]{}” as Klein expresses himself in the Introd. to volume 2 of his Coll. Papers [@Klein-Werke-III_1923 p.5] $\spadesuit$ however it is obvious that this mode of representation is almost forgotten by now and perhaps it could be useful in the future (e.g., to reprove the Rohlin inequality saying that plane dividing curves have at least as many ovals than their half degree, cf., e.g., Gabard 2000 [@Gabard_2000] for more details and the original refs.)\] 

F. Klein, [*Über den Verlauf der Abelschen Integrale bei den Kurven vierten Grades*]{} (Erster Aufsatz), Math. Ann. 10 (1876); also in Ges. math. Abh. II, 99–135. 

F. Klein, [*Über eine neue Art von Riemannschen Flächen*]{} (Zweite Mitteilung), Math. Ann. 10 (1876), also in Ges. math. Abh. II, 136–155. \[$\spadesuit$ p.154 the first place where the dichotomy of “dividing” curves appears, under the designation “Kurven der ersten Art/zweiten Art” depending upon whether its Riemann surface is divided or not by the real locus (this is from where derived the Russian terminology type I/II) \[hopefully Klein came up later with the better terminology ortho- vs. diasymmetric!\] $\spadesuit$ p.154 contains also the first intrinsic proof of the Harnack inequality (1876)\] 

F. Klein, [*Ueber die conforme Abbildung von Flächen*]{}, Math. Ann. 19 (1882), 159–160. \[$\spadesuit$ a lovely announcement of the next item [@Klein_1882], showing a little influence of Schwarz (Ostern 1881). NB: item not reproduced in the Ges. math. Abh.\] 

F. Klein, [*Über Riemann’s Theorie der algebraischen Funktionen und ihrer Integrale*]{} B.G. Teubner, Leipzig, 1882. 60 \[$\spadesuit$ a masterpiece where Klein’s theory reaches full maturity $\spadesuit$ long-distance influence upon Teichmüller 1939 [@Teichmueller_1939] (moduli problems including the case of possibly non-orientable surfaces, alias Klein surfaces since Alling-Greenleaf), and Douglas 1936–39 [@Douglas_1936-Some-new-results; @Douglas_1939-The-most-general] and also Comessatti 1924/26 [@Comessatti_1924/26], Cecioni 1933 [@Cecioni_1933], etc. $\clubsuit$ evident (albeit subconscious) connection with Ahlfors 1950 [@Ahlfors_1950], yet first made explicit (in-print) only by Alling-Greenleaf 1969 [@Alling-Greenleaf_1969] (to the best of the writer’s knowledge)\] 

F. Klein, [*Über eindeutige Funktionen mit linearen Transformationen in sich. Erste Mitteilung.*]{} Math. Ann. 19 (1882); also in [*Gesammelte mathematische Abhandlungen. Dritter Band*]{}. 1923, Reprint Springer-Verlag, 1973, 622–626. 

F. Klein, [*Über eindeutige Funktionen mit linearen Transformationen in sich. Zweite Mitteilung.*]{} Math. Ann. 20 (1882); also in [*Gesammelte mathematische Abhandlungen. Dritter Band*]{}. 1923, Reprint Springer-Verlag, 1973, 627–629. 

F. Klein, [*Neue Beiträge zur Riemannschen Funktionentheorie*]{}, Math. Ann. 21 (1882/83); also in [*Gesammelte mathematische Abhandlungen. Dritter Band*]{}. 1923, Reprint Springer-Verlag, 1973, 630–710. 

F. Klein, [*Über Realitätsverhältnisse bei der einem beliebigen Geschlechte zugehörigen Normalkurve der $\varphi$,*]{} Math. Ann. 42 (1892), 1–29. \[$\spadesuit$ this means the canonical embedding by holomorphic differentials into ${\Bbb P}^{g-1}$, which is like the Gauss map of the Abel embedding normalized through translation within the Jacobi torus $\spadesuit$ an incredible interplay between the intrinsic geometry of the symmetric Riemann surface (including its topological characteristics) and the real enumerative issues allied to the canonical embedding, compare Gross-Harris 1981 [@Gross-Harris_1981] as the most cited best modern counterpart\] 

F. Klein, [*Riemannsche Flächen, I.*]{} Vorlesung, gehalten während des Wintersemester 1891–92, Göttingen 1892, Neuer unveränderter Abdruck, Teubner, Leipzig 1906. 60 

F. Klein, [*Riemannsche Flächen, II.*]{} Vorlesung, gehalten während des Sommersemester 1892, Göttingen 1893, Neuer unveränderter Abdruck, Teubner, Leipzig 1906. 60 \[$\spadesuit$ for those not overwhelmed by German prose and handwritings, these lecture notes gives a very exciting view over Klein’s lectures and a good supplement to his papers. NB: these 2 items are [*not*]{} reprinted in the Ges. math. Abh., and somewhat hard-to-find in Switzerland but well-known in Russia, cf. e.g. Gudkov [@Gudkov_1974/74] and Natanzon 1990 [@Natanzon_1990/90], plus also in some US references, of course\] 

F. Klein, [et al.]{} [*Zu den Verhandlungen betreffend automorphe Funktionen, Karlsruhe am 27. September 1911. Vorträge und Referate von F. Klein, L.E.J. Brouwer, P. Koebe, L. Bieberbach und E. Hilb*]{}. Jahresb. d. Deutsch. Math.-verein. 21 (1912), 153–166. \[$\spadesuit$ an account of the dramatic events occurring in 1911, when Brouwer was able to re-crack the uniformization (of Poincaré-Koebe, at least in the reasonable near to compact context) via topological methods (viz. invariance of domain) implementing thereby the old dream of Klein-Poincaré (or vice versa if you prefer)\] 

F. Klein, [*Gesammelte mathematische Abhandlungen. Zweiter Band*]{}. 1922, Reprint Springer-Verlag, 1973. 60 

F. Klein, [*Gesammelte mathematische Abhandlungen. Dritter Band*]{}. 1923, Reprint Springer-Verlag, 1973. 60, 78 

F. Klein, [*Vorlesungen über die Entwicklung der Mathematik im 19.Jahrhundert, Teil I*]{}. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd.24, Springer-Verlag, Berlin, 1926; Russian transl., [*Lektsii o razvitii matemiki v XIX stoletti*]{}, ONTI, Moscow-Leningrad, 1937. \[$\spadesuit$ where according to the legend Arnold learned all his background about mathematics $\spadesuit$ often cited, e.g. by Arnold, Gudkov, etc.\] 

T. Klotz, [*Imbedding compact Riemann surfaces in $3$-space*]{}, Pacific J. Math. 11 (1961), 1035–1043. \[$\spadesuit$ cited in Garsia 1961 [@Garsia_1961] as follows: “Some interesting results on $C^{\infty}$ imbeddings in the higher genus case have been obtained by T. Klotz in \[10\](=this paper). This author is almost successful in proving that the set of Riemann surfaces of a given genus $g\ge 2$ which can be imbedded in Euclidean space is open \[footnote 2: using the results of Kuiper it could be shown that it is dense.\] in the Teichmüller topology. Perhaps we should point out that from some of the results of the present paper one obtains the arguments that are needed to complete her proof.”\] 

M. Knebusch, [*On real algebraic curves over real closed fields. I*]{}, Math. Z. 150 (1976), 49–70. \[$\spadesuit$\] 

M. Knebusch, [*On real algebraic curves over real closed fields. II*]{}, Math. Z. 151 (1976), 189–205. \[$\spadesuit$\] 

J.T. Knight, [*Riemann surfaces of field extensions*]{}, Proc. Cmabridge Philos. Soc. 65 (1969), 635–650. \[$\spadesuit$ cited in Geyer-Martens?, Monnier 2007\] 

S.-K. Ko, [*Embedding Riemann surfaces in Riemannian manifolds*]{}, University of Connecticut, Dissertation, Aug. 1989. \[$\spadesuit$ it is shown that every compact (=closed) Riemann surface admits a conformal embedding in any preassigned Riemannian manifold of dimension $\ge 3$. Compare also the treatment in Ko 2001 [@Ko_2001]\] 

S.-K. Ko, [*Embedding bordered Riemann surfaces in Riemannian manifolds*]{}, J. Korean Math. Soc. 30 (1993), 465–484. \[$\spadesuit$ §0, Introd.: “Around 1960, A. Garsia (\[6\]=1961 [@Garsia_1961]) proved that every compact Riemann surface can be conformally immersed in Euclidean $3$-space ${\Bbb R}^3$. He stated that he had found a realization of every compact surface as a classical surface although Klein required that classical surfaces be embedded. \[Garsia’s proof uses Teichmüller’s idea, results, and constructions inspired by Nash’s embedding theorem and Brouwer’s fixed point theorem.\][^138]—In 1970, Rüedy extended Garsia’s result to open Riemann surfaces $S$ by applying Garsia’s techniques to compact exhaustions of $S$ (\[16\]=Rüedy 1971 [@Ruedy_1971]) and later[^139] he proved that every compact Riemann surface can be conformally embedded in ${\Bbb R}^3$ (\[17\]=Rüedy 1971 [@Ruedy_1971-BOOK], \[18\]=Rüedy 1968 [@Ruedy_1968]).” $\spadesuit$ \[10.12.12\] It is not clear (to Gabard) if this reflects the real history, for Rüedy himself seems always to ascribe the full embedded result to Garsia, yet perhaps by over-modesty in case Ko’s description is correct!?? $\spadesuit$ next: “In 1989, author apply[^140] Teichmüller theory to prove that we can find a conformally equivalent model surface in an orientable Riemannian manifold $\frak M$ of $\dim \frak M \ge 3$ for every compact Riemann surface (\[8\]=Ko 1989 [@Ko_1989-compact]).—Here we prove the [*extension of the Embedding theorem for compact Riemann surfaces (Ko \[8\]=Ko 1989 [@Ko_1989-compact]) to finite topological type Riemann surfaces in orientable Riemannian manifolds.*]{}\] 

S.-K. Ko, [*Embedding open Riemann surfaces in Riemannian manifolds*]{}, J. Geom. Anal. 9 (1999), 119–141. \[$\spadesuit$ like in the previous entry the author persists in his assertion that Garsia only obtained immersed conformal maps to classical surfaces, while ascribing the embedded results again to Rüedy. $\spadesuit$p.119 (abstract): “Any open Riemann surface has a conformal model in any orientable Riemannian manifold. Precisely, we will prove that, given any open Riemann surface, there is a conformally equivalent model in a prespecified orientable Riemannian manifold \[of $\dim\ge 3$\].”\] 

S.-K. Ko, [*Embedding compact Riemann surfaces in Riemannian manifolds*]{}, Houston J. Math. 27 (2001), 541–577. \[$\spadesuit$ seems to be a published account of the result arrived at in the Ph.D. Dissertation of the writer (Ko 1989 [@Ko_1989-compact]); i.e. p541 (abstract): “Any compact Riemann surface has a conformal model in any orientable Riemannian manifold. Precisely, we will prove that, given any open Riemann surface, there is a conformally equivalent model in a prespecified orientable Riemannian manifold \[of $\dim\ge 3$\]. The techniques we use include Garsia’s Continuity Lemma, Brouwer’s Fixed Point Theorem along with techniques from Teichmüller theory.”\] 

S. Kobayashi, N. Suita, [*On analytic diameters and analytic centers of compact sets*]{}, Trans. Amer. Math. Soc. 267 (1981), 219–228. 47, 50 \[$\spadesuit$ Ahlfors function and the allied conceptions of Vitushkin (analytic diameter and center), plus negative answers to several of Minsker’s questions (cf. Minsker 1974 [@Minsker_1974])\] 

S. Kobayashi, [*On analytic centers of compact sets*]{}, Kodai Math. J. 5 (1982), 318–328. 47, 50 \[$\spadesuit$ second derivative variant of the Ahlfors function developed along conceptions of Vitushkin (analytic diameter and center) and Minsker\] 

B. Köck, D. Singerman, [*Real Belyi theory*]{}, Quarterly J. Math. 58 (2007), 463–478. \[$\spadesuit$ “Abstract. We develop a Belyi-type theory that applies to Klein surfaces, that is (possibly non-orientable) surfaces with boundary which carry a dianalytic structure. In particular, we extend Belyi’s famous theorem from Riemann surfaces to KLein surfaces.”\] 

P. Koebe, [*Über konforme Abbildung mehrfach zusammenhängender ebener Bereiche, insbesondere solcher Bereiche, deren Begrenzung von Kreisen gebildet wird,*]{} Jahresb. d. Deutsch. Math.-Ver. 15 (1906), 142–153. \[$\spadesuit$ special cases of the KN=Kreisnormierung\] 

P. Koebe, [*Über konforme Abbildung mehrfach zusammenhängender ebener Bereiche*]{}, Jahresb. d. Deutsch. Math.-Ver. 16 (1907), 116–130. \[$\spadesuit$ special cases of the KN=Kreisnormierung\] 

P. Koebe, [*Über die Uniformisierung reeller algebraischer Kurven*]{}, Gött. Nachr. (1907), 177–190. \[$\spadesuit$ self-explanatory and relies heavily on Klein’s ortho- and diasymmetry\] 

P. Koebe, [*Über die Uniformisierung beliebiger analytischer Kurven*]{}, Gött. Nachr. (1907), 191–210. \[$\spadesuit$ joint with Poincaré 1907 [@Poincare_1907], the first acceptable and accepted proof of uniformization of open Riemann surfaces (alias analytical curves, by opposition to algebraic reflecting compactness, in the jargon of Fréchet) $\spadesuit$ key ingredient the “Verzerrungssatz”, for which Koebe confess some little “coup de pouce” from the colleague Carathédory\] 

P. Koebe, [*Über die Uniformisierung beliebiger analytischer Kurven, (2. Mitt.)*]{}, Gött. Nachr. (1907), 633–669. \[$\spadesuit$ another proof of the general uniformization inspired by the reading of Poincaré’s account, and using methods of Schwarz (esp. the Gürtelförmigeverschmelzung)\] 

P. Koebe, [*Über die Uniformisierung beliebiger analytischer Kurven, (3. Mitt.)*]{}, Gött. Nachr. (1908), 337–358. \[$\spadesuit$ discusses other types of uniformizations, and put forward the KNP, which he is already able to prove (in finite connectivity, or even in infinite connectivity under special symmetry), but no detailed arguments\] 

P. Koebe, [*Über die Uniformisierung der algebraischen Kurven durch automorphe Funktionen mit imaginärer Substitutionsgruppe*]{}, Gött. Nachr. (1909), 68–76. \[$\spadesuit$ announce other types of uniformization formulated by Klein\] 

P. Koebe, [*Über die Uniformisierung beliebiger analytischer Kurven, (4. Mitt.)*]{}, Gött. Nachr. (1909), 324–361. 

P. Koebe, [*Ueber die Uniformisierung der algebraischen Kurven, I*]{} Math. Ann. 67 (1909), 145–224. \[$\spadesuit$ detailed proof\] 

P. Koebe, [*Über die konforme Abbildung mehrfach zusammenhängender Bereiche*]{} Jahresb. d. Deutsch. Math.-Ver. 19 (1910), 339–348. \[$\spadesuit$ contains the general case of the KN in finite connectivity, via 2 methods: Überlagerungsfläche and the so-called Koebe iteration method $\spadesuit$ again no complete proof but the convergence is ensured by the “Verzerrungssatz” $\spadesuit$ full details only much latter in 1920–22? [@Koebe_1922] (according, e.g., to Bieberbach 1968 [@Bieberbach_1968-Das-Werk-Paul-Koebes]) $\spadesuit$ p.339: “Den Hauptgegenstand dieser und des gegenwärtigen Vortrages bildet das Problem der konformen Abbildung eines $(p+1)$-fach zusammenhängenden Bereiches auf einen von $p+1$ Vollkreisen begrenzten Bereich, ein Problem, welches in der Literatur zuerst bei Schottky (Dissertation, Berlin 1875, umgearbeitet erschienen in Crelle 1877) in seiner bekannten Doktordissertation auftritt, jedoch früher bereits von Riemann in Betracht gezogen worden ist, wie aus seiner nachgelassenen Schriften hervorgeht.”\] 

P. Koebe, [*Über die Hilbertsche Uniformisierungsmethode*]{}, Gött. Nachr. (1910), 59–74. 

P. Koebe, [*Ueber die Uniformisierung der algebraischen Kurven, II*]{} Math. Ann. 69 (1910), 1–81. 

P. Koebe, [*Begründung der Kontinuitätsmethode im Gebiete der konformen Abbildung und Uniformisierung. (Voranzeige)*]{}, Nachr. Königl. Ges. Wiss. Gött., Math.-phys. Kl. (1912), 879–886. \[$\spadesuit$ self-explanatory, but compare the practically simultaneous work of Brouwer 1912 [@Brouwer_1912_top-Schwierig], plus the announcements in 1911 [@Klein-Brouwer-Koebe_1912]\]

P. Koebe, [*Ueber eine neue Methode der konformen Abbildung und Uniformisierung,*]{} Nachr. Königl. Ges. Wiss. Gött., Math.-phys. Kl. (1912), 844–848. \[$\spadesuit$ introduction of the Schmiegungsverfahren (osculation method?)\] 

P. Koebe, [*Begründung der Kontinuitätsmethode*]{}, Ber. Math. Math.-phys. Kl. Sächs. Akad. Wiss. Leipzig 64 (1912), 59–62. 

P. Koebe, [*Ränderzuordnung bei konformer Abbildung*]{}, Gött. Nachr. (1913), 286–288. \[$\spadesuit$ contests the heavy reliance upon Lebesgue’s measure theory in Carathéodory’s proof (1912) of the boundary behavior of the Riemann mapping for Jordan curves, by appealing to a device of Schwarz\] 

P. Koebe, [*Ueber die Uniformisierung der algebraischen Kurven, IV (Zweiter Existenzbeweis der allgemeinen kanonischen uniformisierenden Variablen: Kontinuitätsmethode)*]{}, Math. Ann. 75 (1914), 42–129. 

P. Koebe, [*Abhandlungen zur Theorie der konformen Abbildung, I, die Kreisabbildung des allgemeinsten einfach und zweifach zusammenhängenden schlichten Bereichs und die Ränderzuordnung bei konformer Abbildung*]{}, J. Reine Angew. Math. 145 (1915), 177–223. \[$\spadesuit$ uses the word “Kreisabbildung” which is perhaps first coined in Bieberbach 1914 [@Bieberbach_1914]\] 

P. Koebe, [*Abhandlungen zur Theorie der konformen Abbildung, IV*]{}, Acta Math. 41 (1918), 305–344. \[$\spadesuit$ first existence proof of the circular/radial slit maps for domains of finite connectivity (general case in Grötzsch 1931 [@Groetzsch_1931]); subsequent proof in Reich-Warschawski 1960 [@Reich-Warschawski_1960]\] 

P. Koebe, [*Über die Strömungspotentiale und die zugehörenden konformen Abbildungen Riemannscher Flächen*]{}, Gött. Nachr. (1919), 1–46.

P. Koebe, [*Abhandlungen zur Theorie der konformen Abbildung. VI. (Abbildung mehrfach zusammenhängender schlichter Bereiche auf Kreisbereiche. Uniformisierung hyperelliptischer Kurven. Iterationsmethoden)*]{}, Math. Z. 7 (1920), 235–301. 

P. Koebe, [*Abbildung beliebiger mehrfach zusammenhängender schlichter Bereiche auf Kreisbereichen*]{}, Math. Z. 7 (1922), 116–130. 

P. Koebe, [*????*]{}, Acta Math. ?? (1928), ??–??. 

P. Koebe, [*Das Wesen der Kontinuitätsmethode*]{}, Deutsche Math. 1 (1936), 859–879. 78 \[$\spadesuit$ survey-like with many refs.\] $\bigstar$$\bigstar$$\bigstar$ 

H. Köditz, St. Timmann, [*Ranschlichte meromorphe Funktionen auf endlichen Riemannschen Flächen*]{}, Math. Ann. 217 (1975), 157–159. 78 \[$\clubsuit$ supply a proof of a circle map (without bound) using techniques of Behnke-Stein $\clubsuit$ criticizes and demolishes an earlier argument of Tietz 1955 [@Tietz_1955] intended to give another treatment of the Ahlfors circle map\] 

G. Kokarev, [*Variational aspects of Laplace eigenvalues on Riemannian surfaces* ]{}, arXiv:1103.2448, 2011. \[$\spadesuit$ Abstract: We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem (and its higher eigenvalue versions) via the direct ...\] 

J. Kóllar, [*The topology of real and complex algebraic varieties*]{}, Adv. Stud. Pure Math. 31 (2001), Math. Soc. Japan, 127–145. 

Y. Komatu, [*Identities concerning canonical conformal mappings*]{}, Kōdai math. Sem. Rep. 3 (1953), 77–83. 

W. Koppelman, [*The Riemann-Hilbert problem for finite Riemannian surfaces*]{}, Comm. Pure Appl. Math. 12 (1959), 13–35. \[$\spadesuit$ work oft cited in the investigation of the Slovenian school, see e.g. Černe-Forstnerič 2002 [@Cerne-Forstneric_2002] $\spadesuit$ “The problem of finding a function, analytic in some domain $D$, for a given relation between the limiting values of its real and imaginary parts on the boundary of $D$ was originally mentioned by Riemann in his dissertation \[12\]. Here we shall treat the special case where …”\]$\bigstar$$\bigstar$$\bigstar$ 

A.B. Korchagin, [*$M$-curves of degree $9$: New prohibitions*]{}, Math. Notes 39 (1986), 277–293. \[$\spadesuit$\]$\bigstar$ 

A. Korn, [*Application de la méthode de la moyenne arithmétique aux surfaces de Riemann*]{}, C.R. Acad. Sci. Paris 135 (1902), 94–95. 60 \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$ 

A. Korn, [*Sur le problème de Dirichlet pour des domaines limités par plusieurs contours (ou surfaces)*]{}, C.R. Acad. Sci. Paris 135 (1902), 231–232.$\bigstar$$\bigstar$$\bigstar$60 

A. Korn, [*Über die erste und zweite Randwertaufgabe der Potentialtheorie*]{}, Rend. Circ. Mat. Palermo 35 (1913), 317–323. \[$\spadesuit$ application of the authors’s theory of the asymmetrical kernel to the first and second boundary value problem of potential theory and its resolution by the method of the arithmetical mean (C. Neumann, Robin) leading anew to the solution predicted by Poincaré 1896 [@Poincare_1896], which the author first succeeded in 1901 after appealing to a result of Zaremba (1901)\] 

I. Kra, [*Maximal ideals in the algebra of bounded analytic functions*]{}, ???? 31 (1967), 83–88. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is applied to a characterization of “point-like” maximal ideals in the function algebra $\spadesuit$ more precisely Ahlfors is cited on p.85 as follows (yet without precise control on the degree except for its finiteness): “Lemma 5. Let $X$ be a finite domain of the Riemann surface $W$. Then for each discrete sequence $\{x_n\}\subset X$, there exists an $f\in B(X)$ \[=the ring of bounded holomorphic functions, cf. p.83\] such that $\lim_{n\to \infty} f(x_n)$ does not exist.—[*Proof*]{}. Ahlfors \[1\](=1950 [@Ahlfors_1950]) has shown that there exists a mapping $p$, analytic in a neighborhood of ${\rm Cl} X$, that is an $N$-to-one covering of the closed unit disc, for some positive integer $N$. Moreover $p\vert X$ is an $N$-to-one covering of the interior of the closed unit disc, and $p\vert {\rm Cl} X-X$ is an $N$-to-one covering of the unit circle. Because ${\rm Cl} X$ is compact we may assume (by choosing a subsequence) that $x_n\to x \in {\rm Cl} X-X$. Then $p(x_n)\to 1$ \[modulus missing??\] and $\vert p(x_n)\vert<1$. Again, we may choose a subsequence such that $p(x_n)$ is distinct \[???\] and infinite and constitutes an interpolating sequence (see Hoffman \[6,pp.194–204\]). Choose a bounded analytic function $f$ on the unit disc such that $f(p(x_{2n+1}))=0$ and $f(p(x_{2n}))=1$ for $n=0,1,2,\dots$. Then $f\circ p \in B(X)$, and $\lim_{n\to \infty}(f\circ p)(x_n)$ does not exist. \[q.e.d.\]” $\spadesuit$ p.87: “Theorem 2 is a generalization of Theorem 1, because every boundary point of a finite domain is an essential singularity for some bounded holomorphic function. The unit disc certainly has this property. The general case is reduced to the unit disc [*via*]{} any Ahlfors maps. (See the proof of Lemma 5.)”\] 

I. Kra, [*Automorphic Forms and Kleinian Groups*]{},Benjamin, Reading, Mass., 1972, 464 pp. \[$\spadesuit$\] 

S.G. Krantz, [*The Carathéodory and Kobayashi metrics and applications in complex analysis*]{}, Amer. Math. Monthly 115 (2008), 304–329. \[$\spadesuit$ p.311 brief mention of the Ahlfors function and as it is connected to the Carathéodory metric; for more on the Ahlfors function the reader is referred to Fisher 1983 [@Fisher_1983] or the book Krantz (2006)\] 

V.A. Krasnov, [*Albanese mapping for real algebraic varieties*]{}, Mat. Zametki 32 (1982), 365–374; English transl., Math. Notes 32 (1983), 661–666. \[$\spadesuit$\] 

V.A. Krasnov, [*Albanese map for GM${\Bbb Z}$ varieties*]{}, Mat. Zametki 35 (1984), 739–747; English transl., Math. Notes 35 (1984), 391–396. \[$\spadesuit$\] 

D. Kraus, O. Roth, [*Critical points, the Gauss curvature equation and Blaschke products*]{}, arXiv (2011). 47, 78 \[$\spadesuit$ p.15 the Ahlfors map is mentioned\] 

S. Kravetz, [*On the geometry of Teichmüller spaces and the structure of their modular groups*]{}, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 278 (1959), 1–35. \[$\spadesuit$ according to Natanzon 1999 [@Natanzon_1999-Moduli-real-alg-surf.superanal-differ-spinors p.1101], this Kravetz’s paper was employed in Earle’s 1971 [@Earle_1971-On-the-moduli] description of the topological structure of the components of the moduli space of real algebraic curves as being each diffeomorphic to ${\Bbb R}^{3g-3}/ {\rm
Mod}_{g,r,\varepsilon}$ for a suitable discrete (modular) group. Earle’s proof used the theory of quasiconformal maps, but relied on a Kravetz’s theorem “which turned out latter to be wrong”. Still according to Natanzon () “A correct proof based on the theory of quasiconformal maps was obtained in Seppälä 1978 [@Seppala_1978-Teich-spaces-of-Klein-surfaces].”\] 

?. Krazer, [*Lehrbuch der Thetafuntionen*]{}, Teubner, 1903. \[$\spadesuit$ student of Prym, in turn student of Riemann\] 

I.M. Krichever, S.P. Novikov, [*Virasoro-Gelfand-Fuchs type algebras, Riemann surfaces, operator theory of closed strings*]{}, J. Geom. Phys. 5 (1988), 631–661. \[$\spadesuit$\] 

L. Kronecker, [*Über die Diskriminante algebraischer Funktionen*]{}, Crelles J. 91 (1881). \[$\spadesuit$\] 

L. Kronecker, [*Über einige Interpolationsformeln für ganze Funktionen mehrer Variabeln*]{}, Werke Leipzig 1895, Bd.I, 133–141. \[$\spadesuit$ quoted in Petrovskii 1938 [@Petrowsky_1938] as one of the tool involved in the proof of the Petrovskii’s inequalities, where Kronecker’s work connects to the so-called Euler-Jacobi (interpolation) formula\] 

P.B. Kronheimer, T.S. Mrowka[^141], [*The genus of embedded surfaces in the projective plane*]{}, Math. Res. Letters 1 (1994), 797–808. \[$\spadesuit$ a proof of the Thom conjecture on the genus of smooth surfaces embedded in the complex projective plane, via Gauge theory (Donaldson theory, etc.) $\spadesuit$ this has some modest relevance to Hilbert’s 16th problem, cf. e.g. Theorem \[Thom-Ragsdale:thm\] in this text $\spadesuit$ the special degree $3$ case of Thom’s conjecture was known to Kervaire-Milnor 1961 [@Kervaire-Milnor_1961]\] 

V. Krylov, [*Une application des équations intégrales à la démonstration de certains théorème de la théories des représentations conformes*]{}, (Russian, French Summary) Rec. Math. de Moscou \[Mat. Sb.\] 4 (1938), 9–30. 78 \[$\spadesuit$ Seidel’s summary: the problem of mapping conformally a region of connectivity $n$, bounded by $n$ analytic contours, on various canonical domains is reduced to the problem of solving a system of simultaneous integral equations\] 

T. Kubo, [*Bounded analytic functions in a doubly connected domain*]{}, Mem. Coll. Sci. Univ. Kyoto, A. 26 (1951), 211–223. 

T. Kubota, [*Über konforme Abbildungen. I.*]{}, Science Reports Tôhoku Imperial Univ. ser. I, 9 (1920), 473–490. \[$\spadesuit$ quoted in Grunsky 1932 [@Grunsky_1932 p.135] for the simply-connected case of an extension of Bieberbach’s first Flächensatz\]$\bigstar$$\bigstar$ 

T. Kubota, [*Über konforme Abbildungen. II.*]{}, Science Reports Tôhoku Imperial Univ. ser. I, 10 (1921). \[$\spadesuit$\]$\bigstar$ 

P. Kufareff, [*Über das zweifach zusammenhängende Minimalgebiet*]{}, Bull. Inst. Math. et Mec. Univ. de Tomsk 1 (1935–37), 228–236. \[$\spadesuit$ quoted in Lehto 1949 [@Lehto_1949] and Bergman 1950 [@Bergman_1950], and akin to the works of Zarankiewicz 1934 [@Zarankiewicz_1934; @Zarankiewicz_1934-numerisches] $\spadesuit$ Seidel’s summary: a minimal problem is set up for functions analytic and single-valued in a circular ring and the mapping effected by the minimizing function is discussed\] $\bigstar$ $\bigstar$$\bigstar$ 

R. Kühnau, [*Über die analytische Darstellung von Abbildungsfunktionen, insbesondere von Extremalfunktionen der Theorie der konformen Abbildung*]{}, J. Reine Angew. Math. 228 (1967), 93–132. 78 \[$\spadesuit$ p.95–96 proposes a contribution to a question raised by Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949] related to the representation of the so-called Schottky function (via [*Normalabbildungsfunktionen*]{}) $\spadesuit$ Kühnau alludes to several (subsequent) work of Schottky where the circle maps should appear again? (no precise refs. hence requires some detective work)\] 

R. Kühnau, [*Geometrisch-funktionentheoretische Lösung eines Extremalsproblems der konformen Abbildung, I, II*]{}, J. Reine Angew. Math. 229 (1967), 131–136; 237 (1969), 175–180. 78 \[$\spadesuit$\] 

R. Kühnau, [*Herbert Grötzsch zum Gedächtnis*]{}, Jber. d. Dt. Math.-Verein. 99 (1997), 122–145. \[$\spadesuit$ alas, cited merely for the matter of the “quasi-conformal” jargon, as occuring apparently first (the jargon, not the concept) in Carathéodory 1914 [@Caratheodory_1914]\] 

V.S. Kulikov, [*Epimorphicity of the period map for $K3$ surfaces*]{}, Uspekhi Mat. Nauk 32 (1977), 257–258. (Russian) \[$\spadesuit$ employed in Nikulin’s (1979/80 [@Nikulin_1979/80]) rigid-isotopy classification of plane real sextics\] 

Z. Kuramochi, [*A remark on the bounded analytic function*]{}, Osaka Math. J. 4 (1952), 185–190. 50, 60 \[$\clubsuit$ p.189 seems to reprove the result of Ahlfors 1950 [@Ahlfors_1950] about the existence of a circle map of degree $\le r+2p$ by using the Green’s function (while generalizing a method of Nehari 1951 [@Nehari_1951] for the case of plane domains) $\spadesuit$ unfortunately Kuramochi’s paper is written in some mysterious tongue (the Nipponglish), and despite its moderate size (of ca. 5 pages) it contains several dozens of misprints obstructing seriously its readability $\spadesuit$ despite our critical comments this work is quoted in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] so should probably be not completely science-fictional $\spadesuit$ it would perhaps be desirable (in case this paper emerged from some solid underlying structure) to undertake a polishing of this Kuramochi paper to improve its readability\] 

A. Kuribayashi, [*On analytic families of compact Riemann surfaces with non-trivial automorphisms*]{}, ??? ?? (1966), 119–165. \[$\clubsuit$ Teichmüller theory à la Ahlfors-Bers, plus an influence of Shimura $\spadesuit$ p.133: “Thm 2.17. [*There exists one and only one Riemann surface up to conformal equivalence which has group of automorphism of order 168 among compact Riemann surfaces of genus $3$.”*]{} $\spadesuit$ is this uniqueness new? perhaps already in Hurwitz?\] 

Y. Kusunoki, [*Über die hinreichenden Bedingungen dafür, dass eine Riemannsche Fläche nullberandet ist,*]{} Mem. Coll. Sci. Univ. Kyoto 28 (1952), 99–108. 60 \[also cited in Sario-Nakai [@Sario-Nakai_1970] CHECK $\spadesuit$ an application of Ahlfors 1950 [@Ahlfors_1950] (and the older predecessor Bieberbach 1925 [@Bieberbach_1925]) is given to the type problem\] 

Y. Kusunoki, [*Contributions to Riemann-Roch’s theorem*]{}, Kyoto J. Math. ? (1958), ?–?. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited\] 

Y. Kusunoki, [*Square integrable normal differentials on Riemann surfaces*]{}, J. Math. Kyoto Univ. 3 (1963), 59–69. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited on p.64, in the following connection (as usual many “notatio”): “If $R$ is a bordered surface with $p$ contours, $\{A_n,B_n, C_{\nu}\}_{n=1, \dots, g; \nu 1, \dots,
p-1}$ is admissible for $\Gamma_0=\Gamma_{aS}=\Gamma_{AB}
\oplus\Gamma_C$ and $P_\gamma$ gives a one-to-one mapping of $\Gamma_0=\Gamma_0'$ to $(2g+p-1)$-dimensional vector space by (II) (Ahlfors \[1\](=1950 [@Ahlfors_1950])).”\] 

M.P. Kuvaev, P.P. Kufarev, [*An equation of Löwner’s type for multiply connected regions*]{}, Tomskiĭ gos. Univ. Uč. Zap. Mat. Meh. 25 (1925), 19–34. 78 $\bigstar$ $\bigstar$$\bigstar$ 

J.-L. Lagrange, [*??*]{}, 1779. \[$\spadesuit$ a source often quoted e.g. by Koebe \[ca. 1910, in Math. Ann.\], and Monastyrsky 1987/99 [@Monastyrsky_1987/99/08-even-1979] where one reads (p.15): “It is noteworthy that Joseph-Louis Lagrange (1736–1813) obtained the Cauchy-Riemann equations conditions also in 1779, also in connection with the solution to a cartographic problem.”\] 

J.-L. Lagrange, [*Mecanique analitique*]{}, 2 volumes, Paris, 1788. \[$\spadesuit$\] 

J.-L. Lagrange, [*Theorie des fonctions analytiques*]{}, Paris, 1797, 2ème édition 1813. \[$\spadesuit$\] 

E. Landau, [*Einige Bermerkungen über schlichte Abbildung*]{}, Jahresb. Dt. Math.-Verein. 34 (1926), 239–243. \[$\spadesuit$\] 

H.J. Landau, R. Osserman, [*On analytic mappings of Riemann surfaces*]{}, J. Anal. Math. (1960), 249–279. \[$\spadesuit$ p.266 contains the basic lemma that an analytic map taking the boundary to the boundary is a (full) branched covering (this follows directly from the local behavior of such maps and bears a certain relevance to the Ahlfors circle map) $\spadesuit$ however the paper does not seem to supply an existence proof of the Ahlfors map $\spadesuit$ in fact it is worth reproducing the text faithfully (p.265): “We now turn to the problem of mapping one Riemann surface into another. We shall need a lemma which, in a special case, was proved by Radó \[12\](=Radó 1922 [@Rado_1922-Z-Theorie-mehr]). Let us recall that a sequence of points in a Riemann surface is said to tend to the boundary if the sequence has no limit points in $R$[^142]. We shall say that a map $f$ of one Riemann surface $R_1$ into another, $R_2$, takes the boundary into the boundary if for every sequence of points in $R_1$ which tends to the boundary, the image sequence tends to the boundary of $R_2$. Let us note that if $R_1$ and $R_2$ are relatively compact regions on Riemann surfaces, the above definition coincides exactly with the usual notion of mapping the boundary into the boundary.—[**Lemma 3.1:**]{} [*Let $R_1$ and $R_2$ be any two Riemann surfaces and $f$ an analytic map of $R_1$ into $R_2$ which takes the boundary into the boundary. Then $f$ maps $R_1$ onto $R_2$, and every points of $R_2$ is covered the same number of times, counting multiplicities.*]{}” $\spadesuit$ for this basic lemma see also the treatments in Stoïlow 1938 [@Stoilow_1938-Lecons Chap.VI] and Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.41, 21B.]\]

G. Landsberg, [*Algebraische Untersuchungen über den Riemann-Roch Satz*]{}, Math. Ann. 50 (1898), 333–380. \[$\spadesuit$ drifting the transcendental theory toward arithmetization\]

G. Landsberg, [*Über das Analogon des Riemann-Roch Satz in der Theorie der algebraischen Zahlen*]{}, Math. Ann. 50 (1898), 577–582. \[$\spadesuit$\]

M. Lavrentieff, [*On the theory of conformal mapping*]{}, Trav. Inst. phys.-math. Stekloff 5 (1934), 159–245. \[$\spadesuit$ cited in Schiffer 1950 [@Schiffer_1950-Appendix-Courant]\] $\bigstar$$\bigstar$$\bigstar$ 

P.D. Lax, [*Reciprocal extremal problems in function theory*]{}, Comm. Pure Appl. Math. 8 (1955), 437–453. 78 \[$\spadesuit$ extract from Rogosinski’s review (MathReview): “This principle is dual to one used for similar problems by the reviewer and H.S. Shapiro (=Rogosinski-Shapiro 1953 [@Rogosinski-Shapiro_1953]); both principles are easy interpretations of the Hahn-Banach extension theorem in the complex case. \[…\] This important paper is somewhat marred by numerous misprints and a rather loose presentation.” \] $\bigstar$ 

R.F. Lax, [*On the dimension of the varieties of special divisors*]{}, Illinois J. Math. 19 (1975), 318–324. \[$\spadesuit$ extract from H.H. Martens’s review (MathReview): “The proof is inspired by the methods of T. Meis 1960 [@Meis_1960], and the paper contains, in addition to the author’s results, a very useful review of Meis’ monograph, which is rather difficult to obtain.” \] $\bigstar$ 

R. Le Vavasseur, [*Sur la représentation conforme de deux aires planes à connexion multiple, d’apès M. Schottky*]{}, Ann. Fac. Sci. Toulouse (2) 4 (1902), 45–100. 78 \[$\spadesuit$ re-expose the results of Schottky 1877 [@Schottky_1877]\] 

H. Lebesgue, [*Intégrale, Longueur, Aire*]{}, Annali di Mat. 7 (1902), 231–358. \[$\spadesuit$ Lebesgue’s Thesis, where Lebesgue’s integration and the allied geometry is introduced (yet another descendant of Riemann) $\spadesuit$ Fatou 1906 [@Fatou_1906], F. Riesz 1907 and Carathéodory 1912 [@Caratheodory_1912] are the best illustration of the rôle of measure theory in (complex) function theory, a rôle disputed by Koebe at least in the early steps (compare Gray’s fine analysis [@Gray_1994])\] 

H. Lebesgue, [*Sur le principe de Dirichlet*]{}, Rend. Circ. Mat. Palermo 24 (1907), 371–402. \[$\spadesuit$ extension of Hilbert’s solution to the Dirichlet problem by allowing general boundaries, cf. also Zaremba 1910 [@Zaremba_1910] for possible simplifications and (Beppo Levi 1906 [@Beppo-Levi_1906] and Fubini 1907 [@Fubini_1907] for related contributions of the same period $\spadesuit$ further (quasi-ultimate) simplifications in Perron 1923 [@Perron_1923], in turn simplified in Radó-Riesz 1925 [@Rado-Riesz_1925]\] 

S. Lefschetz, [*On certain numerical invariants of algebraic varieties with applications to Abelian varieties*]{}, Trans. Amer. Math. Soc. 22 (1921), 327–482. \[$\spadesuit$\] 

S. Lefschetz, [*L’analysis situs et la géométrie algébrique*]{}, Gauthier-Villars, 1924. \[$\spadesuit$\] 

J. Lehner, M. Newman, [*On Riemann surfaces with maximal automorphism groups*]{}, Glasgow Math. J. 8 (1967), 102–112. \[$\spadesuit$\] 

O. Lehto, [*Anwendung orthogonaler Systeme auf gewisse funktionentheoretische Extremal- und Abbildungsprobleme*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 59 (1949), 51 pp. \[$\spadesuit$ new existence proof of parallel-slit mappings via the Bergman kernel (and so in particular of RMT, answering thereby the old desideratum of Bieberbach 1914 [@Bieberbach_1914]-Bergman 1922 [@Bergman_1922]-Bochner 1922 [@Bochner_1922]); equivalent work in Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950] $\spadesuit$ p.48 reproves the identity $B(z)=1/S(w(z)-w^{\ast}(z))$ (expressing the least area map as combination of the two Schlitzfunktionen $w$ and $w^{\ast}$) announced by Grunsky 1932 [@Grunsky_1932] $\clubsuit$ p.41 seems to show that the least area map is a circle map\] 

O. Lehto, K.I. Virtanen, [*Quasiconformal mappings in the plane*]{}, 2nd edition, Springer-Verlag, Berlin, 1973. \[$\spadesuit$\] 

O. Lehto, [*Univalent functions and Teichmüller spaces*]{}, Springer-Verlag, Berlin, 1986. \[$\spadesuit$\] 

O. Lehto, [*On the life and work of Lars Ahlfors*]{}, Math. Intelligencer (1998), 4–8. \[$\spadesuit$ p.7: “In this same paper (1953/54), Ahlfors also defined the notion which he called Teichmüller space.”\] 

G.W. Leibniz, [*Characteristica Geometrica*]{}, 1679. \[$\spadesuit$ beside Descartes (pseudo?)-anticipation of the Euler characteristic theorem for spherical polyhedrons ($V-E+F=2$), this is oft regarded as the first “topological” text, summarized as follows in Monastyrsky 1987/99 [@Monastyrsky_1987/99/08-even-1979], p.89: “In 1679 Leibniz published \[t\]his famous book \[…\], in which (in modern terms) he tried to study the topological rather than the metric characteristics of properties of figures. He wrote that, aside from the coordinate representation of figures, ‘we are in need of another analysis, purely geometric or linear, which also defines the position (situs), as algebra defines magnitude.’ It is interesting to note that Leibniz tried to interest Christiaan Huygens (1629–1695) in his work, but the latter showed little enthusiasm. This was the first (albeit unsuccessful) attempt to interest a physicist in topology.” $\spadesuit$ so this Leibniz’s text must be the first place where the term “analysis situs” appears in embryo, then rebaptized “Topologie” in Listing 1847 [@Listing_1847-Vorstudien-zur-Topologie] (yet receiving only slow acceptance, say in the 1920’s, e.g. Riemann, Poincaré, etc. used exclusively the term “analysis situs”).\] 

F. Leja, [*Une méthode de construction de la fonction de Green appartenant à un domaine plan quelconque*]{}, C.R. Acad. Sci. Paris 198 (1934), 231–234. \[$\spadesuit$ Seidels’ summary: a method for constructing the Green’s function of an arbitrary region is given. The approximating functions are closely related to Lagrange polynomials)\] 

F. Leja, [*Construction de la fonction analytique effectuant la représentation conforme d’un domaine plan quelconque sur le cercle*]{}, Math. Ann. 111 (1935), 501–504. \[$\spadesuit$ Seidel’s summary: for a given bounded simply-connected \[sic!?\] region in the plane (of the complex variable $z$), a sequence of elementary functions is constructed which tends to the \[Riemann\] mapping function of the region\] 

F. Leja, [*Sur une suite de polynômes et la représentation conforme d’un domaine plan quelconque sur le cercle*]{}, Annales Soc. Polonaise de Math. 14 (1936), 116–134. \[$\spadesuit$ Seidel’s summary: a set of polynomials is obtained by means of which the mapping function of a region $D$, with $z=\infty$ as interior point, on $\vert w\vert>1$ can be expressed. If $D$ is simply-connected, the map is one-to-one (schlicht) $\spadesuit$ question (of Gabard) and if not, does it relates to the map of Riemann-Bieberbach-Grunsky-Ahlfors (cf. e.g. Bieberbach 1925 [@Bieberbach_1925] and Ahlfors 1947 [@Ahlfors_1947])\]$\bigstar$$\bigstar$ 

J. Lewittes, [*Automorphisms of compact Riemann surfaces*]{}, Amer. J. Math. (1963), 738–752. \[$\spadesuit$\]$\bigstar$ 

J. Lewittes, [*Riemann surfaces and the theta function*]{}, Acta Math. 111 (1964), 37–61. \[$\spadesuit$\] 

B. Levi, [*Sul Principo di Dirichlet*]{}, Rend. Circ. Mat. Palermo (1906). \[$\spadesuit$ cited in Zaremba 1910 [@Zaremba_1910] an extension of Hilbert’s resurrection of the Dirichlet principle\] 

P. Li, S.-T. Yau, [*A new conformal invariant and its application to the Willmore conjecture and the first eigenvalue of a compact surface*]{}, Invent. Math. 69 (1982), 269–291. \[$\spadesuit$ p.272 claims a result along the line of the Witt-Martens mapping theorem for symmetric surfaces without fixed points, but the Li-Yau argument appears as sketchy, or maybe even invalid according to Ross 1997 [@Ross_1997]\] 

J.-L. Lions, [*Remarks on reproducing kernels and some function spaces*]{}, In: Function Spaces, Interpolation Theory and Related topics, Proceedings, Lund, Sweden, 2000, Walter de Gruyter, 2002, 49–59. \[$\spadesuit$ present the definition of the reproducing kernel in the general setting due to Aronszajn (p.50): “This definition is due to N. Aronszajn \[1\](=Aronszajn [@Aronszajn_1950]) who studied general properties of reproducing kernels.—In particular cases, such notions have been introduced by S. Bergman \[2\](=1922 [@Bergman_1922]), G. Szegö \[11\](=1921 [@Szego_1921]), M. Schiffer \[9\](=1946 [@Schiffer_1946]), S. Zaremba \[12\](=1908 [@Zaremba_1908-calcul-numerique]), where the corresponding reproducing kernels are computed and estimated; cf. N Aronszajn, loc.cit., and P. Garabedian \[3\](=1949 [@Garabedian_1949]).” $\spadesuit$ p.56: “All these kernels can be computed by the same strategy as above. But we have not been able to recover by this method the results of P. Garabedian \[3\](=1949 [@Garabedian_1949]), which give the connection between $S(x,b)$ and $B(x,b)$.”\] 

J.B. Listing, [*Vorstudien zur Topologie*]{}, Göttinger Studien, 1847. \[$\spadesuit$ influenced by Descartes, Leibniz 1679 [@Leibniz_1679], Euler, Gauss, etc. and influential upon Riemann (who attended in 1850 a seminar on mathematical physics run by W.Weber, Listing, Stern, and Ulrich), cf. e.g. p.9 of Monastyrsky 1987/99 [@Monastyrsky_1987/99/08-even-1979], as well as upon Tait and Kelvin\] 

M.S. Li Chiavi, [*Sulla rappresentazione conforme delle aree pluriconnesse appartenti a superficie di Riemann su un’opportuna superficie di Riemann su cui siano eseguiti dei tagli paralleli*]{}, Rend. Sem. Mat. Univ. Padova 3 (1932), 95–107. 60 \[$\spadesuit$ Maria Stella Li Chiavi is a student of Cecioni\] 

L. Lichtenstein, [*Randwertaufgaben der Theorie der linearen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus. II*]{}, J. Reine Angew. Math. 143 (1913), 51–105. \[$\spadesuit$ quoted in Nevanlinna 1939 [@Nevanlinna_1939] for Schwarz’s alternating procedure recasted as the solution of an integral equation through successive approximations\] 

L. Lichtenstein, [*Zur Theorie der konformen Abbildung nichtanalytischer, singularitätenfreier Flächenstücke auf ebene Gebieten*]{}, Bull. Internat. Acad. Sci. Cracovie, Cl. Sci. Math. Nat. Ser. A. (1916), 192–217. 60 \[$\spadesuit$ an extension of Gauss 1825 [@Gauss_1825] (local isothermic coordinates), simultaneous work by Korn\] $\bigstar$$\bigstar$$\bigstar$ 

L. Lichtenstein, [*Zur konformen Abbildung einfach zusammenhängender schlichter Gebiete*]{}, Archiv der Math. u. Physik 25 (1917), 179–180. \[$\spadesuit$ Seidels’ summary: the problem of mapping conformally on a circle a simply-connected region bounded by a simple closed curve with continuous curvature is reduced to the solution of a linear integral equation\] $\bigstar$$\bigstar$$\bigstar$ 

L. Lichtenstein, [*Neuere Entwicklung der Potentialtheorie. Konforme Abbildung*]{}, Encykl. d. math. Wiss. II, 3., 1. Hälfte, 177–377. Leipzig, B.G. Teubner, 1919. 78 \[$\spadesuit$ should contain another proof of the Kreisnormierung in finite connectivity, according to Hawley-Schiffer 1962 [@Schiffer-Hawley_1962]\] $\bigstar$ 

B.V. Limaye, [*Blaschke products for finite Riemann surfaces*]{}, Studia Math. 34 (1970), 169–176.\[$\spadesuit$ the paper starts with a little manipulation amounting to annihilate the periods of a conjugate differential, hence quite in line with say Ahlfors 1950 [@Ahlfors_1950], yet does not seem to reprove the existence result of a circle map\] 

B.V. Limaye, [*Ahlfors function on triply connected domains*]{}, J. Indian Math. Soc. 37 (1973), 125–135. 78 $\bigstar$ 

I. Lind, [*An iterative method for conformal mappings of multiply-connected domains*]{}, Ark. Mat. 4 (1963), 557–560. 78 \[$\spadesuit$ another proof of PSM (due to Schottky 1877, Cecioni 1908 [@Cecioni_1908], Hilbert 1909 [@Hilbert_1909], Koebe, etc.) via iterative scheme à la Koebe (who uses rather this device for the harder Kreisnormierung)\] 

E. Lindelöf, [*Memoire sur la théorie des fonctions entières de genre fini*]{}, 1902. \[$\spadesuit$\] 

M. Lindner, [*Über Mannigfalfigkeiten ebener KKurven mit Singularitäten*]{}, Archiv. Math. 28 (1977), 603–606. \[$\spadesuit$ cited e.g. in SHustin 1990/91 [@Shustin_1990/91-Geom-of-discr-alg-curve] $\spadesuit$ seems to contain a variant of the Severi-Brusotti description of the discriminant\] 

F. Lippich, [*Untersuchungen über den Zusammenhang der Flächen im Sinne Riemann’s*]{}, Math. Ann. 7 (1874), 212–230. 60 \[$\spadesuit$ topology of surfaces, overlaps slightly with Möbius and Jordan 1866 [@Jordan_1866], but no cross-citations\] 

K. Löbenstein, [*Über den Satz, dass eine ebene algebraische Kurve 6. Ordnung mit $11$ sich einander ausschliessenden Ovalen nicht existiert*]{}, Inaugural Dissertation, Göttingen, 1910. \[$\spadesuit$ Dissertation under Hilbert, attempting to prohibit the real sextic scheme consisting of 11 unnested ovals $\spadesuit$ considered non-rigorous in Gudkov 1974 [@Gudkov_1974/74], albeit it was in Hilbert’s view (cf. Hilbert 1909 [@Hilbert_1909-Ueber-die-Gestalt-sextic])\] 

O. Lokki, [*Über Existenzbeweise einiger mit Extremaleigenschaft versehenen analytischen Funktionen*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 76 (1950), 15 pp. 60, 78 $\bigstar$ 

B. Lund, [*Subalgebras of finite codimension in the algebra of analytic functions on a Riemann surface*]{}, Pacific J. Math. 51 (1974), 495–497. 60 \[$\spadesuit$ quotes Ahlfors’ existence of a circle map (termed therein unimodular function) and cite also Royden’s proof of 1962 [@Royden_1962] $\spadesuit$ the paper itself is devoted to the following result: if a uniform subalgebra $A$ of $A(R)$ the algebra of all analytic functions on the interior of a compact bordered Riemann surface $\bar R$ and continuous up to its boundary included contains a circle map, then $A$ has finite codimension in $A(R)$ $\spadesuit$ question: what about the converse? If the codimension is zero this boils down to Ahlfors 1950 [@Ahlfors_1950]\] 

J. Lüroth, [*Note über Verzweigungsschnitte und Querschnitte in einer Riemann’schen Fläche*]{}, Math. Ann. 3 (1871), 181–184. 60 \[$\spadesuit$ considered as sketchy by Clebsch 1872 [@Clebsch_1872], and consequently supplemented with more details\] 

A.M. Macbeath, [*On a theorem of Hurwitz*]{}, Proc. Glasgow Math. Assoc. 5 (1961), 90–96. \[$\spadesuit$ construction of infinite families of surfaces for which Hurwitz’s bound $84(g-1)$ is attained\] 

A.M. Macbeath, [*Generators of the linear fractional groups*]{}, Proc. Sympos. Pure Math. (Houston, 1967). \[$\spadesuit$ construction of infinite families of surfaces for which Hurwitz’s bound $84(g-1)$ is attained\] 

A.M. Macbeath, [*The classification of non-euclidean plane crystallographic groups*]{}, Canad. J. Math. 19 (1967), 1192–1205. \[$\spadesuit$\] 

A.J. Macintyre, W.W. Rogosinski, [*Extremum problems in the theory of analytic functions*]{}, Acta Math. 82 (1950), 275–325. \[$\spadesuit$ this enters into our specialized picture as follows: this paper, joint with Rogosinski-Shapiro 1953 [@Rogosinski-Shapiro_1953], and Rudin 1955 [@Rudin_1955-class-Hp] constitutes a stream influencing the production of the paper of Read 1958 [@Read_1958_Acta] and Royden 1962 [@Royden_1962], where a new existence-proof of the Ahlfors map is given via functional analytic tools (Hahn-Banach) $\spadesuit$ challenge \[30.09.12\] upon assuming that Gabard 2006 [@Gabard_2006] is true, prove it via Hahn-Banach (good luck!)\] $\bigstar$ 

F. Maitani, Y. Kusunoki, [*Canonical functions on open Riemann surfaces*]{}, Complex Variables and Elliptic Equations (1992). \[$\spadesuit$ The canonical functions are meromorphic functions with a finite number of poles and their real parts are, roughly speaking, constant on each ideal boundary component of an open Riemann surface. The existence and geometrical: properties of such functions have been ...\] 

B. Manel, [*Conformal mapping of multiply connected domains on the basis of Plateau’s problem*]{}, Revista Univ. Nac. Tucuman 3 (1942), 141–149. 78 \[$\spadesuit$ title essentially self-explanatory modulo the question of knowing which types of mappings are handled: Kreisnormierung, circle map, or some slit mappings?\] $\bigstar$ 

W. Mangler, [*Die Klassen von topologischen Abbildungen einer geschlossenen Fläche auf sich*]{}, Math. Z. 44 (1939), 541–554. \[$\spadesuit$ oft quoted, e.g. by Teichmüller\] 

Yu.I. Manin, [*Superalgebraic curves and quantum strings*]{}, Trudy Mat. Inst. Steklov. 183 (1990), 126–138; English transl., Proc. Steklov Inst. Math. 183 (1991), 149–162. \[$\spadesuit$\] 

A. Marden, [*On homotopic mappings of Riemann surfaces.*]{} Ann of Math. (2) (1969), 1–8. \[$\spadesuit$ Lemma 2 (on unlimited branched covering surfaces) is probably akin to the well-known lemma ascribed to Radó 1922 [@Rado_1922-Z-Theorie-mehr], compare Landau-Osserman 1960 [@Landau-Osserman_1960]\] 

A. Marin, [*Quelques remarques sur les courbes algébriques planes réelles.*]{} In: Séminaire sur la géométrie algébrique réelle. Publ. Math. Univ. Paris VII, 1979, 51–86. \[$\spadesuit$ where the writer (Gabard) learned about the Rohlin inequality $r\ge m/2$, which does not appear in print by Rohlin although quite easily derived from the Rohlin formula in Rohlin 1978 [@Rohlin_1978]. For more details, cf. Gabard 2000 [@Gabard_2000] and the refs. therein\] 

A. Marin, [*Sur un théorème de Cheponkus*]{}, in: Real Analytic and Algebraic Geometry, Proceedings Trento, 1988 (Eds. M Galbiati, A. Tognoli). Springer, Lecture Notes in Math., 1420, 191–193. \[$\spadesuit$ contains a corrected version of a theorem of Cheponkus (unsuitably proved in the original), as well as a proof of Klein’s intuition that curves of type I cannot acquire a new oval by continuous deformation of the coefficients. $\spadesuit$ \[26.03.13\] in fact it seems to me that Marin’s statement is slightly stronger than Klein’s original (unproved) assertion inasmuch as Klein 1876 [@Klein_1876] supposed that the curve traverses the discriminant across an isolated double point (with imaginary conjugate tangents), alias solitary double points in the Russian jargon of Arnold, Viro, etc.\] 

D.E. Marshall, [*An elementary proof of the Pick-Nevanlinna interpolation theorem*]{}, Mich. Math. J. \[$\spadesuit$\] 

D.E. Marshall, [*Removable sets for bounded analytic functions*]{}, In: [*Linear and complex analysis problem book*]{}. Lecture Notes in Mathematics 1043. Springer, Berlin, 1984, 2233–2234. CHECK PAGINATION incompatible with Murai 1990/91 [@Murai_1990/91-ICM] \[$\spadesuit$ if I do not mistake this is the first place where it is explained why Calderón’s achievement (Calderón 1977 [@Calderon_1977] on the $L^p$-continuity of the singular integral operator with a Cauchy kernel on a smooth curve) implies the so-called Denjoy conjecture (Denjoy 1909 [@Denjoy_1909-Painleve/Sur-les-fct-anal-unif-a-sing-discontinues]) about the removability of a closed set lying on a rectifiable curve being equivalent to the vanishing of its length $\spadesuit$ historical detail: Murai 1990/91 [@Murai_1990/91-ICM p.904–905] seems to ascribe the Denjoy conjecture to Calderón-Havin-Marshall using the (cryptical) abbreviation CHM on p.905 (but quotes only the present text of Marshall) $\spadesuit$ the Calderón-to-Denjoy implication is obtained by combining classical results of Garabedian, Havinson with Davie’s reduction of the Denjoy conjecture to the $C^1$-case, completing thereby the proof of Denjoy’s conjecture $\spadesuit$ in fact, Denjoy announced this as a theorem, but his proof turned out to be erroneous (compare Marshall [@Marshall_1978?] or Melnikov 1975/76 [@Melnikov-Sinanyan_1975/76 p.691] or Verdera 2004 [@Verdera_2004 p.29]) $\spadesuit$ alas, people rarely say explicitly who located the gap in Denjoy’s claim (this is a non-trivial historical quiz), but maybe Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950 p.122] are good candidates, yet they do not criticize directly Denjoy but rather establish the special case of Denjoy’s conjecture for linear and then analytic curves\] 

G. Martens, [*Komponentenzerfällende abelsche Erweiterungen reeller algebraischer Funktionenkörper einer Variablen*]{}, Diss. Univ. Heidelberg, 1974. \[$\spadesuit$ cited in Martens 1975 [@Martens_1975], and might be the first place where the phrasing “total reell” appears in print\] 

G. Martens, [*Galoisgruppen über aufgeschlossenen reellen Funktionenkörpern*]{}, Math. Ann. 217 (1975), 191–199. \[$\spadesuit$ p.197: total reality (“total reelle Galois Erweiterungen” are defined in the context of Galois extensions of real function fields)\] 

G. Martens, [*Minimale Blätterzahl bei Überlagerungen Kleinscher Flächen über der projectiven Ebene*]{}, Archiv der Math. 30 (1978), 481–486. \[$\spadesuit$ sharp bound upon the degree of the Witt mapping $\spadesuit$ differential geometric application in Ross 1999 [@Ross_1997]\] 

G. Martens, [*Die Zerlegungscharaktere abelscher total reeller Erweiterungen reeller Funktionenkörper einer Variablen*]{}, J. Reine Angew. Math. ?? (ca. 1978), ??–??. \[$\spadesuit$ quoted in Geyer-Martens 1977 [@Geyer-Martens_1977]\] 

G. Martens, [*Funktionen von vorgegebener Ordnung auf komplexen Kurven*]{}, J. Reine Angew. Math. 320 (1980), 68–85.

H.. Martens, [*A new proof of Torelli’s theorem*]{}, Ann. of Math. (2) 78 (1963), 107–111. \[$\spadesuit$\] 

H.H. Martens, [*On the varieties of special divisors on a curve*]{}, J. Reine Angew. Math. 227 (1967), 111–120. \[$\spadesuit$ self-explanatory title, but do not prove the existence of special divisors\] 

R.S. Martin, [*Minimal positive harmonic functions*]{}, Trans. Amer. Math. Soc. 49 (1941), 137–172. 60 \[$\spadesuit$ plays a fundamental rôle in Heins 1950 [@Heins_1950], who seems to offer an alternative proof of the existence of a circle map as the one of Ahlfors 1950 [@Ahlfors_1950]\] 

M. Maschler, [*Minimal domains and their Bergman kernel function*]{}, Pacific J. Math. 6 (1956), 501–516. \[$\spadesuit$\] 

M. Maschler, [*Classes of minimal and representative domains and their kernel functions*]{}, Pacific J. Math. 9 (1959), 763–782. \[$\spadesuit$ contain (according to entry Maschler 1959 [@Maschler_1959 p.173]) a description of the geometric shapes of minimal domains (i.e. essentially the range of the least area maps) in the case of doubly-connected domains\] 

M. Maschler, [*Analytic functions of the classes $L^2$ and $l^2$ and their kernel functions*]{}, Rend. Circ. Mat. Palermo (2) 8 (1959), 163–177. \[$\spadesuit$ p.173 seems to assert that the range of the least area maps are unknown for domains of connectivity higher than $2$ $\spadesuit$ still on p.173 Kufarev 1935/37 [@Kufareff_1935/37] is credited for establishing that the least area map in the case of doubly-connected domains is not univalent, but schlicht upon a (two sheeted) Riemann surface\] 

B. Maskit, [*The conformal group of a plane domain*]{}, Amer. J. Math. 90 (1968), 718–722. 78 \[$\spadesuit$ proves two results to the effect that any plane domain (resp. Riemann surface of \[finite\] genus $g$) conformally embeds into either the sphere or a closed Riemann surface of the same genus so that, under this embedding, every conformal automorphism of the original surface is the restriction of one of the compactified closed surface $\spadesuit$ the proof proceeds (via exhaustion) by reduction to the case of finite Riemann surfaces, previously established by the author\] 

B. Maskit, [*Moduli of marked Riemann surfaces*]{}, Bull. Amer. Math. Soc. (1974). \[$\spadesuit$\] 

B. Maskit, [*Canonical domains on Riemann surfaces*]{}, Proc. Amer. Math. Soc. 106 (1989), 713–721. \[$\spadesuit$ Kreisnormierung for surfaces supplementing the uniqueness lacking in the existence proof in Haas 1984 [@Haas_1984]\] 

J. Mateu, X. Tolsa, J. Verdera, [*The planar Cantor sets of zero analytic capacity and the local $T(b)$-theorem*]{}, J. Amer. Math. Soc. 16 (2002), 19–28. \[$\spadesuit$ a complete characterization of the sets in the title is given via a little incursion of the Ahlfors function (on p.25) $\spadesuit$ \[23.09.12\] since Vitushkin and especially Garnett 1970 [@Garnett_1970] it is known that the $1/4$-Cantor set has $\gamma=0$ (zero analytic capacity), but positive length. More generally one may consider a $\lambda$-Cantor set $E(\lambda)$ for $0<\lambda<1/2$ (obtained by keeping only the four subsquares of length $\lambda$ pushed to the 4 corners of the unit-square and iterating ad infinitum) and ask about the ‘critical temperature’, i.e. the smallest $\lambda$ such that $\gamma(E(\lambda))>0$ $\spadesuit$ the critical value $\lambda$ is precisely $\lambda=1/4$, as follows from Theorem 1 of the cited work, describing more generally the case of a Cantor set $E(\lambda)$ associated to a sequence $\lambda=(\lambda_n)_{n=1}^{\infty}$ with variable $\lambda_n$ (in the range \]0,1/2\[) $\spadesuit$ naive question: in the case where $\lambda$ is constant (self-similar Cantor set) can we describe the behavior of $\gamma(E(\lambda))$ as a function $]0, 1/2[ \to [0,+\infty)$: is it monotone, bounded, analytic or at least derivable (especially at the critical value)? $\spadesuit$ naive answers: monotone most probably, bounded also certainly namely by $\gamma$ of the unit square corresponding to $E(1/2)$\] 

P. Matildi, [*Sulla rappresentazione conforme di domini appartenenti a superficie di Riemann su di un tipo canonico assegnato*]{}, Ann. Scuola Norm. Super. Pisa (2) 14 (1945), (1948), 81–90. 60 \[$\clubsuit$ this paper (read by writer only the 13.07.12) seems to establish the existence of a circle map ([*cerchio multiplo*]{}) for compact bordered Riemann surface having only [*one*]{} contour. Thus with some imagination this may be regarded as a precursor of the Ahlfors circle map. (Recall that Ahlfors was well aware of this paper at least subsequently for it is cited in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960], alas without detailed comment.) Matildi also proposes a bound on the degree of the mapping whose dependence upon the topology is, however, not made completely explicit. He proposes namely, the degree $\lambda\le n(2n-3)$, where $n$ is the minimum degree of a projective-plane model for the Schottky-double of the given membrane. Perhaps it would be useful to estimate his bound purely in term of the topology (via basic algebraic geometry) $\clubsuit$ an extension of Matildi’s work to the case of membranes having several contours is claimed in Andreotti 1950 [@Andreotti_1950], but it still hard to decide if ti really cover the Ahlfors theorem of 1950\] 

S. Matsuoka, [*Nonsingular algebraic curves in $RP^1\times
RP^1$*]{}, Trans. Amer. Math. Soc. ?? (1991–93 ca. CHECK), ?–?. \[$\spadesuit$ Hilbert’s problem in ${\Bbb P}^1\times {\Bbb P}^1$, while extending to this context the Gudkov-Rohlin congruence and inequalities of Petrovskii and Arnold\] 

S. Matsuoka, [*The configuration of the $M$-curves of degree $(4,4)$ in $RP^1\times RP^1$ and periods of real K3 surfaces*]{}, Hokkaido Math. J. 19 (1990), 359–376. \[$\spadesuit$ Hilbert’s problem in ${\Bbb P}^1\times {\Bbb P}^1$, while extending to this context the Gudkov-Rohlin congruence and inequalities of Petrovskii and Arnold\] 

P. Mattila, [*Smooth maps, null-sets for integralgeometric measure and analytic capacity*]{}, Ann. of Math. (2) 123 (1986), 303–309. \[$\spadesuit$ includes a counterexample to the original formulation of Vitushkin’s conjecture ($E$ removable iff purely unrectifiable, i.e. the intersection with any curve of finite length has zero $1$-dimensional Hausdorff measure $H^1$) $\spadesuit$ Mattila’s counterexample has $H^1(E)=\infty$ (infinite length) $\spadesuit$ for the validation of Vitushkin’s conjecture in the case $H^1(E)<\infty$, see G. David 1998 [@David_1998]\] 

P. Mattila, M.S. Melnikov, J. Verdera, [*The Cauchy integral, analytic capacity and uniform rectifiability*]{}, Ann. of Math. (2) 144 (1996), 127–136. 47 \[$\spadesuit$ analytic capacity, Ahlfors function and a step forward in understanding Painlevé null-sets geometrically $\spadesuit$ for the complete solution see Tolsa 2003 [@Tolsa_2003]\] 

R. Mazzeo, M. Taylor, [*Curvature and uniformization*]{}, Israel J. Math. 130 (2002), 323–346. \[$\spadesuit$ uniformization via Liouville’s equation (Schwarz’s strategy, cf. also Bieberbach 1916 [@Bieberbach_1916-Delta-u-und-die-automorphen-Funkt]), as we know since Koebe (Überlagerungsfläche) this gives then the Kreisnormierung, cf. e.g. Bergman 1946 [@Bergman_1946]\] 

S. McCullough, [*The trisecant identity and operator theory*]{}, Integr. Equat. Oper. Theory 25 (1996), 104–127. \[$\spadesuit$ pp.113–5: discussion of the Ahlfors function along the lines of Fay 1973 [@Fay_1973] and p.125 mentions Bell’s result 1991 [@Bell_1991-Szego] that the zeros of Ahlfors function are distinct if the center $a$ is chosen near enough the boundary $\spadesuit$ \[20.09.12\] as we already observed once, it could be interesting to investigate if Bell’s result extends to bordered surfaces (compact) of positive genus $p>0$. Of course in this case the degree of the Ahlfors map may jump from points to points (within the Ahlfors range $r\le \deg\le r+2p$), and this phenomenology is probably connected with deep algebro-geometric or differential-geometric invariants of the surface (Weierstrass points, etc.), compare the work of Yamada 2001 [@Yamada_2001] and Gouma 1998 [@Gouma_1998] for the hyperelliptic context (phenomenology of the mutation of the Ahlfors maps and their fluctuating degree upon dragging away the basepoint) $\spadesuit$ maybe it could be also worth looking if Bell’s result is somehow connected to Solynin’s result (2007 [@Solynin_2007]) about the confinement of the zeros of the Green’s function inside a compactum when the pole is dragged through the surface $\spadesuit$ at first sight, this looks quite plausible in view of Ahlfors formula (cf. 1947 [@Ahlfors_1947] and 1950 [@Ahlfors_1950]) that if $f$ is a circle map with zeros at $t_1, \dots , t_d$ then $\log \vert f(z)\vert$ matches a superposition of Green’s functions with poles at the $t_i$, i.e. $\log \vert f(z)\vert=\sum_{i=1}^d G(z,t_i)$, since both functions vanish on the contours and present the same singularities at the $t_i$ $\spadesuit$ since it is not the critical points of the individual Green’s function, but those of the superposed Green’s functions which are responsible for the ramification of $f$, a direct application of Solynin looks hazardous $\spadesuit$ still, one may wonder if the ramification of the Ahlfors map stay likewise trapped within a compactum upon dragging the center $a$ of the Ahlfors map $f_a$ through the membrane\] 

A.D. Mednykh, [*Nonequivalent coverings of Riemann surfaces with a prescribed ramification type*]{}, Sibirsk Mat. Zh. 25 (1984), 120–142. \[$\spadesuit$\]$\bigstar$ 

A.D. Mednykh, [*Determination of the number of nonequivalent coverings on a compact Riemann surface*]{}, Doklady SSSR 239 (19??), 269–271. \[$\spadesuit$ cited in Natanzon 1993 [@Natanzon_1993] as akin to Hurwitz 1891 [@Hurwitz_1891-Uber-Riemannsche-Flachen]\]$\bigstar$ 

A.D. Mednykh, [*Branched coverings of Riemann surfaces whose branch orders coincide with the multiplicity*]{}, Comm. in Algebra 18 (1990), 1517–1533. \[$\spadesuit$\]$\bigstar$ 

Th. Meis, [*Die minimale Blätterzahl der Konkretisierung\[en\] einer kompakten Riemannschen Fläche*]{}, Schriftenreihe des Math. Inst. der Univ. Münster, Heft 16 (1960). \[$\spadesuit$ a much quoted—but hard-to-find—source where the gonality of a general closed Riemann surface of genus $g$ is found to be the bound predicted by Riemann, Brill-Noether, etc., namely $[\frac{g+3}{2}]$ $\diamondsuit$ Meis belongs to the Münster school (Behnke–Stein, etc.) $\clubsuit$ \[04.10.12\] it seems probable that the technique employed by Meis (which involves Teichmüller theory according to secondary sources, e.g. H.H. Martens’ MathReview of Gunning 1972 [@Gunning_1972]) could be adapted to the context of bordered surfaces and thus lead to a new proof of the Ahlfors map, even perhaps with the sharp bound given in Gabard 2006 [@Gabard_2006] $\spadesuit$ this seems to us to be a task of primary importance, but lacking a copy of Meis article we were relegated to make some general speculations (cf. Sec.\[sec:question\] which we summarize briefly) $\spadesuit$ the basic idea is to develop a “relative” Teichmüller theory not for pairs of Riemann surfaces of the same topological type (hence relatable by a “möglichst konform” diffeomorphism effecting the minimum distortion upon infinitesimal circles), but for just one Riemann surface which we try to express as a branched cover of the sphere (or the disc) for a fixed mapping degree $d$, while exhibiting the (quasiconformal) map of least distortion. Measuring this least dilatation, we get instead of the usual Teichmüller metric (distance) on the moduli space, a Teichmüller temperature $\varepsilon_d$ (or potential) whose vanishing amounts to the possibility of expressing the given surface as a (conformal!) branched cover of the disc (or the sphere), thereby resolving the Ahlfors mapping problem (or the Riemann-Meis problem) depending on the bordered or closed context. \[As a matter of convention the distortion (eccentricity of infinitesimal ellipses is $\ge 1$ and this is converted in values $\ge 0$ upon taking the logarithm)\] $\spadesuit$ in fact upon looking at the gradient flow of the Teichmüller temperature (trajectories of steepest descent orthogonal to the isothermic hypersurfaces $\varepsilon_d={\rm const.}$) we get a flow on the moduli space ($M_g$, if closed or $M_{p,r}$, if bordered) with the net effect of improving the gonality of each individual surface during its evolution $\spadesuit$ as the Teichmüller space is a cell one can hope to derive the existence of stagnation point of the flow by the usual Poincaré-Brouwer-Hopf index formula giving so an existence-proof of a conformal map. However this is a bit artificial for the existence of low degree maps is usually evident (looking at hyperelliptic surfaces and their bordered avatars). Of course it must perhaps be ensured that the flow only stagnates when the temperature vanishes (i.e. no saddle points nor sinks of positive temperature) $\spadesuit$ in such favorable circumstances any closed surface of genus $g$ would flow toward a hyperelliptic model representing the smallest possible gonality (=two) $\spadesuit$ likewise, in the bordered context one expects that any membrane of type $(p,r)$ converges to a membrane of least possible gonality, that is $r$ (excepted when $r=1$ and $p>0$ where the least topological degree is $2$) $\spadesuit$ admittedly, all this does not readily reprove Meis’ gonality (nor that of Ahlfors-Gabard) but maybe it is a first step toward a solution along this path, which—we repeat—should be found in the work of Meis (which in substance is nothing else than a relative (or ramified) version of classical Teichmüller theory) $\spadesuit$ perhaps the flow we are speaking about is not logically needed in Meis’ proof but it can certainly enhance the game $\spadesuit$ basically for each $d$ the continuity of the temperature function shows that the set of $d$-gonal surfaces is closed in the moduli space $M_g$, and since the set is nonempty as soon as $d\ge 2$ (hyperelliptic models) it suffices to show that it is open when $d$ is appropriately large. The expected value for $d$ is $[\frac{g+3}{2}]$ (resp. $r+p$ in the bordered case of $M_{p,r}$), yet it is precisely here that some idea is required $\spadesuit$ naively if the degree is high enough one disposes of enough free parameters to make variations exploring locally the full moduli space $\spadesuit$ alternatively one can perhaps argue that the temperature function $\varepsilon_d$ is real-analytic on $M_g$ so that it would suffice to check its vanishing on a small parametric (open) ball consisting of Riemann surfaces with explicitly given equations (this resembles perhaps Meis’ approach through the little I know of it via indirect sources, e.g. R.F. Lax 1975 [@Lax_1975-special-divisors-II])\]$\bigstar$ 

M.S. Melnikov, [*Structure of the Gleason part of the algebra $R(E)$*]{}, Funkt. Anal. Prilozhen. 1 (1967), 97–100; English transl. (1968), 84–86. \[$\spadesuit$ p.86, Ahlfors function via Vitushkin 1958 [@Vitushkin_1958] $\spadesuit$ the paper itself is devoted to giving another proof (via the apparatus of analytic capacity) of Wilken’s theorem that the Gleason part of the algebra $R(E)$ (of uniform limits on a compactum $E\subset {\Bbb C}$ of the rational functions of the variable $z$) consists either of one point (and is then a peak point), or it has positive area\] 

M.S. Melnikov, S.O. Sinanyan, [*Aspects of approximation theory for functions of one complex variable*]{}, Itogi Nauki i Tekhniki 4 (1975), 143–250; English transl. in J. Soviet Math. 5 (1976), 688–752. \[$\spadesuit$ Vitushkin’s theory (i.e., uniform approximation by rational functions) and its relation to the Ahlfors function and the allied analytic capacity\] 

M.S. Melnikov, [*Analytic capacity: discrete approach and curvature of measure*]{}, Sb. Math. 186 (1995), 827–846. 47 \[$\spadesuit$ analytic capacity (p.827), Ahlfors function (p.830, 838) and introduction of the concept of the curvature of a (positive Borel) measure in the plane \[Menger curvature\], which enables a new proof of Denjoy’s conjecture (without using Calderón’s $L^2$-estimates for the singular Cauchy integral) $\spadesuit$ this technique of Melnikov is also instrumental in Tolsa’s solution (2003 [@Tolsa_2003]) of the (full) Painlevé problem\] 

M.S. Melnikov, J. Verdera [*A geometric proof of the $L^2$ boundedness of the Cauchy integral on Lipschitz graphs*]{}, Internat. Math. Research Notices 7 (1995), 325–331. \[$\spadesuit$ another approach to Calderón 1977 [@Calderon_1977]\] 

O. Mengoni, [*Die konforme Abbildung, gewisser Polyeder auf die Kugel*]{}, Monatsh. f. Math. u. Phys. 44 (1936), 159–185. \[$\spadesuit$ Seidel’s summary: the paper is a contribution to the problem of conformal mapping of simply-connected closed polyhedra upon the sphere. According to H.A. Schwarz, this problem can be reduced to the determination of a number of constants from a set of transcendental equations. It is shown that the explicit solution can be determined in a number of cases not considered by Schwarz. The paper concludes with a discussion of the results form the viewpoint of numerical computations\] $\bigstar$ 

S.N. Mergelyan, [*Uniform approximations to functions of a complex variable*]{}, Amer. Math. Soc. Transl. 1001 (1954). 

H. Meschkowski, [*Beziehungen zwischen den Normalabbildungsfunktionen der Theorie der konformen Abbildung*]{}, Math. Z. 55 (1951), 114–124. 78 

H. Meschkowski, [*Über die konforme Abbildung gewisser Bereiche von unendlich hohen Zusammenhang auf Vollkreisbereiche, I*]{}, Math. Ann. 123 (1951), 392–405. 60, 78 \[$\spadesuit$ some infinite connectivity cases of KNP, via iterative methods à la Koebe and area estimates due to Rengel 1932/33 [@Rengel_1932-33]\] 

H. Meschkowski, [*Über die konforme Abbildung gewisser Bereiche von unendlich hohen Zusammenhang auf Vollkreisbereiche, II*]{}, Math. Ann. 124 (1952), 178–181. 60, 78 \[$\spadesuit$ sequel of the previous paper, building again over a Rengel (1932/33 [@Rengel_1932-33]) area estimate for 4-gons and using Grötzsch’s (1929 [@Groetzsch_1929]) mapping of a domain of infinite connectivity upon a Kreisschlitzbereich, reducing therefore the general study to this special case\] 

H. Meschkowski, [*Einige Extremalprobleme aus der Theorie der konformen Abbildung*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 117 (1952), 12pp. 60, 78 \[$\spadesuit$ which mappings?, essentially all types, but relies heavily on previous works of Garabedian-Schiffer and Nehari $\spadesuit$ mention the issue that there is no known extremal problem yielding the Koebe Kreisnormierung, for an update on this question see several works of Schiffer via Fredholm eigenvalues\] 

H. Meschkowski, [*Verzerrungssätze für mehrfach zusammenhängende Bereiche*]{}, Compositio Math. 53 (1953), 44–59. 78 \[$\spadesuit$ Kreisbogenschlitz map, and somewhat relevant to the point discussed in Gaier 1978 [@Gaier_1978-JDMV] $\spadesuit$ more precisely shows that the Ahlfors-type problem of maximizing the derivative among [*schlicht*]{} function bounded-by-one gives a conformal map upon a Kreisschlitzbereich (=circular slit disc). See also Reich-Warschawski 1960 [@Reich-Warschawski_1960]\] 

G. Mikhalkin, [*Adjunction inequality for real algebraic curves*]{}, arXiv (1994). \[$\spadesuit$ as pointed out by Th. Fiedler (letter of the 9 March in Sec.\[e-mail-Viro:sec\]), this paper of Mikhalkin seems to be one of the earliest application of the Thom conjecture to Hilbert’s 16th\] 

G. Mikhalkin, [*The complex separation of real surfaces and extensions of Rokhlin congruence*]{}, Invent. Math. 118 (1994), 197–222. \[$\spadesuit$\] 

G. Mikhalkin, [*Adjunction inequality for real algebraic curves*]{}, Math. Res. Lett. 4 (1997), 45–52. \[$\spadesuit$\] 

G. Mikhalkin, [*Real algebraic curves, the moment map and amoebas*]{}, Ann. of Math. (2) 151 (2000), 309–326. \[$\spadesuit$\] 

I.P. Millin, [*The method of areas for schlicht functions in finitely connected domains*]{}, (Russian) Trudy Mat. Inst. Steklov 94 (1968), 90–121. 78 \[$\spadesuit$ cited in Grunsky 1978 [@Grunsky_1978 p.185], hence possibly relevant to the issue discussed in Gaier 1978 [@Gaier_1978-JDMV]\] $\bigstar$$\bigstar$$\bigstar$ 

J. Milnor, [*On spaces having the homotopy type of $CW$complexes*]{}, Trans. Amer. Math. Soc. ? (1959), ?–?. \[$\spadesuit$ states the result that a metric manifolds has always the homotopy type of a $CW$-complex, building over result of the Polish and Swede school (Kuratowski, Borsuk, Hanner), cf. also the more detailed implementation in Palais 1962 $\spadesuit$ be careful about terminology, Milnor states this result under the assumption of separability, yet if this means (as it does presently) existence of a denumerable dense part, then there is some simple counter-example of Prüfer described in Gabard 2006/08 [@Gabard-2006/08] \] 

J. Milnor, [*On the Betti numbers of real varieties*]{}, Proc. Amer. Math. Soc. 15 (1964), 275–280. \[$\spadesuit$ cf. also Thom 1965 [@Thom_1965]\] 

C.D. Minda, [*The Aumann-Carathéodory rigidity constant for doubly connected regions*]{}, Kodai Math. J. 2 (1979), 420–426. 47 \[$\spadesuit$ p.422 an elementary existence-proof of the Ahlfors function is given in the case of an annulus $A:=\{z: 1/R<\vert z\vert < R\}$ conjointly with the fact that the map is uniquely prescribed by its two zeros $a,b$ (up to rotation) subjected to the relation $\vert a
b\vert=1$ $\spadesuit$ one can wonder if in this case the circle maps of minimum degree (here $r=2$, i.e. two contours) coincide exactly with the Ahlfors map $f_a$ maximizing the distortion at $a$ (both depends upon $2$ real parameters) $\spadesuit$ p.424, still in the annulus case an explicit expression of the Ahlfors function is given in terms of the theta function, in a way analogous to work of Robinson 1943 [@Robinson_1943] and Abe 1958 [@Abe_1958] (\[03.10.12\] compare maybe also Golusin 1952/57 [@Golusin_1952/57]) $\spadesuit$ this is then applied to give an explicit formula for the Aumann-Carathéodory rigidity constant (1934 [@Aumann-Caratheodory_1934]) $\spadesuit$ p.420, another proof of the so-called annulus theorem is given (quoting the variety of proofs due to H. Huber 1951, Jenkins 1953, Kobayashi 1970, Landau-Osserman 59/60 [@Landau-Osserman_1960], Reich 1966, Schiffer 1946), but emphasizing that the present proof is patterned along Heins 1941 [@Heins_1941-iteration] showing “that the annulus theorem should properly be traced back to Heins’ work”\] 

C.D. Minda, [*The hyperbolic metric and coverings of Riemann surfaces*]{}, Pacific J. Math. 84 (1979), 171–182. 50 \[$\spadesuit$ Ahlfors 1947 [@Ahlfors_1947] and 1950 [@Ahlfors_1950] are cited as follows (p.180): “A function $\tilde{f}$ in ${\cal B}(X)$ which maximizes $\vert\tilde{f}'(p)\vert$ is called an Ahlfors function (\[1\](=1947), \[2\](=1950)) and $c_B(p)=\max \{
\vert\tilde{f}'(p)\vert : \tilde{f} \in {\cal B}(X)\}$ is called the analytic capacity metric.” $\spadesuit$ from the abstract (freely and perhaps loosely reproduced): given two Riemann surfaces $X,Y$ endowed with their hyperbolic metrics, the principle of hyperbolic metric (aka Schwarz-Pick-Ahlfors lemma) says that any analytic map $f\colon X\to Y$ is a contraction. “Moreover, equality holds if and only if $f$ is an (unbranched, unlimited) covering of $X$ onto $Y$” $\spadesuit$ \[04.10.12\] the latter property is essentially topological so applies to any Ahlfors map (even in the extended sense of—what we call—circle maps). We could then lift the hyperbolic (Riemann-Poincaré) metric on the disc to the bordered surface. Alas the ramification creates singularities (in this metric attached to a circle map), so that we certainly do not recover the hyperbolic metric on the interior of the bordered surface. The other way around we may assume uniformization (recall that the interior of any bordered surface is hyperbolic) and try to investigate the metric properties of varied circle maps. In particular is there any special feature related to the circle maps of smallest degree (alias the separating gonality in Coppens 2011 [@Coppens_2011])? Also, given a point $p\in F$ (in the interior) there is a unique Ahlfors map $f_p$ from $F$ to the disc maximizing the derivative and since $f_p$ is “étale” at $p$ we get the above mentioned capacity metric which is more negatively curved that the hyperbolic metric (cf. Suita and Burbea’s papers). Unfortunately the degree of the Ahlfors function is quite mysterious (being subjected to spontaneous quantum fluctuations), but since everything is encoded in the hyperbolic metric there must be an algorithm which given the input of $F$ with the marked point $p$ computes the degree of $f_p$ in terms of the intrinsic geometry of $F$ $\spadesuit$  Some very vague guesses: given $p$ there is a homology basis consisting of loops all based at $p$, and by compactness a smallest “systolic-type” system of such curves of minimal total length probably individually consisting of geodesics; this gives a real number and \[pure guess\] its integer part is the degree of $f_p$. Variant: there is a compact bordered surface capturing all the homology (plus the given point $p$) whose finite volume(=area) $\spadesuit$ Further once the hyperbolic metric is introduced on $F$ any Ahlfors map at $p$ gives a stretching factor at $p$ which by the principle of contraction is $\le 1$, and we get a (probably continuous) function $\delta\colon F\to ]0,1]$ of $F$ measuring this distortion. Does the function extends to the boundary $\partial F$? (and could it be harmonic??). Intuitively when the degree of $f_p$ is low one may expect that the distortion is high. On the other hand there is largest schlicht disc centered at the origin where $f_a$ is unramified, but beware that ramification may come from another point than $p$ lying above the origin. So the right viewpoint is that there is a maximal disc centered at $p$ which is ramificationless. \] 

C.D. Minda, [*The image of the Ahlfors function*]{}, Proc. Amer. Math. Soc. 83 (1981), 751–756. \[$\spadesuit$ Ahlfors function for domains of infinite connectivity $\spadesuit$ p.751: “Ahlfors \[1\](=1947 [@Ahlfors_1947]) showed that $h(\Omega)=B$ \[i.e. the Ahlfors function is surjective on the disc $B$\] for regions $\Omega$ of finite connectivity that have no trivial boundary components. More precisely, he proved that $h$ expresses $\Omega$ as an $n$-sheeted branched covering of $B$, where $n$ is the order of connectivity of $\Omega$. In the general situation Havinson 1961/64 [@Havinson_1961/64] and Fisher 1969 [@Fisher_1969] demonstrated that $B\setminus
h(\Omega)$ has analytic capacity zero; \[…\]. It is not difficult to give an example of a region $\Omega$ such that $B\setminus h(\Omega)\neq \varnothing$. For example, let $K$ be a closed set of $B$ which has analytic capacity zero and $\Omega=B\setminus K$. If $0\in \Omega$, then the Ahlfors function $h$ for $\Omega$ and $0$ is the identity function, so $h(\Omega)=B\setminus K$. The question of the size of $B\setminus
h(\Omega)$ becomes more interesting if it is required that $\Omega$ be a maximal region for bounded holomorphic functions in the sense of Rudin 1955 [@Rudin_1955]. \[$\to$ Recall Rudin’s definition (p.333): “A boundary point $x$ of $D$ \[=domain in the Riemann sphere\] is said to be [*removable*]{} if for every $f\in B(D)$\[=bounded analytic function\] there exists a neighborhood $V$ of $x$ such that $f$ can be extended to $V$. By an [*essential*]{} boundary point of $D$ we mean one that is not removable. If every boundary point of $D$ is essential, we say that $D$ is [*maximal*]{}.”\] For such a maximal region $\Omega$, Fisher 1972 [@Fisher_1972-The-moduli-of-extremal-fctions] raised the question of whether the Ahlfors function must map $\Omega$ onto $B$. Röding 1977 [@Roeding_1977_Ahlfors] answered this question in the negative by exhibiting a maximal region $\Omega$ and a point $p\in \Omega$ such that the Ahlfors function for $\Omega$ and $p$ omitted two values in $\Omega$. We shall extend Röding’s result by showing that an Ahlfors function for a maximal region can actually omit a fairly general discrete set of values in $B$.” $\spadesuit$ p.755: “Therefore, it is still an open question whether the Ahlfors function for a maximal region can actually omit an uncountable set of zero analytic capacity.” $\spadesuit$ \[05.10.12\] an update (positive answer) is implied by Yamada 1992 [@Yamada_1992-Ahlfors-fct-on-Denjoy] where an example is given where the omitted set of the Ahlfors function has positive logarithmic capacity (hence uncountable, because sets of logarithmic capacity zero are stable under countable unions, see e.g. Tsuji 1959 [@Tsuji_1959-BOOK/Chelsea1975])\] 

C.D. Minda, [*Bloch constant for meromorphic functions*]{}, Math. Z. ?? (1982), ??–??. \[$\spadesuit$ “Our geometric approach to the construction of an upper bound is more elementary and clearly shows the analogy with the Ahlfors-Grunsky example. \[…\] Let $XXXX$ be a compact bordered Riemann surface with genus $g$ and $m$ boundary components.”\] 

S. Minsker, [*Analytic centers and analytic diameters of planar continua*]{}, Trans. Amer. Math. Soc. 191 (1974), 83–93. \[$\spadesuit$ the Ahlfors function is mentioned twice on p.91, 92 $\spadesuit$ the paper itself contains results about analytic centers and analytic diameters (concepts arising in Vitushkin’s work on rational approximation)\] 

N.M. Mishachev, [*Complex orientations of plane $M$-curves of odd degree*]{}, Funkt. Anal. Prilozhen. 9 (1975), 77–78; English transl., Funct. Anal. Appl. 9 (1975), 342–345. \[$\spadesuit$ adaptation of Rohlin’s complex orientation formula to odd degrees\] 

I.P. Mitjuk, [*The principle of symmetrization for multiply connected regions and certain of its applications*]{}, (Russ.) Ukrain. Mat. Ž 17 (1965), 46–54; Amer. math. Soc. Transl. 73, 73–85. 78 $\bigstar$ 

I.P. Mitjuk, [*The inner radius of a region and various properties of it*]{}, (Russ.) Ukrain. Mat. Ž 17 (1965), 117–122; Amer. math. Soc. Transl. ??, ??–??. \[$\spadesuit$ a formula expressing the inner radius $r(G,0)$ of a domain $G$ containing the origin and bounded by $n$ analytic Jordan curves is given in terms of the Ahlfors function $F_G(z,0)$ (normalized as usual by $F_G(0,0)=0$ and $F'_G(0,0)>0$) and the Green’s function $g_G(z,0)$ $\spadesuit$ the formula reads $r(G,0)=\frac{1}{F'_G(0,0)}\exp(\sum_{k=1}^{n-1}g_G(0,z_k))$, where $z_k$ are the $n-1$ extra zeros of the Ahlfors function $\spadesuit$ related material in Bandle-Flucher 1996 [@Bandle-Flucher_1996]\] $\bigstar$$\bigstar$$\bigstar$ 

I.P. Mitjuk, [*Extremal properties of meromorphic functions in multiply connected domains*]{}, Ukrain. Mat. Ž 20 (1968), 122–127; Amer. math. Soc. Transl. 76, 116–120. 47 \[$\spadesuit$ Ahlfors’ function occurs thrice: twice on p.116 and once on p.117 and is applied to obtain a connection between the inner radius and the transfinite diameter\] 

Y. Miyahara, [*On relations between conformal mappings and isomorphisms of spaces of analytic functions on Riemann surfaces*]{}, J. Math. Soc. Japan 31 (1979), 373–389. 50 \[$\clubsuit$ on p.375, Ahlfors 1950 [@Ahlfors_1950] is cited for a result on the existence of a basis of analytic Schottky differentials whose periods along a canonical homology basis are calibrated to Kronecker’s delta. Hence the discussion is not directly relevant to the circle map, yet the general construction is quite akin (Green’s function, period of the conjugate differential, etc.) $\spadesuit$ p.380, one reads: “Let $g$ be a nonconstant function in $A(S')$ satisfying $\vert g\vert=1$ on $\partial S'$. (This is a so-called inner function.)” $\spadesuit$ the existence of a such a map follows (perhaps) from Ahlfors 1950, and if so the author perhaps fails to emphasize this issue adequately\] 

Y. Miyahara, [*On local deformations of a Banach space of analytic functions on a Riemann surface*]{}, J. Math. Soc. Japan 40 (1988), 425–443. 50 \[$\clubsuit$ on p.436, Ahlfors 1950 [@Ahlfors_1950] is cited in essentially the same context as for the previous entry, i.e. Miyahara 1979 [@Miyahara_1979]\] 

H. Mizumoto, [*On conformal mapping of a Riemann surface onto a canonical covering surface*]{}, Kōdai Math. Sem. Rep. 12 (1960), 57–69. 50, 78 \[$\clubsuit$ an essentially topological proof of (Ahlfors) circle maps is given, recovering the same degree $r+2p$ as Ahlfors 1950 [@Ahlfors_1950] $\clubsuit$ for a (possible) improvement to $r+p$, cf. Gabard 2006 [@Gabard_2006] $\spadesuit$ in case Mizumoto’s argument is solid, this seems to be a much underestimated paper as it is quoted by $0$ according to the electronic counters, but it is in Grunsky 1978 [@Grunsky_1978]\] 

A.F. Möbius, [*Über die Grundformen der Linien der dritten Ordnung*]{}, ?? (18??), ??–??. (Möbius Werke II, S.89). \[$\spadesuit$ cited in Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege] for the elementary property of what we call now “ovals” versus “pseudolines” (then called “paare und unpaare Curvenzüge”) and also in Harnack 1876 (p.190), but not in Zeuthen 1874 [@Zeuthen_1874] who only cites von Staudt 18XX [@von-Staudt_18XX]\] 

A.F. Möbius, [*Theorie der elementaren Verwandschaft*]{}, Ber. Verhandl. Königl. Sächs. Gesell. d. Wiss., mat.-phys. Klasse 15 (1863), 18–57. (Möbius Werke II). \[$\spadesuit$ a revolutionnary paper fixing the bases of “Morse theory” and classifying en passant the closed orientable surfaces, $\spadesuit$ followed by Jordan 1866 [@Jordan_1866], and vital to Klein’s theory of symmetric surfaces. Of course according to Klein (cf. 1892 [@Klein_1892_Vorlesung-Goettingen]), this topological classification must have been known to Riemann\] 

M. Monastyrsky, [*Riemann, Topology and Physics*]{}, 1st edition 1987; Second Edition 1999, Reprint in 2008, Modern Birkhäuser Classics (Part I of the 1st ed., Moscow, 1979). \[$\spadesuit$ p.20: “The concluding remarks in the dissertation show that the general nature of the problem of analytic functions on arbitrary multiconnected domains was already clear to Riemann.” $\spadesuit$ compare for similar remarks Klein 1882 [@Klein_1882], Klein 1892 [@Klein_1892_Vorlesung-Goettingen] $\clubsuit$ p.72: “Riemann’s note, “Equilibrium of electricity on circular cylinders”, evidently dates to this same period. The problem of the distribution of electrical charge in cylindrical conductors leads to the purely mathematical problem of solving Laplace’s equation in a simply connected[^143] domain with prescribed boundary condition. Here for the first time automorphic functions arise.” $\spadesuit$ \[26.12.12\] quoting Weierstrass: “To the question, Can one really obtain anything directly applicable from those abstract theories with which today’s contemporary mathematicians occupy themselves?, I can answer that Greek mathematicians studied the properties of conic sections in a purely theoretical way long before the time when anyone could foresee that these curves represent the paths along which the planets move. I believe that many more functions with such properties will be found; for example, the well-known $\theta$-functions of Jacobi make it possible, on the one hand, to find the number of squares into which any given number decomposes, thereby making it possible to rectify an arc of ellipse, and, on the other hand, they make it possible to find the true law of the oscillations of a pendulum.”\] 

A.F. Monna, [*Dirichlet’s principle. A mathematical comedy of errors and its influence on the development of analysis*]{}, Oosthoek, Scheltema, and Holkema, Utrecht, 1975. $\bigstar$ 

J.-P. Monnier, [*Divisors on real curves*]{}, Adv. Geom. 3 (2003), 339–360. \[$\spadesuit$ compare for a partial rejection of Monnier’s conjecture the discussion in Coppens-Martens 2010 [@Coppens-Martens_2010]\] 

P. Montel, [*Sur les suites infinies de fonctions*]{}, Ann. École Norm. Sup. (3) 4 (1907), 233–304. \[$\spadesuit$ Montel’s Thesis building over Arzelá, Vitali, Lebesgue, etc. leading to the concept of “normal families”, pivotal in the resolution of extremal problems involving bounded functions (e.g. the so-called Ahlfors function) $\spadesuit$ the nomenclature “normal families” was coined afterwards in Montel 1913 [@Montel_1913-CRAS] $\spadesuit$ simultaneous related work appeared independently by Koebe ca. 1907 in relation with his distortion theorem, compare e.g. the historical analysis of Bieberbach 1968 [@Bieberbach_1968-Das-Werk-Paul-Koebes p.150–151] who writes: “Beim Beweis wird nun neben dem Viertelsatz ein allgemeiner Konvergenzsatz benutzt. Das ist nichts anderes als das, was man in Montels Theorie der Normalen Funktionenfamilien, heute kurz den [*Vitalischen Reihensatz*]{} nennt. Koebe hat ihn selbständig entdeckt \[11\](=Koebe 1908 UbaK3 [@Koebe_1908_UbaK3]). Er leitet ihn aus der Wurzel ab, die auch den anderen Forschern die Anregung gab: Hilberts Arbeit über das Dirichletsche Prinzip (1901) und die vierte Mitteilung über Integralgleichung (1906) des gleichen Forschers. \[…\]”\] 

P. Montel, [*???*]{}, C.R. Acad. Sci. Paris 153 (1911), 996–998. \[$\spadesuit$ where the nomenclature “normal families” appears first in the literature\] 

P. Montel, [*Leçons sur les familles normales de fonctions analytiques et leurs applications*]{}, Gauthier-Villars, Paris, 1927. \[$\spadesuit$ Montel’s treatise on the subject which appeared 20 years after the subject began\] 

G. Moore, N. Seiberg, [*Classical and quantum conformal field theory*]{}, Commun. Math. Phys. 123 (1989), 177–254. \[$\spadesuit$ p.178: “The Riemann surface can be formed by sewing a number of three holed spheres (a.k.a. trinions).” \[this jargon is due to Möbius 1860/63 [@Moebius_1863]\]\] 

J.W. Morgan, Z. Szabó, C.H. Taubes, [*The generalized Thom conjecture*]{}, Preprint 1995; cf. also next entry \[$\spadesuit$ cited in Kirby’s list 1970–95 [@Kirby_1970--95], as another (independent of Kronheimer-Mrowka’s) proof of the Thom conjecture\] 

J.W. Morgan, Z. Szabó, C.H. Taubes, [*A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture*]{}, J. Differ. Geometry 44 (1996), 706–788. \[$\spadesuit$ proof of a generalized Thom conjecture for smooth holomorphic curves with $C\cdot C\ge 0$ (nonnegative self-intersection) in an arbitrary Kähler surface (or even a symplectic $4$-manifolds) $\spadesuit$ p.707: “The Thom conjecture and very similar generalizations of it have been established independently by Kronheimer-Mrowka; see \[4\](=Kronheimer-Mrowka 1994 [@Kronheimer-Mrowka_1994]).”\] 

A. Mori, [*Conformal representation of multiply connected domain on many-sheeted disc*]{}, J. Math. Soc. Japan 2 (1951), 198–209. 60, 78 \[$\clubsuit$ reprove the circle map (ascribed to Bieberbach/Grunsky) via potential theory (Green’s function), plus a mixture of linear algebra and topology (homology) $\spadesuit$ Lemma 1 gives also an “iff” condition for a group of points in the interior to be the fibre of a circle map (in terms of harmonic measure) (compare Fedorov 1991 [@Fedorov_1991] for a similar game) $\spadesuit$ \[26.09.12\] it would be nice(?) to extend such a characterization to the positive genus case, and try to recover the Gabard bound $r+p$ by this procedure\] 

C.B. Morrey, [*Multiple Integrals in the Calculus of Variations*]{}, Grundlehren der math. Wissenschaften 130, Springer-Verlag, Berlin, 1966. \[$\spadesuit$ includes a proof of Koebe’s Kreisnormierung via a Plateau-style approach (extending thereby Douglas’ derivation (1931 [@Douglas_1931-Solution]) of the RMT) $\spadesuit$ however some little gaps in the execution are noticed (but filled) by Jost 1985 [@Jost_1985], cf. also Hildebrandt-von der Mosel 2009 [@Hildebrandt-von-der-Mosel_2009] $\spadesuit$ \[07.10.12\] it is tempting to conjecture that the Plateau-style approach should also have something to say about the Ahlfors circle maps (cf. Courant 1939 [@Courant_1939] for the planar case, i.e. the Bieberbach-Grunsky theorem), however to (my knowledge) it was never attempted to tackle the case of positive genus ($p>0$)\] 

M. Morse, [*?*]{}, Trans. Amer. Math. Soc. 27 (1925?). \[$\spadesuit$ crtical point theory, cited in Petrowsky 1938 [@Petrowsky_1938] as one of the tool used in the proof of the Petrovskii’s inequalities\] 

M. Morse, M. Heins, [*Topological methods in the theory of a function of a complex variable*]{}, Bull. Amer. Math. Soc. ?? (1947), 1–14. \[$\spadesuit$ p.1: “The modern theory of meromorphic functions has distinguished itself by the fruitful use of the instruments of modern analysis and in particular by its use of the theories of integration. It success along the latter line has perhaps diverted attention from some of the more finitary aspects of the theory which may be regarded as fundamental.”\] 

M. Morse, [*La construction topologique d’un réseau isotherme sur une surface ouverte*]{}, J. Math. Pures Appl. (9) 35 (1956), 67–75. 60 $\bigstar$ 

D. Mumford, [*Theta characteristics of an algebraic curve*]{}, Ann. Sci. École Norm. Sup. (4) 4 (1971), 181–192. \[$\spadesuit$\] 

D. Mumford, [*Curve and Their Jacobians*]{}, The University of Michigan Press, Ann Arbor, 1975, 104 pp; reprinted in the “Red Book of Varieties and Schemes”, 2nd Edition, Lecture Notes in Math. 1358, Springer, 1999. \[$\spadesuit$\] 

D. Mumford, [*Complex algebraic varieties*]{}, ??, Springer, 197X. \[$\spadesuit$\] 

T. Murai, [*Construction of $H^1$ functions concerning the estimate of analytic capacity*]{}, Bull. London Math. Soc. 19 (1987), 154–160. \[$\spadesuit$ p.154 mentions the Ahlfors function (via Garnett’s book 1972 [@Garnett_1972 p.18]) and its indirect rôle in Garnett’s 1970 [@Garnett_1970] exposition (of Vitushkin’s 1959 example [@Vitushkin_1959] of a set of positive length but vanishing analytic capacity), but then Murai prefers to switch to the so-called Garabedian function to derive a direct proof of the vanishing of the analytic capacity\] 

T. Murai, [*Analytic capacity for arcs*]{}, In: Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990, 901–911, The Mathematical Soc. of Japan, 1991. 47 \[$\spadesuit$ 3 occurrences of the Ahlfors function, on p.902 (via Garnett 1970 [@Garnett_1970 p.24]), p.904, p.905 $\spadesuit$ seems to ascribe the Denjoy conjecture to Calderón-Havin-Marshall using the (cryptical) abbreviation CHM on p.905 (but quotes only Marshall [@Marshall_1978?])\] 

T. Murai, [*The arc-length variation of analytic capacity and a conformal geometry*]{}, Nagoya Math. J. 125 (1992), 151–216. 47 \[$\spadesuit$ 4 occurrences of the Ahlfors function, on p.152, 159, 191, 199 $\spadesuit$ analytic capacity (of a compact plane set) and its variation under a small change of the compactum $E$ (theory of Hadamard-Schiffer), with apparently a connection to Löwner’s differential equation\] 

P.J. Myrberg, [*Über die Existenz der Greenschen Funktionen auf einer gegebenen Riemannschen Fläche*]{}, Acta Math. 61 (1933), 39–79. 60 \[$\spadesuit$\] 

S. Nagura, [*Kernel functions on Riemann surfaces*]{}, Kōdai Math. Sem. Rep. (9) 35 (1951), 73–76. 60 \[$\spadesuit$ theory of the Bergman kernel on a Riemann surface using an exhaustion by compact bordered subregions with analytic boundaries\] 

M. Nakai, [*The corona problem on finitely sheeted covering surfaces*]{}, Nagoya Math. J. 92 (1983), 163–173. 50 \[$\spadesuit$ p.164: “As is well known these surfaces \[=finite open Riemann surfaces\] are represented as unbounded finitely sheeted covering surfaces of the unit disk $\Delta$ (cf. e.g. Ahlfors \[1\](=1950 [@Ahlfors_1950])).” $\spadesuit$ comment of Gabard \[12.09.12\]: it may appear as a bit unfair that Alling’s works are omitted in the bibliography of this work, and more specifically Gamelin’s accreditation of the bordered corona on p.164 looks historically erroneous in view of the earlier work of Alling 1964 [@Alling_1964], and Alling 1965 [@Alling_1965] (for full details)\] 

M. Nakai, 1985 see Hara-Nakai 1985 [@Hara-Nakai_1985].

M. Nakai, [*Valuations on meromorphic functions of bounded type*]{}, Trans. Amer. Math. Soc. 309 (1988), 231–252. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited in the following context (of valuation-theoretic stability) on p.240: “The following is due to Frank Forelli \[4\](=private communication) to whom the author is very grateful for many valuable suggestions and information:—[Example 1.]{} [*Any finitely sheeted disc is stable*]{}.—The result follows immediately from Theorem 1 \[[*an unlimited finite covering surface is stable iff its base is*]{}\] and Theorem 2 \[[*the open unit disc is stable*]{}\]. Plane regions bounded by finitely many mutually disjoint nondegenerate continua are finitely sheeted disks by the Bieberbach-Grunsky theorem (cf. e.g. \[16\](=Tsuji 1959/75 [@Tsuji_1959-BOOK/Chelsea1975])) or more generally finite open Riemann surfaces are finitely sheeted disks by the Ahlfors theorem \[1\](=Ahlfors 1950 [@Ahlfors_1950]). Here a finite open Riemann surface is a surface obtained from a closed surface by removing a finite number of mutually disjoint nondegenerate continua. Hence as a special case of the above example we have—[Corollary.]{} [*Finite open Riemann surfaces are stable.*]{}” $\spadesuit$ the notion of stability involved is the following (p.231): “Any valuation on the field $M(W)$ of single-valued meromorphic functions on a Riemann surface $W$ is a point valuation (Iss’ssa 1966). What happens to valuations on subfields of $M(W)$? An especially interesting subfield in this context is the field $M^{\infty}(W)$ of meromorphic functions of bounded type on $W$ (cf. \[2\](=Alling 1968)) $\spadesuit$ the exact definition is given on p.232: “A single-valued meromorphic function $f$ on a Riemann surface $W$ is said to be [*of bounded type*]{} if $f=\frac{g}{h}$ on $W$ where $g$ and $h$ are bounded holomorphic functions on $W$ with $h \not\equiv 0$.” $\spadesuit$ p.232/4: “We say that a Riemann surface $W$ is [*stable*]{} if $M^{\infty}(W)$ is nontrivial and any valuation on $M^{\infty}(W)$ is a point valuation.” $\spadesuit$ \[29.09.12\] roughly it seems that this notion of stability leads to a theory quite parallel to that of the corona problem, for the above positive (finitistic) result of Nakai is quite parallel to that of Alling 1964 [@Alling_1964] in the “coronal realm” and further the open question are similar e.g. p.241: “[Open problem 2]{}. [*Is there any stable plane region of infinite connectivity?*]{}” $\spadesuit$ however in the Corona problem it is still an open problem whether any plane region satisfies the corona theorem, but here Nakai (p.241) gives a nonstable plane region “obtained from the punctured open unit disc $\Delta_0$ by removing a sequence of mutually disjoint closed disks with centers on the positive real axis that accumulates only at $z=0$ (a \[so-called\] Zalcman $L$-domains \[17\](=Zalcman 1969 [@Zalcman_1969-TAMS]))\] 

M. Namba, [*Geometry of Projective Algebraic Curves*]{}, Marcel Dekker, New York and Basel, 1984. \[$\spadesuit$ a textbook on curves via a mixture of transcendant and algebro-geometric recipes (browsed through it ca. 1998–2000, so cannot remember exactly the content)\] 

D. Nash, [*Representing measures and topological type of finite bordered Riemann surfaces*]{}, Trans. Amer. Math. Soc. 192 (1974), 129–138. (Dissertation Berkeley, Advisor: Sarason) 50 \[$\spadesuit$ cite Ahlfors 1950 [@Ahlfors_1950], yet apparently not within the main-body of the text $\spadesuit$ given $\overline R$ a finite bordered surface, let $A$ be the usual hypo-Dirichlet algebra consisting of functions continuous on the bordered surface and holomorphic on its interior $R$. For a point $a\in R$, let $e_a$ be the corresponding evaluation. A [*representing measure for $e_a$*]{} is a positive Borel measure $m$ of total mass one supported on $\partial R$ such that $f(a)=\int_{\partial R} f
dm$ for all $f\in A$. The collection of all such measures form a compact convex set ${\frak M}_a$. The paper shows some connections between the topology and even the conformal type of the surface $R$ and the geometry of the convex body ${\frak
M}_a$ of representing measures. It is shown that if ${\frak
M}_a$ has an isolated extreme point, then $R$ must be a planar surface. $\spadesuit$ let $g$ be the genus of $R$ and $s$ the number of contours, Theorem 1.2 states: “If $g=0$ and $s=3$, then ${\frak M}_a$ has precisely four extreme points if $a$ lies on one of three distinguished analytic arcs, and ${\frak
M}_a$ is strictly convex if $a$ lies off these arcs. If $g=s=1$, then ${\frak M}_a$ is strictly convex for all $a\in
R$.” $\spadesuit$ \[28.09.12\] it seems evident that this article (using such concepts as harmonic measure, Green’s function, Schottky differentials, convex bodies, etc.) must bear some close connection with Ahlfors 1950 [@Ahlfors_1950], and it would be nice if the degree of the Ahlfors map $f_a$ (at $a$) could somehow be related to the geometry of the body ${\frak M}_a$\] 

S.M. Natanzon, [*Invariant lines of Fuchsian groups and moduli of real algebraic curves*]{}, Candidate (Ph.D.) dissertation, Moscow, 1974. (Russian) \[$\spadesuit$\] 

S.M. Natanzon, [*Moduli of real algebraic curves*]{}, Uspekhi Mat. Nauk 30 (1975), 251–252. (Russian) \[$\spadesuit$ it is shown (in line with Klein’s intuition or Teichmüller’s work 1939 [@Teichmueller_1939]) that all real algebraic curves of a given topological type $(g,k,\varepsilon)$ (viz. genus, invariant “ovals” and the “dividing” type) form a connected space of dimension $3g-3$ $\spadesuit$ for an English translation see also Natanzon 1978/80 [@Natanzon_1978/80]\] 

S.M. Natanzon, [*Automorphisms of the Riemann surface of an $M$-curve*]{}, Funct. Anal. Appl. 12 (1978), 228–229. \[$\spadesuit$\] 

S.M. Natanzon, [*Moduli spaces of real curves*]{}, Trudy Moskov. Mat. Obshch. 37 (1978), 219–253; English transl., Trans. Moscow Math. Soc. 37 (1980), 233–272. \[$\clubsuit$ modernized account of the theory of Klein 1882 [@Klein_1882] and Teichmüller 1939 [@Teichmueller_1939] $\spadesuit$ compare also nearly parallel work by Seppäla 1978 [@Seppala_1978-Teich-spaces-of-Klein-surfaces]\]$\bigstar$$\bigstar$ 

S.M. Natanzon, [*Spaces of real meromorphic functions on real algebraic curves*]{}, Dokl. Akad. Nauk SSSR 279 (1984), 803–805; English transl., Soviet. Math. Dokl. 30 (1984), 724–726. \[$\clubsuit$ contains a topological description of real meromorphic function, cf. also the subsequent note Natanzon 1987/88 [@Natanzon_1987/88] and full details in Natanzon 1993 [@Natanzon_1993]\] 

S.M. Natanzon, [*Topological classification of pairs of commuting antiholomorphic involutions of Riemann surfaces*]{}, Russian Math. Surveys 41 (1986), 159–160. \[$\spadesuit$ p.159: “It is well known that the topological equivalence class of a pair $(P, \alpha)$ consisting of a compact \[orientable\] surface $P
$ and an orientation-reversing involutory homeomorphism $\alpha\colon P \to P$ is determined by the genus $g=g(P)$ of $P$, the number of ovals $k= \Vert P^\alpha \Vert$, and whether the set $P-P^\alpha$ is connected $(\varepsilon=0)$ or not $(\varepsilon=1)$. The triple $(g,k,\varepsilon)$ is called the [*topological type*]{} of $(P, \alpha)$. For such triples the Weichold \[read Klein to be slightly more accurate\] relation hold (see \[4\](=Weichold 1883 [@Weichold_1883]), \[5\](=Natanzon 1978 [@Natanzon_1978/80])):—(1) $0\le k \le g$ when $\varepsilon=0$,—(2) $1\le k\le g+1$ and $k \equiv g+1 \pmod 2$ when $\varepsilon=1$.\] 

S.M. Natanzon, [*Real meromorphic functions on real algebraic curves*]{}, Dokl. Akad. Nauk SSSR 297 (1987), ?–?; English transl., Soviet. Math. Dokl. 36 (1988), 425–427. \[$\clubsuit$ contains a fine topological study of real meromorphic functions (using the method of Clebsch 1873 [@Clebsch_1872]), yet (apparently) without reproving Ahlfors theorem $\spadesuit$ \[30.12.12\] the proofs seem only sketched,but it is of utmost interest to assimilate better this and subsequent works by Natanzon (e.g. Natanzon 1993 [@Natanzon_1993])\] 

S.M. Natanzon, [*Finite groups of homeomorphisms of surfaces and real forms of complex algebraic curves*]{}, Trudy Moskov. Mat. Obsh. 51 (1988), 3–53. \[$\spadesuit$ inspiration Clebsch 1873 [@Clebsch_1872] and Hurwitz 1891 [@Hurwitz_1891-Uber-Riemannsche-Flachen]\]$\bigstar$$\bigstar$ 

S.M. Natanzon, [*Spinor bundles over real algebraic curves*]{}, Uspekhi Mat. Nauk 44 (1989), 165–166; English transl., Russian Math. Surveys 44 (1989), 208–209. \[$\clubsuit$\] 

S.M. Natanzon, [*Prymians of real curves and their applications to the effectivization of Schrödinger operators*]{}, Funkt. Anal. Prilozhen. 23 (1989), 41–56; English. transl., Funct. Anal. Appl. 23 (1989), 33–45. \[$\clubsuit$\]$\bigstar$$\bigstar$ 

S.M. Natanzon, [*Klein surfaces*]{}, Uspekhi Mat. Nauk 45 (1990), 47–90; English transl., Russian Math. Surveys 45 (1990), 43–108. \[$\clubsuit$ contains an extensive bibliography, through which—if I remember accurately—I discovered circa 2001 the papers Alling-Greenleaf 1969 [@Alling-Greenleaf_1969] and Geyer-Martens 1977 [@Geyer-Martens_1977] which pointed out to me the connection between Klein’s dividing curves and the Ahlfors map of Ahlfors 1950 [@Ahlfors_1950] (i.e. circle maps) $\spadesuit$ “The structure of a Klein surface is an analogue of the complex-analytic structure for surfaces with boundary and non-orientable surfaces. Similar to the way in which the theory of compact Riemann surfaces gives an adequate language for the description of complex …”\] 

S.M. Natanzon, [*Topology of $2$-dimensional coverings and meromorphic functions on real and complex algebraic curves*]{}, Selecta Math. (formerly Sovietica) 12 (1993), 251–291; Originally published in: Trudy Sem. Vektor. Tenzor. Anal. 23 (1988), 79–103; and ibidem 24 (1991), 104–132. \[$\spadesuit$\] 

S.M. Natanzon, [*Moduli of Riemann surfaces, Hurwitz-type spaces, and their superanalogs*]{}, Uspekhi Mat. Nauk 54 (199?), 61–116; English transl., Russian Math. Surveys 54 (1999), 61–117. \[$\spadesuit$\] 

S.M. Natanzon, [*Moduli of real algebraic surfaces, and their superanalogues. Differentials, spinors, and Jacobian of real curves*]{}, Uspekhi Mat. Nauk 54 (199?), 3–60; English transl., Russian Math. Surveys 54 (1999), 1091–1147. \[$\spadesuit$ real algebraic curves (à la Klein-Weichold), antiholomorphic involution and its action upon all structures allied to the Riemann surface (vector bundles, Jacobians, Prymians and so on), topological invariants and the corresponding moduli spaces, inspiration=mathematical physics (solitons, string theory, etc.) $\spadesuit$ p.1092: “According to standard definitions, a [*real algebraic curve*]{} is a pair $(P,\tau)$, where $P$ is a complex algebraic curve (that is, a compact Riemann surface) and $\tau\colon P\to P$ is an antiholomorphic involution. The category of real algebraic curves is isomorphic to the category of Klein surfaces \[1\](=Alling-Greenleaf 1971 [@Alling-Greenleaf_1971]), \[35\](=Natanzon 1990 [@Natanzon_1990/90]). Investigations of real algebraic curves were started by Klein \[25\] (=1892=Vorles. Gött.[^144] [@Klein_1891--92_Vorlesung-Goettingen], [@Klein_1892_Vorlesung-Goettingen]) and Weichold 1883 [@Weichold_1883]. For a long time thereafter researchers studied only plane algebraic curves[^145], that is, real curves embedded in ${\Bbb R P}^2$. The systematic study of “general” real algebraic curves was renewed only in the seventies \[1\](=Alling-Greenleaf 1971 [@Alling-Greenleaf_1971]), \[16\](=Earle 1971 [@Earle_1971-On-the-moduli]), \[20\](=Gross-Harris 1981 [@Gross-Harris_1981]), \[31\]–\[33\](=Natanzon 1974 [@Natanzon_1974-PhD], 1975 [@Natanzon_1975], 1978/80 [@Natanzon_1978/80]), \[48\](=Seppälä 1978 [@Seppala_1978-Teich-spaces-of-Klein-surfaces]). The method of algebraic-geometric integration of works by S.P. Novikov and his school, posed a number of new problems in the theory of real curves and significantly stimulated the development of this theory \[10\](=Cherednik 1980 [@Cherednik_1980/80]), \[12\]–\[14\](=Dubrovin 1987/88 [@Dubrovin_1987/88], Dubrovin-Natanzon 1982 [@Dubrovin-Natanzon_1982/82], Dubrovin-Natanzon 1988 [@Dubrovin-Natanzon_1988/89]), \[34\](=Natanzon 1989 [@Natanzon_1989/89]), \[37\](=Natanzon 1992), \[42\](=Natanzon 1995). Conformal field theory and, in particular, string theory \[9\](=Carey-Hannabuss 1996 [@Carey-Hannabuss_1996]), \[23\](=Jaffe-Klimek-Lesniewski 1990 [@Jaffe-Klimek-Lesniewski_1990]), \[24\](=Karimipour-Mostafazadeh 1997 [@Karimipour-Mostafazadeh_1997]), \[49\](=Vajsburd-Radul 1991 [@Vajsburd-Radul-1991]) has become another area of applications of real curves.” \] 

S.M. Natanzon, B. Shapiro, A. Vainshtein, [*Topological classification of generic real rational functions*]{}, arXiv (2001) or J. Knot Theory Ramif. 11 (2002), 1063–1075. \[$\clubsuit$ §3.1, p.7 (arXiv pagination) titled “On the space of branched covering of a hemisphere by a Riemann surface with boundary” should evidently bears some strong connection with Ahlfors theory. In fact the authors describe the “set ${\cal
H}_{g,m}^k$ of all generic degree $m$ branched coverings of the form $f\colon P \to \Lambda^{+}$” where $P$ is a topological surface of genus $g$ with $k$ contours and $\Lambda^{+}$ is the upper hemisphere $\{ z \in \overline {\Bbb C}: {\rm Im} (z)\ge 0
\}$. $\spadesuit$ \[21.10.12\] this space is of course thought of as a Hurwitz space and it may be partitioned according to the varied multi-degrees of the restricted maps along the $k$ contours, which are indexed by partitions $(m_1,\dots, m_k)$ of $m$. The corresponding subspace of the Hurwitz space having fixed bordered degree $(m_1, \cdot, m_k)$ is shown to be connected (via an extension of the Lüroth-Clebsch theorem). $\spadesuit$ alas, it is not clear to me (Gabard) if the article shows an Ahlfors-type existence result, amounting to the non-emptiness of ${\cal
H}_{g,m}^k$ for $m$ sufficiently large (cf. Ahlfors 1950 [@Ahlfors_1950], or Gabard 2006 [@Gabard_2006]). But note that the surface is here only topological, so that the viewpoint is different! Yet perhaps compatible if one lifts the complex structure of the disk/hemisphere via all topological maps obtaining a “variable” Riemann surface with enough free moduli to realize them all, recovering so perhaps Ahlfors’ theorem via an Hurwitz-type strategy. (I clearly remember to have discussed this idea with Natanzon in a 2001 Rennes conference, but as yet never managed to deduce an existence proof corroborating either Ahlfors 1950 or Gabard 2006.) The argument could start as follows: set ${\cal H}_{g}^k$ the set of all branched covers of the disc (without specified degree). Lifting the complex structure, gives a map ${\cal H}_{g}^k \to M_{g,k}$ to the moduli space of bordered surfaces of type $(g,k)$ (=genus, number of contours). The latter is probably continuous and one would like to show (by a topological argument akin to the continuity method made rigorous by Brouwer-Koebe) that the map is onto when restricted to the Hurwitz space of degree $m$, for some suitable value of $m$. Of course the lack of compactness of the moduli space may suggest to invoke a Deligne-Mumford compactification? Alternatively one can maybe avoid compactification via a clopen argument based on Brouwer’s invariance of the domain\] 

Z. Nehari \[né Willi Weisbach\], [*Analytic functions possessing a positive real part*]{}, Duke Math. J. 15 (1948), 165–178. 78 \[$\spadesuit$ cites the result of Bieberbach 1925 [@Bieberbach_1925], Grunsky 1937–41 [@Grunsky_1937; @Grunsky_1941_KA], Ahlfors 1947 [@Ahlfors_1947], i.e. only planar domains via extremal methods\] 

Z. Nehari, [*The kernel function and canonical conformal maps*]{}, Duke Math. J. 16 (1949), 165–178. 60, 78 \[$\spadesuit$ integral representation of the varied slit-mappings (parallel/circular slits or circular holes) via the Bergman kernel\]$\bigstar$$\bigstar$  Z. Nehari, [*The radius of univalence of an analytic function*]{}, Amer. J. Math. 71 (1949), 845–852. 78 \[$\spadesuit$ application of the Ahlfors function 1947 [@Ahlfors_1947] and of Garabedian’s identity $2\pi F'(z)=K(z,z)$ (Szegö kernel) to the problem of determining the radius of univalence to some families of analytic functions on multi-connected domains, generalizing thereby sharp estimates of Landau for bounded functions in the unit-circle\] 

Z. Nehari, [*On bounded analytic functions*]{}, Proc. Amer. Math. Soc. 1 (1950), 268–275. 60, 78 \[$\spadesuit$ alternative (simplified, but lucky-guess type) derivation of Ahlfors 1947 [@Ahlfors_1947] and Garabedian 1949 [@Garabedian_1949] results around the Schwarz’s lemma via potential theory (Green’s function) and the Szegö kernel\] 

Z. Nehari, [*Conformal mapping of open Riemann surfaces*]{}, Trans. Amer. Math. Soc. 68 (1950), 258–277. 60, 78 \[$\clubsuit$ the paper starts with the historically interesting fact that the main result in Ahlfors 1950 [@Ahlfors_1950] was already presented in Spring 1948 at Harvard (multiply-covered circle with number of sheets not exceeding $(r+2p)$) $\clubsuit$ contains various type of slit mappings (parallel vs. circular or radial), where the first type is given an elementary proof whereas the second requires Jacobi inversion (cf. Ahlfors’ in MathReviews) \[Incidentally one may wonder whether the first (parallel-slit) result is not already implicit in Hilbert 1909 [@Hilbert_1909]?\] $\clubsuit$ $\spadesuit$ p.267: “Representation of the Ahlfors mapping in terms of the kernel function.” $\spadesuit$ NB: some part of this paper are criticized by Tietz 1955 [@Tietz_1955], but himself is critiqued later so it is not clear who (and what) is right and how reliable those papers are $\clubsuit$ the writer asserted in Gabard 2006 [@Gabard_2006 p.946], that Nehari and Tietz may have conjectured the improved bound $r+p$ upon the degree of a circle map, yet on more mature thought this assignment may be a bit cavalier. We leave the competent readers make their own opinion\] 

Z. Nehari, [*Bounded analytic functions*]{}, Bull. Amer. Math. Soc. 57 (1951), 354–366. 50, 78 \[$\clubsuit$ an interesting survey of the Ahlfors’ extremal function (the name appears on p.357) emphasizing its relation to other domain functions such as the kernel functions and the Green’s function\] 

Z. Nehari, [*Extremal problems in the theory of bounded analytic functions*]{}, Amer. J. Math. 73 (1951), 78–106. 78 \[$\spadesuit$ only multiply-connected domains, but the methodology is extended to the positive genus case by Kuramochi 1952 [@Kuramochi_1952], which seems to recover Ahlfors’ 1950 result [@Ahlfors_1950] with the same upper-bound\] 

Z. Nehari, [*Conformal Mapping*]{}, Mac Graw-Hill, New York, 1952. (Dover reprint 1975.) 60, 78 \[$\spadesuit$ only the planar case (domains)\] 

Z. Nehari, [*Some inequalities in the theory of functions*]{}, Trans. Amer. Math. Soc. 75 (1953), 256–286. 78 \[$\spadesuit$ p.264–65 another derivation of the fact (ascribed to Grötzsch 1928 [@Groetzsch_1928] and Grunsky 1932 [@Grunsky_1932]) that the mapping maximizing the derivative at some inner point of a multi-connected domain amongst schlicht functions bounded-by-one (i.e. $\vert f \vert
\le 1$) is a circular slit mapping\] 

Z. Nehari, [*An integral equation associated with a function-theoretic extremal problems*]{}, J. Anal. Math. 4 (1955), 29–48. \[not quoted in 60 nor in 78\] \[$\spadesuit$ p.36 cite Bieberbach 1925 [@Bieberbach_1925] (i.e. existence of a circle map of degree equal to the number of contours for a planar domain) and find a brilliant application of it to bound the the number of linearly independent solutions of a certain extremal problem. It seems realist to expect that this Nehari argument could be widely generalized by using Ahlfors 1950 [@Ahlfors_1950] (and optionally Gabard 2006 [@Gabard_2006]) in place of Bieberbach 1925 (). However the writer \[Gabard, 30.07.12\] does not understand why the inequality advanced by Nehari on p.36 ought to be strict (as the integration is taking place within the contours where the modulus of the Bieberbach(-Ahlfors) function is unity! Hence try to locate the bug... $\spadesuit$ in fact helped by an article of Leung 2007 (On an isoperimetric …) it seems that Nehari’s argument is hygienical modulo correcting the misprint on p.29 that $C_1$ should be a subset of the (open) domain $D$ (instead of the asserted contour $C$) \[this is in agreement with the reviews generated by MR and ZB\] $\spadesuit$ then everything looks more plausible, and there is some hope to extend Nehari’s arguments to the more general setting of bordered surfaces—compare our treatment in Sec.\[Nehari-digression:sec\]\] 

E. Neuenschwander, [*Lettres de Bernhard Riemann à sa famille*]{}, Cahiers du séminaire d’histoire des mathématiques 2 (1981), 85–131. 

E. Neuenschwander, [*Über die Wechselwirkungen zwischen der französischen Schule, Riemann und Weierstra[ß]{}. Eine Übersicht mit zwei Quellenstudien*]{}, Arch. History Exact Sci. 24 (1981), 221–255. \[$\spadesuit$ Cauchy, Puiseux 1850, Weierstrass and the geometrization by Riemann\] 

C. Neumann, [*Das Dirichletsche Prinzip in seiner Anwendung auf die Riemannschen Flächen*]{}, Leipzig bei B.G. Teubner, 1865. $\bigstar$ \[$\spadesuit$ probably—together with the next item—one of the first place where the jargon “Riemann surface” is used in history\] 

C. Neumann, [*Vorlesungen über Riemanns Theorie der Abelschen Integrale*]{}, Leipzig bei B.G. Teubner, 1865. \[$\spadesuit$ (For the Zweite Auflage, cf. 1884 [@Neumann_1884]. $\spadesuit$ seems to post- resp. anti-cipate what is called nowadays the Riemann-Hurwitz relation (cf. e.g. the discussion in Scholz 1999 [@Scholz_1999])\] 

C. Neumann, [*Neumann’s Untersuchungen über das Logarithmische und Newton’sche Potential*]{}, (Referat des Verfasser). Math. Ann. 13 (1878), 255–300. 

C. Neumann, [*Vorlesungen über Riemanns Theorie der Abelschen Integrale*]{}, Zweite Auflage, 1884, 472 pp. 60 $\bigstar$ \[$\spadesuit$ contains, e.g., the first purely topological proof of the (so-called) Riemann-Hurwitz relation, according to Laugel’s French translation of Riemann’s Werke, p.164.\] 

C. Neumann, [*Über die Methode des arithmetischen Mittels insbesondere über die Vervollkommnungen, welche die betreffende Poincaré’schen Untersuchungen in letzter Zeit durch die Arbeiten von A. Korn und E.R. Neumann erhalten haben*]{}, Math. Ann. 54 (1900), 1–48. 60 $\bigstar$ 

R. Nevanlinna, [*Ueber beschränkte analytische Funktionen, die in gegebenen Punkten vorgeschriebene Werte annehmen*]{}, Ann. Acad. Sci. Fenn. BXV (1919), 71pp. \[$\spadesuit$ Nevanlinna’s first paper on the so-called Pick-Nevanlinna interpolation $\spadesuit$ for a connection with the Ahlfors map (or generalization thereof) cf. e.g. Jenkins-Suita 1979 [@Jenkins-Suita_1979] $\spadesuit$ as to Pick’s work cf. Pick 1916 [@Pick_1916-U-d-Beschraekungen]\] 

R. Nevanlinna, [*Ueber beschränkte analytische Funktionen*]{}, Comm. in honorem Ernesti Leonardi Lindelöf, Ann. Acad. Sci. Fenn. A XXXII (1929), 75pp. \[$\spadesuit$ Nevanlinna’s second paper on the so-called Pick-Nevanlinna interpolation $\spadesuit$ same comment as for the previous entry [@Nevanlinna_1919-U-beschr-anal-Funkt]\] 

R. Nevanlinna, [*Das harmonische Mass von Punktmengen und seine Anwendung in der Funktionentheorie*]{}, C.R. Huitième Congr. Math. Scand., Stockholm, 1934, 116–133. 60 $\bigstar$ \[$\spadesuit$ presumably the first place where the name “harmonic measure” appears, the concept going back at least to H.A. Schwarz (compare, e.g. Sario-Nakai 1970 [@Sario-Nakai_1970])\] 

R. Nevanlinna, [*Eindeutige analytische Funktionen*]{}, 1936. 60 \[$\spadesuit$\]$\bigstar$$\bigstar$ 

R. Nevanlinna, [*Über die Lösbarkeit des Dirichletschen Problems für eine Riemannsche Fläche*]{}, Nachr. zu Gött. 1 (1939), 181–193. \[$\spadesuit$ cited in Brelot-Choquet 1951 [@Brelot-Choquet_1951], but the case of open Riemann surfaces\]$\bigstar$$\bigstar$\[ZB OK\] 

R. Nevanlinna, [*Über das alternierende Verfahren von Schwarz*]{}, J. Reine Angew. Math. 180 (1939), 121–128. \[$\spadesuit$ Seidel’s summary: the convergence of the alternating procedure of Schwarz is proved under more general conditions on the boundary of the region than those considered by Schwarz and the problem is reformulated as a method of successive approximation applied to a certain integral equation\] 

R. Nevanlinna, [*Quadratisch integrierbare Differentiale auf einer Riemannschen Mannigfaltigkeit*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 1 (1941), 34 pp. 60 \[$\spadesuit$ an indispensible prerequisite to understand Kusunoki 1952 [@Kusunoki_1952]: application of the Ahlfors mapping to the type problem.\] R. Nevanlinna, [*Über die Neumannsche Methode zur Konstruktion von Abelschen Integralen*]{}, Comment. Math. Helv. 22 (1949), 302–316. 60 R. Nevanlinna, [*Uniformisierung*]{}, Zweite Auflage, Grundlehren der math. Wiss. 64, Springer, 1953, 391 pp. (The Second edition to which we refer, published in 1967) 60 \[$\spadesuit$ p.148–150, contains a very illuminating implementation of Schwarz’s alternating method applied to the problem of constructing harmonic functions with prescribed singular behavior, in particular the Green’s function of a compact bordered surface\] 

D.J. Newman, [*???*]{}, Trans. Amer. Math. Soc. 92 (1959), 501–507. \[$\spadesuit$ like the very deep corona problem, Newman’s characterization of interpolating sequence (also studied by Carleson, cf. e.g. Hoffman 1962 [@Hoffman_1962] for more historical details) is yet another paradigm which can be lifted from the disc to more general finite bordered Riemann surface via appeal to the Ahlfors map, as shown by Stout, cf. e.g. his second implementation in Stout 1967 [@Stout_1967-Interpolation]\] 

I. Newton, [*The method of fluxions and infinite series with applications to the geometry of curves*]{}, in: The Mathematical Papers of Isaac Newton, Cambridge Univ. Press, 1967. \[$\spadesuit$\] 

V.V. Nikulin, [*Integral symmetric bilinear forms and some of their applications*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177; English transl., Math. USSR Izv. 14 (1980), 103–167. \[$\spadesuit$ cited by many (e.g. Fiedler 1982/83 [@Fiedler_1982/83-Pencil p.168]) for “the strict/rigid-isotopy classification of curves of degree six” showing that the real scheme enhanced by the type in the sense of Klein 1876 (and Rohlin 1978) affords a complete invariant of the rigid-isotopy class of sextics $\spadesuit$ the proof employs the apparatus of complex K3 surfaces, especially the version of Torelli’s theorem due to Pyatetsky-Shapiro–Shafarevich 1971/71 [@Pyatetsky-Shapiro-Shafarevich_1971/71] as well via remarks of Kharlamov the profound description in Rohlin 1978 [@Rohlin_1978] of complex topological characteristics (i.e. Klein’s orthosymmetry) in the realm of real plane sextic $\spadesuit$ quite strangely Rohlin’s 1978 paper is not even cited in Nikulin’s albeit it is logically used (for the assertion made on p.107), namely: “As a supplement to Gudkov’s isotopic classification \[42\](=Gudkov-Utkin 1969 [@Gudkov-Utkin_1969/78]) of plane sextics, we shall show that this classification differs from the coarse projective classification(=rigid-isotopy) only for the following sequence of ovals(=real schemes): $\frac{8}{1}, \frac{4}{1}4, 9, \frac{5}{1}1,
\frac{3}{1}3, \frac{1}{1}5, \frac{4}{1}, \frac{2}{1}2$, while each of these listed ovals corresponds to precisely two coarse projective equivalence classes (see \[…\]).” $\spadesuit$ this is, of course, precisely the list of indefinite schemes as listed in Rohlin 1978 [@Rohlin_1978] (upon which Nikulin rests without reproving it)\] 

V.V. Nikulin, [*Involutions of integral quadratic forms and their applications to real algebraic geometry*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), 109–188; English transl., Math. USSR Izv. 22 (1984), 99–172. \[$\spadesuit$ just cited for the nomenclature “separating” (on p.158)\] 

T. Nishino, [*L’existence d’une fonction analytique sur une variété analytique complexe à deux dimensions*]{}, Publ. RIMS, Kyoto Univ. 18 (1982), 387–419. 50 \[$\spadesuit$ applies Ahlfors 1950 [@Ahlfors_1950] to complex surfaces (4 real dimensions), and specifically the existence of an analytic function under a suitable assumption $\spadesuit$ Nishino’s result was quickly extended by himself to arbitrary dimensions, yet during the process it seems that the relevance of Ahlfors 1950 [@Ahlfors_1950] disappeared\] 

M. Noether, [*Zur Grundlegung der Theorie der algebraischen Raumcurven*]{}, Verlag d. Königl. Akad. d. Wiss., Berlin, 1882, 120 pp. \[$\spadesuit$ price shared with Halphen 1882 [@Halphen_1882]\] 

W. Nuij, [*A note on hyperbolic polynomials*]{}, Math. Scand. 23 (1968), 69–72. \[$\spadesuit$ proof that two smooth plane curves with a deep nest are rigidly isotopic in the space of all algebraic curves $\spadesuit$ cited in Vinnikov 1993 [@Vinnikov_1993], who point out also the proof of Dubrovin 1983 [@Dubrovin_1983/85] and also in Viro 1986/86 [@Viro_1986/86-Progress p.74]: “In conclusion I state an old result on rigid-isotopy, which for a long time was not known to experts in the topology of real algebraic manifolds. In 1968, Nuij \[24\](=this entry) proved that any two hypersurfaces of degree $m$ in ${\Bbb R}P^n$ containing $[m/2]$ spheres totally ordered by inclusion are rigidly isotopic. Recently Dubrovin \[5\](=1983 [@Dubrovin_1983/85]) obtained this result for the case of plane curves by a different method.” $\spadesuit$ (from an e-mail of Shustin \[26.01.13\]) By the way, another (well) known connected chamber consists of hyperbolic curves (i.e. those which have totally real intersection with lines of certain pencil) - this is a consequence of Nuij W. A note on hyperbolic polynomials. Math. Scandinavica 23 (1968), no. 1, 69–72. \] 

B.G. Oh, [*A short proof of Hara and Nakai’s theorem*]{}, Proc. Amer. Math. Soc. 136 (2008), 4385–4388. 50 \[$\spadesuit$ Ahlfors 1950’s result on circle maps is used in a quantitative version of the corona $\spadesuit$ question of the writer (since Sept. 2011): is it possible to exploit the improved bound of Gabard 2004/06 [@Gabard_2006] in this sort of game $\spadesuit$ p.4387, Ahlfors 1950 [@Ahlfors_1950] is cited as follows: “[**Theorem 3**]{} (Ahlfors \[1\](=Ahlfors 1950 [@Ahlfors_1950])). [*Suppose $R$ is a finitely[^146] bordered Riemann surface with $g(R)=g$ and $b(R)=b$. Then there exists an $m$-sheeted branched covering map $f\colon R \to {\Bbb D}$, called the [*A*hlfors map]{}, such that $b\le m\le 2g+b$.*]{}”\] 

M. Ohtsuka, [*Dirichlet problems on Riemann surfaces and conformal mappings*]{}, Nagoya Math. J. 3 (1951), 91–137. 60 \[$\spadesuit$\] 

O.A. Oleinik, [*Estimates of the Betti numbers of real algebraic hypersurfaces*]{}, Mat. Sb. 28 (1951), 635–640 (Russian). \[$\spadesuit$\] 

O.A. Oleinik, [*On the topology of real algebraic curves on an algebraic surface*]{}, Mat. Sb. 29 (1951), 133–156 (Russian). \[$\spadesuit$\] 

B.V. O’Neill, Jr., J. Wermer [*Parts as finite-sheeted coverings of the disk*]{}, Amer. J. Math. 90 (1968), 98–107. 50 \[$\spadesuit$ p.98, the paper is started by citing Ahlfors 1950 [@Ahlfors_1950] and mentions the alternative proof of Royden 1962 [@Royden_1962] $\spadesuit$ the Ahlfors’ function is given an application to Gleason parts (certain analytic discs in the maximal ideal space) extending thereby a previous disc-result of Wermer 1964 $\spadesuit$ p.98, it is emphasized that E. Bishop 196 5 [@Bishop_1965] gave an abstract version of Ahlfors’ extremal problem in the context of function algebra on a compact space $X$ (i.e. an algebra of complex-valued continuous functions containing the constants, separating the points, and closed under uniform convergence)\] 

S.Yu. Orevkov, [*Link theory and oval arrangements of real algebraic curves*]{}, Topology 38 (1999), 779–810. \[$\spadesuit$ can be realized holomorphically, pseudo-holomorphically, holomorphically, pseudo-holomorphically, etc.\] 

S.Yu. Orevkov, [*Riemann existence theorem and construction of real algebraic curves*]{}, Ann. Fac. Sci. Toulouse Math. 12 (4) (2003), 517–531. \[$\spadesuit$\] 

S.Yu. Orevkov, [*Proper analytic embedding of ${\Bbb CP}^1$ minus a Cantor set into ${\Bbb C}^2$*]{}, Uspehki Math. Nauk 63 (2008), 155–156; English transl.: Russian Math. Surveys 63 (2008), 168–169. \[$\spadesuit$ shows the result of the title, but as pointed out in Forstnerič-Wold 2012 [@Forstneric-Wold_2012 p.17] it is an open problem whether this holds for each Cantor set.\] 

D. Orth, [*On holomorphic families of holomorphic maps*]{}, Nagoya Math. J. 39 (1970), 29–37. \[$\spadesuit$ p.33, Ahlfors 1950 is cited as follows: “Ahlfors \[1\](=1950 [@Ahlfors_1950]) has shown the existence of a holomorphic map $f$ from a bordered Riemann surface with finite genus and a finite number of boundary components onto a full covering surface $S\buildrel{\pi}\over{\longrightarrow} D$ of the unit disk. N. Alling \[2\] has shown that $\pi\circ f\vert U$ is a covering map of $D$ near $\partial D$ for some open neighborhood $U$ of $\partial X$. Theorem 2.–4. can be thought of as concerning holomorphic families of such maps.”\] 

B. Osgood, [*Notes on the Ahlfors mapping of a multiply connected domain*]{}, Unpublished (?) manuscript (available from the web), undated (estimated date in the range 1993/2005). \[$\spadesuit$ pleasant re-exposition of the neo-expressionist sort of the Ahlfors-Garabedian theory (inspired by Bell, Kerzman-Stein, etc.), in particular the formula for the Ahlfors function as the ratio of the Szegö kernel divided by the Garabedian kernel\] 

W.F. Osgood, [*On the existence of the Green’s function for the most general simply connected plane region*]{}, Trans. Amer. Math. Soc. 1 (1900), 310–314. 60 

W.F. Osgood, [*Jordan curve of positive area*]{}, Trans. Amer. Math. Soc. 4 (1903), 107–112. \[$\spadesuit$ shows how pathological Jordan curve can be\] 

W.F. Osgood, E.H. Taylor, [*Conformal transformations on the boundary of their regions of definition*]{}, Trans. Amer. Math. Soc. 14 (1913), 277–???. 

W.F. Osgood, [*Existenzbeweis betreffend Funktionen, welche zu einer eigentlichen diskontinuierlichen automorphen Gruppe gehören*]{}, Palermo Rend. 35 (1913), 103–106. 60 

R. Osserman, [*Riemann surfaces of class $A$*]{}, Trans. Amer. Math. Soc. 82 (1956), 217–245. \[$\spadesuit$\] 

R. Osserman, [*A hyperbolic surface in $3$-space*]{}, Proc. Amer. Math. Soc. 7 (1956), 54–58. 60 \[$\spadesuit$ example of a function ${\Bbb R}^2\to {\Bbb R}$, whose graph (endowed with the Euclidean metric) defines a surface of hyperbolic type, i.e. conformally equivalent to the disc, answering thereby a question of Ch. Loewner, reported by L. Bers in 1951 on the occasion of the 100th Birthday of Riemann’s Thesis\] 

A. Ostrowski, [*Mathematische Miszellen XV. Zur konformen Abbildung einfach zusammenhängender Gebiete*]{}, Jahresb. Deutsch. Math.-Ver. 38 (1929), 168–182. \[$\clubsuit$ omitted in both 60 and 78; however this (joint with Carathéodory 1928 [@Caratheodory_1928]) is the simply-connected version of the Ahlfors map\] 

K. Ott, [*Über die Konstruktion monogener analytischer Funktionen mit vorgegebenen Unstetigkeitsstellen auf der Riemann’schen Fläche*]{}, Monatsh. Math. 4 (1893), 367–375. 60 $\bigstar$ 

M.P. Ovchintsev, [*Optimal recovery of functions of class $E_p$, $1\le p\le \infty$, in multiply connected domains*]{}, Siberian Math. J. 37 (1996), 288–307. \[$\clubsuit$ p.293, three occurrences of “Ahlfors function” for $m$-connected domains; in particular Prop.1 asserts the existence of neighborhoods of the boundary contours such that if $z_0$ lies in one of these neighborhood then the extra zeros of the Ahlfors function lie one-by-one in the other domains; in particular it seems likely that such neighborhoods can be chosen pairwise disjoint, in which case we recover a result of Bell 1991 [@Bell_1991-Szego]\] 

M. Ozawa, [*On bounded analytic functions and conformal mapping, I*]{}, Kōdai Math. J. (1950), 33–36. 78 

M. Ozawa, [*A supplement to “Szegö kernel function on some domains of infinite connectivity”*]{}, Kōdai Math. J. 13 (1961), 215–218. 78 \[$\spadesuit$ p.215: “Let $D$ be an $n$-ply connected analytic domain and ${\frak B}(D) $ be the class of regular functions in $D$ whose moduli are bounded by the value $1$. In ${\frak B}(D) $ there exists, up to rotation, a unique extremal function by which the maximum $\max_{{\frak B}(D) } \vert f'(z_0)\vert$ for a fixed point is attained. This extremal function $F(z, z_0)$ maps $D$ onto the $n$ times covered unit disc \[1\](=Ahlfors 1947 [@Ahlfors_1947]), \[3\](=Garabedian 1949 [@Garabedian_1949]), \[4\](=Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950]), \[9\](=Nehari 1950 [@Nehari_1950-PAMS-On-bounded-anal-fcts]), \[11\](=Schiffer 1950 [@Schiffer_1950-Duke]). In ${\frak B}(D) $ there exists an infinite number of essentially different functions which map $D$ onto the $n$ times covered unit disc \[2\](=Bieberbach 1925 [@Bieberbach_1925]), \[5\](=Grunsky 1937 [@Grunsky_1937]), \[8\](=Mori 1951 [@Mori_1951]).”\] 

P. Painlevé, [*Sur les lignes singulières des fonctions analytiques*]{}, Ann. Fac. Sci. Toulouse 2 (1888), 130 pp. 78 \[$\spadesuit$ the classical Painlevé problem, interest revived through the work of Ahlfors 1947 [@Ahlfors_1947] and complete solution in Tolsa 2003 [@Tolsa_2003]\] 

P. Painlevé, [*Sur la théorie de la représentation conforme*]{}, C. R. Acad. Sci. Paris 112 (1891), 653–657. \[$\spadesuit$ one of the first study of the boundary behavior of the Riemann mapping for a domain bounded by a smooth Jordan curve $\spadesuit$ same holds true for a general (topological) Jordan domain, cf. Osgood and Carathéodory\] 

H. Pajot, [*Analytic capacity, rectifiability, Menger curvature and the Cauchy integral*]{}, Lecture Notes in Math. 1799, Springer-Verlag, Berlin, 2002. \[$\spadesuit$\]$\bigstar$ 

P. Parenti, [*Combinatorics of dividing $T$-curves*]{}, Tesi di dottorato, Pisa, (1996), 133 pp. Tutori: Galbiati, Itenberg \[$\spadesuit$ combinatorial construction of curves with a control of the type, building upon Viro’s method (early 1980’s) and the special case thereof called the $T$-construction $\spadesuit$ contains a combinatorial version of Rohlin’s formula for $T$-curves\]$\bigstar$ 

S. Paris, [*An extremal property of Rokhlin’s inequality for real algebraic curves*]{}, Math. Ann. 304 (1996), 613–620. \[$\spadesuit$\]$\bigstar$ 

J. Parkkonen, V. Ruuska, [*Finite degree holomorphic covers of compact Riemann surfaces*]{}, Acta Math. Sinica, English Ser. 23 (2007), 89–94. \[$\spadesuit$ “A conjecture of Ehrenpreis (1970) states that any two compact Riemann surfaces of genus $\ge 2$ have finite degree unbranched holomorphic covers that are arbitrarily close in moduli space. Here we prove a weaker result …”\] 

M. Parreau, [*Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann*]{}, Ann. Inst. Fourier (Grenoble) 3 (1951), 103–197. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is briefly cited in two footnotes $\spadesuit$ the work also contains a study of Hardy classes on Riemann surfaces extending the classical Hardy-Riesz’s brothers theory for the disc, and some overlap is to be found with the (subsequent) work of Rudin 1955 [@Rudin_1955-class-Hp]\]$\bigstar$ 

O. Perron, [*Eine neue Behandlung der ersten Randwertaufgabe fur $\Delta u=0$*]{}, Math. Z. 18 (1923), 42–54. 60 \[$\spadesuit$ a new solution to the Dirichlet problem (using Poisson and Lebesgue’s integration) yielding the result in the same generality on the boundary (cf. p.53–54) as those obtained by Lebesgue 1907 [@Lebesgue_1907], Courant 1914 [@Courant_1914] and Lichtenstein (1916), but further very much simplified in Radó-Riesz 1925 [@Rado-Riesz_1925] (according to e.g., Carathéodory 1937 [@Caratheodory_1937-On-Dirichlet's-problem p.710]) $\spadesuit$ the paper is concluded by the simple remark (already made by Zaremba 1910 [@Zaremba_1910]) that the Dirichlet problem does not permit isolated boundary component (reducing to an isolated point), e.g. the punctured disc with boundary prescription $1$ on the circumference and $0$ at the center does not admit a harmonic extension, since otherwise the mean value property would be violated (intuitively a punktförmig radiator is too insignificant to induce a heat flow equilibrium) $\spadesuit$ on the other hand this paper tolerates non-schlicht surfaces covering multiply the plane and therefore may be regarded as a suitable treatment of the Dirichlet problem on a compact bordered Riemann surface (given abstractly à la (Riemann-Prym-Klein)-Weyl-Radó), compare for this well-known affiliation the following ref. given backwardly in time: Radó 1925 [@Rado_1925], Weyl 1913 [@Weyl_1913], and Klein 1882 [@Klein_1882]\] 

K. Petri, [*Über die invariante Darstellung algebraischer Funktionen einer Veränderlichen*]{}, Math. Ann. 88 (1923), 242–289. \[$\spadesuit$\] 

I.G. Petrovsky \[Petrovskii\], [*Sur la topologie des courbes réelles et algébriques*]{}, C.R. Acad. Sci. Paris 197 (1933), 1270–1273. \[$\spadesuit$ announcement of results with proofs detailed in the next entry (Petrowsky 1938 [@Petrowsky_1938])\] 

I.G. Petrowsky \[Petrovskii\], [*On the topology of real plane algebraic curves*]{}, Ann. of Math. (2) 39 (1938), 189–209. (in English of course.) \[$\spadesuit$ where the jargon $M$-curve is coined, and where some obstruction is given (using the Euler-Jacobi interpolation formula), yielding perhaps the first proof, e.g. of the fact (first enunciated by Hilbert, Rohn, etc.) that a plane sextic cannot have 11 unnested ovals $\spadesuit$ note however that Petrovskii validates Rohn’s proof of 1911 by writing on p.189: “After a series of attempts the above mentioned theorem announced by Hilbert was at last proved in 1911 by K. Rohn(=Rohn 1911 [@Rohn_1911]).” This contrast with Gudkov’s latter diagnostic (e.g. in Gudkov 1974 [@Gudkov_1974/74]) that even Rohn’s proof was not logically complete, though the method fruitful when suitably consolidated with Russian conceptions of roughness.\] 

I.G. Petrovskii, [*On the diffusion of waves and the lacunas for hyperbolic equations*]{}, Mat. Sb. 17 (1945), 289–370. (in English.) \[$\spadesuit$ cited in Viro 1986/86 [@Viro_1986/86-Progress p.58] as a forerunner of Rohlin’s complex orientations for dividing curves\] 

I. Petrovskii, O.A. Oleinik, [*On the topology of real algebraic surfaces*]{}, Izv. Akad. Nauk SSSR, Ser. Mat. 13 (1949), 389–402. (Russian) \[$\spadesuit$\] 

P. del Pezzo, [*Sulle superficie di Riemann relative alle curve algebrice*]{}, Palermo Rend. 6 (1892), 115–126. 60 \[$\spadesuit$ presumably one among the first reaction to the reality works of F. Klein outside his direct circle of student (Harnack, Hurwitz, Weichold)\] 

A. Pfluger, [*Ein alternierendes Verfahren auf Riemannschen Flächen*]{}, Comment. Math. Helv. 30 (1956), 265–274. 60 \[$\spadesuit$\] 

A. Pfluger, [*Theorie der Riemannschen Flächen*]{}, Grundlehren der math. Wiss. 89, Springer, Berlin, 1957, 248pp. 50, 60, 78 \[$\spadesuit$ quotes the article Ahlfors 1950 [@Ahlfors_1950] at several places (p.126, 181, 185, 202) yet never in close connection with the circle map paradigm $\spadesuit$ of course the book itself is a masterpiece of Swiss-German architecture and we do not attempt to summarize its broad content\] 

E. Picard, [*Sur une propriété des fonctions entières*]{}, C.R. Acad. Sci. Paris 88 (1879), 1024–1027. \[$\spadesuit$ where the famous Picard theorem appears first (a nonconstant entire function (on ${\Bbb C}$) omits at most one value, for otherwise lifting to the universal covering $\Delta$ of $S^2-\{3 {rm pts}\}$ we get ${\Bbb C}\to \Delta$ a bounded analytic function violating Liouville’s theorem) $\spadesuit$ widespread influence over Borel 1896, Schottky, Landau 1904, Lindelöf 1902 [@Lindeloef_1902], Phragmén, Iversen, Montel, Bloch, Littlewood, Nevanlinna 1923, Ahlfors, Sario, etc. $\spadesuit$ \[07.10.12\] since ${\Bbb C}$ is the punctured sphere and Liouville’s theorem may be interpreted as Riemann’s removable singularity for bounded analytic function, one can also state that any analytic function defined on a punctured closed Riemann surface omits at most 3 values, but this is completely wrong for the monodromy principle does not apply anymore\] 

E. Picard, [*De l’équation $\Delta u=k e^u$ sur une surface de Riemann fermée*]{}, J. Math. Pures Appl. (4) 9 (1893), 273–291. 60 \[$\spadesuit$ supply an attempt to uniformize via the so-called Liouville equation, such a strategy seems to follow a problem suggested by H.A. Schwarz; for a modern execution of this programme cf. Mazzeo-Taylor 2002 [@Mazzeo-Taylor_2002] (and also a related work of Bieberbach 1916 [@Bieberbach_1916-Delta-u-und-die-automorphen-Funkt])\] 

E. Picard, [*Traité d’analyse, Vol.II, Fonctions harmoniques et fonctions analytiques. Introduction à la théorie des équations différentielles, intégrales abéliennes et surfaces de Riemann*]{}, Gauthier-Villars, Paris 1892. Reedited 1926, 624pp. 60 \[$\spadesuit$ contains a treatment of Schottky’s theory of 1877 (cited e.g. in Le Vavasseur 1902 [@Le-Vavasseur_1902], Cecioni 1908 [@Cecioni_1908] and Schiffer-Spencer 1954 [@Schiffer-Spencer_1954])\] 

E. Picard, [*Sur la représentation conforme des aires multiplement connexes*]{}, Ann. École Norm. (3) 30 (1913), 483–488. 78 \[$\spadesuit$ a brilliant re-exposition of Schottky 1877 [@Schottky_1877], which was much appreciated by Julia 1932 [@Julia_1932]\] 

E. Picard, [*?????*]{}, Ann. École Norm. (3) 30 (1915), 483–488. \[$\spadesuit$ yet another brilliant re-exposition of the Riemann mapping theorem via the Green’s function\] 

G. Pick, [*Ueber die Beschränkungen analytischen Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden*]{}, Math. Ann. 77 (1916), 7–23. \[$\spadesuit$ the beginning of so-called Pick-Nevanlinna interpolation, and see Heins 1975 [@Heins_1975] or Jenkins-Suita 1979 [@Jenkins-Suita_1979] for an extension to finite bordered Riemann surface offering an overlap (indeed an extension) of the Ahlfors map\] 

U. Pinkall, [*Hopf tori in $S^3$*]{}, Invent. Math. 81 (1985), 379–386. \[$\spadesuit$ p.379: “Corollary. [*Every compact Riemann surface of genus one can be conformally embedded in ${\Bbb
R}^3$ as an algebraic surface of degree $8$.*]{}—Garsia \[2\](=1962/63 [@Garsia_1962/63-algebraic-surfaces]) had shown that every compact Riemann surface (of any genus) can be conformally embedded in ${\Bbb R}^3$ as an algebraic surface, but his method of proof was not constructive and he therefore did not obtain bounds for the degree of this surface.” $\spadesuit$ this result does not seem to answer the Garsia question (1962/63 [@Garsia_1962/63-algebraic-surfaces]) if the image can always be chosen among torus of revolution twisted by an affine transformation of $3$-space. In this case the degree would be four. $\spadesuit$ for each genus $g$ we can define the Garsia degree $d(g)$ as the smallest integer $d$ such that each surface $F_g$ conformally embeds as an algebraic surface of degree $\le
d$. In fact from Garsia’s theorem (1962/63 ) it is not perfectly clear that there is a uniform bound depending only on the topology. (So in general $d(g)$ is possibly ill-defined.) Of course $d(0)=2$ (every sphere is conformal to the round $2$-sphere, Riemann, Schwarz 1870); $d(1)\le 8$ (Pinkall 1985, [*op.cit.*]{}), but is this sharp?, in general do somebody know a bound on $d(g)\le ???$\] 

U. Pirl, [*Über isotherme Kurvenscharen vorgegebenen topologischen Verlaufs und ein zugehöriges Extremalproblem der konformen Abbildung*]{}, Math. Ann. 133 (1957), 91–117. 78 \[$\spadesuit$\] (another well-known student of Herbert Grötzsch)

J.A.F. Plateau, [*Statique expérimentale et théorétique des liquides soumis aux seules forces moléculaires*]{}, Gauthier-Villars, Paris, 1873. \[$\spadesuit$\] 

J. Plemelj, [*Ein Ergänzungssatz zur Cauchy’schen Integraldarstellung analytischer Funktionen, Randwerte betreffend*]{}, Monats. f. Math. u. Phys. 19 (1908), 205–210. \[$\spadesuit$ quoted in Nehari 1955 [@Nehari_1955]\] 

J. Plücker, [*System der analytischen Geometrie*]{}, Berlin, 1835. \[$\spadesuit$ quoted in Brieskorn-Knörrer 1981/86 [@Brieskorn-Knörrer_1981/1986]\]

J. Pücker, [*Theorie der algebraischen Curven*]{}, Bonn 1839. \[$\spadesuit$ cited by all the masters, e.g. Zeuthen 1874 [@Zeuthen_1874 p.415], Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn], Gudkov 1974/74 [@Gudkov_1974/74] $\spadesuit$ according to Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn] might be one of the first place where the method of small perturbation is mentioned $\spadesuit$ p.253, contains a conjecture on the number of real bitangents to a quartic as taking only the values $28, 16, 8,
4,0$, the last case of which was prohibited in Zeuthen 1874 [@Zeuthen_1874]\] 

H. Poincaré, [*Mémoire sur les fonctions fuchsiennes*]{}, Acta Math. 1 (1882), 193–294. 60 

H. Poincaré, [*Sur un théorème général de la théorie des fonctions*]{}, Bull. Soc. Math. France 11 (1883), 112–125. 78 \[$\spadesuit$ proposes (and succeeds partially) to uniformize not only algebraic, but also analytic curves (=open, a priori highly transcendental, Riemann surfaces). Programm completed in Poincaré 1907 [@Poincare_1907], independently Koebe 1907 [@Koebe_1907_UbaK1].\] 

H. Poincaré, [*Sur les groupes des équations linéaires*]{}, Acta Math. 4 (1884), 201–311. \[$\spadesuit$\] 

H. Poincaré, [*Sur les équations aux dérivées partielles de la physique mathématique*]{}, Amer. J. Math. 12 (1890), 211–294. \[$\spadesuit$ where the [*méthode du balayage*]{} is first introduced\] 

H. Poincaré, [*Analysis Situs*]{}, J. École Polytechnique 1 (1895), 1–121. \[$\spadesuit$ embryo of modern homology theory, quite relevant to problems of conformal mappings (especially circle maps), e.g. in Gabard 2006 [@Gabard_2006]\] 

H. Poincaré, [*Sur la méthode de Neumann et le problème de Dirichlet*]{}, C.R. Acad. Sci. Paris 120 (1895), 347–352. 60 $\spadesuit$ 

H. Poincaré, [*La méthode de Neumann et le problème de Dirichlet*]{}, Acta Math. 20 (1896), 59–142. 60 \[$\spadesuit$ it seems that the method in question, may in turn goes back to Gauss 1839 [@Gauss_1839]\] $\spadesuit$ 

H. Poincaré, [*Sur l’uniformisation des fonctions analytiques*]{}, Acta Math. 31 (1907), 1–63. 60, 78 \[$\spadesuit$ simultaneously with Koebe 1907 [@Koebe_1907_UbaK1] uniformize arbitrary complex analytic curves (equivalently open Riemann surfaces), completing the 1883 desideratum of Poincaré in [@Poincare_1883], revived in Hilbert’s 22th problem\] 

G.M. Polotovskii, [*Problem of topological classification of the disposition of ovals of nonsingular algebraic curves in the projective plane*]{}, in: Methods of the Qualitative Theory of Differential Equations \[in Russian\], Vol.1, Gorki (1975), 101–128. \[$\spadesuit$\] 

G.M. Polotovskii, [*A catalogue of $M$-reducible curves of order $6$*]{}, Dokl. Akad. Nauk SSSR 236 (1977), 548–551; English transl., Soviet Math Dokl. 18 (1977), 1241–1245. \[$\spadesuit$\] 

G.M. Polotovskii, [*$(M-2)$-curves of order $8$ and some conjectures*]{}, Uspekhi Mat. Nauk SSSR 36 (1981), 235–236. \[$\spadesuit$ contains some observation on Rohlin’s conjecture, that were ultimately employed in Shustin 1985/85 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin] to disprove one implication of Rohlin’s conjecture (in degree 8)\]$\bigstar$$\bigstar$$\bigstar$ 

G.M. Polotovskii, [*Dimitrii Andreevich Gudkov*]{}, in: Topology of Real Algebraic Varieties and Related Topics, Amer. MAth. Soc. Transl. 173, 1996, 1–9. \[$\spadesuit$ survey of Gudkov’s contributions with an exhaustive list of his scientific works\] 

Ch. Pommerenke, [*Über die analytische Kapazität*]{}, Archiv der Math. 11 (1960), 270–277. \[$\spadesuit$ some estimates of the analytic capacity (defined as in Ahlfors 1947 [@Ahlfors_1947]) and its connection to Schiffer’s span 1943 [@Schiffer_1943] $\spadesuit$ uses heavily Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950] and Nehari 1952 [@Nehari_1952-BOOK]\] 

H. Poritsky, [*Some industrial applications of conformal mapping*]{}. In: [*Construction and Applications of Conformal Maps*]{}, Proc. of a Sympos. held on June 22–25 1949, Applied Math. Series [*18*]{}, 1952, 207–213. \[$\spadesuit$ quoted for a joke about free-hand drawings\] 

R. de Possel, [*Sur le prolongement des surfaces de Riemann*]{}, C.R. Acad. Sci. Paris 186 (1928), 1092–1095. 60 \[$\spadesuit$ problem of deciding when an (open) Riemann surface can be continued to a larger one $\spadesuit$ relates to work of Radó 1924 [@Rado_1924-Uber-nicht-fortsetzbare], and Bochner 1927 [@Bochner_1927]\] 

R. de Possel, [*Sur le prolongement des surfaces de Riemann*]{}, C.R. Acad. Sci. Paris 187 (1929), 98–100. 60 \[continuation of the previous work in the spirit of Radó and Bochner\] 

R. de Possel, [*Zum Parallelschlitztheorem unendlich-vielfach zusammenhängender Gebiete*]{}, Gött. Nachr. (1931), 199–202. 60, 78 \[$\spadesuit$ proof of the parallel-slit mapping à la Schottky 1877 [@Schottky_1877]-Cecioni 1908 [@Cecioni_1908]-Hilbert 1909 [@Hilbert_1909]-Koebe 1910 [@Koebe_1910_Hilbert]-Courant 1910/12 [@Courant_1912], via an extremal problem (method analogous to Carathéodory 1928 [@Caratheodory_1928], but uses also the Flächensatz of Bieberbach) $\spadesuit$ of course Schottky-Cecioni are not cited as they only treats the case of finite connectivity $\spadesuit$ it is noteworthy that the similar problem for the Kreisnormierung is still unsolved in full generality. This supports once more the philosophy advanced by Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950] that parallel-slit mappings are easier than circle maps\]

R. de Possel, [*Quelques problèmes de représentation conforme*]{}, J. École Polytech. (2) 30 (1932), 1–98. 60, 78 \[$\spadesuit$ parallel (as well as radial) slit maps in the case of domains via an extremal problem $\spadesuit$ some little details of it seem to be criticized in Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950]\] 

R. de Possel, [*Sur quelques propriétés de la représentation conforme des domaines multiplement connexes, en relation avec le théorème des fentes parallèles*]{}, Math. Ann. 107 (1932), 496–504. 60, 78 \[$\spadesuit$ again parallel-slits via an extremal problem, overlap with work by Grötzsch\] 

R. de Possel, [*Sur les ensembles de type maximum, et le prolongement des surfaces de Riemann*]{}, C.R. Acad. Sci. Paris 194 (1932), 98–100. 60 \[$\spadesuit$ still relates to work of Radó, and Bochner and reports some mistakes in the previous notes\] 

R. de Possel, [*Sur la représentation conforme d’un domaine à connexion infinie sur un domaine à fentes parallèles*]{}, J. Math. Pures Appl. (9) 18 (1939), 285–290. 60, 78 \[$\spadesuit$ as noted in Burckel 1979 [@Burckel_1979], this de Possel paper affords a trick to circumvent the reliance upon RMT in his 1931 proof of the PSM through an extremum problem, similar trick in Garabedian 1976 [@Garabedian_1976]\] $\bigstar$$\bigstar$$\bigstar$ 

W. Pranger, [*Extreme points in the Hardy class $H^1$ of a Riemann surface*]{}, Canad. J. Math. 23 (1971), 969–976. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is quoted twice: on p.975 for certain decompositions and on p.976: “On a compact bordered surface $R$ the periods of the conjugate of a function which is harmonic on $R$ and continuous on its closure may be specified arbitrarily (see \[1, p.110\]=Ahlfors 1950 [@Ahlfors_1950 p.110])\] 

F.E. Prym, [*Zur Integration der Differentialgleichung $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial
y^2}=0$*]{}, J. Reine Angew. Math. 73 (1871), 340–364. \[$\spadesuit$ p.361–364, an example is given of a continuous function on the unit-circle whose harmonic extension to the disc has infinite Dirichlet integral! (The existence of such an extension is established directly in the first part of the paper, independently of Schwarz’s 1870 [@Schwarz_1870-Zurich-15ter-Jahrgang] solution based upon Poisson’s integral.) This Prym’s example is nothing less than a counterexample to the Dirichlet principle (as formulated, e.g., in Grube’s text [@Dirichlet_1840-1876]=redaction of Dirichlet’s lectures). Compare Elstrodt-Ulrich 1999 [@Elstrodt-Ullrich_1999 p.285]. Prym emphasizes at the end of his paper (p.364) that Riemann himself never committed such a “basic” mistake, but (still on p.364) Prym formulates an implicit critique to all contemporary attempts to rescue the Dirichlet principle based on the tacit assumption of finiteness of the Dirichlet integral, presumably including the one of Weber 1870 [@Weber_1870] (who is however not directly attacked for diplomatique reasons) $\spadesuit$ a related example (where any continuous function matching the boundary data has infinite Dirichlet integral) is due to Hadamard 1906 [@Hadamard_1906] $\spadesuit$ such counter-example affects directly the Dirichlet-Riemann argument of minimizing the Dirichlet integral, and seems to destroy as well H. Weber’s attempt (1870 [@Weber_1870]) to consolidate Riemann’s proof $\diamondsuit$ student of Riemann, who played a pivotal rôle as well in explaining to Klein, that Riemann himself did not confined his attention to surfaces spread over the plane but included a more organical mode leading to the “abstract” Riemann surfaces, compare Klein 1882 [@Klein_1882]\] 

P.M. Pu, [*Some inequalities in certain nonorientable Riemannian manifolds*]{}, Pacific J. Math. 2 (1952), 55–71. \[$\spadesuit$ includes a proof of the isosystolic estimates for the projective plane ${\Bbb R}P^2$ (by adapting the method of Loewner 1949 for the torus) stating that the round elliptical metric has the best systolic ratio (i.e. is the more robust less susceptible to dye from a “Herzinfarkt”: that is $sys^2/area\le
(\pi)^2/2 \pi=\pi/2=1.570796327\dots$) $\spadesuit$ this results implies directly the Gromov filling conjecture for genus $p=0$, upon cross-capping the boundary contour (cf. Gromov 1983 [@Gromov_1983])\] 

V.A. Puiseux, [*Recherches sur les fonctions algébriques*]{}, Journal de Math. 1 15 (1850), 365–480. \[$\spadesuit$ study of the algebraic equation $f(z,u)=0$ ($f$ a polynomial), poles, branch points, concept of essential singularities (where the Laurent series expansion contains an infinity of negative terms, e.g. $e^{1/z}$ at $z=0$) $\spadesuit$ independent investigations of the same material by Weierstra[ß]{}\] 

I.I. Pyatetsky-Shapiro, I.R. Shafarevich, [*A Torelli theorem for algebraic surfaces of type K3*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572; English transl., Math. USSR Izv. 5 (1971), ?–?. \[$\spadesuit$ used by Nikulin 1979/80 [@Nikulin_1979/80] in his rigid-isotopy classification of plane sextic via Klein-Rohlin’s type (I/II=ortho- vs. dia-symmetry)\] 

T. Radó, [*Zur Theorie der mehrdeutigen konformen Abbildung*]{}, Acta Szeged 1 (1922), 55–64. 78 \[$\spadesuit$ quoted in Landau-Osserman 1960 [@Landau-Osserman_1960], who ascribe to Radó the basic fact that an analytic map taking boundary to boundary is a full covering surface taking each value a constant number of times $\spadesuit$ hence this Radó bears an obvious connection to the Ahlfors map, albeit it does not reprove its existence when the target surface is the disc\] 

T. Radó, [*Über die Fundamentalabbildung schlichter Gebiete*]{}, Acta Sci. Math. Szeged 1 (1922/23), 240–251; cf. also Fejér’s Ges. Arb. 2, 841–842. 78 \[$\clubsuit$ supplies in print an argument of Fejér-Riesz proving RMT via an extremal problem (maximization of the derivative), perfected in Carathéodory 1928 [@Caratheodory_1928] and Ostroski 1929 [@Ostrowski_1929] $\clubsuit$ this constitutes the underlying background for the extremal methods used by Grunsky and Ahlfors, leading ultimately to Ahlfors 1950 [@Ahlfors_1950]\] 

T. Radó, [*Bemerkung zu einem Unitätssatz der konformen Abbildung*]{}, Acta litt. ac. scient. Univ. Hung. 1 (1923), 101–103. 78 

T. Radó, [*Über die konforme Abbildung schlichter Gebiete*]{}, Acta litt. ac. scient. Univ. Hung. 2 (1924), 47–60. 78 

T. Radó, [*Über eine nicht fortsetzbare Riemannsche Mannigfaltigkeit*]{}, Math. Z. 20 (1924), 1–6. 60 

T. Radó, [*Über den Begriff der Riemannschen Fläche*]{}, Acta Szeged 2 (1925), 101–121. 60 \[$\spadesuit$ aside from Weyl 1913 [@Weyl_1913] (“sheaf theoretic”) this supplies the first (modern) definition of an “abstract” Riemann surface, modulo Klein who anticipated the “atlas” idea quite explicitly in [@Klein_1891--92_Vorlesung-Goettingen] (“Dachziegelige Überdeckung”). \[CHECK, pages\] Klein knew it essentially since Prym indicated him how Riemann saw the story, as reported, e.g., in the introduction of Klein 1882 [@Klein_1882]. $\spadesuit$ compare also the discussion in Remmert 1998 [@Remmert_1998] $\spadesuit$ besides the article contains a bunch of results: triangulability of surfaces (via what is nowadays known as the Schoenflies theorem), existence of non-metric surfaces following the (unpublished) construction of Prüfer (ca. 1922)\] 

T. Radó, F. Riesz, [*Über die erste Randwertaufgabe für $\Delta u=0$*]{}, Math. Z. 22 (1925), 41–44. \[$\spadesuit$ supplies drastic simplifications over Perron’s method (Perron 1923 [@Perron_1923]) according to Carathéodory 1937 [@Caratheodory_1937-On-Dirichlet's-problem p.710]\] 

T. Radó, [*Subharmonic functions*]{}, Berlin, 1937. 

V. Ragsdale, [*On the arrangement of the real branches of plane algebraic curves*]{}, Amer. J Math. 28 (1906), 377–404. \[$\spadesuit$ formulation of the Ragsdale conjecture saying that if $m=2k$, and $p,n$ are the number of even resp. odd ovals then $p\le \frac{3k(k-1)}{2}+1$ and $n\le \frac{3k(k-1)}{2}$. This conjecture was disproved by Itenberg in 2000 using Viro’s patchworking (in degree 10)\]$\bigstar$ 

Z. Ran, [*Families of plane curves and their limits: Enriques’ conjecture and beyond*]{}, Ann. of Math. (2) 130 (1989), 121–157. \[$\spadesuit$\] 

H.E. Rauch, [*Weierstrass points, branch points, and the moduli of Riemann surfaces*]{}, Comm. Pure Appl. Math. 12 (1959), 543–560. \[$\spadesuit$\]$\bigstar$ 

H.E. Rauch, [*A transcendental view of the spaces of algebraic Riemann surfaces*]{}, Bull. Amer. Math. Soc. 71 (1965), 1–39. \[$\spadesuit$ the cream of the theory (Riemann, Teichmüller, Ahlfors, etc. revisited)\] 

A.H. Read, [*Conjugate extremal problems of class $p=1$*]{}, Ann. Acad. Sci. Fenn., A.I., 250/28 (1958), 8 pp. 60, 78 

A.H. Read, [*A converse to Cauchy’s theorem and applications to extremal problems*]{}, Acta Math. 100 (1958), 1–22. 50, 78 \[$\clubsuit$ an alternative proof of Ahlfors 1950 [@Ahlfors_1950] is given via Hahn-Banach $\clubsuit$ subsequent work via a similar approach in Royden 1962 [@Royden_1962] $\diamondsuit$ we probably do not need to recall that both Royden and Read were students of Ahlfors\] 

E. Reich, S.E. Warschawski, [*On canonical conformal maps of regions of arbitrary connectivity*]{}, Pacific J. Math. 10 (1960), 965–985. 78 \[$\spadesuit$ like Meschkowski 1953 [@Meschkowski_1953] (which is not cited!) shows that the Ahlfors-type problem of maximizing the derivative among [*schlicht*]{} function bounded-by-one gives a conformal map upon a Kreisschlitzbereich (=circular slit disc). This analysis is also based upon Rengel’s inequality, or a variant thereof closer to Grunsky’s Thesis 1932\] 

H.J. Reiffen, [*Die differentialgeometrischen Eigenschaften der invarianten Distanzfunktion von Carathéodory*]{}, Schrift Math. Inst. Univ. Münster 26 (1963). \[$\spadesuit$ quoted e.g. in Burbea 1977 [@Burbea_1977-Caratheodory]\]$\bigstar$ 

R. Remmert, [*Funktionentheorie 2*]{}, Grundwissen Mathematik [*6*]{}, Springer-Lehrbuch, 1991. (1. unveränderter Nachdruck 1992 der 1. Auflage.)

R. Remmert, [*From Riemann surfaces to complex spaces*]{}, Séminaire et Congrès 3, Société Math. de France, 1998, 203–241. 

E. Rengel, [*Über einige Schlitztheoreme der konformen Abbildung.*]{} (Diss.), Schriften math. Semin., Inst. angew. Math. d. Univ. Berlin 1 (1932/33), 140–162. 60, 78 $\bigstar$ 

E. Rengel, [*Existenzbeweise für schlichte Abbildungen mehrfach zusammenhängender Bereiche auf gewisse Normalbereiche*]{}, J.-Ber. Deutsche Math.-verein. 44 (1934), 51–55. 60, 78 \[$\spadesuit$ via the extremal problem method in vogue at the time obtain the exitence of the circular/radial slit maps for domain of finite connectivity (cf. also de Possel, and Grötzsch) $\spadesuit$ the terminology “Normalbereiche” goes back to Weierstrass, compare Schottky’s Thesis 1877 [@Schottky_1877] $\spadesuit$ this paper shows the existence of a schlicht mapping of a finitely-connected domain upon a circular slit disk $\spadesuit$ antecedent in Koebe 1918, see also Reich-Warschawski 1960 [@Reich-Warschawski_1960]\] 

H. Renggli, [*Zur konformen Abbildung auf Normalgebiete*]{}, (Diss. ETH Zürich) Comment. Math. Helv. 31 (1956), 5–40 60, 78 \[$\spadesuit$ limited to plane domains, where the various slit mappings are reproved via an extremal problem involving the extremal length, Montel’s normal families are used\] 

M. von Renteln, [*Friedrich Prym (1841–1915) and his investigations on the Dirichlet problem*]{}, Suppl. Rend. Circ. Mat. Palermo 44 (1996), 43–55 \[$\spadesuit$ detailed discussion of Prym’s counterexample to the (naive) Dirichlet principle (compare Prym 1871 [@Prym_1871])\]$\bigstar$$\bigstar$$\bigstar$ 

S. Richardson, [*Hele-Shaw flows with time-dependent free boundaries involving a multiply-connected fluid region*]{}, European J. Appl. Math. 12 (2001), 571–599 \[$\spadesuit$\] 

B. Riemann, [*Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse*]{}. Inauguraldissertation Göttingen, 1851. In: Ges. math. Werke 1876/1892/1990 [@Riemann_1990]. \[$\clubsuit$ first proof of RMT, some bad tongues claim that the proof is dubious (even abstraction made of the difficulty allied to the Dirichlet principle), whereupon Riemann reacted with [@Riemann_1857-DP]\] 

B. Riemann, [*Über die Hypothesen, welche der Geometrie zugrunde liegen*]{}. Habilitationsvortrag 1854, first published in: Abhandlungen der Königl. Ges. d. Wiss. Göttingen 13 (1867), reproduced in Ges. math. Werke [@Riemann_1990]. \[$\spadesuit$ a breathtaking generalization of geometry, ramifying to the eclectic topic of Riemannian geometry, Dedekind, Beltrami, Ricci, etc., up to Gromov, Perelman, etc.\] 

B. Riemann, [*Fragment aus der Analysis Situs*]{}. circa 1852/53. Published In: Ges. math. Werke [@Riemann_1990]. \[$\spadesuit$ a first attempt to generalize the connectivity number to high-dimensional manifolds, leads to the work of Betti, and Poincaré 1895 [@Poincare_1895-Analysis-Situs]\] 

B. Riemann, [*Theorie der Abel’schen Functionen*]{}, Crelle J. Reine Angew. Math. 54 (1857), ?–?. In: Ges. math. Werke [@Riemann_1990 88–142]. \[$\spadesuit$ contains in particular the statement that any (or at least one with general moduli?) closed Riemann surface of genus $g$ maps conformally to the sphere with $\le[\frac{g+3}{2}]$ sheets $\spadesuit$ this assertion not accepted by modern geometers until Meis 1960 [@Meis_1960] $\clubsuit$ p.116, some historical hints given by Riemann shows an involvement with conformal maps of multiply-connected regions (maybe even surfaces) as early as Fall 1851 (up to Begin 1852), but then he was sidetracked to another topic\] 

B. Riemann, [*Bestimmung einer Function einer veränderlichen complexen Grösse durch Grenz- und Unstetigkeitsbedingungen*]{}, Crelle J. Reine Angew. Math. 54 (1857), 111–114. \[$\clubsuit$ after Riemann 1851 [@Riemann_1851] the second (more solid, but less romantic) proof of RMT, of course in retrospect not sound until Hilbert’s resurrection of the Dirichlet principle\] 

B. Riemann, [*Gleichgewicht der Electricität auf Cylindern mit Kreisförmigem Querschnitt und parallelen Axen. Conforme Abbildung von durch Kreise begrenzten Figuren*]{} (Nachlass XXVI). In: Ges. math. Werke [@Riemann_1990 p.472–476]. 78 \[$\clubsuit$ the first version of the “Ahlfors map” in the planar case (perhaps confined to the case of circular domains) $\spadesuit$ for subsequent works see primarily Schottky 1875/77 [@Schottky_1877], Bieberbach 1925 [@Bieberbach_1925], Grunsky 1937–41/40–42, Ahlfors 1947–50 [@Ahlfors_1950]\] 

B. Riemann, [*Ueber das Verschwinden der Thetafunctionen*]{}, ?? ?? (1865), ?–?. In: Ges. math. Werke [@Riemann_1990 ?–?]. \[$\spadesuit$ another complete solution to the problem of inverting Abelian integrals $\spadesuit$ Schottky’s problem (1903): what conditions must be imposed on the Riemann matrices to arise as period matrices $\spadesuit$ full effective solution in Shiota 1986\] 

B. Riemann, [*Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachträge,*]{} Nach der Ausgabe von H. Weber und R. Dedekind, Teubner, Leipzig, 1876; neu herausgegeben von R. Narasimhan, Springer-Verlag, Berlin, 1990. \[$\spadesuit$ the first edition 1876 (as well as the subsequent editions) contains the first publication of Riemann’s Nachlass ([@Riemann_1857_Nachlass] estimated 1857/58), where existence of circle maps is proven for planar surfaces, especially in the case of a domain bounded by circles\] 60 

F. Riesz, [*Ueber Potenzreihen mit vorgeschriebenen Anfangsgliedern*]{}, Math. Z. 18 (1923), 87–95. \[$\spadesuit$ cited in Heins 1975 [@Heins_1975], who employs a Riesz variational formula to derive another proof of Ahlfors’ circle maps with upper control upon the degree $\spadesuit$ in fact the cited variational formula of F. Riesz, was given by him for the case $p=1$ (Hardy classes index) and for the disc $\Delta$. However it is available suitably modified for any (finite) bordered Riemann surface and all possible Hardy classes indexes $1\le p <\infty$. (source=p.20 of the just cited Heins work, where for details one must probably browse Heins 1969 [@Heins_1969-LNM-Hardy])\] 

F. Riesz, [*Über die Randwerte einer analytische Funktion*]{}, Math. Z. 18 (1923), 87–95. \[$\spadesuit$\] 

F. Riesz, [*Sur les fonctions subharmoniques et leur rapport à la théorie du potentiel*]{}, Acta Math. 54 (1930), 321–360. \[$\spadesuit$ somehow inspired by Perron 1923 [@Perron_1923]\] 

J.-J. Risler, [*Les nombres de Betti des ensembles algébriques réels, une mise au point*]{}, Gazette des math. ?? (1992), ?–?. \[$\spadesuit$\] 

E. Ritter, [*Die multiplicativen Formen auf algebraischem Gebilde beliebigen Geschlechts mit Anwendung auf die Theorie der automorphen Formen*]{}, Math. Ann. 44 (1894), 261–374. \[$\spadesuit$\] 

W. Ritz, [*???*]{}, ??? ?? (1908), ??–??. \[$\spadesuit$ inspired Bieberbach 1914 [@Bieberbach_1914], who in turn inspired Bergman 1922 [@Bergman_1922], which had some influence on Ahlfors 1950 [@Ahlfors_1950] $\spadesuit$ for a brilliant discussion of Ritz work [*per se*]{} cf. Gander-Wanner 2012 [@Gander-Wanner_2012]\] 

??. Robertson, [*On the theory of univalent functions*]{}, Ann. of Math. 37 (1936), 374–408. \[$\spadesuit$ contains a simple derivation of the Bieberbach conjecture $\vert a_n\vert \le n
\vert a_1 \vert$ for starlike regions via the Schwarz-Christoffel formula\] 

G. Robin, [*Sur la distribution de l’éléctricité à la surface des conducteurs fermés et des conducteurs ouverts*]{}, Ann. Sci. École Norm. Sup. 3 (1886), 3–58. \[$\spadesuit$\] 

R.M. Robinson, [*Analytic functions in circular rings*]{}, Duke Math. J. 10 (1943), 341–354. 78 \[$\spadesuit$ quoted in Minda 1979 [@Minda_1979] in connection with the theta function expression of the Ahlfors function of an annulus $\spadesuit$ for this see also Golusin 1952/57 [@Golusin_1952/57] $\spadesuit$ also quoted in Jenkins-Suita 1979 [@Jenkins-Suita_1979]\] 

R.M. Robinson, [*Hadamard’s three circles theorem*]{}, Bull. Amer. Math. Soc. 50 (1944), 795–802. 78 \[$\spadesuit$\]$\bigstar$$\bigstar$ 

G. Roch, [*Ueber die Anzahl der willkürlichen Constanten in algebraischen Functionen*]{}, Crelle J. Reine Angew. Math. 64 (1865), 372–376. 

R. Rochberg, [*Almost isometries of Banach spaces and moduli of Riemann surfaces*]{}, Duke Math. J. ?? (1973), ??–??. \[$\spadesuit$ compact bordered Riemann surfaces\] 

R. Rochberg, [*Deformation of uniform algebras on Riemann surfaces*]{}, Pacific J. Math. 121 (1986), 135–181. 50 \[$\spadesuit$ on p.142 Ahlfors 1950 [@Ahlfors_1950] is cited as follows: Ahlfors has shown that given $S$ in $\cal S$ \[the set of all connected finite bordered Riemann surfaces, cf. p.135\] and $x,y$ in $S\setminus
\partial S$ there is a function $F=F_{x,y}$ in $A(S)$ which has $\vert F \vert=1$identically on $\partial S$, $F(x)=0$, $F(y)\neq 0$, and $F$ maps $S$ onto the closed unit disk in an $m$ to one manner (counting multiplicity). Furthermore, if $g$ denotes the genus of $S$ and $c$ the number of components of $\partial S$, then $F$ can be selected so that $m$ satisfies $c\le m \le 2g+c$. $\spadesuit$ on the same page the Ahlfors’ bound ($r+2p$ in our notation) is applied to a problem a bit too technical to be summarized here, and naively one could ask if the improved bound $r+p$ of Gabard 2006 [@Gabard_2006] could be applied to Rochberg’s work. This is not evident because a lowest possible degree map does not a priori separates two points prescribed in advance (hence we have not pursued the issue further)\] 

B. Rodin, L. Sario, [*Principal functions*]{}, Princeton, van Nostrand, 1968. 78 $\bigstar$$\bigstar$

B. Rodin, [*The method of extremal length*]{}, Bull. Amer. Math. Soc. 80 (1974), 587–606. 78 \[$\spadesuit$ p.590 Teichmüller listed (without reference!) amongst the contributor to the Löwner-Pu systolic inequality? $\spadesuit$ if this is true it would be nice to localize the precise source\] 

B. Rodin, D. Sullivan [*The convergence of circle packings to the Riemann mapping*]{}, J. Differ. Geom. 26 (1987), 349–360. \[$\spadesuit$ building over work of Koebe 1936 (not cited), Andreev 1970 and Thurston 1985, develop a convergence proof of (finitistic) approximation by circle packings of the Riemann mapping $\clubsuit$ an obvious desideratum would be to implement a similar proof for the case of the Ahlfors function on compact bordered Riemann surface\] 

E. Röding, [*Konforme Abbildung endlicher Riemannscher Flächen auf kanonische Überlagerungsflächen der Zahlenkugel*]{}, Diss. Würzburg, 1972, 71 S. 78 \[$\spadesuit$ this entry is cited on the “critical” page 198 of Grunsky 1978 [@Grunsky_1978], according to which it gives a generalization to Riemann surfaces of the Bieberbach-Grunsky theorem (i.e. circle map in the planar case) $\spadesuit$ in particular, it could be the case that Röding reproves the existence of an Ahlfors circle map, yet probably this is not the case $\spadesuit$ perhaps this aspect has been subsequently published in Röding 1977 [@Roeding_1977_mero]\] $\bigstar$ 

E. Röding, [*Nichtschlichte konforme Abbildung\[en\] unendlich vielfach zusammenhängender Teilgebiete der Ebene*]{}, Arch. d. Math. 26 (1975), 391–397. 78 \[$\spadesuit$ infinite connectivity analog of the “Riemann-Bieberbach” mapping theorem.\] 

E. Röding, [*Über die Wertannahme der Ahlforsfunktion in beliebigen Gebieten*]{}, Manuscr. Math. 20 (1977), 133–140. 78 

E. Röding, [*Über meromorphe Funktionen auf endlichen Riemannschen Flächen vom Betrag eins auf den Randlinien*]{}, Math. Nachr. 78 (1977), 309–318. 78 

M. Roggero, [*Real divisors on real curves*]{}, Le Matematiche 54 (1999), 67–76. \[$\spadesuit$ “...every divisor \[on a smooth real algebraic curve having a nonempty real part\], which is linearly equivalent to its conjugate, is also equivalent to a divisor supported on a set of real points.” $\spadesuit$ this resembles slightly the reformulation of Ahlfors theorem given in Gabard 2006 [@Gabard_2006], but differs substantially for Roggero’s result applies also to diasymmetric curves (with real points) $\spadesuit$ p.75–76: an example is given of a smooth real plane quartic such that every line intersect the (supposed nonempty) real locus in at most 2 points; evidently such a curve has at most one oval and another such example is the Fermat quartic $x^4+y^4=1$\] 

W.W. Rogosinski, H.S. Shapiro, [*On certain extremum problems for analytic functions*]{}, Acta Math. 90 (1953), 287–318. \[$\spadesuit$ this article pertains to our topic (of the Ahlfors map) inasmuch as it may have influenced some new generation existence-proof (of “abstract” functional analytic character) of the Ahlfors map (where Hahn-Banach takes over the rôle of Euler-Lagrange), like those of Read 1958 [@Read_1958_Acta], and the popular version of Royden 1962 [@Royden_1962]\] 

V.A. Rohlin, [*New results in the theory of $4$-dimensional manifolds*]{}, Dokl. Akad. Nauk SSSR 84 (1952), 221–224; French transl. available in Guillou-Marin 1986 [@Guillou-Marin_1986]. \[$\spadesuit$ seminal result of 4D-differential topology on the divisibility by 16 of a simply-connected manifold with even intersection form $\spadesuit$ this (suitably generalized) turned out to be relevant to Hilbert’s 16th problem yielding a proof of the Gudkov hypothesis, cf. Rohlin 1972/72 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov]\] 

V.A. Rohlin, [*Proof of a conjecture of Gudkov*]{}, Funkt. Anal. Prilozhen. 6 (1972), 62–64; English transl., Funct. Anal. Appl. 6 (1972), 136–138. \[$\spadesuit$ the congruence in question (nowadays known as the Gudkov-Rohlin congruence) states that a plane $M$-curve of order $2k$ satisfies $p-n\cong k^2 \pmod 8$ $\spadesuit$ when particularized to degree 6 it affords a new “elementary” solution to Hilbert’s 16th problem (free from the vicissitudes allied to the Hilbert-Rohn-Gudkov method) $\spadesuit$ alas Rohlin’s first proof contains a little flaw (cf. next $\spadesuit$) though being essentially correct using the seminal Rohlin’s divisibility by 16 of the signatures of spin $4$-manifolds (even forms of intersection on the $2$-dimensional homology) $\spadesuit$ from Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91 p.361]: “Three proof of the Gudkov-Rohlin congruence have been published. They are due to V.A. Rohlin \[16\](=1972/72 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov]=Proof of Gudkov’s hypothesis), \[17\](=[@Rohlin_1972/72-Cong-mod-16]=Congruence modulo $16$ in Hilbert’s 16th problem) and A. Marin \[12\](=1979/80 [@Marin_1979]). The third \[12\](=Marin ) appears to be an improvement of the first. The example considered by Marin \[12\](=) seems to show that there is no correct proof of (1.A)\[=Gudkov’s hypothesis\] which is closer to Rohlin’s argument than Marin’s proof.—Marin’s \[12\] and Rohlin’s second \[17\] approaches \[are\] based on quite different techniques. Rohlin’s proof work in any dimension while no generalization of Marin’s proof to higher dimensions is known. Nevertheless the approaches seem to be closely related. Rohlin asked his students to find a relation and said that an understanding of it might lead to essential progress.” $\spadesuit$ from Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.736]: “In Rokhlin’s first paper \[97\](=this entry) there is a mistake in the proof of Gudkov’s conjecture. However the approach in the paper, namely, using characteristic surfaces in a $4$-manifold to evaluate the signature $\mod 16$, became a powerful method in the study of real algebraic curves. It was used by Marin, who together with Guillou (see \[46\](=Guillou-Marin 1977 [@Guillou-Marin_1977])) extended Rokhlin’s signature formula to non-orientable characteristic surfaces and thus corrected the mistake.”\] 

V.A. Rohlin, [*Congruence modulo 16 in Hilbert’s sixteenth problem*]{}, Funkt. Anal. Prilozhen. 6 (1972), 58–64; English transl., Funct. Anal. Appl. 6 (1972), 301–306. \[$\spadesuit$ severe restriction upon the isotopy classification of $M$-curves reinforcing earlier work of Petrovskii 1938 [@Petrowsky_1938] and Arnold 1971 [@Arnold_1971/72]\] 

V.A. Rohlin, [*Congruence modulo 16 in Hilbert’s sixteenth problem, II*]{}, Funkt. Anal. Prilozhen. 7 (1973), 91–92; English transl., Funct. Anal. Appl. ? (197?), ?–?. \[$\spadesuit$\] 

V.A. Rohlin, [*Complex orientations of real algebraic curves*]{}, Funkt. Anal. Prilozhen. 8 (1974), 71–75; English transl., Funct. Anal. Appl. 8 (1974), 331–334. \[$\spadesuit$ present a general method of closing the one half of a dividing real plane curve by piecing together real discs to construct a closed membrane, whose (fundamental) homology class yields via intersection theory a certain numerical relation known as Rohlin’s complex orientation formula. The latter implies the striking fact that a dividing curve exhibits at least as many ovals as the half value of its degree(=order). This answers a question of Klein, made more explicit in Gross-Harris 1981 [@Gross-Harris_1981]. Compare Gabard 2000 [@Gabard_2000] for more details. NB: In this seminal paper, Rohlin treats only the case of $M$-curve(=Harnack-maximal) (the general formula being written down in the next entry Rohlin 1978 [@Rohlin_1978], but the proof is easy to adapt). $\spadesuit$ Rohlin’s formula also prohibits many (but not all) $M$-schemes of sextics (e.g. that consisting of eleven unnested ovals) supplying so a 5 minutes proof of the tricky theorem of Hilbert (1891–00–08), which he was never able to complete himself (or with his numerous students)\] 

V.A. Rohlin, [*Complex topological characteristics of real algebraic curves*]{}, Uspekhi Mat. Nauk. 33 (1978), 77–89; translation: Russian Math. Surveys 33 (1978), 85–98. \[$\spadesuit$ shows strikingly that Rohlin discovered Klein’s work at a very late stage (despite the fact that Klein is generously quoted e.g. in Gudkov 1974 [@Gudkov_1974/74]), but with great happiness apparently (p.85): “As I learned recently, more than hundred years ago, the problems of this article occupied Klein, who succeeded in coping with curves of degree $m\le 4$ (see \[4\](=Klein 1922 [@Klein-Werke-II_1922]), p.155).” $\spadesuit$ p.93–94 prove the result that a real plane curve with a nest of maximal depth is dividing, via an argument which (in our opinion) can be slightly simplified as follows $\spadesuit$ given $C_m\subset {\Bbb P}^2$ a nonsingular curve of degree $m$ with a deep nest then projecting the curve from any point chosen in the innermost oval gives a morphism $C_m \to {\Bbb
P}^1$ whose fibers over real points are totally real. Hence there is an induced map between the imaginary loci $C_m({\Bbb
C})-C_m({\Bbb R}) \to {\Bbb P}^1({\Bbb C})-{\Bbb P}^1({\Bbb R})$ and it follows that $C_m$ is dividing (just by using the fact that the image of a connected set is connected). q.e.d. (this argument avoids the consideration of the canonical fibering ${\rm pr}\colon
{\Bbb C} P^2- {\Bbb R} P^2 \to S^2$ envisaged by Rohlin) $\spadesuit$ p.94: “If $A_1$ and $A_2$ belong to type I, then the question is rather complicated, in general, but Fiedler first noted that everything is radically simplified when $s=m_1 m_2$ \[i.e. all intersections are real\]. Namely, in this situation, $A$ belongs to type I” $\spadesuit$ some interesting question is raised on p.95: “[**3.9 A conjecture about real schemes of type I**]{}. A study of the available factual material suggests that possibly a real scheme belongs to type I iff it is [*maximal*]{}, that is, it is not part of a larger real scheme of the same degree. This conjecture is true for $m\le 6$, and there is much to be said in its favour for $m>6$. There is an allusion to it in Klein: see \[4\], p.155 (=Klein 1922=Ges. Math. Abh. II [@Klein-Werke-II_1922]).” \[31.12.12\] Gabard’s guess: perhaps this conjecture of Klein-Rohlin follows from Ahlfors theorem translated in terms of total reality (intuitively having a total pencil, no real circuit can be added for otherwise Bézout would be corrupted, yet perhaps this is too naive, cf. our Sec.\[Klein-Rohlin-conj:sec\]) $\spadesuit$ [*Warning.*]{} p.788 of Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000] one reads: “[**Digression: real rational curves**]{}. As far as we know, the following problem is still open: is it possible to draw an irreducible real rational curve (or more precisely a connected component of it) of degree $q$ through any set of $3q-1$ real points in general position? In \[99\](=Rohlin 1978 [@Rohlin_1978]) the question is answered in the affirmative; however, the proof has never been published; possibly it contained a gap.”\] 

V.A. Rohlin, [*New inequalities in the topology of real plane algebraic curves*]{}, Uspekhi Mat. Nauk. 14 (1980), 37–43; translation: Russian Math. Surveys ?? (198?), ??–??. \[$\spadesuit$\] 

V.A. Rohlin, [*Two aspects of the topology of real algebraic curves*]{}, Proc. Leningrad Internat. Topology Conf., Nauka, Leningrad, 1983; (translation available?). \[$\spadesuit$ cited in Viro 1986/86 [@Viro_1986/86-Progress]\] 

K. Rohn, [*Flächen vierter Ordnung hinsichtlich ihrer Knotenpunkte und ihrer Gestaltung*]{}, Preisschriften der Fürstlich Jablonowskischen Gesellschaft, Leipzig, 1886; also in Math. Ann. 29 (1887), 81–96. \[$\spadesuit$\] 

K. Rohn, [*Die ebenen Kurven 6. Ordnung mit elf ovalen*]{}, Leipzig Ber. Dezember 1911. \[$\spadesuit$ cited in Petrovsky 1938 [@Petrowsky_1938] and considered there as the first rigorous proof of Hilbert’s announced theorem that an $M$-sextic cannot have all its $11$ ovals lying unnested. However Gudkov (e.g. in 1974 [@Gudkov_1974/74]) is more severe and does not consider Rohn’s proof as complete. $\spadesuit$ \[18.03.13\] perhaps nowadays the most expediting way to prove this Hilbert-Rohn theorem is via Rohlin’s formula for complex orientations, which proves more generally that any $M$-curve (of even degree) has some nesting provided its degree $m=2k\ge 6$. The first proof of this statement (and much more) goes really back to Petrovskii’s seminal inequalities of 1938, cf. Petrovskii 1933/38 [@Petrowsky_1938]\] 

K. Rohn, [*Die Maximalzahl und Anordnung der Ovale bei der ebenen Kurve 6. Ordnung und bei der Fläche 4. Ordnung*]{}, Math. Ann. 73 (1913), 177–229. \[$\spadesuit$\] 

H. Röhrl, [*Unbounded coverings of Riemann surfaces and extensions of rings of meromorphic functions*]{}, Trans. Amer. Math. Soc. 107 (1963), 320–346. \[$\spadesuit$ cited in Alling 1965 [@Alling_1965], and one may wonder about a connection with Ahlfors 1950, i.e. the “unbounded covering” in question (cf. definition on p.328) are probably related to circle maps, at least extended versions thereof where the target is not necessarily the unit disc of course Röhrl’s notion is quite standard, albeit the terminology is far from uniformized, cf. e.g. Ahlfors-Sario’s “complete covering surfaces” (in 1960=[@Ahlfors-Sario_1960 p.42] themselves patterned after Stoilow’s “total coverings” $\spadesuit$ alas, it does not seem that Röhrl reproves Ahlfors result (which would have been pleasant in view of Röhrl great familiarity with Meis’ work 1960 [@Meis_1960])\] 

F. Ronga, [*Analyse réelle post-élémentaire*]{}, Presses polytechniques romandes, 1999. \[$\spadesuit$ cited for the picture Fig.\[ItenbergViroRiem:fig\]d\] 

P.C. Rosenbloom, [*Quelques classes de problèmes extrémaux*]{}, Bull. Soc. Math. France 80 (1952), 183–215. \[$\spadesuit$ this worked is cited in Forelli 1979 [@Forelli_1979], where it is employed to derive another existence-proof of circle-maps with the same control upon the degree as in Ahlfors 1950 [@Ahlfors_1950]\] 

M. Ross, [*The second variation of nonorientable minimal submanifolds*]{}, Trans. Amer. Math. Soc. 349 (1997), 3093–3104. \[$\spadesuit$ p.3097 criticizes the argument of Li-Yau 1982 [@Li-Yau_1982] for the Witt-Martens mapping $\spadesuit$ gives differential geometric application of it to (non-orientable) minimal surfaces\] 

H.L. Royden, [*Harmonic functions on open Riemann surfaces*]{}, Trans. Amer. Math. Soc. 73 (1952), 40–94. 50 \[$\clubsuit$ this is, in substance, the author’s Thesis \[Harvard University, 1951\] (under Ahlfors) $\spadesuit$ it contains very deep material “sufficient condition for the hyperbolic type in term of a triangulation of the surface” (causing a great admiration by Pfluger, etc.), yet from our finitistic perspective the paper seems to contain little about the Ahlfors map, for this issue see rather the subsequent paper Royden 1962 [@Royden_1962]\] 

H.L. Royden, [*Rings of meromorphic functions*]{}, Proc. Amer. Math. Soc. 9 (1958), 959–965. \[$\clubsuit$ this article is often credited by Alling to be the first employment of Ahlfors map as a technique to lift truths from the disc to more general finite bordered surfaces, e.g. in the Acknowledgements of Alling 1965 [@Alling_1965] or in Alling’s review of Stout 1965 [@Stout_1965] one reads: “The third technique is dependent on the existence of the Ahlfors map $P$ (=1950 [@Ahlfors_1950]), which maps a compact bordered Riemann surface $\overline R$, finite-to-one, onto $\overline U$. This gives rise to the algebraic approach, for the adjoint of $P$ is an isomorphism of $H_{\infty}(U)$ into $H_{\infty}(R)$, the extension being finite and very tractable. This approach was apparently first used by Royden 1958 \[=this entry=[@Royden_1958]\]. Later it was utilized extensively by the reviewer, who working independently of the author\[=Stout\], announced his extension of Carleson’s corona result to $R$ \[$\dots$\]”\]$\bigstar$$\bigstar$ 

H.L. Royden, [*The boundary values of analytic and harmonic functions*]{}, Math. Z. 78 (1962), 1–24. \[$\clubsuit$ re-prove the existence and properties of the Ahlfors function via Hahn-Banach, along the path of Read 1958 [@Read_1958_Acta]\] 

L.A. Rubel, J.V. Ryff, [*The bounded weak-star topology and the bounded analytic functions*]{}, J. Funct. Anal. 5 (1970), 167–183. 47, 50 \[$\clubsuit$\]$\bigstar$$\bigstar$$\bigstar$NY(only-MR) 

L.A. Rubel, [*Bounded convergence of analytic functions*]{}, Bull. Amer. Math. Soc. 77 (1971), 13–24. 47, 50 \[$\clubsuit$ p.18 the two works of Ahlfors 1947 [@Ahlfors_1947], 1950 [@Ahlfors_1950] are quoted in connection with the following problem about inner functions: “In the case of the general region $G$ \[supposed (cf. p.17) to support nonconstant bounded analytic functions and to enclose no removable singularities for all bounded analytic functions\], one would guess that the solution, known to exist, of any of several extremal problems would be inner, and consequently hypo-inner. For example, choose a point $z_0\in
G$ and consider $f\in B_H(G)$ \[i.e. the space of bounded analytic function\] so that $\| f\|_{\infty}\le 1$ and $f(z_0)=0$, and maximize $\vert f'(z_0)\vert$. The extremal function is the so-called Ahlfors function, and in case $G$ is finitely connected, it is known \[2\](=Ahlfors 1947 [@Ahlfors_1947]), \[3\](=Ahlfors 1950 [@Ahlfors_1950]) to be inner.” $\spadesuit$ let us recall definitions (cf. p.17–18): a bounded analytic function on the disc $F\in
B_H(D)$ is [*inner*]{} if $\|F\|_{\infty}\le 1$ and if its Fatou radial limit function $F^{\ast}(e^{i\theta})=\lim_{r\to
1} F(re^{i\theta})$ has unit modulus for almost all $\theta$ (w.r.t. usual arc length). It is said to be [*hypo-inner*]{} if the Fatou limit has unit modulus for a set of $\theta$ of positive measure. For a function on a general domain $G$, $f\in B_H(G)$, the notions of inner and hypo-inner are transposed via precomposition with the universal covering map $D \to G$. $\spadesuit$ now as to Rubel’s guess, it seems to be answered in the negative in Gamelin 1973 [@Gamelin_1973-BAMS p.1107], with details to be found in Gamelin 1974 [@Gamelin_1974-Shilov]\] 

L.A. Rubel, [*Some research problems about algebraic differential equations*]{}, Trans. Amer. Math. Soc. 280 (1983), 43–52. \[$\clubsuit$ p.47 the Ahlfors function is mentioned as follows: “To prepare the way for the next problem, we shall define the [*Ahlfors function*]{}. If $G$ is a (presumably multiply connected) region and $z_0$ is a point in $G$, we define the Ahlfors function $\alpha_{z_0}$ with [*basepoint*]{} $z_0$ as the (unique)solution of the following extremal problem: (i) $\alpha(z_0)=0$, (ii) $\vert \alpha(z)\vert \le 1$ for all $z\in
G$, (iii) $\alpha'(z_0)$ is as large as it can be for the class of functions satisfying (i) and (ii). In case $G$ is simply-connected, $\alpha_{z_0}$ becomes the Riemann map of $G$ onto $D$ that takes $z_0$ to $0$, with positive derivative there. [*Problem*]{} 11. [*Suppose $\alpha_{z_0}$ is hypotranscendental, and let $z_1\in G$ be another base point. Must $\alpha_{z_1}$ be hypotranscendental too?*]{}\] 

W. Rudin, [*Some theorems on bounded analytic functions*]{}, Trans. Amer. Math. Soc. 78 (1955), 333–342. 47, 78 \[$\spadesuit$ new (simpler) proof of an (unpublished) theorem of Chevalley-Kakutani stating that a plane domain $B$ such that for each of its boundary-point $p$ there is a bounded analytic function on $B$ possessing at $p$ a singularity is determined (modulo a conformal transformation) by the ring of all bounded analytic functions on $B$ $\spadesuit$ the proof makes uses of general results of Ahlfors 1947 [@Ahlfors_1947], yet apparently no use is made of the Ahlfors function\] 

W. Rudin, [*Analytic functions of class $H^p$*]{}, Trans. Amer. Math. Soc. 78 (1955), 46–66. 47 \[$\spadesuit$\] 

W. Rudin, [*The closed ideals in an algebra of continuous functions*]{}, Canad. J. Math. 9 (1957), 426–434. \[$\spadesuit$ proof of an unpublished result of Beurling describing the ideal theory of the algebra $A(\overline
\Delta)$ of continuous function on the closed disc analytic on its interior $\spadesuit$ for extensions of this Beurling-Rudin result to compact bordered surfaces, cf. Voichick 1964 [@Voichick_1964], Limaye’s Thesis 1968 and Stanton 1971 [@Stanton_1971] (who makes use of the Ahlfors map) $\spadesuit$ for an extension to non-orientable Klein surfaces (where no Ahlfors map are available!), see Alling-Limaye 1972 [@Alling-Limaye_1972]\] 

W. Rudin, [*Pairs of inner functions on finite Riemann surfaces*]{}, Trans. Amer. Math. Soc. 140 (1969), 423–434. \[$\spadesuit$ inner function as a synonym of the (Ahlfors) circle maps\] 

W. Rudin, [*Real and complex analysis*]{}, McGraw-Hill. \[$\spadesuit$\] 

L. Rudolph, [*Some topologically locally-flat surfaces in the complex projective plane*]{}, Comment. Math. Helv. 59 (1984), 592–599. \[$\spadesuit$ locally-flat counterexamples to Thom’s conjecture (based upon work of Freedman) $\spadesuit$ p.593 contains the sharpest historical information I am aware of about the terminology “Thom conjecture”, namely: “Professor Thom has remarked (personal communication, November 19, 1982) that the conjecture perhaps more properly belongs to folklore.” $\spadesuit$ As far as I know the designation “Thom conjecture” appears first in Kirby’s problem list (1970) [@Kirby_1970--95]\] 

R. Rüedy, [*Einbettungen Riemannscher Flächen in den dreidimensionalen euklischen Raum*]{}, Comment. Math. Helv. 43 (1968), 417–442. \[$\spadesuit$ p.417: “Flächen im Sinne der elementaren Differentialgeometrie können zu Riemannschen Flächen gemacht werden, indem man die iostheremen Parameter als lokale Koordinaten benutzt. Diese Struktur nennt man die [*natürliche*]{}, weil genau diese lokalen Darstellung winkeltreu sind.—F. Klein warf schon 1882 in seiner Schrift [*Über Riemanns Theorie der algebraischen Funktionen und ihrer Integrale*]{} das Problem auf, ob sich jede Riemannsce Fläche konform und bijectiv auf eine solche differentialgeometrische Fläche abbilden lasse.—Der Weg zu diesem überraschend schwierig zugänglichen Problem wurde durch die fundamentalen Arbeiten von Teichmüller geöffnet; aber erst um 1960 gelang der Beweis für den folgenden Satz:—[Einbettungssatz von Garsia.]{} [*Jede kompakte Riemannsche Fläche ist konform äquivalent zu einer differentialgeometrischen Fläche, die reelle-algebraisch im dreidimensionalen euklidischen Raum eingebettet ist.*]{}”\] 

R. Rüedy, [*Embeddings of open Riemann surfaces*]{}, Comment. Math. Helv. 46 (1971), 214–225. \[$\spadesuit$ p.214: “In the final section of his famous thesis Riemann states that in his investigations the branched covering surfaces of the plane could be replaced by smooth orientable surfaces embedded in Euclidean $3$-space. \[…\] In his lectures Felix Klein emphasized the concept of viewing classical surfaces as Riemann surfaces, …. It was also he who asked in 1882 if every Riemann surface were conformally equivalent to a classical surface. \[F. Klein, [*Ges. math. Abh.,*]{} Bd.3 (Springer 1923), p.502 and p.635.\]—For a long time the only result in this direction were that every compact Riemann surface of genus zero is conformally equivalent to the sphere, every non-compact planar (schlichtartig) surface is conformally equivalent to a subregion of the plane, and a compact Riemann surface of genus $1$ is conformally equivalent to a ring surface provided its modulus is purely imaginary (see \[16\]=Weyl 1913/65 [@Weyl_1913], 3.Auflage).—The first result beyond these facts was obtained by Teichmüller in \[15\](=1944 [@Teichmueller_1944-Beweis-der-analytischen-Abhaengigkeit]), where he applied his theory of spaces of Riemann surfaces to the embedding problem. He could show that not all compact embedded surfaces of genus $1$ are conformally equivalent to ring surfaces. More important than this result was the method by which he obtained it: He deformed an embedded surface by moving each point along the normal line and studied the dependence of the modulus of the deformed surface on the deformation.—Around 1960 Garsia constructed a surprisingly large class of compact Riemann surfaces whose moduli could be determined (\[5\](=1960 [@Garsia_1960-Pacific]),\[6\](=1960 [@Garsia_1960-Rend.])). But he succeeded in answering Klein’s question in the affirmative for all compact Riemann surfaces only when he abandoned his beautiful models and embarked on Teichmüller’s road. His proof in \[7\](=Garsia-Rodemich 1961 [@Garsia-Rodemich_1961]) and \[8\](=Garsia 1961 [@Garsia_1961]) is an ingenious combination of Teichmüller’s ideas and results, constructions inspired by Nash’ isometric embeddings, and Brouwer’s fixed point theorem.—We will see in this paper that his methods are even strong enough to prove this theorem for noncompact surfaces too. \[…\], we may formulate our theorem as follows:—[Embedding theorem.]{} [*Every Riemann surface $R$ is conformally equivalent to a complete classical surface. A model can be constructed by deforming any topologically equivalent complete classical surface $X$ in the direction of the normals. [$X$ is complete, if $X$ is a closed subset of Euclidean space.]{}*]{}—A nontrivial corollary (due to R. Osserman) follows, if $R$ is the unit disc and $X={\Bbb
C}$: For a suitable real-valued $C^{\infty}$-function $f$ the classical surface represented by $(x,y)\to(x,y,f(x,y))$, $x+iy\in{\Bbb C}$, is hyperbolic.”\] 

R. Rüedy, [*Deformations of embedded Riemann surfaces*]{}, Ann. of Math. Studies 66, 1971. \[$\spadesuit$\] 

S. Saitoh, [*The kernel functions of Szegö type on Riemann surface*]{}, Kodai Math. Sem. Rep. 24 (1972), 410–421. \[$\spadesuit$ Bergman kernel on compact bordered Riemann surfaces\] 

S. Saitoh, [*The exact Bergman kernel and the kernels of Szegö*]{}, Pacific J. Math. 71 (1977), 545–557. \[$\spadesuit$ Bergman kernel on compact bordered Riemann surfaces\] 

S. Saitoh, [*The Bergman norm and the Szegö norm*]{}, Trans. Amer. Math. Soc. 249 (1979), 261–279. \[$\spadesuit$ Bergman kernel on compact bordered Riemann surfaces\] 

S. Saitoh, [*A characterization of the adjoint $L$-kernel of Szegö type*]{}, Pacific J. Math. 96 (1981), 489–493. \[$\spadesuit$ compact bordered Riemann surfaces, Green’s function and reproducing kernel\] 

S. Saitoh, [*Theory of reproducing kernels and its applications*]{}, Pitman Res. Notes in Math Series 189, 1988. x+157pp. \[$\spadesuit$ reproducing kernel in the abstract united exposition of Aronszajn 1950 [@Aronszajn_1950], followed by a specialization to the case of multiply connected plane domains (esp. Garabedian’s $L$-kernel as the solution to an extremal problem for the Dirichlet integral)\] 

S. Saitoh, [*Theory of reproducing kernels; applications to approximate solutions of bounded linear operator equations on Hilbert spaces*]{}, Amer. Math. Soc. Transl., 2010. \[$\spadesuit$ mentions the “Ahlfors function”\] 

M. Sakai, [*On constants in extremal problems of analytic functions*]{}, Kodai Math. Sem. Report 21 (1969), 223–225. \[$\spadesuit$ p.223 seems to consider the problem of minimizing the Dirichlet integral $D[f]=\int\int_W df \cdot
\overline{df^{\ast}}$ among the analytic functions $f$ on a Riemann surface $W$ normalized by $f(t)=0$ and $f'(t)=1$ (w.r.t. some local uniformizer) \[see also Schiffer-Spencer 1954 [@Schiffer-Spencer_1954]\] $\spadesuit$ alas nothing seems to be asserted about the range of the least area mapping (in particular we still wonder if it is a circle map as looks plausible in view of the simply-connected case treated in Bieberbach 1914 [@Bieberbach_1914])\] 

T. Salvemini, [*Sulla rappresentazione conforme delle aree piane pluriconnesse su una superficie di Riemann di genere zero in cui sono siano eseguiti dei tagli paralleli*]{}, Ann. Scuola Norm. Super. Pisa (1) 16 (1930), 1–34. \[$\spadesuit$ just cited to mention that Schottky’s proof of PSM relied on a parameter count not completely justified at his time\] 

M.V. Samo\[k\]hin, [*On some questions connected with the problem of existence of automorphic analytic functions with given modulus of boundary values*]{}, Mat. Sb. 111 (1980); English transl.: Math. USSR Sbornik 39 (1981), 501–518. \[$\spadesuit$ p.505 occurrence of the Ahlfors function as an example of non-constant function in $H^{\infty}$ whose Gelfand transform is unity on the Šilov boundary of $H^{\infty}$, p.509: “We used an Ahlfors function to “knock down” the growth of the function…”, p.512: another occurrence of the Ahlfors function\] 

M.V. Samokhin, [*Cauchy’s integral formula in domains of arbitrary connectivity*]{}, Sb. Math. 191 (2000), 1215–1231. \[$\spadesuit$ From the Abstract: An example of a simply-connected domain with boundary of infinite length is constructed such that for fairly general functionals on $H^{\infty}$ no extremal function (including the Ahlfors function) can be represented as a Cauchy potential\] 

D. Sarason, [*Representing measures for $R(X)$ and their Green’s functions*]{}, J. Funct. Anal. 7 (1971), 359–385. \[$\spadesuit$ some questions asked in the paper are answered in Nash 1974 [@Nash_1974]\] 

L. Sario, [*A linear operator method on arbitrary Riemann surfaces*]{}, Trans. Amer. Math. Soc. 72 (1952), 281–295. 50 \[$\spadesuit$ perhaps first a general remark about Sario: to the best of my knowledge none of Sario’s papers (or books) works out a reproof of Ahlfors circle maps, albeit he is often gravitating around closely related or even more grandiose (i.e. foundational) paradigms. Quite ironically, much of the impulse and modernity along the Nevanlinna-Sario tradition finds its starting point in the Schwarz alternating method (which seemed outdated after Hilbert 1900 [@Hilbert_1900] “direct” resolution (=resuscitation) of the Dirichlet principle) $\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited in the bibliography yet apparently not within the text\] 

L. Sario, [*Extremal problems and harmonic interpolation on open Riemann surfaces*]{}, Trans. Amer. Math. Soc. 79 (1955), 362–377. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited on p.364 as follows: “Concerning extremal problems for differentials, the reader is referred to the comprehensive study \[1\](=1950 [@Ahlfors_1950]) by Ahlfors.” $\spadesuit$ “The ultimate purpose of the present paper is to study interpolation of harmonic and analytic functions on open Riemann surfaces $W$. We shall, however, first take a less restricted viewpoint and consider, in general, extremal problems on Riemann surfaces.” $\spadesuit$ the bulk of the paper is a reduction of a certain extremal problem over very general open Riemann surfaces to the special case of compact bordered surface (with analytic contours) via the usual exhaustion trick\] 

L. Sario, [*Strong and weak boundary components*]{}, J. Anal. Math. 5 (1956/57), 389–398. \[$\spadesuit$ quoted in Reich-Warschawski 1960 [@Reich-Warschawski_1960] for another proof of Grötzsch’s extension to infinite connectivity of the Kreisbogenschlitztheorem\] 

L. Sario, K. Oikawa, [*Capacity Functions*]{}, Grundlehren d. math. Wiss. 149, Springer, Berlin, 1969. 47, 50, 78 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited at three places: pp.46, 110, 175, where this last citation come closest to our interest (but the discussion seems to be confined to plane regions $W$, cf. p.175 (top)). $\spadesuit$ We cite the relevant extract (p.175): “Concerning the quantity $c_B$, Schwarz’s lemma give us the unique function minimizing $M[F]$ if $W$ is simply-connected. The problem has not been solved completely for an arbitrary region $W$. However, Carleson \[1\](=1968 [@Carleson_1967-book]) established the uniqueness of the minimizing function if $c_B>0$. For a regular region $W$ (in which case $c_B>0$), further results have been obtained by Ahlfors \[1\](=1947 [@Ahlfors_1947]), \[2\](=1950 [@Ahlfors_1950]), Garabedian \[1\](=1949 [@Garabedian_1949]), and Nehari \[2\](=1951 [@Nehari_1951-survey-BAMS]), \[3\](=1952 [@Nehari_1952-BOOK]). For the function minimizing $M[F]$, they obtained a characterization which in particular implies that the function maps $W$ onto an $n$-sheeted disk of radius $1/c_B$, where $n$ is the connectivity of $W$; note that this property does not in turn characterize the minimizing function. Garabedian and Nehari further derived a relationship with Szegö’s kernel function (Szegö \[1\](=1921 [@Szego_1921]), Schiffer \[5\](=1950 [@Schiffer_1950-Duke])). However, we shall not go into a more detailed discussion of these interesting results.”\] 

L. Sario, M. Nakai, [*Classification Theory of Riemann Surfaces*]{}, Grundlehren d. math. Wiss. 164, Springer, Berlin, 1970, 446 pp. 47, 50, 78 \[$\spadesuit$ cite the work Ahlfors 1950 [@Ahlfors_1950] in the bibliography (p.412), but not in the main body of the text (sauf erreur!) $\spadesuit$ p.452, the article by Kusunoki 1952 [@Kusunoki_1952] (where the Ahlfors map of a bordered surface is applied to the so-called “type problem”) is cited (and as far as I know this is the [*unique*]{} citation of Kusunoki’s work throughout the world literature). Alas, Kusunoki’s work does not seem to be quoted inside the main body of the text. $\spadesuit$ p.332: “The concept of harmonic measure was introduced by Schwarz \[1\](=Ges. math. Abh. 1890) and effectively used by Beurling \[1\](=1935 [@Beurling_1935-These]). Nevanlinna \[1\](=1934 [@Nevanlinna_1934]) coined the phrase “harmonic measure” and introduced the class of “nullbounded” surfaces characterized by the vanishing of the harmonic measure. That this class coincides with the class $O_G$ of “parabolic” surfaces was shown by Myrberg \[2\](=1933 [@Myrberg_1933]) for surfaces of finite genus.”\] 

S. Scheinberg, [*Hardy spaces and boundary problems in one complex variables*]{}, Ph.D. Thesis, Princeton University, 1963. \[$\spadesuit$ includes a proof of the corona theorem on annuli, cf. also Stout 1965 [@Stout_1965]\]$\bigstar$ 

L. Schläfli, [*On the distribution of surfaces of the third order into species, in reference to the absence or presence of singular points, and the reality of their lines*]{}, Philos. Trans. Roy. Soc. London 153 (1863), 195–241. \[$\spadesuit$ the heaviest brain ever met ca. 1.9 kg?\] 

E. Schmidt, [*Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener ???*]{}, Math. Ann. 63 (1907), 433–476. \[$\spadesuit$ quoted in Nehari 1955 [@Nehari_1955]\] 

E. Schmidt, [*Zur Theorie der linearen und nichtlinearen Integralgleichungen*]{}, Math. Ann. 64 (1907), 161–174. \[$\spadesuit$ quoted in Bergman 1950 [@Bergman_1950]\] 

M. Schiffer, [*Sur les domaines minima dans la théorie des transformations pseudo-conformes*]{}, C.R. Acad. Sci. Paris 207 (1938), 112–115. \[$\spadesuit$ quoted in Maschler 1956 [@Maschler_1956 p.506] for the issue that minimal domains satisfy the mean value property; thus perhaps if ranges of least area maps are minimal domains we may hope that by virtue of a theorem of XXX-Schiffer (cited in the introd. of Aharonov-Shapiro 1976 [@Aharonov-Shapiro_1976]) the least area map is a circle map \[02.08.12\] $\spadesuit$ also quoted in Bergman 1947 [@Bergman_1947 p.32] for the issue that for a proof of the partial result that for starlike domains the least area map effects the Riemann mapping upon the circle\]$\bigstar$$\bigstar$$\bigstar$ 

M. Schiffer, [*Sur un théorème de la représentation conforme*]{}, C.R. Acad. Sci. Paris 207 (1938), 520–522. 60, 78 \[$\spadesuit$ located via Reich-Warschawski 1960 [@Reich-Warschawski_1960], who cite the paper for another proof of Grötzsch’s extension 1929–1931 [@Groetzsch_1931] to infinite connectivity of the Kreisschlitzbereich mapping of Koebe 1918 [@Koebe_1918] $\spadesuit$ contains indeed a proof based upon an extremal problem of the circular slit map, yet the argument seems to depend upon the longer paper Schiffer 1937/38 [@Schiffer_1937/38]\] 

M. Schiffer, [*A method of variation within the family of simple functions*]{}, Proc. London Math. Soc. (2) 44 (1937/38), 432–449. 60, 78 \[$\spadesuit$ principle of areas (Flächensatz) of Bieberbach-Faber $\spadesuit$ quotes Grötzsch 1930 and extends a result of Marty 1934\] 

M. Schiffer, [*The span of multiply connected domains*]{}, Duke Math. J. 10 (1943), 209–216. 60, 78 $\bigstar$ 

M. Schiffer, [*The kernel function of an orthonormal system*]{}, Duke Math. J. 13 (1946), 529–540. 78 $\bigstar$ $\bigstar$$\bigstar$ \[$\spadesuit$ establish for domains an identity relating the Bergman kernel to the Green’s function\] 

M. Schiffer, [*An application of orthonormal functions in the theory of conformal mapping*]{}, Amer. J. Math. 70 (1948), 147–156. 60, 78 \[$\spadesuit$ new derivation via the Bergman kernel of inequalities of Grunsky’s Thesis 1932, which were previously derived by variational methods\] 

M. Schiffer, [*Various types of orthogonalization*]{}, Duke Math. J. 17 (1950), 329–366. $\bigstar$$\bigstar$$\bigstar$ 

M. Schiffer, [*Some recent developments in the theory of conformal mapping*]{}, Appendix to R. Courant, 1950 [@Courant_1950], 249–324. \[$\spadesuit$ an extremely readable survey of several trends in potential theory, including the Green-Dirichlet yoga, the kernel method and some of the allied extremal problems, plus the method of extremal length and schlicht functions\] 

M. Schiffer, [*Variational methods in the theory of conformal mapping*]{}, Proc. Internat. Congr. Math., Cambridge, Mass., 1950, (1952), 233–240. 78 \[$\spadesuit$ survey of variational methods\] 

M. Schiffer, D.C. Spencer, [*Functionals of Finite Riemann Surfaces*]{}, Princeton Mathematical Series, Princeton University Press, 1954. \[$\clubsuit$ \[noticed the 26.07.12\] on p.135 the authors consider the problem of the least-area map (normed at a point $q$)for a compact bordered Riemann surface $\clubsuit$ it would be extremely desirable to know if the extremal map is a circle map, and if it relates to the Ahlfors function described in Ahlfors 1950 [@Ahlfors_1950]\]

M. Schiffer, [*Extremum problems and variational methods in conformal mapping*]{}, Proc. Internat. Congr. Math., Stockholm, 1958, 211–231. 78 \[$\spadesuit$ p.229 suggest a new proof (via Fredholm) of Schottky’s famous circular mapping (i.e. Kreisnormierung): details to be found in the next voluminous paper\] 

M. Schiffer, [*Fredholm eigenvalues of multiply connected domains*]{}, Pacific J. Math. 9 (1959), 211–269. 78 \[$\spadesuit$ includes a new proof (via an extremum problem involving the Fredholm determinant) of the Schottky-Koebe Kreisnormierung; for yet another proof cf. the next item [@Schiffer-Hawley_1962]\] 

M. Schiffer, N.S. Hawley, [*Connections and conformal mapping*]{}, Acta Math. 107 (1962), 175–274. 78 \[$\spadesuit$ p.183–189 includes yet another proof of the Schottky-Koebe Kreisnormierung (finite-connectivity) via an extremum problem of the Dirichlet type\] 

M. Schiffer, [*Fredholm eigenvalues and conformal mapping*]{}, Rend. Mat. e Appl. (5) 22 (1963), 447–468. 78 \[$\spadesuit$ which mappings? the method must be the same as the previous item\] 

M. Schiffer, G. Springer, [*Fredholm eigenvalues and conformal mapping of multiply connected domains*]{}, J. Anal. Math. 14 (1965), 337–378. 78 

M. Schiffer, [*Half-order differentials on Riemann surfaces*]{}, J. SIAM Appl. Math. 14 (1966), 922–934. 60, 78 \[$\spadesuit$ summary of research joint with Hawley, $\spadesuit$ immediate generalization for the Bergman kernel for any closed Riemann surface to be found in Schiffer-Spencer 1954 [@Schiffer-Spencer_1954] $\spadesuit$ contour integration introduced by Riemann himself\] 

M. Schiffman, [*The Plateau problem for non-relative minima*]{}, Ann. of Math. (2) 40 (1939), 834–854. \[$\spadesuit$ Seidel’s summary: the problem of mapping a region bounded by a simple closed curve with a continuously turning tangent is reduced to that of minimizing a functional, somewhat similar to that of Douglas (cf. Douglas 1931 [@Douglas_1931-Solution]). This functional has an electrostatic interpretation which may provide an effective mechanical method for the determination of conformal maps\] 

M. Schiffman, [*Uniqueness theorems for conformal mapping of multiply connected domain*]{}, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 137–139. 78 \[$\spadesuit$ quoted in Bergman 1950 [@Bergman_1950]\] 

P. Schmutz, [*Riemann surfaces with shortest geodesics of maximal length*]{}, Geom. Funct. Anal. 3 (1993), 564–631. \[$\spadesuit$\] 

P. Schmutz-Schaller, [*Geometry of Riemann surfaces based on closed geodesics* ]{}, Bull. Amer. Math. Soc. 35 (1998), 193–214. \[$\spadesuit$... Extremal problems have been considered in similar contexts; see in particular Bollobás \[9\] for extremal graphs and Ahlfors \[3\] for extremal problems in conformal geometry. ...\] 

A. Schönflies, [*Über gewisse geradlinig begrenzte Stücke Riemann’scher Flächen*]{}, Nachr. Akad. Wiss. Göttingen (1892), 257–267. $\bigstar$ \[$\spadesuit$ detected via AS60.\] 

E. Scholz, [*Geschichte des Mannigfaltigkeitsbegriff von Riemann bis Poincaré*]{}. Birkäuser, 1980. \[$\spadesuit$\] 

E. Scholz, [*The concept of manifold, 1850–1950.*]{} Chapter 2, in: History of Topology, 25–64. Elsevier, 1999 \[$\spadesuit$ p.26: “Also leading mathematicians like Cauchy and Gauss started to use geometrizing language in ${\Bbb R}^n$ in publications (Cauchy, 1847) or lecture courses (Gau[ß]{}, 1851/1917). Gauss, in his lecture courses, even used the vocabulary of [*$(n-k)$-dimensional manifolds (Mannigfaltigkeiten)*]{}, but still restricted in his context to affine subspaces of the $n$-dimensional real space (Gau[ß]{}, 1851/1917, pp.477ff.). There is no reason to doubt that Riemann got at least some vague suggestion of how to generalize the basic conceptual frame for geometry along these lines from Gauss and developed it in a highly independent way.” $\spadesuit$ p.36: “In geometric function theory divers authors contributed to a refined understanding of the rôle of topological concepts, in particular C. Neumann with his calculation of the connectivity of a Riemann surface from the winding orders of branch points \[Neumann 1865 [@Neumann_1865-Vorlesungen]\], Lüroth, Clebsch and Clifford with their normalized representation during the 1870-s for branched coverings of $P_1({\Bbb C})$, which represent a Riemann surface with given number of leaves, given loci and winding numbers of branch points.”\] 

F. Schottky, [*Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen*]{}, (Diss. Berlin 1875) Crelle J. für die Math. 83 (1877), 300–351. 60, 78 \[$\clubsuit$ after Riemann 1857/58 [@Riemann_1857_Nachlass], the first existence proof of the Ahlfors map in the planar case $\spadesuit$ contains in germ all type of mapping like the Circle mapping, the Kreisnormierung plus the parallel-slit maps $\spadesuit$ the only drawback is a certain confinement to planar regions, but this will be quickly relaxed in Klein 1882 [@Klein_1882] $\spadesuit$ regarding the rigor of proofs, the appreciation are rather random, compare Cecioni 1908 [@Cecioni_1908] and Grunsky 1978 who ascribes the first rigorous proof of the PSM to Cecioni\] 

F. Schottky, [*Ueber eindeutige Functionen mit linearen Transformationen in sich. (Auszug aus einem Schreiben an Herrn F. Klein.)*]{} Math. Ann. 19 (1882), ?–?. 

F. Schottky, [*Ueber eindeutige Functionen mit linearen Transformationen in sich,*]{} Math. Ann. 20 (1882), 293–300. 60 $\bigstar$ 

F. Schottky, [*Zur Frage: Haben die Klassenfunktionen Differentialgleichungen,*]{} Sitz.-Ber. Peu[ß]{}. Akad. Wiss., math.-phys. Kl. (1922), 414–423. \[$\spadesuit$ cited in Grunsky 1978 [@Grunsky_1978 p.197] as follows: “Considering meromorphic functions on $D$ with real boundary values (which he later called “Klassenfunktionen”, \[476\](=this entry); now they are called Schottky functions) and proving the existence of a real algebraic relation between any two of them, he disclosed an intimate relation between problems in multiply connected domains and the theory of algebraic functions. The most concise expression of this relation is the idea of the “Schottky double” of a multiply connected domain (or of any finite Riemann surface) with analytic boundary; this is a compact Riemann surface, gained by identifying boundary points of two replicas …”\] $\bigstar$ 

O. Schramm, [*Conformal uniformization and packings*]{}, Israel J. Math. 93 (1996), 399–428. \[$\spadesuit$ new proof of the Brandt-Harrington (1980 [@Brandt_1980] and 1982 [@Harrington_1982]) generalization of Koebe’s KNP via a topological method (mapping degree), plus the PSM (parallel slit maps) and some other gadgets\] 

K. Schüffler, [*Zur Fredholmtheorie des Riemann-Hilbert-Operator*]{}, Arch. Math. 47 (1986), 359–366. \[$\spadesuit$ p.359: “Ausgehend von dem bekannten klassischen Riemann-Hilbert Randwertproblem \[8,S.181ff\](=Vekua 1963 [@Vekua_1963]) betrachten wir den Operator $RH\colon
A^m(\Omega)\to H^{m-1/2}(\partial \Omega, {\Bbb R})$, $RH(f):={\rm
Re}(\bar \lambda f)\vert_{\partial \Omega}$. \[…\] das Symbol “$A^m$” bezeichne den Sobolevraum $H^{m,2}$ der auf $\Omega$ holomorphen Funktionen, $m\ge 2$; die komplexwertige Funktion $\lambda$ sei nullstellenfrei (auf $\partial \Omega$) und o.E. glatt.—Es ist bekannt, da[ß]{} der Operator $RH$ für glattberandete, endliche Riemannsche Flächen ein Fredholmoperator ist. Sien Index hängt sowohl von der Topologie von $\Omega$ (Anzahl der Randkomponenten und Geschlecht) als vom “geometrischen Index, dem Argumentzuwachs $\kappa(\lambda)=\Delta
\arg (\lambda)/ 2\pi \in {\Bbb Z}$ von $\lambda$ beim positiven Durchlaufen von $\partial \Omega$ ab (siehe \[8, S.189\]=Vekua 1963 [@Vekua_1963])” $\spadesuit$ \[17.10.12\] this seems connected to the Ahlfors map, by taking $\lambda$ its boundary restriction\] 

H.A. Schwarz, [*Ueber einige Abbildungsaufgaben*]{}, Crelle J. für die Math. 70 (1869), 105–120. \[$\spadesuit$ introduces the principle of symmetry $\clubsuit$ solves special case of the RMT by hand\] 

H.A. Schwarz, [*Zur Theorie der Abbildung*]{}, Züricher Vierteljahrsschrift (1869/70); also (theilweise umgearbeitet ca. 1890) in Ges. Abh. II, 108–132. 

H.A. Schwarz, [*Ueber die Integration der partiellen Differentialgleichung $\frac{\partial^2 u}{\partial
x^2}+\frac{\partial^2 u}{\partial y^2}=0$ für die Fläche eines Kreises*]{}, Züricher Vierteljahrsschrift (1870), 113–128; reprinted (or rather integrated) in the longer paper Schwarz 1872 [@Schwarz_1872]. \[$\spadesuit$ this entry is the first rigorous solution to the Dirichlet problem to have appeared in print (for the very special case of the disc) and via usage of the Poisson integral (occurring in several publications dated 1820–23–27–29–31–35) $\spadesuit$ see also Prym 1871 [@Prym_1871] for an essentially simultaneous resolution (which however turned out to have less impact on the future events)\] 

H.A. Schwarz, [*Ueber einen Grenzübergang durch alternirendes Verfahren*]{}, Züricher Vierteljahrsschrift (1870), 272–286; also in Ges. Abh. II, 133–143. 

H.A. Schwarz, [*Ueber die Integration der partiellen Differentialgleichung $\frac{\partial^2 u}{\partial
x^2}+\frac{\partial^2 u}{\partial y^2}=0$ unter vorgeschriebenen Grenz- und Unstetigkeitsbedingungen*]{}, Berliner Monatsb. (1870), 767–795; or Ges. Abh. Bd. II, 144–171 \[$\spadesuit$ p.167–170 uniqueness of the conformal structure on the 2-sphere\] 

H.A. Schwarz, [*Zur Integration der partiellen Differentialgleichung $\frac{\partial^2 u}{\partial
x^2}+\frac{\partial^2 u}{\partial y^2}=0$*]{}, Crelle J. für die Math. 74 (1872), 218–253; or Ges. Abh. II, 175–210.

A. Sebbar, Th. Falliero, [*Equilibrium points of Green’s function for the annulus and Eisenstein series*]{}, Proc. Amer. Math. Soc. 135 (2007), 313–328. \[$\spadesuit$ p.314: “By the classical Hopf’s lemma, the normal derivative of the Green’s function is positive on the boundary \[of a multi-connected domain\], and one may ask if there is a compact set \[in the domain\], independent of the pole, containing all the equilibrium points of the Green’s function.” $\spadesuit$ a positive answer to this problem is supplied by Solynin 2007 [@Solynin_2007]\] 

B. Segre, [*Sui moduli delle curve poligonali, e sopra un complemento al teorema di esistenza di Riemann*]{}, Math. Ann. 100 (1928), 537–551. 

W. Seidel, [*Bibliography of numerical methods in conformal mapping*]{}. In: [*Construction and Applications of Conformal Maps*]{}, Proc. of a Sympos. held on June 22–25 1949, Applied Math. Series [*18*]{}, 1952, 269–280. \[$\spadesuit$ a useful compilation of (old) conformal maps literature emphasizing the numerical methods, and out of which we borrowed several summaries\] 

H.L. Selberg, [*Ein Satz über beschränkte endlichvieldeutige analytische Funktionen*]{}, Comment. Math. Helv. 9 (1937), 104–108. \[$\spadesuit$ quoted in Hayashi-Nakai 1988\]$\bigstar$$\bigstar$$\bigstar$\[CHECK\] 

M. Seppälä, [*Teichmüller spaces of Klein surfaces*]{}, Ann. Acad. Sci. Fenn. Ser.AI Math. Dissertationes 15 (1978), 1–37. \[$\spadesuit$\] 

M. Seppälä, [*Quotient of complex manifolds and moduli spaces of Klein surfaces*]{}, ?? (198?), ?–?. \[$\spadesuit$\] 

M. Seppälä, R. Silhol, [*Moduli spaces for real algebraic curves and real abelian varieties*]{}, Math. Z. 201 (1989), 151–165. \[$\spadesuit$ modernization of Klein’s resp. Comessatti’s theories\] 

M. Seppälä, [*Real algebraic curves in the moduli spaces of complex curves*]{}, Compos. Math. 74 (1990), 259–283. \[$\spadesuit$\] 

M. Seppälä, [*Moduli spaces of stable real algebraic curves*]{}, Ann. Sci. Éc. Norm. Sup. 24 (1991), 519–544. \[$\spadesuit$\] 

M. Seppälä, [*Computation of period matrices of real algebraic curves*]{}, Discr. Comput. Geom. (1994). \[$\spadesuit$ Abstract. In this paper we derive a numerical method which allows us to compute periods of differentials on a real algebraic curve with real points. This leads to an algorithm which can be implemented on a computer and can be used to study the Torelli mapping numerically.\] 

F. Severi, [*Vorlesungen über algebraische Geometrie*]{}, Leipzig, Teubner, 1921. \[$\spadesuit$ p.159 re-proves the upper bound for the gonality of a complex curve (according to Segre 1928 [@Segre_1928]), but for the “modern standards” the first accepted proof is that of Meis 1960 [@Meis_1960] $\spadesuit$ contains a brief discussion of Klein’s theory of real algebraic curves $\spadesuit$ Anhang F also contains the complex case of Brusotti’s theorem (1921 [@Brusotti_1921]) on the independence of smoothing nodal curves $\spadesuit$ the same ideas where used in Harris’ proof on the irreducibility of the variety of plane curves of fixed degree and prescribed genus\] 

F. Severi, [*Sul teorema di esistenza di Riemann*]{}, Rend. Circ. Mat. Palermo 46 (1922), 105–116. 

G.B. Shabat, V.A. Voevodsky, [*Equilateral triangulations of Riemann surfaces and curves over algebraic number fields*]{}, Doklady SSSR 304 (1989), 265–268; Soviet Math. Dokl. 39 (1989), 38–41. \[$\spadesuit$ geometric translation of Belyi-Grothendieck’s theorem that a curve is defined over $\Qbar$ iff it ramifies only over 3 points of the sphere. Question: can one extend this to Ahlfors maps in the bordered case cf. Sec.\[sec:Belyi-Grothendieck\] for a pessimist answer, yet probably all real curves are to be integrated. So what about real Riemann surfaces with an equilateral triangulation invariant under complex conjugation. So the vertices occurs as ${\Bbb Q}$-ratioanl points? etc.\]$\bigstar$ 

G.B. Shabat, V.A. Voevodsky, [*Drawing curves over number fields*]{}, in: Grothendieck Festschrift, Birkhäuser. 

I.R. Shafarevich, [*Basic Algebraic Geometry*]{}, NAuka, Moscow, 1972; English. transl., Die Grundlehren der math. Wiss. in Einzeldarstellungen, Bd.213, Springer-Verlag, Berlin, 1974. (many subsequent reeditions) \[$\spadesuit$ contain a lovely picture of a real elliptic curve (with two real circuits) acted upon by complex conjugation (I confess that this little picture is actually, besides some theory told by Felice Ronga and Daniel Coray, the very origin of my modest involvement with the topic of real algebraic curves)\] 

C.S. Sheshadri, [*Space of unitary vector bundles on a compact Riemann surface*]{}, Ann. of Math. 85 (1967). \[$\spadesuit$\] 

M. Shiba, K. Shibata, [*Singular hydrodynamical continuations of finite Riemann surfaces*]{}, Kyoto J. Math. (1985). \[$\spadesuit$ The present study arose, in close relationships to a series of our investigations \[16\],\[17\] and \[18\], from an attempt to embed an arbitrary open Riemann surface of finite genus into another closed Riemann surface of the same genus, so that the prolongation of the ...\] 

G.E. Shilov, [*On rings of functions with uniform convergence*]{}, Ukrain. Mat. Z. 3 (1951), 404–411. \[$\spadesuit$\]$\bigstar$ 

V.V. Shokurov, [*The Noether-Enriques theorem on canonical curves*]{}, Mat. Sb. Nov. Ser. 86 (1971), 367–408; English transl., Math. USSR Sb. 15 (1972), 361–401. \[$\spadesuit$\]$\bigstar$ 

E.I. Shustin, [*The Hilbert-Rohn method and bifurcation of complicated singular points of curves of degree $8$*]{}, Uspekhi Mat. Nauk 38 (1983), 157–158; English transl., ??. \[$\spadesuit$\]$\bigstar$ 

E.I. Shustin, [*Gluing of singular algebraic curves*]{}, in: Methods of qualitative theory. Gorky Univ. Press, Gorky, 1985, 116–128 (Russian). \[$\spadesuit$\]$\bigstar$ 

E.I. Shustin, [*Independent removal of singular points and new $M$-curves of degree $8$*]{}, Uspekhi Mat. Nauk 40 (1985), ?–?; English transl., ??. \[$\spadesuit$\]$\bigstar$ 

E.I. Shustin, [*The Hilbert-Rohn method and smoothing of singular points of real algebraic curves*]{}, Dokl. Akad. Nauk SSSR 281 (1985), 33–36; English transl., Soviet Math. Doklady 31 (1985), 282–286. \[$\spadesuit$\]$\bigstar$ 

E.I. Shustin, [*Counterexamples to a conjecture of Rokhlin*]{}, Funkt. Anal. Prilozhen 19 (1985), 94–95; English transl., Funct. Anal. Appl. 19 (1985), 162–163. \[$\spadesuit$ counterexamples in degree 8 to Rohlin’s conjecture (type I iff maximal), based on earlier work by Polotovskii [@Polotovskii_1981] compare the discussion in Viro 1986/86 [@Viro_1986/86-Progress] $\spadesuit$ \[25.01.13\] this note also implies a counterexample to an Ansatz of Klein 1876 [@Klein_1876], to the effect that nondividing curves could always win a supplementary oval by crossing a solitary node. In fact Shustin’s note uses predominantly a Bézout-like obstruction for $M$-octics due to Viro 1983 [@Viro_1983/84-new-prohibitions] extending the one of Fiedler 1982/83 [@Fiedler_1982/83-Pencil]. In Shustin’s disproof, the counterexample is an $(M-2)$-curve, which (either itself or more likely one of its $(M-1)$-enlargements) is maximal but of type II. $\spadesuit$ \[25.01.13\] I do not know if Klein’s Ansatz has some chance to be true in degree $7$. $\spadesuit$ nor do I know if it could old for $(M-2)$-curves, or more generally all curves except possibly $(M-1)$-curves. Compare Orevkov’s remark in Sec.\[Orevkov:sec\], which seems to prompt that there is some open problem here.\] 

E.I. Shustin, [*A new $M$-curve of degree $8$*]{}, Mat. Zametki 42 (1987), 180–186; English transl., Math. Notes 42 (1987), 606–610. \[$\spadesuit$ this is perhaps the paper to which Orevkov is referring to in Sec.\[Orevkov:sec\]\] 

E.I. Shustin, [*Versal deformations in the space of a fixed degree curves*]{}, Funct. Anal. Appl. 21 (1987), 90–91. \[$\spadesuit$\] 

E.I. Shustin, [*Smoothness and irreducibility of varieties of singular algebraic curves*]{}, in: Arithmetic and geometry of algebraic varieties. Saratov Univ. Press/Kuibyshev branch, Kuibyshev, 1989, 102–117. \[$\spadesuit$\] 

E.I. Shustin, [*Geometry of discriminant and topology of algebraic curves*]{}, in: Proc. Internat. Congr. Math., Kyoto, Japan, 1990, Math. Soc. Japan. (1991), 559–567. \[$\spadesuit$ p.566: “\[…\] the complete description of discriminant in the space of plane real quartics curves and complete classification of inflexion point arrangements on these curves \[9\](=Gudkov 1988 [@Gudkov_1988-quartic]).” $\spadesuit$ p.566: “It should also be noted that there is $M$-curve of degree $8$, whose constructions does not satisfy conditions of Viro method and is based on Theorem 4 \[22\](=Shustin 1987 [@Shustin_1987/87-a-new-M-curve-of-deg-8]).”\] 

E.I. Shustin, [*On manifolds of singular algebraic curves*]{}, Selecta Math. Soviet. 10 (1991), 27–37; this is the English transl., of the Russian original dating back to 1983. \[$\spadesuit$\] 

R.J. Sibner, [*Uniformization of symmetric Riemann surfaces by Schottky groups*]{}, (Diss.) Trans. Amer. Math. Soc. 116 (1965), 79–85. 78 \[$\clubsuit$ new proofs of the Rückkehrschnitttheorem (retrosection theorem) and the Kreisnormierung=KNP via quasiconformal mappings techniques (Ahlfors-Bers)=Teichmüller modernized; as oft emphasized in our text (cf. Sec.\[sec:question\]) thiis might be the route through which one can hope to reprove the Ahlfors mapping via the original method of Klein (as cryptically asserted in Teichmüller 1941 [@Teichmueller_1941])\] 

R.J. Sibner, [*Symmetric Fuchsian groups*]{}, Amer. J. Math. 90 (1968), 1237–1259. \[$\spadesuit$\] 

R.J. Sibner, [*Remarks on the Koebe Kreisnormierungsproblem*]{}, Comment. Math. Helv. 43 (1968), 289–295. 78 \[$\clubsuit$ quasiconformal reduction of KNP: can every plane domain be deformed quasiconformally onto a circle domain? (still open today June 2012)\] 

R.J. Sibner, [*An elementary proof of a theorem concerning infinitely connected domains*]{}, Proc. Amer. Math. Soc. 37 (1973), 459–461. 78 \[$\spadesuit$ simplifies by circumventing the usage of quasi-conformal techniques (normal family proof instead) an earlier proof of the fact that any domain of infinite connectivity admits a conformally equivalent model bounded by analytic contours (Jordan curves) $\spadesuit$ as probably just a matter of nomenclature it is not perfectly clear (to the writer) if this is obtained for all domains (as stated e.g. in Grunsky’s review (1978) [@Grunsky_1978 p.196] of this work) or if the assertion is only established in the case of countably many boundary components (cf. the parenthetical proviso on p.459 of [*opera cit.*]{}) $\spadesuit$ of course the real dream of Koebe (Kreisnormierung) would be that all these Jordan contours are ultimately circles!\] 

, Ueber eine neue analytische Behandlungsweise der Brennpunkte. [*J. Reine Angew. Math. 64*]{} (1865), 175–182. \[$\spadesuit$\] 

L. Siebenmann, *The Osgood-Schoenflies theorem revisited*, Russian Math. Surveys [ 60]{} (2005), 645–672. See also the online version available in the Hopf archive: http://hopf.math.purdue.edu/cgi-bin/generate?/Siebenmann/Schoen-02Sept2005 (from which a number of the editors misprints have been removed.)\[$\spadesuit$ contains a brilliant historical discussion of the contribution due to the complex analytic community (Osgood, Carathéodory) upon the so-called Schoenflies theorem about the bounding disc property of plane Jordan curves\] 

C.L. Siegel, *Topics in Complex Function Theory*, Vols.I–III. John Wiley and Sons, Inc., New York, 1960, 1971, 1973. \[$\spadesuit$\] 

J.-C. Sikorav, [*Proof that every torus with one hole can be properly holomorphically embedded in ${\Bbb C}^2$*]{}, preprint, October 1997 (unpublished). \[$\spadesuit$ self-explanatory title, and see Černe-Forstnerič for an extension of Sikorav’s result\]$\bigstar$ 

R. Silhol, [*Real algebraic surfaces*]{}, Lecture Notes in Math. 1392, Springer-Verlag, 1989. \[$\spadesuit$\] 

R. Silhol, [*Compactifications of moduli spaces in real algebraic geometry*]{}, Invent. Math. (1992). \[$\spadesuit$ It is probably useful to begin this paper by explaining why an approach, specific to real algebraic geometry, is necessary for Moduli problems. We will only be concerned in this paper, with the moduli problems for curves and abelian varieties (but the remarks we ...\]$\bigstar$ 

R. Silhol, [*The Schottky problem for real genus 3 $M$-curves*]{}, Math. Z. 236 (2001), 841–881. \[$\spadesuit$\]$\bigstar$ 

R.R. Simha, [*The Carathéodory metric of the annulus*]{}, Proc. Amer. Math. Soc. 50 (1975), 162–166. 50, 78 \[$\spadesuit$ write down everything (Ahlfors function, Carathéodory metric) in the case of an annulus\] 

S.O. Sinanjan, [*Approximation by polynomials in the mean with respect to area*]{}, Mat. Sbornik 82 (1970); English transl.: Math USSR Sbornik 11 (1970), 411–421. \[$\spadesuit$ p.416: “Let $\phi(z)$ be an Ahlfors $p$-function of the set $E$: $\gamma_p(E,\phi)=\gamma_p(E)$, $\phi \in A^p_E$. Such a function exists due to the compactness of the set $A^p_E$.” $\spadesuit$ p.420 one further occurrence of the Ahlfors function\] 

D. Singerman, [*Automorphisms of compact non-orientable Riemann surfaces*]{}, Proc. London Math. Soc. (3) 10 (1969), 376–394. 78 \[$\spadesuit$ “Using the definition of a Riemann surface, as given for example by Ahlfors-Sario, one can prove that all Riemann surfaces are orientable. However by modifying their definition one can obtain structures on non-orientable surfaces. In fact non-orientable Riemann surfaces have been considered by Klein and Teichmüller amongst others. The problem we consider here is to look for the largest possible groups of automorphisms of compact non-orientable Riemann surfaces and we find that this throws light on the corresponding problem for orientable Riemann surfaces, which was first considered by Hurwitz \[1\](=1893 [@Hurwitz_1893-U-algebr-Gebilde-m-eind-Transf-in-sich]). He showed that the order of a group of automorphisms of compact orientable Riemann surface of genus $g$ cannot be bigger than $84(g-1)$. This bound he knew to be attained because Klein had exhibited a surface of genus $3$ which admitted $PSL(2,7)$ as its automorphism group, and the order of $PSL(2,7)$ is $168=84(3-1)$. More recently Macbeath \[5,3\](=1967 [@Macbeath_1967],=1961 [@Macbeath_1961]) and Lehner and Newman \[2\](=1967 [@Lehner-Newman_1967]) have found infinite families of compact orientable surfaces for which the Hurwitz bound is attained, and in this paper we shall exhibit some new families.”\]$\bigstar$$\bigstar$$\bigstar$ 

D. Singerman, [*Mirrors on Riemann surfaces*]{}, Contemp. Math. 184 (1995), 411–417. \[$\spadesuit$\]$\bigstar$ 

V. Singh, [*An integral equation associated with the Szegö kernel function*]{}, Proc. London Math. Soc. (3) 10 (1960), 376–394. 78 \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$ 

E.P. Smith, [*The Garabedian function of an arbitrary compact set*]{}, Pacific J. Math. 51 (1974), 289–300. 78 \[$\spadesuit$ Ahlfors function is mentioned in its usual connection with the analytic capacity (p.289, 290) $\spadesuit$ Gamelin’s summary in 1973 [@Gamelin_1973-BAMS]: “Recently E. Smith \[43\](=the present entry Smith 1974 [@Smith_1974]) settled a problem left open by S.Ya. Havinson \[26\](=Havinson 1961/64 [@Havinson_1961/64]), proving that if domains $D_n$ with analytic boundaries increase to an arbitrary domain $D$, then the Garabedian functions of the $D_n$ converge normally. The limiting function depends only on $D$ and on the point $z_0$, and it is accordingly called the Garabedian function of $D$. In order to study the subadditivity problem for analytic capacity, Smith had been led to investigate the dependence of the Szegö kernel on certain perturbations of domains with analytic boundaries. The result on the Garabedian function dropped out as a special dividend. There is now the problem of simplifying Smith’s proof, and of freeing the result from Hilbert space considerations, in order to extend the theorem to more general extremal problems. [Added in proof]{}. A simple proof, which still depends on Hilbert space considerations, has been given by N. Suita 1973 (=[@Suita_1973])”\] 

A.Yu. Solynin, [*A note on equilibrium points of Green function*]{}, Proc. Amer. Math. Soc. 136 (2008), 1019–1021. \[$\spadesuit$ given a finitely connected planar domain, it is shown that there is a “universal” compactum (inside the domain) containing all critical points of the Green’s functions $G(z,t)$ whatever the location of the pole $t$ is (answering thereby a question of Sebbar-Falliero 2007 [@Sebbar-Falliero_2007]) $\clubsuit$ question (Gabard \[11.08.12\]): does this fantastic result extends to (compact) bordered surfaces $\spadesuit$ further there must be a minimum Solynin’s compactum $K$, what can be said about its shape, area, etc. $\spadesuit$ considering the example of a ring (annulus, say circular to simplify) it seems evident that upon dragging the pole around the hole the unique critical point of Green will rotate (being roughly located at the “antipode”), thus it seems that Solynin’s compactum will be a sub-annulus it this case $\spadesuit$ maybe in general the inclusion of $K$ into the domain is a homotopy equivalence\] 

A.J. Sommese, [*Real algebraic spaces*]{}, Ann. Scuola Norm. Sup. Pisa 4 (1977), 599–612. \[$\spadesuit$\] 

A. Speiser, [*Über symmetrische analytische Funktionen*]{}, Comment. Math. Helv. 16 (1944), 105–114. 60 \[$\spadesuit$ not symmetric in the “reality” sense of Felix Klein, so a priori no link with Ahlfors 1950 [@Ahlfors_1950]\] 

G. Springer, [*Introduction to Riemann surfaces*]{}, Addison-Wesley, Reading, Mass., 1957, 307 pp. \[$\spadesuit$ contains a discussion of the Schottky double, the Prüfer surface, etc.\] 

Ch.M. Stanton, [*The closed ideals in a function algebra*]{}, Trans. Amer. Math. Soc. 154 (1971), 289–300. 50 \[$\spadesuit$ a clear-cut application of the Ahlfors function (mapping) is given to a “bordered surface” extension of a “disc result” of Beurling (unpublished)—Rudin (1957) (telling that—in the function algebra $A(W)$ of functions analytic in the interior and continuous up to the boundary—[*every closed ideal is the closure of a principal ideal*]{}) $\spadesuit$ this extension was actually first derived by Voichick 1964 [@Voichick_1964], via the more complicated universal covering whose uniformizing map presents rather complicated boundary behavior $\spadesuit$ p.293, as Royden’s student, the author points out that Ahlfors result is re-proved in Royden 1962 [@Royden_1962] (this is not an isolated attitude, cf. Sec.\[dissident:sec\] for an exhaustive list) $\spadesuit$ p.289, the author remarks that similar use of Ahlfors’ theorem was initiated by Alling 1965 [@Alling_1965] and Stout (in the corona realm) $\spadesuit$ naive question of the writer \[08.08.12\]: is it reasonable to expect that the same Ahlfors-Alling lifting procedure conducts to an extension of Fatou’s theorem about existence of radial limits a.e. from the disc to a bordered surface: the notion of radiality is simple to define (orthogonality to the boundary), yet a function on the bordered surface does not descend to one on the disc via the Ahlfors branched covering (thus rather a method of localization is required, and the problem is surely well treated by several authors (e.g. Heins, Voichick 1964, Gamelin (localization of the corona), etc.) UPDATE \[12.09.12\]: see also Alling 1966 [@Alling_1966 p.345], who claims that Fatou is trivial to extend upon appealing to the Ahlfors map $\spadesuit$ \[08.08.12\] in the same vein it should be noted that the Ahlfors function shows some weakness for instance in the problem of solving the Dirichlet problem which in the disc-case can be cracked via the Poisson formula (H.A. Schwarz’s coinage) and one could hope to lift the solution to the bordered surface via the Ahlfors map. Alas, for given boundary values along the contours of the bordered surface there is no naturally defined procedure to descend the data along the boundary of the disc (implying a failure of the naive lifting trick). Consequently, the Dirichlet problem (for a bordered surface) lies somewhat deeper than the Ahlfors function, since it is probably well-known that the Ahlfors function may be derived from Dirichlet (or its close avatar the Green’s functions), see our Sec.\[Green:sec\] where we shall attempt to redirect to the first-hand sources implementations (Grunsky (planar case), Ahlfors, maybe Cecioni’s students, and Heins 1950 [@Heins_1950]).\] 

Ch.M. Stanton, [*Bounded analytic functions on a class of open Riemann surfaces*]{}, Pacific J. Math. 59 (1975), 557–565. \[$\spadesuit$ p.559 uses the terminology Myrberg surface for a concept closely allied to the Ahlfors function in the sense of our circle maps\] 

K. Stein, [*Topics on holomorphic correspondences*]{}, Rocky Mountain J. Math. 2 (1972), 443–463. \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited on p.457: “By a theorem of Ahlfors \[1\](=1950 [@Ahlfors_1950]) there is always a meromorphic function $\varphi\colon \widehat{R_0} \to
\overline{\Bbb C}$ \[from the double of a bordered surface to the sphere\] such that $R_0=\{ \xi \in \widehat{R_0} : \vert
\varphi(\xi)\vert <1$; hence $R_0$ is a distinguished polyhedral domain in $\widehat{R_0}$.”\] 

S. Stoïlow, [*Leçons sur les principes topologiques de la théorie des fonctions analytiques*]{}, Gauthier-Villars, Paris 1938. (Second edition in 1956,; Russian translation 1964.) \[$\spadesuit$ includes in particular a notion of “total Riemann covering”, defined by asking that any sequence tending to the boundary has an image tending to the boundary. This topological behaviour subsumes of course those of Ahlfors circle maps. $\spadesuit$ of course Stoïlow’s concept is also implicit in Radó 1922 [@Rado_1922-Z-Theorie-mehr], as one sees e.g. from Landau-Osserman’s account (1960 [@Landau-Osserman_1960]) $\spadesuit$ from Grunsky’s Review (JFM): “in die Definition der Mannigfaltigkeit wird dabei kein Abzählbarkeitsaxiom aufgenommen; es folgt der Beweis des Brouwerschen Satzes von der Invarianz des inneren Punktes nach [*Lebesgue-Sperner*]{}. \[$\dots$\] Zur Verdeutlichung dienen Beispiele nicht orientierbarer Fläche sowie ein von [*Prüfer*]{} stammendes Beispiel einer nicht triangulieren zweidimensionalen Mannigfaltigkeit (in der Formel Zeile 11 v.u. S.72 findet sich ein störender Druckfehler: \[$\dots$\]).” $\spadesuit$ In fact most relevant to our purpose (of the Ahlfors map) is Chap.VI of the book, which Grunsky (loc. cit.) summarizes as follows: “Ferner werden innere Abbildungen einer Riemannschen Fläche $R$ auf eine andere, $S$, betrachtet. Eine solche hei[ß]{}t eine totale Überdeckung von $S$ durch $R$, wenn jede Punktfolge aus $R$, die keine kompakte Teilfolge enthält (die “gegen den Rand strebt”) in eine ebensolche übergeht. Die Überdeckung ist dann auch vollständig, d.h. [*jeder*]{} Punkt von $S$ wird überdeckt, und au[ß]{}erdem auch jeder gleich oft.”\] 

S. Stoïlow, [*Sur les surfaces de Riemann normalement exhaustibles et sur le théorème des disques pour ces surfaces*]{}, Compositio Math. 7 (1940), 428–435. \[$\spadesuit$\] 

S. Stoïlow, [*Einiges über topologische Funktionentheorie auf nicht orientierbaren Flächen*]{}, Rev. Roumaine Math. Pures Appl. 19 (1974), 503–506. \[$\spadesuit$\] 

E.L. Stout, [*Bounded holomorphic functions on finite Riemann surfaces*]{}, Trans. Amer. Math. Soc. 120 (1965), 255–285. 50 \[$\spadesuit$ on p.263 (and 272), Ahlfors 1950 [@Ahlfors_1950] is quoted as follows (without precise bound): “In order to establish our result, we shall need to make use of a result of Ahlfors \[1\](=Ahlfors 1950 [@Ahlfors_1950]). For an alternative proof, one may consult Royden \[15\](=Royden 1962 [@Royden_1962]). Theorem 3.1 [*There exists a function $P$ holomorphic on a neighborhood of $\bar R$ which maps $R$ onto the open unit disc in an one-to-one manner for some $n$ and which satisfies $\vert P \vert =1$ on $\partial R$.*]{}” $\spadesuit$ first it is evident that “one-to-one” is a misprint that should be read as “$n$-to-one” $\spadesuit$ the paper addresses primarily the corona problem (overlapping with Alling 1964 [@Alling_1964]) and the allied interpolation, notably an extension of the celebrated results of Carleson and Newman on interpolation sets for the disc (i.e. those subsets enjoying the property that every bounded complex-valued function on $E$ can be extended to a bounded analytic function on the disc)\] 

E.L. Stout, [*On some algebras of analytic functions on finite open Riemann surfaces*]{}, Math. Z. 92 (1966), 366–379; with Corrections in: Math. Z. 95 (1967), 403–404. 50 \[$\spadesuit$ cite Ahlfors 1950 [@Ahlfors_1950] twice, on p.366: “Let $R$ be a finite open Riemann surface whose boundary $\Gamma$ consists of $N$ analytic, pairwise disjoint, simple closed curves. Let $\eta$ be an analytic mapping from $R$ onto $U$, the open unit disc which is holomorphic on a neighborhood of $\overline R$ and which is of modulus one on $\Gamma$. That such functions exists was first established by Ahlfors \[1\](=Ahlfors 1950 [@Ahlfors_1950]); another proof of their existence is in the paper \[12\](=Royden 1962 [@Royden_1962]).” Then on p.375: “Ahlfors \[1\] has shown that if $z_0, z_1$ are distinct points of $R$ (neither in $\Gamma$), then any solution of the extremal problem $\sup\{\vert f(z_0): f \textrm{ in } H_{\infty}[R], f(z_1)=0,
\|f \|\le 1\}$ is an inner function in $A[R]$. Thus inner functions separate points on $R$. …” $\spadesuit$ quoted by Fedorov, for using “inner function” as a synonym of “circle map”\] 

E.L. Stout, [*Interpolation on finite open Riemann surfaces*]{}, Proc. Amer. Math. Soc. 18 (1967), 274–278. 50 \[$\spadesuit$ p.274, Ahlfors 1950 is quoted as follows: “It is convenient to make use of an [*Ahlfors map*]{} for $R$, i.e., a function continuous on $\overline R$ and holomorphic in $R$ which is constantly of modulus one on $\Gamma$. The existence of such function was established by Ahlfors in \[1\](=Ahlfors 1950 [@Ahlfors_1950]); an alternative proof of their existence is in \[4\](=Royden 1962 [@Royden_1962])” $\spadesuit$ The Ahlfors map (and the machinery of uniformization) are again utilized to lift the characterization of interpolating sets for the disc (available from the celebrated results of Carleson, Newman, cf. also Hoffman 1962 [@Hoffman_1962]). The main theorem states that a subset $E\subset R$ of a finite open Riemann surface is an interpolating set for $R$ [*iff*]{} $\inf_{z\in E} d_R(z,
E)>0 $, where $d_R(z,E):=\sup \{ \vert f(z) \vert : f\in
H_{\infty}(R), f_{\vert E-\{z\}}=0, \|f\|_R \le 1 \}$. For convenience, recall that the subset $E$ is called an interpolation set for $R$ if every bounded complex-valued function on $E$ can be extended to a bounded analytic function on $R$.\] 

E.L. Stout, [*Inner functions, doubles and special analytic polyhedra*]{}, Amer. J. Math. 94 (1972), 343–365. 50 \[$\spadesuit$ p.345 credits Heins 1950 [@Heins_1950] for another (beside Ahlfors’ 1950 [@Ahlfors_1950]) eleg[*r*]{}ant \[sic\] construction of inner functions on compact bordered surfaces\] 

E. Study, W. Blaschke, [*Vorlesungen über ausgewählten Gegenstände der Geometrie*]{}, vol.2, Konforme Abbildung einfach zusammenhängender Bereiche, Teubner, Leipzig, 1912. \[$\spadesuit$ closely related to Carathéodory’s seminal study of the boundary behaviour of the Riemann map along an arbitrary Jordan curve and the more general theory of prime ends\] 

A. Stray, [*Approximation by analytic functions which are uniformly continuous on a subset of their domain of definition*]{}, Amer. J. Math. 99 (1977), 787–800. \[$\spadesuit$ p.797 brief apparition of the Ahlfors function via cross-reference to Gamelin 1969 [@Gamelin_1969]\] 

K. Strebel, [*Über das Kreisnormierungsproblem der konformen Abbildung*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 101 (1951), 22 pp. 60, 78 \[$\diamondsuit$ Kurt Strebel is a student of R. Nevanlinna (who teached frequently in Zürich)\] $\bigstar$ 

K. Strebel, [*Über die konforme Abbildung von Gebieten unendlich hohen Zusammenhangs, (I. Teil)*]{}, Comment. Math. Helv. 27 (1952), 101–127 78 \[$\clubsuit$ partial results on the Kreisnormierung in infinite connectivity\] 

K. Strebel, [*Ein Klassifizierungsproblem für Riemannsche Fläche vom Geschlecht 1* ]{}, Arch. Math. 48 (1987), 77–81. \[$\clubsuit$ p.77: “Herr K. Schüffler benötigt in seiner Arbeit \[2\] zur Theorie der Minimalflächen vom Geschlecht 1 den Satz, da[ß]{} [*jeder $p$-fach gelochte Torus auf einen ebensolchen mit kreisförmigen Löchern konform abgebildet werden kann, und da[ß]{} eine solche Abbildung durch diese geometrische Forderung im wesentlichen eindeutig bestimmt ist.*]{} Dabei wird der Torus durch die komplexe Ebene ${\Bbb C}$ modulo einer Translationsgruppe dargestellt, und die Kreisförmigkeit der Löcher ist ebenfalls in ${\Bbb C}$ gemeint.” $\spadesuit$ \[17.10.12\] one naturally wonders about higher genuses than one (where one must probably interpret the Kreisförmigkeit within the hyperbolic plane/disc), and it seems that such positive genus instances of the Kreisnormierung are also handled in Haas 1984 [@Haas_1984]\] 

V. Strehl, [*Minimal transitive products of transpositions–the reconstruction of a proof by A. Hurwitz*]{}, Sem. Lothar. Combinat. 37 (1996), Art.B37c, 12pp. \[$\spadesuit$ modern reconstruction of Hurwitz’s count of the number of Riemann surfaces having prescribed ramification, cf. also Ekedahl-Lando-Shapiro-Vainshtein 2001 [@Ekedahl-Lando-Shapiro-Vainshtein_2001]\] 

D.J. Struik, [*Outline of a history of differential geometry II*]{}, Isis 20 (1933), 161–191. \[$\spadesuit$ Gauss 1844 (and even F.T. Schubert) are credited for the nomenclature “conformal” as follows, p.164: “Of Gauss’ contribution to notation and nomenclature we mention the symbols $E,F,G,D,D',D''$ for what we now call the coefficients of the first and second fundamental differential form, and the word “conformal”. (6a)=footnote=(6a) In the first paper on higher geodesy, 1844: “ich werde daher dieselben conforme Abbildungen oder Übertragungen nennen, indem ich diesem sonst vagen Beiworte eine mathematisch scharf bestimmte Bedeutung beilege” \[Werke IV, p.262\]. The word is indeed, already used by F.T. Schubert, “De projectione sphaeroidis ellipticae geographica”, [*Nova Acta Petr.*]{}, p.130–146, see Cantor IV, p.575.”\] 

T. Sugawa, [*Unified approach to conformally invariant metrics on Riemann surfaces*]{}, Proc. of the Second ISAAC Congress, Vol.2 (Fukuoka, 1999), 1117–1127, Int. Soc. Anal. Appl. Comput., 8, Kluwer Acad. Publ., Dordrecht, 2000. \[$\spadesuit$ the Ahlfors function is mentioned on p.5: “The quantity $c_R(p)$ is sometimes called the analytic capacity. An extremal function $f\colon R \to \Bbb D$ satisfying $\vert df \vert(p)=c_R(p)$ is usually called the [*Ahlfors function*]{} at $p$ and known to be unique up to unimodular constants (see \[4\](=Fisher 1983 [@Fisher_1983])). We remark that the condition $c_R(p)=0$ at some point $p$ need not imply that $c_R(p)=0$ at every point $p$ in the case that $R$ is non-planar. A counterexample was constructed by Virtanen \[13\](=Virtanen 1952 [@Virtanen_1952]) (see also \[10,X.2K\]=Sario-Oikawa 1969 [@Sario-Oikawa_1969]).” $\spadesuit$ the article as whole present an unified framework to the interplay between conformally invariant metrics and extremal problems emphasizing the contractive property of holomorphic maps (à la Schwarz-Pick-Ahlfors) $\spadesuit$ more precisely several metrics are presented culminating to their comparison as $$a\;\le \;s  \!\!\!\!\buildrel{\rm AB50S69}\over{\le}\!\!\! c
\le
\begin{Bmatrix} \;\;\;\le \;\;r\buildrel{\rm Bu79}\over{\le} k \\
\buildrel{\rm HeSu72}\over{\le}\!\!\! b\;\;\; \le\;\, q
\end{Bmatrix}\le h,$$ where $a$ stands for Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950], $s$ for span (or Schiffer!), $c$ for Carathéodory(-Reiffen) (or for analytic capacity), $r$ for Robin (or logarithmic capacity), $k$ for Kobayashi, $b$ for Bergman, $q$ for quadratic differentials (Grötzsch-Teichmüller!), $h$ for Hahn $\spadesuit$ the inequality AB50S69 is due to Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950] for the planar case and in general to Sakai 1969/70 [@Sakai_1969] $\spadesuit$ inequality Bu79 is due to Burbea 1979 [@Burbea_1979-Schwarzian] $\spadesuit$ inequality HeSu72 is due to Hejhal 1972 [@Hejhal_1972-Memoirs-AMS p.106] (case of finite bordered surface) and Suita 1972 [@Suita_1972] in general\] 

T. Sugawa, [*An explicit bound for uniform perfectness of the Julia sets of rational maps*]{}, Math. Z. 238 (2001), 317–333. \[$\spadesuit$ the Ahlfors map is briefly mentioned as follows: “In fact, for a finitely connected planar domain $U$ whose boundary consists of Jordan curves, it is known that there exists a branched holomorphic covering map from $U$ onto the unit disk (e.g. the Ahlfors map). Thus $L_U$ cannot be estimated from below by only the data of $W$ (in this case $L_W=+\infty$).”\] 

N. Suita, [*Capacities and kernels on Riemann surfaces*]{}, Arch. Rat. Mech. Anal. 46 (1972), 212–217. \[$\spadesuit$\] 

N. Suita, [*On a metric induced by analytic capacity*]{}, Kodai Math. Sem. Report 25 (1973), 215–218. 78 \[$\spadesuit$ Ahlfors function à la Havinson 1961/64 [@Havinson_1961/64], i.e. for domains $D\notin O_{AB}$ (i.e. supporting nonconstant bounded analytic functions), analytic capacity and conformal metrics $\spadesuit$ the metric in question is also known as the Carathéodory metric (cf. e.g., Grunsky 1940 [@Grunsky_1940])\] 

N. Suita, [*On a class of analytic functions*]{}, Proc. Amer. Math. Soc. 43 (1974), 249–250. 78 \[$\spadesuit$ p.249, the Ahlfors function is discussed as follows: “If $\Omega \notin O_{AB}$ \[i.e. $\Omega$ is a plane region having a nonconstant bounded analytic function\], there exist the extremal functions $A(z)$ which maximize $\vert
f'(z_0)\vert$ in ${\frak B}_0$ \[the class of analytic functions $f$ such that $f(z_0)=0$ and $\vert f(z) \vert \le
1$\]. Those functions are called the [*Ahlfors functions*]{} which are unique save for rotations \[3\](=Havinson 1961/64 [@Havinson_1961/64]).” $\spadesuit$ the note includes a counterexample to an (erroneous) claim made by Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950] about the compactness of the class ${\frak E}_0$ of those analytic functions in a plane region $\Omega \notin O_{AB}$ vanishing at $z_0\in \Omega$ and such that $1/f$ omits a set of of values of area $\ge \pi$\] 

N. Suita, [*On a metric induced by analytic capacity, II*]{}, Kodai Math. Sem. Report 27 (1976), 159–162. 47 \[$\spadesuit$ the Ahlfors function appears on p.160 and 161 $\spadesuit$ for a plane region $\Omega\notin O_{AB}$ (i.e. supporting nonconstant bounded analytic functions) it was known (Suita 1973 [@Suita_1973] via “making use of a supporting metric due to Ahlfors 1938”) that the curvature $\kappa(\zeta)$ of the metric $ds_B=c_B(\zeta) \vert
d\zeta\vert$ induced by analytic capacity $c_B(\zeta)=\sup
\vert f'(\zeta)\vert$ in the class of functions bounded-by-one (=stretching factor of the Ahlfors function at $\zeta$) is $\le -4$ $\spadesuit$ the present article rederives this estimate ($\kappa \le -4$) by a limiting/exhaustion argument reducing to the case of a regularly bounded finitely connected domain which is analyzed via Bergman’s method of minimal integrals, but making also extensive use of Garabedian’s sharp analysis (our opinion!) $\spadesuit$ the novelty of the present article is that the ‘Bergman-Garabedian method’ gives the “more precise estimation $\kappa(\zeta)<-4$” for regions with more than one contour $\spadesuit$ paraphrase (p.161): “the equality $\kappa(\zeta)=-4$ at one point $\zeta\in
\Omega$ implies that $\Omega$ is conformally equivalent to the unit disc.” $\spadesuit$ \[23.09.12\] maybe it would be worth looking if Suita’s work extends to finite bordered surfaces (the problem being that quantity $\vert f'(\zeta)\vert$ depends on a local uniformizer), yet it seems that the theory is extensible (cf. e.g. Sugawa 1999/00 [@Sugawa_1999/00])\] 

N. Suita, A. Yamada, [*On the Lu Qi-keng conjecture*]{}, Proc. Amer. Math. Soc. 59 (1976), 222–224. \[$\spadesuit$ “We shall give a complete answer to the Lu Qi-keng conjecture for finite Riemann surfaces. Our result is that every finite Riemann surface which is not simply-connected is never a Lu Qi-keng domain, i.e. the Bergman kernel $K(z,t)$ of it has zeros for suitable $t$’s.”\] 

G. Szegö, [*Über orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene gehören*]{}, Math. Z. 9 (1921), 218–270 78 \[$\spadesuit$ Szegö kernel representation of the Riemann mapping (p.245) $\spadesuit$ like Bergmann 1922 [@Bergman_1922] or Bochner 1922 [@Bochner_1922] it is confessed (p.249) that the method does not duplicate a new existence proof of the Riemann mapping (this had to wait upon Garabedian and Lehto 1949 [@Lehto_1949]) $\spadesuit$ what is the geometric interpretation (i.e. the allied extremal problem): answer of course it is just that of minimizing the integral $\int_C \vert f(z) \vert^2 ds$, where integration is taken along the contour $C$ of the domain (and $ds$ is its Bogenelement)\] 

G. Szegö, [*Über die Randwerte einer analytischer Funktion*]{}, Math. Ann. 84 (1921), 232–244 \[$\spadesuit$\] 

G. Szegö, [*Verallgemeinerung des ersten Bieberbachschen Flächensatzes auf mehrfach zusammenhängende Gebiete*]{}, Sitz.-Ber. Preu[ß]{}. Akad. d. Wiss., math.-phys. Kl. (1928), 477–481 78 \[$\spadesuit$ can we do the same on a Riemann surface? and relate this to a Bergman-style proof of the Ahlfors map?\] $\bigstar$$\bigstar$$\bigstar$ 

G. Szegö, [*Inequalities for certain eigenvalues of a membrane of given area*]{}, J. Rat. Mech. Anal. 3 (1954), 343–356 \[$\spadesuit$ one of the early implementation of the conformal transplantation method to vibratory/elasticity problem; for wide extensions cf. Hersch 1970 [@Hersch_1970], Yang-Yau 1980 [@Yang-Yau_1980] and Fraser-Schoen 2011 [@Fraser-Schoen_2011], the last article effecting the junction with the Ahlfors map\] 

J. Tagamlizki, [*Zum allgemeinen Kreisnormierungsprinzip der konformen Abbildung*]{}, Ber. Verhandl. Sächs. Akad. Wiss., math.-phys. Kl. 95 (1943), 111–132. 78 $\bigstar$ 

M. Taniguchi, [*Bell’s result on, and representations of finitely connected planar domains*]{}, Some Japanese fonts 1352 (2004), 47–53. \[$\spadesuit$ survey of several results of Bell on the Ahlfors function and concludes by some questions about Bell representations, i.e. a certain family of canonical domains admitting an evident proper holomorphic map to the disc\] $\bigstar$ 

T.J. Tegtmeyer, A.D. Thomas, [*The Ahlfors map and Szegö kernel for an annulus*]{}, Rocky Mountain J. Math. 29 (1999), 709–723. \[$\spadesuit$ contains some lovely pictures of Ahlfors function in the case of an annulus\] 

O. Teichmüller, [*Eine Verschärfung des Dreikreisesatzes*]{}, Deutsche Math. 4 (1939), 16–22. 78 \[$\spadesuit$ quoted (joint with Carlson 1938 [@Carlson_1938]) in Grunsky 1940 [@Grunsky_1940] as a forerunner of the extremal problem for bounded analytic functions $\diamondsuit$ Oswald Teichmüller (1913–1943) is formally a student of Hasse, but his interest shifted to function theory (presumably due to lectures held in Göttingen ca. 1935 by R. Nevanlinna) and then joined ca. 1937 Berlin where Bieberbach was located\] 

O. Teichmüller, [*Extremale quasikonforme Abbildungen und quadratische Differentiale*]{}, Abh. Peu[ß]{}. Akad. Wiss. math.-naturw. Kl. 22 (1939), 1–197; also in the Collected Papers, 335–531. 60, 78 \[$\spadesuit$ discusses in details the Klein dictionary between symmetric surfaces and bordered Riemann surfaces through the [*Verdoppelung*]{} (=Schottky-Klein double) $\spadesuit$ discusses moduli in a way quite anticipated in Klein 1882 [@Klein_1882], modulo of course the usual Riemann-style heuristics\] 

O. Teichmüller, [*Über Extremalprobleme der konformen Geometrie*]{}, Deutsche Math. 6 (1941), 50–77; also in Collected Papers, 554–581. 60, 78 \[$\clubsuit$ a mention is made (without proof and a cryptical unreferenced allusion to Klein) of a statement which could be interpreted as a forerunner of the Ahlfors circle map $\spadesuit$ despite long searches, the writer (Gabard) was unable—on the basis of printed evidence—to adhere conclusively to Teichmüller’s accreditation of the result to Klein, compare Sec.\[sec:Teichmueller\] for more tergiversations $\spadesuit$ the original Teichmüller text reads as follows (p.554–5): “Wir beschäftigen uns nur mit [**orientierten endlichen Riemannschen Mannigfaltigkeiten.**]{} Diese können als Gebiete auf geschlossenen orientierten Riemannschen Flächen erklärt werden, die von endlich vielen geschlossenen, stückweise analytischen Kurven begrenzt werden. Sie sind entweder geschlossen, also selbst geschlossene orientierte Riemannsche Flächen, die man sich endlichvielblättrig über eine $z$-Kugel ausgebreitet vorstellen darf, oder berandet. Im letzteren Falle, kann man sie nach Klein[^147] durch konforme Abbildung auf folgende Normalform bringen: ein endlichvielblättriges Flächenstück über der oberen $z$-Halbebene mit endlich vielen Windungspunkten, das durch Spiegelung an der reellen Achse eine symmetrische geschlossene Riemannsche Fläche ergibt; \[…\] —(So lä[ß]{}t sich z.B. jedes Ringgebiet, d.h. jede schlichtartige endliche Riemannsche Mannigfaltigkeit mit zwei Randkurven, konform auf eine zweiblättrige Überlagerung der oberen Halbebene mit zwei Verzweigungspunkte abbilden.)” $\spadesuit$ Another puzzle would be to know if Teichmüller’s text exerted some influence over Ahlfors subsequent findings (1950 [@Ahlfors_1950]). Possibly yes, but note the absence of cross-citation until Ahlfors-Sario 1960 [@Ahlfors-Sario_1960]. All this should by no mean palish the originality of Ahlfors achievement which looks substantially sharper by controlling the mapping degree.\] 

O. Teichmüller, [*Beweis der analytischen Abhängigkeit des konformen Moduls einer analytischen Ringflächenschar von den Parametern*]{}, Deutsche Math. 7 (1944), 309–336; also in Collected Papers, ??–??. \[$\spadesuit$ quoted in Rüedy 1971 [@Ruedy_1971] as the technological forerunner of the Garsia embedding result\] 

O. Teichmüller, [*Gesammelte Abhandlungen, Collected Papers*]{}, Herausgegeben von L.V. Ahlfors und F.W. Gehring, Springer Verlag, Berlin, 1982. 

R. Thom, [*Sur l’homologie des variétés algébriques réelles*]{}, in: Differential and Combinatorial Topology, Symposium in honor of Marston Morse, Princeton Univ. Press, Princeton, N.J., 1965, 255–265. \[$\spadesuit$ cf. also a related work by J. Milnor 1964 [@Milnor_1964]\]

W. Thomson (later Lord Kelvin), [*Sur une équation aux différences partielles qui se présente dans plusieurs questions de physique mathématique*]{}, J. Math. Pures Appl. 12 (1847), 493–496. \[$\spadesuit$ one of the early apparition of the Dirichlet principle, cf. also Green 1928 [@Green_1828], Gauss 1839 [@Gauss_1839], Kirchhoff 1850 [@Kirchhoff_1850], Riemann 1851–57–57 [@Riemann_1851; @Riemann_1857; @Riemann_1857-DP] and Dirichlet as edited by Grube 1876 [@Dirichlet_1840-1876]\]

W. Thurston, [*The geometry and topology of $3$-manifolds*]{}, Princeton University Notes, Princeton, N.J., 1979. \[$\spadesuit$ circle packing theorem, cf. precise citations e.g. in He 1990 [@He_1990], i.e. especially Corollary 13.6.2 and Theorem 13.7.1 (circle packing theorem)\] 

H. Tietz, [*Eine Normalform berandeter Riemannscher Flächen*]{}, Math. Ann. 129 (1955), 44–49. 50, 60, 78 \[$\clubsuit$ cite Ahlfors 1950 [@Ahlfors_1950] and Nehari 1950 [@Nehari_1950], then criticizes the arguments of the latter $\clubsuit$ seems to reprove a sort of circle map for bordered surfaces inspired by Ahlfors (but with the desideratum of schlichtness along the boundary), alas Tietz’s argument is criticized (and apparently destroyed) in Köditz-Timmann 1975 [@Koeditz-Timmann_1975] $\spadesuit$ Grunsky 1978 [@Grunsky_1978 p.198] also seems to approve the Köditz-Timmann critique for he cites the (present) paper Tietz 1955 [@Tietz_1955], but right after add the parenthetical proviso “(cf. \[266\])”, that is Köditz-Timmann $\spadesuit$ despite those defects the prose of the introduction is brilliant and worth quoting (especially as it emphasizes the historical rôle of Ahlfors 1950 [@Ahlfors_1950], note however that Tietz seems to neglect both the Italian works as well as the cryptical allusion in Teichmüller 1941 [@Teichmueller_1941]): “Die Existenz eindeutiger analytischer Funktionen auf Riemannschen Flächen[^148] bedeutet, da[ß]{} jede Klasse konformäquivalenter Riemannscher Flächen “realisiert” werden kann durch Überlagerungsflächen der Zahlenebene. Damit stellt sich die Frage nach besonders einfachen Realisierungen oder Normalformen[^149].—Das wichtigste Ergebnis zu dieser Frage ist der Riemannsche Abbildungssatz, der sie für einfach-zusammenhängende Riemannsche Flächen beantwortet[^150]. Einen Schritt weiter gehen die Schlitztheoreme, die von den topologischen Voraussetzungen des Riemannschen Abbildungssatzes nur die Schlichtartigkeit der Riemannschen Fläche beibehalten. Hierher gehört auch der Satz, da[ß]{} jede berandete schlichtartige Riemannsche Fläche einem mehrfach überdeckten Kreis mit geeigneten Verzweigungsschnitten, die den Rand nicht treffen, konformäquivalent ist[^151].—Die Frage nach kanonischen Riemannschen Flächen im Falle höheren Geschlechtes is erst in letzter Zeit von Herrn [Ahlfors]{} \[1\](=1950 [@Ahlfors_1950]) angeschnitten und von Herrn [Nehari]{} \[2\](=1950 [@Nehari_1950]) systematisch behandelt worden:—Herr [Ahlfors]{} zeigt, da[ß]{} jede berandete Riemannsche Fläche realisiert werden kann als mehrfach überdeckter Einheitskreis, während Herr [Nehari]{} die Schlitztheoreme auf diesen Fall überträgt[^152]. \[…\]—Es erscheint wünschenswert, eine Normalform für berandete Riemannschen Flächen zu besitzen, die—im Gegensatz zur [Ahlfors]{}schen—sicherstellt, da[ß]{} das Bild jeder einzelnen Randkurve schlicht über die Linie des Einheitskreises liegt. \[…\]” $\clubsuit$ Tietz concludes his paper (p.49) as follows: “Die selben Überlegungen, die zu unserem Abbildungssatz führten, ermöglichen auch einen neuen Existenzbeweis für die Ahlforsche Normalform, wiederum jedoch ohne eine Schranke für die Anzahl der benötigten Blätter zu ergeben.” so this would be another (weak) version of Ahlfors, alas it seems that Tietz’s arguments where the object of critics, cf. Köditz-Timmann 1975 [@Koeditz-Timmann_1975]\] 

H. Tietz, [*Zur Realisierung Riemannscher Flächen*]{}, Math. Ann. 128 (1955), 453–458. 60 \[$\spadesuit$ with corrections in the next entry [@Tietz_1955_Berechtigung]\] 

H. Tietz, [*Berechtigung der Arbeit “Zur Realisierung Riemannscher Flächen”*]{}, Math. Ann. 129 (1955), 453–458. 60

St. Timmann, [*Kompakte berandete Riemannsche Flächen*]{}, Diss. Hannover, 1969, 56 S. 78 \[$\spadesuit$ this entry is cited on the “critical” page 198 of Grunsky 1978 [@Grunsky_1978], according to which it gives a generalization to Riemann surfaces of the Bieberbach-Grunsky theorem (i.e. circle map in the planar case) $\spadesuit$ in particular, it could be the case that Timmann’s reproves the existence of an Ahlfors circle map, yet probably this is not the case\] $\bigstar$ 

X. Tolsa, [*Painlevé’s problem and the semiadditivity of analytic capacity*]{}, Acta Math. 190 (2003), 105–149. 47 \[$\spadesuit$ complete solutions of both problems of the title are given, the first being usually regarded as implicitly posed in Painlevé 1888 [@Painleve_1888] (albeit nobody was ever able to locate the precise place, see e.g. Rubel 1971 [@Rubel_1971] or Verdera 2004 [@Verdera_2004] for why) and the second emanated from Vitushkin’s advanced studies in the 1960’s $\spadesuit$ the introduction contain a historical sketch, from Ahlfors 1947 [@Ahlfors_1947], Vitushkin 1950’s to Murai 1988 [@Murai_1987], Melnikov 1995 [@Melnikov_1995] (curvature of measures), G. David 1998 [@David_1998] (solution of Vitushkin’s conjecture), etc.\] 

G. Toumarkine, S. Havinson, [*Propriétés qualitatives des solutions des problèmes extrémaux de certains types*]{}, In: Fonctions d’une variable complexe. Problèmes contemporains. Paris 1962, p.73. \[$\spadesuit$ survey containing a quite complete bibliography\] 

S. Treil, [*Estimates in the corona theorem and ideals of $H^{\infty}$: a problem of T. Wolff*]{}, J. Anal. Math. 87 (2002), 481–495. \[$\spadesuit$ improved lower estimates for the solution of the corona problem, but with still a large gap up the upper bound of Uchiyama 1980 (cf. esp. p.494)\] 

C.L. Tretkoff, M.D. Tretkoff, [*Combinatorial group theory, Riemann surfaces and differential equations*]{}, In: [*Contribution to Group Theory*]{}, Contemp. Math. 33, 467–519. Amer. Math. Soc., Providence, 1984. \[$\spadesuit$\]$\bigstar$ 

A. Tromba, [*On Plateau’s problem for minimal surfaces of higher genus in ${\Bbb R}^n$*]{}, SFB 72-Preprint 580, Bonn, 1983. \[$\spadesuit$ doubts expressed about the validity of Douglas and Courant for the Plateau problem in the case of higher topological structure, compare Jost 1985 [@Jost_1985]\] $\bigstar$$\bigstar$$\bigstar$ 

A. Tromba, [*Dirichlet’s energy on Teichmüller’s moduli space and the Nielsen realization problem*]{}, Math. Z. 222 (1996), 451–464. \[$\spadesuit$\] $\bigstar$$\bigstar$ 

V.V. Tsanov, [*On hyperelliptic Riemann surfaces and doubly generated function algebras*]{}, C.R. Acad. Bulgare Sci. 31 (1978), 1249–1252. \[$\spadesuit$ quoted in Černe-Forstnerič 2002 [@Cerne-Forstneric_2002]\]$\bigstar$$\bigstar$ 

M. Tsuji, [*A simple proof of Bieberbach-Grunsky’s theorem*]{}, Comment. math. Univ. St. Paul 4 (1956), 29–32. 78 \[$\spadesuit$ Nehari’s review (in MR): “A new proof of the classical result that there exists a $(1,n)$ conformal mapping of a plane domain $D$ of connectivity $n$ onto the unit circle which carries a given point on each of the boundary components of $D$ into the same point of the unit circumference.”\] $\bigstar$ 

M. Tsuji, [*Potential theory in modern function theory*]{}, Tokyo, Maruzen, 1959. (Chelsea edition 1975.) $\bigstar$ 78 \[$\spadesuit$ contains apparently yet another proof of the Bieberbach-Grunsky theorem, perhaps the same as in the previous item\]$\bigstar$ 

A.W. Tucker, [*Branched and folded coverings*]{}, Bull. Amer. Math. Soc. 42 (1936), 859–862. \[$\spadesuit$\]$\bigstar$ 

G. Tumarkin, see Toumarkine.

N.X. Uy, [*On Riesz transforms of bounded functions of compact support*]{}, Michigan Math. J. 24 (1977), 169–175. \[$\spadesuit$ p.170 the Ahlfors function (referenced via Gamelin’s book 1969 [@Gamelin_1969]) is involved in a theorem involving the Riesz transform\] 

L. Vajsburd, A. Radul, [*Non-orientable strings*]{}, Comm. Math. Phys. 135 (1991), 413–420. \[$\spadesuit$ real algebraic (diasymmetric) curves as applied to string theory, more related refs. in Natanzon 1999 [@Natanzon_1999-Moduli-real-alg-surf.superanal-differ-spinors]\] 

Ch. de la Vallée Poussin, [*Sur la représentation conforme des aires multiplement connexes*]{}, Ann. École Norm. (3) 47 (1930), 267–309 78 

J. Verdera, [*Removability, capacity and approximation*]{}, in: Complex Potential Theory, NATO ASI Series, Kluwer Acad. Publ., Dordrecht, 1994, 419–473. \[$\spadesuit$\] 

J. Verdera, [*The $L^2$ boundedness of the Cauchy integral and Menger curvature*]{}, Contemp. Math. 277 (2001), 139–158. \[$\spadesuit$\] 

J. Verdera, [*Ensembles effaçables, ensembles invisibles et le problème du voyageur de commerce, ou comment l’analyse réelle aide l’analyse complexe*]{}, Gazette des Math. 101 (2004), 21–49 47 \[$\spadesuit$ a thorough survey about Painlevé null-sets including the following points: $\spadesuit$ Painlevé’s problem about searching a geometric characterization of null-sets (nobody ever found an explicit formulation in Painlevé’s writings, but Ahlfors 1947 [@Ahlfors_1947] may be considered as the father of the modern era (introduction of the analytic capacity and insistance upon pure geometric conditions) $\spadesuit$ Tolsa’s resolution (ca. February 2003) of Painlevé’s problem (via bilipchitzian invariance of analytic capacity) is mentioned $\spadesuit$ p.29: the Denjoy conjecture (i.e., a compactum of a rectifiable curve is a (Painlevé) null-set iff its length is zero). This conjecture was cracked by the seminal work of Calderón 1977 [@Calderon_1977] as was made explicit in a note of Marshall $\spadesuit$ the (Vitushkin)-Garnett 1970 [@Garnett_1970] example of the $1/4$-Cantor set is discussed: this has positive length (because a certain projection is a full segment) but is a null-set (removable) $\spadesuit$ this is used to motivate Besicovitch’s notion of “invisible sets”, i.e. those projecting to sets of zero-length along almost every angular direction $\spadesuit$ Vitushkin’s conjecture: a compactum of the plane is a null-set iff it is invisible (alas, there is counter-examples of Mattila, and Jones-Murai 1988 [@Jones-Murai_1988]), yet the direct sense is true if finite length (as follows from the Denjoy conjecture solved since Calderón), hence $\spadesuit$ weak Vitushkin conjecture (1967): among compacta of finite length, the null-sets coincide with the invisible sets. This was completed in G.David 1998 [@David_1998] upon combining a chain of contributions: Christ 1990, Mattila-Melnikov-Verdera 1996 [@Mattila-Melnikov-Verdera_1996] and Jones 1990\] 

I.N. Vekua, [*Generalized analytic functions*]{}, Pergamon Press, Oxford, 1962. \[$\spadesuit$ an account of the theory of the Beltrami equation, with roots going back to Gauss, Korn, Lichtenstein, Morrey, Lavrentiev, Bojarski, Lehto, Ahlfors and Bers, etc.\] 

I.N. Vekua, [*Verallgemeinerte analytische Funktionen*]{}, Berlin 1963 \[$\spadesuit$ Riemann-Hilbert problem on finite bordered Riemann surfaces (and the allied Fredholm theory), cf. also Koppelman 1959 [@Koppelman_1959] Schüffler 1986 [@Schueffler_1986]\] 

H. Villat, [*Le problème de Dirichlet dans une aire annulaire*]{}, Rend. Circ. Mat. Palermo 33 (1912), 149 \[$\spadesuit$ a brief proof of Villat’s formula in Komatu (1945)\] 

V. Vinnikov, [*Self-adjoint determinantal representations of real plane curves*]{}, Math. Ann. 296 (1993), 453–479. \[$\spadesuit$ a brilliant presentation of the theory of Klein-Weichold of real curves and simplified proof of results of Dubrovin-Natanzon, discuses complex orientations (à la Rohlin) $\spadesuit$ mentions the result that a real plane curve with a nest of maximal depth is dividing (via Rohlin 1978 [@Rohlin_1978 p.93]), whose argument can (in our opinion) can be slightly simplified as follows $\spadesuit$ given $C_m\subset {\Bbb P}^2$ a nonsingular curve of degree $m$ with a deep nest then projecting the curve from any point chosen in the innermost oval gives a morphism $C_m \to {\Bbb
P}^1$ whose fibers over real points are totally real, hence there is an induced map between the imaginary loci and it follows that $C_m$ is dividing (just by using the fact that the image of a connected set is connected). q.e.d. (N.B.: this is exactly Rohlin’s argument except that we avoid the consideration of the canonical fibering ${\rm pr}\colon {\Bbb
C} P^2-{\Bbb R} P^2 \to S^2$ envisaged by Rohlin) $\spadesuit$ p.478 mentions the result of Nuij 1968 [@Nuij_1968]: “any two real smooth plane curves of degree $n$ having a nest of of ovals of maximal depth are [*rigidly isotopic*]{} (i.e. belongs to the same component in the space of all real smooth plane curves of degree $n$)” $\spadesuit$ \[30.09.12\] I vaguely remember of a sharper question (result?) asking if the space of deeply nested curves is not even a (contractible) cell $\spadesuit$ \[02.10.12\] probably this question was rather asked for ovalless real curves, yet the idea (coming to me only today) is that the $\pi_1$ (fundamental group) of any chamber (=component of the complement of the discriminant hypersurface $D\subset \vert {\cal O}_{{\Bbb P}^2}(m) \vert=\vert mH\vert$ consisting of all singular curves) must act on the set of ovals of any fixed plane curves. Hence when there is no oval or a nest (not necessarily of maximal depth) then the induced (monodromy) permutation must be trivial and consequently there is no obstruction to the chamber having a simple topology. More generally this applies when there are several nests of different depths (then again nothing can be permuted). In contrast when there is collection of non-nested ovals (or two nests of the same depth) then there is no obstruction to there permutability (e.g. imagine a quartic with $4$ ovals resulting from the smoothing of two conics then by rotating the plane we can achieve a transitive permutation of cyclic type). But probably the monodromy group of this quartic is bigger. How large exactly? $\spadesuit$ a problem would be to count the number of component of $\vert mH\vert-D$ and if possible to describe the complex encoding the adjacency relation between the different chambers $\spadesuit$ of course in the general question of describing the monodromy of a given curve, one can exploit Rohlin’s idea of the complex orientation in the case where the curve is dividing, as the latter must probably be conserved during an isotopy-loop (up to reversion). If so then for the 4 ovals quartic we get an obstruction to there complete permutability, and the monodromy group is not the full symmetric group ${\frak S}_4$. Naively two ovals gyrate clock-wise and two anti-clock-wise (draw the complex orientations by doing sense preserving smoothings), yet since ${\Bbb R}P^2$ is nonorientable nothing is secure (i.e. the clockwise can continuously mutate in the anti-clock-wise)? (of course all this must be described somewhere with more care!) $\spadesuit$ as in Nuij’s result one can ask when the real scheme (Rohlin’s jargon) determine unambiguously the isotopy type (or what is the same a unique chamber). A naive (probably wrong) guess is that if the monodromy is trivial, then the chamber is unique\] 

V. Vinnikov, [*Commuting operators and function theory on a Riemann surface*]{}, In: Holomorphic spaces (Berkeley 1995), MSRI Publications 33, 1998. 50 \[$\spadesuit$ p.468, Ahlfors 1950 is briefly cited as a mapping onto the upper half-plane, and is applied to problems of operator theory and maybe as well to a generalization of the Riesz-Nevanlinna-Smirnov factorization $\spadesuit$ compare optionally Havinson 1989/89[@Havinson_1989/89] where a similar desideratum was found to be difficult (and unsolved?) $\spadesuit$ from the abstract: “In the late 70’s M.S. Livsic has discovered that a pair of commuting nonselfadjoint operators in a Hilbert space, with finite nonhermitian ranks, satisfy a polynomial equation with constant (real) coefficients; …”, whence the link with real curves (à la Klein) and therefore with Ahlfors\] 

O.Ya. Viro, [*Construction of multicomponent real algebraic surfaces*]{}, Dokl. Akad. Nauk SSSR 248 (1979), 279–282; English transl., Soviet Math. Dokl. 20 (1979), 991–995. \[$\spadesuit$\] 

O.Ya. Viro, [*Construction of $M$-surfaces*]{}, Funkt. Anal. i Prilozhen. 13 (1979), 71–72; English Transl. in: ???. \[$\spadesuit$\] 

O.Ya. Viro, [*Curves of degree $7$, curves of degree $8$, and the Ragsdale conjecture*]{}, Dokl. Akad. Nauk SSSR 254 (1980), 1305–1310; English Transl., ???. \[$\spadesuit$\] 

O.Ya. Viro, [*Gluing of plane real algebraic curves and construction of curves of degree $6$ and $7$*]{}, in: Topology, Proc. Leningrad 1982, Lect. Notes in Math. 1060, 1984, 185–200. \[$\spadesuit$\] 

O. Viro, [*Progress over the last five years in the topology of real algebraic varieties*]{}, Proc. Internat. Congr. of Mathematicians, Warsaw 1983, 525–611. \[$\spadesuit$ a more expanded version of the same material in Viro 1986/86 [@Viro_1986/86-Progress]\] 

O. Viro, [*Real varieties with prescribed topological properties*]{}, Doct. Thesis, Leningrad Univ., 1983. \[$\spadesuit$ under the direction of V.A. Rohlin\] 

O.Ya. Viro, [*??*]{}, Izv. Akad. Nauk SSSR, Ser. Mat. 47 (1983), 1135–1150; English Transl., ???. \[$\spadesuit$ cited in Shustin 1985/85 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]\] 

O.Ya. Viro, [*Real plane curves of degree $7$ and $8$: new prohibitions*]{}, Izv. Akad. Nauk SSSR, Ser. Mat. 47 (1983), 1135–1150; English Transl., Math. USSR Izv. 23 (1984), 409–422. \[$\spadesuit$ cited in Shustin 1985/85 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin] for his counterexample to Rohlin’s maximality conjecture $\spadesuit$ another point of this paper is that it completes the isotopy classification of septics, thereby cracking the next case of Hilbert’s 16th problem. This is based on work of 1979 by Viro, where after a bunch of construction it remained him to prohibit the scheme $\langle J \sqcup 1 \langle 14 \rangle \rangle$. This was done using auxiliary curves of degree $2$ and the theory of complex orientations. The resulting classification involves $121=11^2$ real schemes (cf. e.g. Viro 1989/90 [@Viro_1989/90-Construction p.1124]). $\spadesuit$ \[24.01.13\] try at the occasion to draw the corresponding pyramid. Another idea try to prohibit Viro’s scheme $\langle J \sqcup 1
\langle 14 \rangle \rangle$ via CCC (collective contraction conjecture of empty ovals, cf. Sec.\[CCC:sec\]), yet looks difficult unless another idea appears $\spadesuit$ by Marin 1979 [@Marin_1979] (or Fiedler) we know that such schemes even when enriched by the type (or the stronger complex orientations) do not encodes unambigously the rigid-isotopy class, hence the rigid-isotopy classification is even much harder and probably still unsolved, compare e.g. Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical]. As a pure guessing (of Gabard) one could expect something like $512=2^9$ chambers???\] 

O.Ya. Viro, [*Gluing of algebraic hypersurfaces, smoothing of singularities and construction of curves*]{}, Proc. Leningrad Int. Topology Conf., 1983, 149–197; English Transl., ???. \[$\spadesuit$ cited in Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91]\] 

O. Viro, [*Progress in the topology of real algebraic varieties over the last six years*]{}, Uspekhi Mat. Nauk 41 (1986), 45–67; English Transl., Russian Math. Surveys 41 (1986), 55–82. \[$\spadesuit$ “Contents. Introduction 55 §1. Real algebraic curves as complex objects 57 §2. Numerical characteristics and encoding of schemes of curves 59 §3. Old restrictions on schemes of curves 60 §4. New restrictions on schemes of curves 63 §5. Klein’s assertion 67 §6. …”\] 

O.Ya. Viro, [*Real algebraic plane curves: constructions with controlled topology*]{}, Alg. i Analiz 1 (1989), 1–73; English transl., Leningrad Math. J. 1 (1990), 1059–1134. \[$\spadesuit$ includes a complete solution of Hilbert’s 16th problem for $M$-curves up to orders $m\le 7$\] 

O.Ya. Viro, [*Patchworking real algebraic varieties*]{}, preprint: http://www.math.uu.se/$\sim$oleg \[$\spadesuit$\] 

O.Ya. Viro, S.Yu. Orevkov, [*Congruence modulo $8$ for real algebraic curves of degree $9$*]{}, Uspekhi Mat. Nauk 56 (2001), 137–138; English transl., Russian Math. Surveys 56 (2001), 770–771. \[$\spadesuit$\] 

O.Ya. Viro, [*From the sixteenth Hilbert problem to tropical geometry*]{}, Japan. J. Math. 3 (2008), 185–214. \[$\spadesuit$ includes a complete solution of Hilbert’s 16th problem for $M$-curves up to orders $m\le 7$\] 

K.I. Virtanen, [*Über die Existenz von beschränkten harmonischen Funktionen auf offenen Riemannschen Flächen*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 75 (1950), 8 pp. 50, 60 \[cite Ahlfors 1950 [@Ahlfors_1950] in a footnote (p.6) as follows: “Zusatz b.d. Korr.: Die Extremalfunktion $\eta_n$ findet sich auch bei Ahlfors 1950 (=[@Ahlfors_1950]).” $\spadesuit$ yet this function is only a harmonic function, hence not the (analytic) Ahlfors map we are focused upon. In particular Virtanen’s paper does not reproves the existence of the Ahlfors maps, its main purpose being rather to establish the inclusion $O_{HB}\subset O_{HD}$ in the so-called classification theory of open Riemann surfaces\] 

K.I. Virtanen, [*Über Extremalfunktionen auf offenen Riemannschen Flächen*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 141 (1952), 7 pp. 60 \[Ahlfors 1950 [@Ahlfors_1950] is cited maybe?\] 

A.G. Vitushkin, [*Analytic capacity of sets and some of its properties*]{}, Dokl. Akad. Nauk SSSR 123 (1958), ?–?. (Russian) \[$\spadesuit$ cited in Melnikov 1967 [@Melnikov_1967] for the definition of the Ahlfors function\] 

A.G. Vitushkin, [*Example of a set of positive length but zero analytic capacity*]{}, Dokl. Akad. Nauk SSSR 127 (1959), 246–249. (Russian) \[$\spadesuit$ compare also the (simplified) construction in Garnett 1970 [@Garnett_1970], who warns us that Vitushkin’s paper contains many typographical errors $\spadesuit$ the basic implication “zero analytic capacity whenever zero linear measure” is a classical theorem of Painlevé (cf. e.g. Ahlfors 1947 [@Ahlfors_1947 p.2], a simple application of Cauchy’s formula)\] 

A.G. Vitushkin, [*Analytic capacity of sets in problems of approximation theory*]{}, Uspekhi Mat. Nauk 22 (1967), 141–199; English transl.: Russian Math. Surveys 22 (1967), 139–200. 47 \[$\spadesuit$ Ahlfors function appears on p.142 $\spadesuit$ formulation of the problem of the semi-additivity of analytic capacity solved (jointly with the older Painlevé problem on the geometric characterization of removable singularities) in Tolsa 2003 [@Tolsa_2003]\] 

Vo Dang Thao, [*Über einige Flächeninhaltsformeln bei schlichtkonformer Abbildung von Kreisbogenschlitzgebieten*]{}, Math. Nachr. 74 (1976), 253–261. \[$\spadesuit$ cited in Alenicyn 1981/82 [@Alenicyn_1981/82]\] $\bigstar$$\bigstar$$\bigstar$ 

M. Voichick, [*Ideals and invariant subspaces of analytic functions*]{}, Trans. Amer. Math. Soc. 111 (1964), 493–512. \[$\spadesuit$ bounded analytic functions, nontangential boundary values (almost everywhere), inner function, Beurling’s invariant subspace theorem extended to finite Riemann surfaces (tools: Harnack’s principle, Fatou’s theorem, plus Read 1958 [@Read_1958_Acta] and Royden 1962 [@Royden_1962] (both direct descendants of Ahlfors 1950 [@Ahlfors_1950]), but the link if any is masked behind “une propice brume d’analyse fonctionnelle”) $\spadesuit$ similar work by Hasumi 1966 [@Hasumi_1966] $\spadesuit$ Voichick’s work also contains a “bordered” extension of the Beurling-Rudin description of closed ideals in the disc algebra, for which result Stanton 1971 [@Stanton_1971] proposes another route hinging on the use of the Ahlfors map\] 

M. Voichick, L. Zalcman, [*Inner and outer functions on Riemann surfaces*]{}, Proc. Amer. Math. Soc. 16 (1965), 1200–1204. \[$\spadesuit$ factorization theory in the Hardy classes $H^p$ for finite bordered Riemann surfaces extending the classical theory (Hardy and the Riesz brothers) on the disc (antecedent by Parreau, Rudin 1955 [@Rudin_1955-class-Hp], and Royden 1962 [@Royden_1962]), inner function, Blaschke product, Green’s function, etc. $\spadesuit$ naively speaking one could hope that the Ahlfors function alone is a sufficient tool to lift the truth from the disc to the bordered surface, yet the implementation usually diverge slightly (here by using the universal covering to effect the reduction to the classical disc case)\] 

M. Voichick, [*Extreme points of bounded analytic functions of infinitely connected regions*]{}, Proc. Amer. Math. Soc. 11 (1966), 83–86. 50, 78 \[$\spadesuit$ p.1369, cite Ahlfors 1950 [@Ahlfors_1950] for the existence of a negative harmonic function whose harmonic conjugate has prescribed periods $\spadesuit$ this page contains an acrobatical implementation of the usual yoga attempting to annihilate periods to ensure single-valuedness (hence quite close to Ahlfors’ existence-proof of a circle map) $\spadesuit$ p.1367: “It should be noted that Gamelin in \[2\](=to appear=and seems to have appeared under extended coauthoring, namely Gamelin-Voichick 1968 [@Gamelin-Voichick_1968]) characterized the extreme points of the unit ball of $H^{\infty}(R)$ when $R$ is a finite bordered Riemann surface.”\] 

M. Voichick, [*Invariant subspaces on Riemann surfaces*]{}, Canad. J. Math. 18 (1966), 399–403. \[$\spadesuit$\]$\bigstar$$\bigstar$ 

V. Volterra, [*Sul Principo di Dirichlet*]{}, Palermo Rend. 11 (1897), 83–86. 60 \[$\spadesuit$\] 

B.L. van der Waerden, [*Topologie und Uniformisierung der Riemannschen Flächen*]{}, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl. 93 (1941), 147–160. 60 \[$\spadesuit$ cf. also Carathéodory 1950 [@Caratheodory_1950] and ref. therein, esp. to Reichardt.\] $\bigstar$ 

B.L. van der Waerden, [*Einführung in die algebraische Geometrie*]{}, Die Grundlehren der math. Wiss. in Einzeldarstellungen, Bd.51, Springer-Verlag, Berlin, 1973. (Zweite Auflage of the 1939 original). \[$\spadesuit$ p.223, Riemann-Roch in der Brill-Noetherschen Fassung, etc.\] 

W. von Dyck, [*Beiträge zur Analysis situs. I Aufsatz, Ein- und zwidimensionale Mannigfaltigkeiten*]{}, Math. Ann. (1888), 457–512. \[$\spadesuit$ contains an account of what was known at that time (nearly definitive results) on the topology of surfaces, as well a historical account of the theory of foliation. The sole possible forerunner of that period is Poincaré 1885, which in our opinion (albeit not perfectly organized) is sometimes more digest than Dyck’s account, especially when it comes to the “Poincaré” index formula, which can perhaps only be regarded as anticipated by masters like Cauchy, Gauss, Riemann, Kronecker\] 

?. von Staudt, [*Geometrie der Lage*]{}, ??. \[$\spadesuit$ often cited by early worker in the topology of real curves, for the notion of ovals and pseudo-line, i.e. isotopy classification of a circle in the real projective plane $\spadesuit$ so cited e.g. in Harnack 1876 [@Harnack_1876], Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege]\] 

J. Wahl, [*Deformations of plane curves with nodes and cusps*]{}, Amer. J. Math. 96 (1974), 529–577. \[$\spadesuit$ cited e.g. in Shustin 1990/91 [@Shustin_1990/91-Geom-of-discr-alg-curve]\] 

R.J. Walker, [*Algebraic Curves*]{}. Dover Publications, Inc., New York, 1962; unabridged and corrected reprint of the work first published as Princeton Mathematical Series [*13*]{}. Princeton University Press, Princeton, N.J., 1950.\[$\spadesuit$ often cited e.g. by Gudkov\] 

J.L. Walsh, [*The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions*]{}, Bull. Amer. Math. Soc. 35 (1929), 499–544. \[$\spadesuit$ quoted via Axler’s review (BAMS) of Fisher’s book, for the harmonic conjugate as generally multiple-valued with periods\]$\bigstar$$\bigstar$$\bigstar$ 

J.L. Walsh, [*Interpolation and functions analytic interior to the unit circle*]{}, Trans. Amer. Math. Soc. 34 (1932), 523–556. \[$\spadesuit$ Pick-Nevanlinna like still in the disc but see Heins 1975 [@Heins_1975] for an extension subsuming (in principle) the theory of the Ahlfors map\]$\bigstar$ 

J.L. Walsh, [*Approximation by polynomials in the complex domain*]{}, Mémorial des Sci. Math. 73 (1935), 1–72. \[$\spadesuit$ formulates a general formalism of best approximation which encloses as special cases the least area interpretation of the Riemann mapping of Bieberbach 1914 [@Bieberbach_1914], as well as generalizations of Julia, and many other workers including Kubota, Wirtinger, Kakey, F. Riesz (cf. esp. p.61) $\spadesuit$ further on p.64 it is emphasized that (at time) virtually nothing was known for multiply-connected regions (this had to wait over Grunsky, Ahlfors, etc.)\] 

J.L. Walsh, [*On the shape of level curves of Green’s function*]{}, Amer. Math. Monthly 44 (1937), 202–213. \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$ 

J.L. Walsh, [*The critical points of linear combinations of harmonic functions*]{}, Bull. Amer. Math. Soc. ?? (1948), 196–205. 47 \[$\spadesuit$ p.196: “In various extremal problems of function theory the critical points of linear combinations of Green’s functions and harmonic measures are of significance (See for instance M. Schiffer 1946; L.V. Ahlfors 1947 [@Ahlfors_1947].) $\spadesuit$ p.205: “In connection with the methods we are using, a remark due to Bôcher (1904) is appropriate: “The proofs of the theorems which we have here deduced from mechanical intuition can readily be thrown, without essentially modifying their character, into purely algebraic form. The mechanical problem must nevertheless be regarded as valuable, for it suggests not only the theorems but also the method of proof.” ”\] 

J.L. Walsh, [*The location of critical points*]{}, Amer. Math. Soc. Colloq. Publ. 34, 1950. \[$\spadesuit$ Chap.VII is quoted in Jones-Marshall 1985 [@Jones-Marshall_1985] “for more information on the location of the critical points” \[of the Green’s function\]\] 

J.L. Walsh, [*Note on least-square approximation to an analytic function by polynomials, as measured by a surface integral*]{}, Proc. Amer. Math. Soc. 10 (1959), 273–279. 

J.L. Walsh, [*History of the Riemann mapping theorem*]{}, Amer. Math. Monthly 80 (1973), 270–276. \[$\spadesuit$ a brilliant essay, which on p.273 mentions briefly the counterexamples to the “naive” Dirichlet principle cooked by Prym 1871 and Hadamard 1906 (the precise links are not given but are Prym 1871 [@Prym_1871] and Hadamard 1906 [@Hadamard_1906])\] 

S. Warschawsky, [*Über einige Konvergenzsätze aus der Theorie der konformen Abbildung*]{}, Nachr. Ges. Wiss. Göttingen (1930), 344–369. 60 $\bigstar$ 

H. Weber, [*Note zu\[m\] Riemann’s Beweis des Dirichlet’schen Prinzips*]{}, J. Reine Angew. Math. 71 (1870), 29–39. 60 \[$\spadesuit$ an attempt is made to complete the reasoning of Riemann to establish the Dirichlet principle $\spadesuit$ this work is quoted in Ahlfors-Sario’s masterpiece [@Ahlfors-Sario_1960], but Weber’s work seems to be subjected to serious objections (according to Zaremba 1910 [@Zaremba_1910]) including the basic one of Weierstrass about the existence of a minimum value for the Dirichlet integral $\spadesuit$ further \[as our attempt to make Zaremba’s objections more explicit\] on p.30 (line 4) Weber makes the tacit assumption that he can find a function $u$ matching the boundary values and of [*finite*]{} Dirichlet integral: this is however violently attacked by the Hadamard 1906 [@Hadamard_1906] counterexample of a boundary data all of whose matching functions explode to infinite Dirichlet integral $\spadesuit$ a weaker result of this type was already obtained by Prym 1871 [@Prym_1871] who gave a continuous function on the unit-circumference whose harmonic extension to the disc (existence via e.g. Poisson) has infinite Dirichlet integral $\spadesuit$ can we characterize such exploding functions? Maybe in terms of wild oscillations (can a such be differentiable (probably recall the wild functions à la Köpcke–Denjoy, etc.), $C^1$ (=continuously derivable), etc.) $\diamondsuit$ H. Weber albeit not a direct student of Riemann, was regarded as one of the efficient successor (e.g. by Thieme, compare Elstrodt-Ulrich [@Elstrodt-Ullrich_1999]). Weber played a pivotal rôle (joint with Dedekind) in editing Riemann’s Werke (including the Nachlass [@Riemann_1857_Nachlass]), and replaced Clebsch who desisted from this task due to health problems\] 

H. Weber, [*Lehrbuch der Algebra*]{}, Bd.I und II. Friedrich Vieweg und Sohn Verlag, Braunschweig, 1898/99. \[$\spadesuit$ Galois theory made in Germany, etc.\] 

H. Weber, [*Lehrbuch der Algebra*]{}, Bd.III. Friedrich Vieweg und Sohn Verlag, Braunschweig, 1908. \[$\spadesuit$ Vorwort (p.VII): “Dagegen habe ich, einem mehrfach an mich herangetretenen Wunsche entschprechend, einen Abri[ß]{} der Theorie der algebraischen Funktionen auf arithmetischer Grundlage beigefügt, der sich im wesentlichen an die Abhandlung von Dedekind und mir im 92.Bande von Crelles Journal anschlie[ß]{}t, aber durch Anwendung der Theroie dr Funktionale, auf die ich im zweiten Bande der Theorie der algebraischen Zahlen gegründet habe, wie mir scheint, eine Vereinfachung erreicht.”\] 

G. Weichold, [*Ueber symmetrischen Riemann’sche Flächen und die Periodicitätsmoduln der zugehörigen Abel’schen Normalintegrale erster Gattung*]{}, Z. Math. Phys. 28 (1883), 321–351. \[$\spadesuit$ exposes the theory of Klein’s symmetric surfaces in full detail (basing the topological study upon the Möbius-Jordan classification [@Jordan_1866]), and do some more subtle things with period matrices $\spadesuit$ this latter object is re-treated in Klein 1892 [@Klein_1892_Realitaet], and will influence the work of Comessatti 1924/26 [@Comessatti_1924/26] $\diamondsuit$ Guido Weichold was a student of Klein, who seems to have been strongly attracted to the topic of symmetric Riemann surfaces through Klein’s lectures. Apparently, Weichold did not pursued his research on this topic\] 

K. Weierstrass, [*Über das sogenannte Dirichletsche Princip*]{}. In: Werke vol. 2, Mayer & Müller, 49–54, 1895. gelesen in der Königl. Akademie der Wissenschaften am 14. Juli 1870. \[$\spadesuit$ a little objection to the Dirichlet principle, yet with desastrous repercussions $\spadesuit$ resurrection by Hilbert 1900, etc. [@Hilbert_1900] $\diamondsuit$ Karl Weierstrass needs not to be introduced. Formally a student of Gudermann, he came across the problem of Jacobi inversion, but unfortunately never published his solution (probably being slightly devanced by Riemann 1857 in this respect). Of course as the whole Riemann approach was for a long time subjected to critics, it would have been of prior interest to know what can be achieved through the pure Weierstrass conceptions collapsing to a sort of arithmetics of power series\] 

K. Weierstrass, [*Vorlesungen 1875/76*]{}. In: Werke, Bd.IV. \[$\spadesuit$ algebraic and Abelian functions\] 

A. Weil, [*The field of definition of a variety*]{}, Amer. J. Math. 78 (1956), 509–524. \[$\spadesuit$\] 

A. Weil, [*Modules des surfaces de Riemann*]{}, Sém. Bourbaki, Mai (1958). \[$\spadesuit$ Teichmüller et cie.\] 

G.G. Weill, [*Reproducing kernels and orthogonal kernels for analytic differentials on Riemann surfaces*]{}, Pacific J. Math. 12 (1962), 729–767. \[$\spadesuit$ refers to Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] for the Bergman kernel on Riemann surfaces, other source includes Schiffer-Spencer 1954 [@Schiffer-Spencer_1954] $\diamondsuit$ Weill is a student of Sario (Ph.D.) ca. 1962\] 

H.F. Weinberger, [*An isoperimetric inequality for the $N$-dimensional free membrane problem*]{}, J. Rat. Mech. Anal. 5 (1956), 633–636. \[$\spadesuit$ inspired by Szegö, but starts to give a more topological argument but the existence of balanced test functions; culminate to Fraser-Schoen 2011 [@Fraser-Schoen_2011], where the junction with the Ahlfors map is made explicit\] 

R. Weinstock, [*Inequalities for a classical eigenvalue problem*]{}, J. Rat. Mech. Anal. 3 (1954), 745–753. \[$\spadesuit$ inspired by Szegö, but Steklov eigenvalues; culminate in Fraser-Schoen 2011 [@Fraser-Schoen_2011], where the junction with the Ahlfors map is made explicit\] 

G. Weiss, [*Complex methods in harmonic analysis*]{}, Amer. Math. Monthly 77 (1970), 465–474. \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$ 

J. Wermer, [*Function rings and Riemann surfaces*]{}, Ann. of Math. (2) 67 (1958), 45–71. \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$ 

J. Wermer, [*Rings of analytic functions*]{}, Ann. of Math. (2) 68 (1958), 550–561. \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$ 

J. Wermer, [*Analytic disks in maximal ideal spaces*]{}, Amer. J. Math. 86 (1964), 161–170. \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$ 

H. Weyl. [*Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen*]{}, Math. Ann. 71 (1912), 441–479. \[$\spadesuit$ the so-called Weyl’s (asymptotic) law asserting that one can hear the area of a drum $\spadesuit$ naive conjecture \[ca. Mai 2011\] can this Weyl’s law be employed as tool to prove the Gromov filling area conjecture (eventually in conjunction with an Ahlfors map to make the usual conformal transplantation of vibratory modes, cf. e.g. Fraser-Schoen 2011 [@Fraser-Schoen_2011] for the first implementation of Ahlfors’ circle maps in spectral theory)\] 

H. Weyl. [*Die Idee der Riemannschen Fläche*]{}, B.G. Teubner, Leipzig und Berlin 1913. 

H. Weyl. [*Ueber das Pick-Nevanlinnasche Interpolationsproblem und sein infinitesimales Analogon*]{}, Ann. of Math. (2) 36 (1935), 230–254. \[$\spadesuit$\] 

H. Weyl. [*The method of orthogonal projection in potential theory*]{}, Duke Math. J. 7 (1940), 411–440. 60 $\bigstar$ 

H. Whitney, [*Complex analytic varieties*]{}, Addison Wesley Publ. Company, Reading, Mass. 1972. \[$\spadesuit$\] 

H. Widom. [*Extremal polynomials associated with a system of curves in the complex plane*]{}, Adv. Math. 3 (1969), 127–232. \[$\spadesuit$\]$\bigstar$$\bigstar$ 

H. Widom, [*$H_p$ sections of vector bundles over Riemann surfaces*]{}, Ann. of Math. (2) 94 (1971), 304–324. 60 \[$\spadesuit$ the geometric quintessence of the paper seems to be Lemma 6 (p.320), created with apparently some helping hand from Royden, and amounting to prescribe (modulo $2\pi$) the periods of the conjugate differential of a superposition of (modified) Green’s functions $\spadesuit$ albeit Ahlfors 1950 [@Ahlfors_1950] is not directly cited, a certain technological “air de famille” transpires throughout the execution $\spadesuit$ alas, Widom’s argument (pp.320–1) seems to give only a poor control upon the number of poles $\zeta_k$ required, and is therefore unlikely to reprove Ahlfors 1950 [@Ahlfors_1950] by specializing to the trivial line bundle case $\spadesuit$ but of course, Widom do something quite grandiose and so the real depth of the work cannot be appreciated by focused comparison with Ahlfors 1950 [@Ahlfors_1950] $\spadesuit$ in particular Widom (re)discover a certain class of open Riemann surfaces (alias of Parreau-Widom) type which are characterized by a moderate growth of the Betti number during the cytoplasmic expansion generated by levels of the Green’s function, which turns out to be a very distinguished class of Riemann surfaces where paradigms like the corona, etc. extends reasonably\] 

R.J. Wille, [*Sur la transformation intérieure d’une surface non orientable dans le plan projectif*]{}, Indagationes Math. 56 (1953), 63–65. \[$\spadesuit$ probably a nonorientable avatar of Stoïlow’s work, and maybe related to Witt 1934 [@Witt_1934]\]$\bigstar$$\bigstar$$\bigstar$ 

G. Wilson, [*Hilbert’s sixteenth problem*]{}, Topology 17 (1978), 53–73. \[$\spadesuit$ discusses Klein’s orthosymmetry (as dividing curves) and ask whether the dividing character of a real plane curve may be recognized by sole inspection of its real locus, p.67: “I do not know if one can tell whether or not $X$ divides by examining only the real part $X_{\Bbb R}\subset {\Bbb R}P^2$.” $\spadesuit$ Our partisan answer (compare Gabard 2004 [@Gabard_2004 p.7]) is a decided [*yes*]{}, posited by Ahlfors theorem $\spadesuit$ however this is pure existence theory and some algorithmic recipes still deserve to be implemented at the occasion. For related efforts cf. e.g. Kalla-Klein 2012 [@Kalla-Klein_2012]\] 

A. Wiman, [*Über die reellen Züge der ebenen algebraischen Kurven*]{}, Math. Ann. 90 (1923), 222–228. \[$\spadesuit$\] 

J. Winkelmann, [*Non-degenerate maps and sets*]{}, Math. Z. (2005), 783–795. 50 \[$\spadesuit$ \[27.09.12\] Ahlfors 1950 [@Ahlfors_1950] is cited, yet not within the main-body of the text, but its companion Bell 1992 [@Bell_1992-Book] is cited for the same purpose. In fact Winkelmann’s article only uses the planar case of the Ahlfors function, hence citing Ahlfors 1947 [@Ahlfors_1947] may have been more appropriate (yet recall that the latter article contains a little gap fixed in Ahlfors 1950 [@Ahlfors_1950 p.123, footnote]) $\spadesuit$ the author gives the following lovely application of the Ahlfors map of a plane bounded domain $\spadesuit$ call a holomorphic map [*dominant*]{} if it has dense image, and a complex manifold universally dominant (UDO) if it admits a dominant map to any irreducible complex space. The author shows first that the unit disc $\Delta$ is UDO (Cor.3, p.786), and via the Ahlfors function this implies more generally that any complex manifold admitting a nonconstant bounded analytic function (BAF) is UDO. Here are the details. $\spadesuit$ first if the complex manifold is UDO, then it dominates the unit disc $\Delta$, and so it carries a nonconstant BAF. Conversely, let $f\colon X \to {\Bbb
C}$ be a nonconstant BAF then $f(X)$ is a bounded domain. (It is crucial here to assume $X$ connected, for $X$ the disjoint union of say two Riemann spheres carries a nonconstant BAF, yet fails to be UDO.) Now observe the following fact. [**Lemma.**]{} [*The Ahlfors map $f_a$ at $a$ of the bounded domain $G\ni a$ is dominant.*]{}—[*Proof.*]{} If not, then the map $f_a\colon G\to
\Delta$ misses a little disc $D\subset \Delta$ not overlapping the origin (recall that $f_a(a)=0$). Since the identity map restricted to the ring $\Delta-\overline{D}$ is bounded-by-one (hence admissible in the extremal problem), it follows that the Ahlfors map for the ring centered at $0$, say $g_0$, has a derivative with modulus strictly larger than unity, i.e. $\vert g'_0(0)\vert > \vert
(id)' (0) \vert=1$. But then the composed map $(g_0 \circ
f_a)$ effects the stretching $\vert (g_0 \circ f_a )'(a)
\vert=\vert (g_0'(f_a(a)) \cdot f_a'(a) \vert=\vert g_0'(0)
\cdot f_a'(a) \vert> \vert f_a'(a) \vert$, violating the extremal property of $f_a$. q.e.d.—$\spadesuit$ At this stage it may be observed that the Ahlfors map of a bounded domain needs not be surjective. Consider indeed the unit disc $\Delta$ punctured at say $1/2$, then the Ahlfors function of $\Delta-\{1/2\}$ centered at $0$, denoted $f_0$, is the identity (up to a rotation). Indeed, since a (pointlike) puncture is a removable singularity for BAF any function admitted in the extremal problem extends analytically across the whole unpunctured disc. More generally, the Ahlfors map is insensitive to the puncturing of a removable singularity (alias Painlevé null sets), e.g. Cantor’s $1/4$-set described in Garnett 1970 [@Garnett_1970] $\spadesuit$ back to Winkelmann’s argument, the above lemma applied to $G:=f(X)$ gives a dominant map $f_a$ to the disc, hence a dominant map $X\to f(X)\to \Delta$. Summarizing: [*any complex manifold $X$ supporting a nonconstant BAF dominates the disc.*]{} $\spadesuit$ Perhaps one could try to improve this by using the surjectivity of the Ahlfors map for a domain of finite connectivity (without pointlike boundaries), assuming e.g. that $X$ has a finitely generated fundamental group $\pi_1$. Alas, this does not seem to imply automatically that $\pi_1(f(X))$ is of finite generation and we need of course to control the shape of the image $f(X)$, which has to be a finite region bounded by Jordan curves $\spadesuit$ finally since the disc $\Delta$ dominates any irreducible complex space $Y$ (of course the definition of the latter must be calibrated so as to avoid non-metric complex manifolds of Calabi-Rosenlicht of the Prüfer type, at least those specimens which are not separable), the composition $X\to f(X) \to \Delta \to Y$ yields the desired dominant map showing that $X$ is UDO. This completes Winkelmann’s proof.\] 

W. Wirtinger, [*Untersuchungen über Thetafunktionen*]{}, Teubner, 1895. 

W. Wirtinger, [*Algebraische Funktionen und ihre Integrale*]{}, Enzykl. d. math. Wiss. $2_2$ (1902), 115–175. 60 

W. Wirtinger, [*Über die konforme Abbildung der Riemannschen Flächen durch Abelsche Integrale besonders bei $p=1,2$*]{}, Denkschr. Wien (1909), 22pp. 60 

W. Wirtinger, [*Über eine Minimalaufgabe im Gebiete der analytischen Funktionen*]{}, Monatsh. Math. u. Phys. 39 (1932), 377–384. \[$\clubsuit$ quoted p.269 of Schiffer 1950 [@Schiffer_1950-Appendix-Courant] for a the first notice of a certain reproducing kernel property, also quoted in Bergman 1950 [@Bergman_1950] $\spadesuit$ poses (and solves via the Green’s function) the problem of the best analytic approximation $f$ in $L^2$-norm $\int\int_B \vert f-
\Phi \vert^2 d \omega$ of a given continuous function $\Phi$\] 

W. Wirtinger, [*Zur Theorie der konformen Abbildung mehrfach zusammenhängender ebener Flächen*]{}, Abh. Preu[ß]{}. Akad. Wiss. math.-nat. Kl. 4 (1942), 1–9. 60, 78 \[$\clubsuit$ reproves the theorem of Riemann-Schottky-Bieberbach-Grunsky(=RSBG), i.e. the schlicht(artig) case of the Ahlfors map, via algebraic functions (i.e. Riemann-(Roch) essentially)\] 

E. Witt, [*Zerlegung reeller algebraischer Funktionen in Quadrate*]{}, J. Reine Angew. Math. 171 (1934), 4–11. \[$\clubsuit$ contains a sort of non-orientable version of the Riemann/Ahlfors map. Subsequent developments in Geyer 1964/67 [@Geyer_1964-67] and Martens 1978 [@Martens_1978]\] 

E. Witten, [*Two dimensional gravity and intersection theory on the moduli space*]{}, Survey in Differential Geometry, Leigh Univ., 1991, 243–310. \[$\spadesuit$\] 

J.J. Wolfart, [*The ’obvious’ part of Belyi’s theorem and Riemann surfaces with many automorphisms* ]{}, In: [*Geometric Galois Actions, 1*]{}, London Math. Soc. Lecture Note Series 242, 1997. \[$\clubsuit$\] 

S. Wolpert, [*The length spectra as moduli for compact Riemann surfaces*]{}, Ann. of Math. (2) (1979). \[$\clubsuit$\] 

D.V. Yakubovich, [*Real separated algebraic curves, quadrature domains, Ahlfors type functions and operator theory*]{}, J. Funct. Anal. 236 (2006), 25–58. 50 \[$\clubsuit$ contains also (after Alling-Greenleaf [@Alling-Greenleaf_1969]) a clear-cut formulation of the Klein-Ahlfors correspondence: i.e. a curve is dividing/separating iff it maps to the line in a totally real fashion (i.e. real fibres are entirely real)\] 

A. Yamada, [*On the linear transformations of Ahlfors functions*]{}, Kōdai Math. J. 1 (1978), 159–169. 50 \[$\clubsuit$ evaluate the degree of the Ahlfors function at the Weierstrass points of a non-planar hyperelliptic membrane as taking the maximum value permissible, i.e. $r+2p=g+1$\] 

A. Yamada, [*A remark on the image of the Ahlfors function*]{}, Proc. Amer. Math. Soc. 88 (1983), 639–642. \[$\spadesuit$ domains of infinite connectivity $\spadesuit$ p.639 (abstract extract): “By an example we show that the complement in the unit disc of the image of the Ahlfors function for $ \Omega $ and $ p$ can be a fairly general set of logarithmic capacity zero.”\] 

A. Yamada, [*Ahlfors functions on Denjoy domains*]{}, Proc. Amer. Math. Soc. 115 (1992), 757–763. \[$\spadesuit$ domains of infinite connectivity $\spadesuit$ p.757: “The main result of our paper gives a necessary and sufficient condition for a subset of the unit disc to be the omitted set of the Ahlfors function $F$ for some maximal Denjoy domain and $\infty$ such that $F$ is a covering onto its image. As a corollary we give examples of omitted sets of Ahlfors functions that have positive logarithmic capacity.” $\spadesuit$ \[05.10.12\] if I don’t mistake Yamada’s example thus answers a question by Minda 1981 [@Minda_1981-image-Ahlfors-fct p.755] about knowing if the Ahlfors function can “omit an uncountable set”. (Recall indeed that sets of zero logarithmic capacity are stable under countable unions, cf. p.762, where Yamada refers to Tsuji 1959 [@Tsuji_1959-BOOK/Chelsea1975 p.57].)\] 

A. Yamada, [*Ahlfors functions on compact bordered Riemann surfaces*]{}, J. Math. Soc. Japan 53 (2001), 261–283. 50 \[$\clubsuit$ establish a conjecture of Gouma 1998 [@Gouma_1998] to the effect that the Ahlfors degree of a hyperelliptic membrane centered outside the Weierstrass points always degenerates to the minimum value $2$\] 

P.C. Yang, S.-T. Yau, [*Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds*]{}, Ann. Sc. Norm. Sup. di Pisa (4) 7 (1980), 55–63. \[$\spadesuit$ applies conformal branched covering of [*closed*]{} Riemann surfaces to the sphere and the trick of conformal transplantation to generate test functions yielding an estimate of the first three Laplace eigenvalues of a closed Riemann surface considered as a vibrating membrane. Inspiration Szëgo, Hersch 1970 [@Hersch_1970], but goes somewhat deeper as there is no fear of multi-sheetedness $\spadesuit$ for an adaptation of Yang-Yau’s method to bordered surfaces via the Ahlfors map see Fraser-Schoen 2011 [@Fraser-Schoen_2011], or some derived products like Gabard 2011 [@Gabard_2011] or Girouard-Polterovich 2012 [@Girouard-Polterovich_2012]\] 

O. Yavuz, [*Invariant subspaces for Banach space operators with a multiply connected spectrum*]{}, Integr. Equ. Oper. Theory 58 (2007), 433–446. \[$\spadesuit$ p.439–440, the Ahlfors function (via Fisher’s book 1983 [@Fisher_1983]) is employed to extend a result on the existence of invariant subspaces for operators with a multiply-connected spectrum (previously known when the spectrum contained the unit-circle)\] 

O. Yavuz, [*A reflexivity result concerning Banach space operators with a multiply connected spectrum*]{}, Integr. Equ. Oper. Theory 68 (2010), 473–485. \[$\spadesuit$ p.475–6, Ahlfors function via Fisher’s book 1983 [@Fisher_1983]\] 

N.X. Yu, [*On Riesz transforms of bounded function of compact support*]{}, Michigan Math. J. 24 (1977), 169–175. \[$\spadesuit$ p.170, Ahlfors function via Carleson’s book 1967 [@Carleson_1967-book ChapterVIII]\] 

L. Zalcman, [*Analytic capacity and rational approximation*]{}, Lecture Notes in Math. 50, Springer-Verlag, Berlin–New York, 1968. 50 \[$\spadesuit$\] 

L. Zalcman, [*Analytic functions and Jordan arcs*]{}, Proc. Amer. Math. Soc. 19 (1968), 508. \[$\spadesuit$\] 

L. Zalcman, [*Bounded analytic functions on domains of infinite connectivity*]{}, Trans. Amer. Math. Soc. 144 (1969), 241–270. \[$\spadesuit$\] 

K. Zarankiewicz, [*Sur la représentation conforme d’un domaine doublement connexe sur un anneau circulaire*]{}, C.R. Acad. Sci. Paris 198 (1934), 1347–1349. \[$\spadesuit$ Seidel’s summary: method for the effective construction of the conformal mapping of a doubly connected domain upon a circular ring, via orthogonal systems (Bergman kernel) $\spadesuit$ consider (with Bergman \[no precise cross-ref.\]) the problem of maximizing the modulus of $f(t)$ among functions with $L^2$-norm bounded by $1$: $\int\int_B
{\vert f(z)\vert}^2\le 1$\] 

K. Zarankiewicz, [*Über ein numerisches Verfahren zur konformen Abbildung zweifach zusammenhängender Gebiete*]{}, Zeitschr. f. angew. Math. u. Mech. 14 (1934), 97–104. \[$\spadesuit$ Seidel’s summary: a detailed account is given of the method indicated in Zarankiewicz 1934 [@Zarankiewicz_1934], i.e. Bergman kernel style numerical device to compute the conformal map of a doubly connected domain $\spadesuit$ oft quoted e.g. in Lehto 1949 [@Lehto_1949], Bergman 1950 [@Bergman_1950]\]$\bigstar$$\bigstar$$\bigstar$ 

S. Zaremba, [*Sur le calcul numérique des fonctions demandées dans le problème de Dirichlet et le problème hydrodynamique*]{}, Bull. Inst. Acad. Sci. Cracovie (1908), 125–195. \[$\spadesuit$ quoted (e.g.) in Lions 2000/02 [@Lions_2000/02] as one of the very early apparition of the notion of reproducing kernel\] 

S. Zaremba, [*Sur le principe de Dirichlet*]{}, Acta Math. ? (1910), 293–316. \[$\spadesuit$ Hadamard’s 1906 [@Hadamard_1906] counterexample to the Dirichlet principle is cited (but not the earlier one of Prym 1871 [@Prym_1871]) and further (p.294) asserts that Weber’s 1869/70 [@Weber_1870] attempt to consolidate Riemann’s proof is subjected to serious objections $\spadesuit$ unfortunately, Zaremba does not make explicit any objection, but it is implicit that he has in mind the Weierstrass critique (of a functional not achieving a minimum) and further Weber’s tacit assumption that the Dirichlet integral is finite is violently attacked by the Hadamard 1906 [@Hadamard_1906] counterexample of a boundary data all of whose matching functions have infinite Dirichlet integral (of course, Prym 1871 [@Prym_1871] is a sufficient torpedo to destroy completely Weber’s argumentation) $\spadesuit$ notice that Arzelà 1897 [@Arzela_1897] has to be counted as a forerunner of Hilbert’s triumph of all the difficulty of the question (in certain particular cases), and mentions the remarkable extensions due to B. Levi 1906 [@Beppo-Levi_1906], Fubini 1907 [@Fubini_1907] and Lebesgue 1907 [@Lebesgue_1907], while proposing to recover those results through a simpler method without loosing anything essential to their generality\] 

R. Zarrow, [*Anticonformal automorphisms of compact Riemann surfaces*]{}, Proc. Amer. Math. Soc. 54 (1976), 162–164. \[$\spadesuit$ cf. little corrections in Costa 1996 [@Costa_1996-PAMS]\] 

H.G. Zeuthen, [*Sur les formes différentes des courbes du quatrième ordre*]{}, Math. Ann. 7 (1874), 410–432. (+Tafel I, II, Fig.1,2,3,4,5) \[$\spadesuit$ a work who inspired much of Klein investigation $\spadesuit$ cite von Staudt (Geometrie der Lage) $\spadesuit$ uses the term “ovale” $\spadesuit$ p.411: “Une courbe du quatrième ordre ([*quartique*]{}) a, au plus, quatre branches externes l’une à l’autre, ou deux branches dont l’une se trouve dans la partie du plan interne à l’autre, et dans ce dernier cas la branche interne ne peut avoir des tangentes doubles ou d’inflexion.—Car s’il en était autrement on pourrait construire des coniques rencontrant la courbe en plus de 8 points, ou des droites la rencontrant en plus de 4 points.” (This is the sort of Bézout-type argument out of which will emerge the Harnack inequality 1876 [@Harnack_1876]), yet the full intrinsic grasp (especially the interpretation via Riemann surfaces) will be effected through Klein’s work 1876 [@Klein_1876] $\spadesuit$ p.412: “Nous appelons ici réelle toute courbe dont l’équation ne contient que des coefficients réels.” $\spadesuit$ p.428, cite Geiser and the yoga between cubic surface and quartic curves, which is instrumental in Klein 1876 to make the rigid-isotopy classification of quartic curves.\] 

H.G. Zeuthen, [*Études des propriétés de situation des surfaces cubiques*]{}, Math. Ann. 8 (1874/75), 1–30. \[$\spadesuit$ also quoted by Klein 1876, as to complete the rigid-isotopy classification of quartics by reduction to the case of cubics (Schläfli 1863 [@Schlaefli_1863] and Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn])\] 

M. Zhang, Y. Li, W. Zeng, X. Gu, [*Canonical conformal mapping for high genus surfaces with boundaries*]{}, Computers Graphics 36 (2012), 417–426. \[$\spadesuit$ completely in line with our present topic, and use high powered machinery like Koebe’s iteration and (Yau-Hamilton’s) Ricci flow for conformal theoretic purposes $\spadesuit$ can we adapt such algorithms to the (Ahlfors) circle map\] 

V.A. Zmorovič, [*The generalization of the Schwarz formula for multiply connected domains*]{}, (Ukrainian) Dokl. Akad. Nauk Ukrain. SSR 7 (1962), 853–856. \[$\spadesuit$ quoted in Khavinson 1984 [@Khavinson-Dimitri_1984]\]$\bigstar$$\bigstar$ 

V.I. Zvonilov, [*Complex topological characteristics of real algebraic curves on surfaces*]{}, Funkt. Anal. Prilozhen. 16 (1982), 56–57; English transl. in Funct. Anal. Appl. 16 (1982), 202–204. \[$\spadesuit$\]$\bigstar$ 

V.I. Zvonilov, [*Complex orientations of real algebraic curves with singularities*]{}, Dokl. Akad. Nauk SSSR 268 (1983), 22–26; English transl., Soviet Math. Dokl. 27 (1983), 14–17. \[$\spadesuit$\]$\bigstar$ 

V.I. Zvonilov, [*Complex topological characteristics of real algebraic curves on a hyperboloid and an ellipsoid*]{}, Funkt. Anal. Prilozhen. ?? (1986), ?–?; English transl. in Funct. Anal. Appl. ?? (1986), ?–?. \[$\spadesuit$\]$\bigstar$ 

Zal68 L. Zalcman, [*Analytic capacity and rational approximation*]{}, Lecture Notes in Math. 50, Springer-Verlag, Berlin–New York, 1968. 50 \[$\spadesuit$\] 

Zal68b L. Zalcman, [*Analytic functions and Jordan arcs*]{}, Proc. Amer. Math. Soc. 19 (1968), 508. \[$\spadesuit$\] 

Zal69 L. Zalcman, [*Bounded analytic functions on domains of infinite connectivity*]{}, Trans. Amer. Math. Soc. 144 (1969), 241–270. \[$\spadesuit$\] 

Zar34 K. Zarankiewicz, [*Sur la représentation conforme d’un domaine doublement connexe sur un anneau circulaire*]{}, C.R. Acad. Sci. Paris 198 (1934), 1347–1349. \[$\spadesuit$ Seidel’s summary: method for the effective construction of the conformal mapping of a doubly connected domain upon a circular ring, via orthogonal systems (Bergman kernel) $\spadesuit$ consider (with Bergman \[no precise cross-ref.\]) the problem of maximizing the modulus of $f(t)$ among functions with $L^2$-norm bounded by $1$: $\int\int_B
{\vert f(z)\vert}^2\le 1$\] 

Zar34b K. Zarankiewicz, [*Über ein numerisches Verfahren zur konformen Abbildung zweifach zusammenhängender Gebiete*]{}, Zeitschr. f. angew. Math. u. Mech. 14 (1934), 97–104. \[$\spadesuit$ Seidel’s summary: a detailed account is given of the method indicated in Zarankiewicz 1934 [@Zarankiewicz_1934], i.e. Bergman kernel style numerical device to compute the conformal map of a doubly connected domain $\spadesuit$ oft quoted e.g. in Lehto 1949 [@Lehto_1949], Bergman 1950 [@Bergman_1950]\]$\bigstar$$\bigstar$$\bigstar$ 

Zar08 S. Zaremba, [*Sur le calcul numérique des fonctions demandées dans le problème de Dirichlet et le problème hydrodynamique*]{}, Bull. Inst. Acad. Sci. Cracovie (1908), 125–195. \[$\spadesuit$ quoted (e.g.) in Lions 2000/02 [@Lions_2000/02] as one of the very early apparition of the notion of reproducing kernel\] 

Zar10 S. Zaremba, [*Sur le principe de Dirichlet*]{}, Acta Math. ? (1910), 293–316. \[$\spadesuit$ Hadamard’s 1906 [@Hadamard_1906] counterexample to the Dirichlet principle is cited (but not the earlier one of Prym 1871 [@Prym_1871]) and further (p.294) asserts that Weber’s 1869/70 [@Weber_1870] attempt to consolidate Riemann’s proof is subjected to serious objections $\spadesuit$ unfortunately, Zaremba does not make explicit any objection, but it is implicit that he has in mind the Weierstrass critique (of a functional not achieving a minimum) and further Weber’s tacit assumption that the Dirichlet integral is finite is violently attacked by the Hadamard 1906 [@Hadamard_1906] counterexample of a boundary data all of whose matching functions have infinite Dirichlet integral (of course, Prym 1871 [@Prym_1871] is a sufficient torpedo to destroy completely Weber’s argumentation) $\spadesuit$ notice that Arzelà 1897 [@Arzela_1897] has to be counted as a forerunner of Hilbert’s triumph of all the difficulty of the question (in certain particular cases), and mentions the remarkable extensions due to B. Levi 1906 [@Beppo-Levi_1906], Fubini 1907 [@Fubini_1907] and Lebesgue 1907 [@Lebesgue_1907], while proposing to recover those results through a simpler method without loosing anything essential to their generality\] 

Zeu74 H.G. Zeuthen, [*Sur les formes différentes des courbes du quatrième ordre*]{}, Math. Ann. 7 (1874), 410–432. (+Tafel I, II, Fig.1,2,3,4,5) \[$\spadesuit$ a work who inspired much of Klein investigation $\spadesuit$ cite von Staudt (Geometrie der Lage) $\spadesuit$ uses the term “ovale” $\spadesuit$ p.411: “Une courbe du quatrième ordre ([*quartique*]{}) a, au plus, quatre branches externes l’une à l’autre, ou deux branches dont l’une se trouve dans la partie du plan interne à l’autre, et dans ce dernier cas la branche interne ne peut avoir des tangentes doubles ou d’inflexion.—Car s’il en était autrement on pourrait construire des coniques rencontrant la courbe en plus de 8 points, ou des droites la rencontrant en plus de 4 points.” (This is the sort of Bézout-type argument out of which will emerge the Harnack inequality 1876 [@Harnack_1876]), yet the full intrinsic grasp (especially the interpretation via Riemann surfaces) will be effected through Klein’s work 1876 [@Klein_1876] $\spadesuit$ p.412: “Nous appelons ici réelle toute courbe dont l’équation ne contient que des coefficients réels.”\] 

Zha12 M. Zhang, Y. Li, W. Zeng, X. Gu, [*Canonical conformal mapping for high genus surfaces with boundaries*]{}, Computers Graphics 36 (2012), 417–426. \[$\spadesuit$ completely in line with our present topic, and use high powered machinery like Koebe’s iteration and (Yau-Hamilton’s) Ricci flow for conformal theoretic purposes $\spadesuit$ can we adapt such algorithms to the (Ahlfors) circle map\] 

Zmo62 V.A. Zmorovič, [*The generalization of the Schwarz formula for multiply connected domains*]{}, (Ukrainian) Dokl. Akad. Nauk Ukrain. SSR 7 (1962), 853–856. \[$\spadesuit$ quoted in Khavinson 1984 [@Khavinson-Dimitri_1984]\]$\bigstar$$\bigstar$ 

Alexandre Gabard

Université de Genève

Section de Mathématiques

2-4 rue du Lièvre, CP 64

CH-1211 Genève 4

Switzerland

alexandregabard@hotmail.com

[^1]: =Fusion in Klein’s prose when viewing all his work (and that of Sophus Lie) as being merely a Galois-Riemann synthesis.

[^2]: Best example thereof, read Borsuk’s article ca. 1936 where a contractible compactum lacking the fixed-point property is presented. If you have just the boring (unreadable) formulas of Borsuk you understand nothing, but if you know the picture that the space in question is a crumpled-cube spiraling twice around itself as pictured by Bing, you start to believe why the fact holds true.

[^3]: Source=H. Cremer, Erinnerungen an Paul Koebe, Jahresber. DMV, 1968, p.160. (Mitteilung von Heinrich Behnke).

[^4]: For more historical details on the theory of quasiconformal mappings compare Ahlfors 1984 [@Ahlfors_1984-The-Joy] or Lehto 1998 [@Lehto_1998]. \[02.10.12\] Alas we were not as yet able to show any deep connection between the theory of Ahlfors circle maps and that of quasiconformal maps, yet it is not unlikely that such a connection is worth studying, more in Section \[sec:question\].

[^5]: Prose borrowed by Louis de Branges.

[^6]: This nomenclature is used by Hajek, compare some arXiv preprints of the author joint with Gauld.

[^7]: There is a letter form Jordan to Lebesgue saying roughly: “Perséverez dans vos recherches mathématiques, vous allez y éprouver beaucoup de plaisirs, mais il va vous falloir apprendre à y gouter seul, car en général les géomètres ne se lisent même pas entre eux-mêmes.” (quoted by pure memory, hence highly unreliable).

[^8]: Apparently via H.A. Schwarz, compare Bieberbach 1925 [@Bieberbach_1925].

[^9]: Notice that Ahlfors never quote Teichmüller 1981 [@Teichmueller_1941], except in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960], where also all the Italian works of Matildi 1945/48 [@Matildi_1945/48] and Andreotti 1950 [@Andreotti_1950] are cited.

[^10]: Gabard micro-comment: Here the last edition of Riemann’s Werke contains a little misprint $F$ instead of the obvious $T$, not present e.g., in the French translation of Riemann by Laugel, Paris 1898.

[^11]: Joke of Ivan Babenko, yet irritating the western auditors coming down from the “alpage”.

[^12]: This is not explicitly specified in the paper, but is the (common) jargon in Klein surface theory, probably due to Alling-Greenleaf 1971 [@Alling-Greenleaf_1971].

[^13]: Of course this can hardly be taken seriously, in view of the messy nature of the present text!

[^14]: Perhaps it would be more corrected to say “along” here. Compare in this respect Ahlfors, p.108, the text just preceding footnote 3)

[^15]: Here our argument shorten slightly the prose of Ahlfors, hopefully without loosing in precision?!

[^16]: In the combinatorial sense, by opposition to the topological sense.

[^17]: Since Petrovskii 1938 [@Petrowsky_1938] it is customary to call an $M$-curve, any curve realizing Harnack’s bound $r\le g+1$ of 1876.

[^18]: This concept is not really meaningful for $M$-curves.

[^19]: \[28.03.13\] I would personally be much interested, if someone can guess more explicitly what Rohlin had in mind at this place!

[^20]: This shows that Klein anticipated the phenomenon of total reality.

[^21]: Alas the word “oval” is quite ambiguous, as it is either just a real component of the abstract curve, or sometimes used in the much more specific sense of a component of a plane curve which is null-homotopic (equivalently bounds a disc) in ${\Bbb R}P^2$. Whenever we use the term oval in this abstract sense, due to a lack of better synonym (in German there is a good one “Zug/Züge”), we write it “entre guillemet” (=inverted comma in English, according to my Dictionary).

[^22]: This appellation is now a common joke in Geneva, based on a mixture of the writer’s name with themuch more eminent Paul Garabedian, the notorious student of Ahlfors, Schiffer, Bergman, who seems to have played a pivotal rôle in the ultimate shape of Ahlfors theorem, as published in 1950 [@Ahlfors_1950].

[^23]: \[30.03.13\] Strictly speaking I do not know how to proof this, but hope this to be a triviality of basic algebraic geometry.

[^24]: Unpublished, but see Rohn 1911 [@Rohn_1911], 1913 [@Rohn_1913], yet not judged complete by Gudkov 1974.

[^25]: Prose borrowed by Jack Milnor, when he speaks about non-metric manifold, cf. his preprint on foliated bundles.

[^26]: I.e. identity.

[^27]: Thom learned Sard from de Rham, cf. the 1954 Commentarii article.

[^28]: \[29.03.13\] This is just a very special case of a more general satellite principle, cf. Sec.\[satellite-total-reality:sec\].

[^29]: Omit this bracketing for it is just to refer to Gudkov’s notation.

[^30]: I.e. the present text as it was on the date of the 09.01.13, meanwhile pagination may have changed.

[^31]: This is not perfectly true, as I exposed Ahlfors theorem to Oleg Viro during its last visit in Geneva (ca. 2010–11, his talk on fields of char 1, Connes, tropical geometry, etc.), yet probably my explanations where so obscure that Oleg immediately forgot about it.

[^32]: =Vladimir Abramovich Rohlin, of course.

[^33]: Sic, singular or plural? Not so important of course.

[^34]: i.e. Lemma \[Klein-Marin:lem\], warning meanwhile the numbering may changes, but the one in this footnote is automatic (hence the right one)

[^35]: “isotopie” of course.

[^36]: Meanwhile this numbering may have changed into Quote \[Klein\_1876-niemals-isolierte:quote\].

[^37]: Not clear how to interpret this? Does it mean that Shustin’s claim is wrong, or simply that this scheme is an $(M-1)$-scheme. My question was whether this $(M_1)$-scheme is realized algebraically, of course. Yet, I admit that my question was a bit ill posed.

[^38]: \[24.01.13\] On reading Viro’ survey 1989/90 [@Viro_1989/90-Construction p.1085, 2.5.H], one should easily locate the source for this assertion. References seems to be Viro 1983 [@Viro_1983/84-new-prohibitions], and the survey Viro 1986 [@Viro_1986/86-Progress]. As explained there (Viro 1989/90, p.1085) this is a prohibition not coming from topology, but from Bézout. In fact this result is mentioned again in Viro 1989/90 [@Viro_1989/90-Construction p.1126, 5.3.E], with the exact cross-reference as being Viro 1983 [@Viro_1983/84-new-prohibitions]

[^39]: \[30.01.13\] This is another very interesting (inedited?) piece of information, as to my knowledge this never appeared under the (printed) pen of V.A. Rohlin. So here Viro tell us something very interesting not yet available in print (as far as I know).

[^40]: \[30.01.13\] This is probably true yet this requires a more highbrow dissipation theory, than in Gudkov’s second existence proof which apart from the trick with Cremona uses only dissipation of ordinary nodes à la Brusotti 1921 (so-called $A_1$-singularities).

[^41]: \[30.01.13\] This alas turned out to be quite foiled as the invisible part of the discriminant as real codimension 2.

[^42]: \[26.03.13\] Alas as spotted by Th. Fiedler this turned out to be wrong as it was based on the tacit supposition that the Arnold surface is always orientable, compare Fiedler’s letter dated \[21.03.13\].

[^43]: \[29.03.13\] A nearby corroboration of Rohlin’s claim is now available in Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics].

[^44]: \[30.03.13\] This is not exactly Viro’s opinion, cf. his letter in Sec.\[e-mail-Viro:sec\].

[^45]: I presume this list is not exhaustive, as Shustin’s scheme above ought to be also realized? If I have well understood the former “It is.”.

[^46]: This is Viro’s notation, and mean 6 ovals enveloped in one, and this thrice. So 21 ovals.

[^47]: \[24.01.13\] The exact reference for this result is Viro 1983 [@Viro_1983/84-new-prohibitions]

[^48]: Our (non-standard) terminology: V. Kharlamov explained us (cf. Sec.\[e-mail-Viro:sec\]) that this phenomenon was quite crucial a motivation when Morosov suspected some anomaly in Gudkov’s initial solution to Hilbert’s 16th problem along the lines expected by Hilbert. Compare for more Viatcheslav’s e-mail in Sec.\[e-mail-Viro:sec\], and his terminology “partner relationship”. \[01.04.13\] It would be interesting to know if the low Gudkov-schemes $\frac{5}{1}4$, and $\frac{5}{1}3$ can be constructed from their mirrors.

[^49]: This conclusion actually holds true unconditionally as follows from Rohlin’s formula $2(\Pi^{+}-\Pi^{-})=r-k^2$.

[^50]: We often commit an abuse of language, as we should say one chamber residual to the discriminant. Such an abuse is harmless like when speaking of the group of a knot, when it is really that of its complement.

[^51]: After several permutation in this text, it is not clear anymore what was the “previous section”, but checking dates and contents this can be almost surely identified as Sec.\[application-of-CCC:sec\].

[^52]: I should acknowledge the assistance of my cousin Élias Boulé-Schneider for several discussions on such topics, like topographical discussions about Hilbert’s 16th.

[^53]: We use “smooth” as an abridgement of nonsingular in the algebro-geometric sense, or if you prefer smoothness of the complexification, but not merely of the real locus. So a curve may look smooth while having imaginary conjugate singularities. Such a curve is not considered as smooth by us.

[^54]: I borrow this jargon from Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000].

[^55]: Indeed, we remember well, cf. e.g. EDM=Encyclopaedic Dictionary of Mathematics 1968/87 [@EDM_1968/87 p.46, Art. 13 G] “\[…\] the form $f$ is anisotropic, i.e. the homogeneous equation $f=0$ has no solution other than zero in $k$”. Or cf. Serre’s “Cours d’arithmétique”, 1970–1977, who seems clever enough to avoid the jargon, yet speaks of isotropic for quadratic forms. I don’t know who coined the term (in arithmetics), maybe Minkowski, Hilbert, Weil? Ask a competent arithmetician.

[^56]: Joke of Misha Gromov (yet another notorious student of V.A. Ro\[k\]hlin).

[^57]: Student of E. Schmidt, himself student of D. Hilbert. So we are not to far apart form the 16th problem.

[^58]: But widely anticipated by Poincaré, Tietze, Brouwer, and many others combinatorial topologists of the early 20th century.

[^59]: Laboureur means nearly laborieux in French, and was Feldbau’s pseudonym to publish Comptes Rendus notes during the German occupation of France (World War II, 1939–45). Alas, it did not helped to save his life from the Nazi persecutions. Another notorious victim of the genocide soon afterwards was F. Hausdorff, 1944. Why so much dramas in the human history is a puzzle to each philosopher. Materialism, capitalism, caused by the ontological existential fears ought to be the cause of such disasters. We can only hope that the Riemann(=woman) surface will quickly lead us to stabler psychological comforts (immortality, and global resurrection as to repair such disasters).

[^60]: Without this proviso it is false, e.g. the tangent bundle to the simply-connected Prüfer surface is not trivial, for otherwise the manifold could be given a Riemann metric tensor, hence be metricized. Compare Radó 1925 (publishing a contribution of Heinz Prüfer 1922), Calabi-Rosenlicht 1953, Spivak 1970 (Vol.I, Appendix of Differential Geometry), or ask Mathieu Baillif why. The latter’s e-mail is: labaffle(at)gmail.com

[^61]: Peintre américain d’origine russe (Dvinsk 1903–New York 1970). Il est célèbre pour la formule d’abstraction chromatique qu’il a établie vers 1950. (Source=Larousse Dictionnary, 1991).

[^62]: Read “chamber” if you prefer.

[^63]: Read “isotopy classes”, if you like.

[^64]: This seems to just mean non-singular curve, cf. footnote in GMA.

[^65]: If you are called Felix Klein!, else it may be tricky especially if our previous theorem is right in which case Klein-Rohlin are false! But of course, it is more likely that Gabard missed something. \[06.04.13\] More seriously, it would be interesting if a detailed account of this direct proof (alluded to by Klein) has been worked out meanwhile. Some details are perhaps gleanable from Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000].

[^66]: This is a slight abuse of language to suit Russian jargon (coined by Petrovskii 1938).

[^67]: \[08.04.13\] Compare with Le Touzé’s article 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics], where it is asserted that this was implicitly conjectured by Rohlin, in 1978. Recall that Le Touzé’s husband Fiedler is a direct student of Rohlin, and so this may also be based upon some oral tradition, in case Rohlin was too cautious to put crazy ideas on the paper.

[^68]: Tartaglia, also known as Niccolo Fontana (1500?–1557) won in 1535 a mathematical contest by solving many different cubics, and gave his solution to Cardano (1501–1576), who published in 1539 “Artis Magnae” alias the “Great Art, or the Rules of Algebra”, where complex numbers are used in Cardano’s formula to express the real roots of cubics.

[^69]: Again this claim is a mistake: an obstruction follows from Thom’s conjecture, meanwhile the theorem of Kronheimer-Mrowka 1994 [@Kronheimer-Mrowka_1994].

[^70]: \[08.03.13\] Fortunately this schemes is not realized.

[^71]: \[06.03.13\] This special treatment can be dispensed as this scheme is prohibited by Thom conjecture, cf. Theorem \[Thom-Ragsdale:thm\], or more elementarily by Rohlin’s formula.

[^72]: \[02.03.13\] This is a misconception of mine and Le Touzé’s statement is finer and so caricatural (or strong) as I misinterpreted her announcement, more discussion about this soon. However I still do not know whether the strong caricatural statement is wrong or not.

[^73]: The official name of this author is Fiedler-Le Touzé, yet as well-known née Le Touzé and abridged as a such in the sequel.

[^74]: This seems alas to be the bitter state of affairs as follows from a recent consultation (January 2013) of the leading experts (Viro, Marin, Kharlamov, Fiedler, Le Touzé, etc.); compare e-mails gathered in Sec.\[e-mail-Viro:sec\]

[^75]: I agree, but the argument is nearly trivial in the sense that it just uses the fact that the image of a connected set is connected (Marin’s oral remark during my talk in Grenoble ca. 1999). Personally, I knew this argument since 1999 (arguing with pathes prior to Marin’s oral simplification of it). We cannot record if we rediscovered it independently of Rohlin 1978 (but do record that we may have found some indirect inspiration from Gross-Harris 1981, who treat the case of hyperelliptic curves $y^2=f(x)$, with $f(x)>0$ throughout). At any rate it is evident that Rohlin’s argument can be drastically simplified. Rohlin uses a lot a certain fibering while it is plain that it suffices to use the map to the equatorial (orthosymmetric) sphere, cf. e.g. Gabard 2006 [@Gabard_2006].

[^76]: My opinion was always that a positive answer should be a trivial consequence of Ahlfors theorem (cf. e.g. Gabard’s Thesis 2004 [@Gabard_2004 p.7]). However since Marin warned me in January 2013 (cf. Sec.\[e-mail-Viro:sec\]) it may be the case that the transition from the abstract to the embedded viewpoints is not so easy. Yet I am still confident that it holds true. The point is primarily a matter of projective algebraic geometry, namely the question if any abstract morphism on a concrete plane curve is induced by a pencil of ambient curves. This is either trivially true or trivially wrong, but alas I do not know the answer due to failing memory about the foundations of algebraic geometry. \[12.04.13\] Additionally, it can also be that sometimes imaginary basepoints have to be allowed, and so total reality really the mobile part of the pencil. We hope to be capable presenting this more clearly in the future, but see perhaps already (\[Le-Touzé-scholium-deg-6:lem\]).

[^77]: Whether this is implicit or not is an interpretation-matter, unless of course some direct contact with Rohlin (e.g. via the husband) testifies such a conjecture of Rohlin. Again it could be the case that Ahlfors theorem nearly trivially implies this (novel) conjecture of Rohlin. Even if so in abstracto then the game is probably far from finished as one would like to get synthetic descriptions of the total pencils. It seems quite likely that this game can keep busy several generation of workers. \[12\]

[^78]: \[12.04.13\] (Please skip this footnote, if you believe in capitalism).—We invented this exoplanet metaphor in 2004, as to sell our postdoc-research programme to an FNS-administrator (FNS=SNF=Schweizerische National Fond), specialized in astronomy (at some Geneva observatory). The success was very limited, no funding were ever obtained and much energy and time wasted for nothing. Some few weeks later another Swiss cooperative stole me 15’000 Euros of economies. Life then started to require environmental punch (nutrition in the containers, and other pleasant duties like bicycling the heavy nutriments over steep mountains). Can we develop a more tolerant science enrolling more people on less restrictive financial constraints, especially more modest retribution of the workers? Ahlfors is far from a hero in this respect (elitist attitude than looks much overdone in view of the little originality of his contributions to science, compare what he borrowed from Courant, Hurwitz, Riemann, Grötzsch, Teichmüller, etc.). The real question is of course: can we get rid off of capitalism, granting the fact that a sufficient motor of life is to reach immortality (for free and for all), as it was ever encoded in our genes since the amoebic morphogenesis.

[^79]: \[27.03.13\] Meanwhile the simplest counterexample, I was able to find is the Itenberg-Viro curve constructed on Fig.\[Itenberg:fig\].

[^80]: \[21.03.13\] Find accurate references, by Thom, Cerf, Hirzebruch, Milnor, Wall, etc. I confess that I lack a precise reference.

[^81]: That is the cautious Petrovskii version of Ragsdale’s estimates.

[^82]: It seems to me that the estimates on $p$ follows from Thom’s conjecture, as explained in Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]

[^83]: Tarik Garidi (aus der Nordseeküste) is a well-known scientist in Geneva (student of Piron), specialized in anti-de-Sitter and notorious for having introduced a mass concept which can be negative-valued, like the signed difference $\Delta \Pi:=\Pi^+-\Pi^-$ of Rohlin.

[^84]: \[18.03.13\] I think that (modest) Theorem \[Alsatian-schemes:thm\] below corrupts this belief of Th. Fiedler (who left the subject a long time ago), yet this does not jeopardize at all his invaluable help (and incredible memory!) in view of all the crucial corrections he took care to make on the present text.

[^85]: Our notational trick is to denote with parenthesis the degree of the scheme, since without parenthesis e.g. in an $M$-scheme the magnitude in front is traditionally not the degree but the number of ovals of the scheme.

[^86]: The usual notation is $n^-$ at least to be conform with Rohlin 1978 [@Rohlin_1978 p.86–87], but I keep it so to stay faithful to the message of Fiedler.

[^87]: As far as I am informed the general coinage of this trichotomy is due to Felix Klein (in geometry) and Dubois-Reymond (in PDE’s).

[^88]: From the Greek “poros”=“hole” (aping a bit Grothendieck’s “topos” or “topoi”).

[^89]: It is crucial here to adopt the modern convention regarding the sign of Euler’s $\chi$. This is courtesy of Michel Kervaire, that turned out to be correct when looking at old texts, like perhaps Listing, von Dyck 1988, Poincaré 1885–1895, etc., where the opposite sign convention was used!

[^90]: \[19.03.13\] The answer is no and follows from Petrovskii’s inequalities.

[^91]: As we shall soon recall, since Thom/Kronheimer-Mrowka 1995 we may replace “nicht immer” by “never”!!! (provided $m\ge 6$) and $m$ even. \[17.03.13\] In fact this was known much earlier since Petrovskii 1938, cf. Lemma \[Hilbert’s-nesting-intuition:lem\] below. An elementary proof also follows from Rohlin’s formula as reminded in the same lemma. It would be interesting to say more on the odd degree case (again Petrovskii 1938 [@Petrowsky_1938] should suffice to corroboration “Hilbert thesis” for $m\ge 7$).

[^92]: So-called because V.A. Rohlin was born in Bakou, from parents themselves coming from Odessa, and if I am not wrong in Geography Bakou belongs to Caucasus.

[^93]: This French prose is borrowed by René Thom, in a letter to André Haefliger reproduced in part in a recent issue of L’Enseign. Math., ca. 2011–12.

[^94]: I.e. the present text as it was on the date of the 09.01.13, meanwhile pagination may have changed.

[^95]: This is not perfectly true, as I exposed Ahlfors theorem to Oleg Viro during its last visit in Geneva (ca. 2010–11, his talk on fields of char 1, Connes, tropical geometry, etc.), yet probably my explanations where so obscure that Oleg immediately forgot about it.

[^96]: =Vladimir Abramovich Rohlin, of course.

[^97]: Sic, singular or plural? Not so important of course.

[^98]: i.e. Lemma \[Klein-Marin:lem\], warning meanwhile the numbering may changes, but the one in this footnote is automatic (hence the right one)

[^99]: “isotopie” of course.

[^100]: \[26.03.13\] This “evidently” sounds to me a bit sloppy, as the full credit for this remark goes to Viro.

[^101]: Meanwhile this numbering may have changed into Quote \[Klein\_1876-niemals-isolierte:quote\].

[^102]: Not clear how to interpret this? Does it mean that Shustin’s claim is wrong, or simply that this scheme is an $(M-1)$-scheme. My question was whether this $(M_1)$-scheme is realized algebraically, of course. Yet, I admit that my question was a bit ill posed.

[^103]: \[24.01.13\] On reading Viro’ survey 1989/90 [@Viro_1989/90-Construction p.1085, 2.5.H], one should easily locate the source for this assertion. References seems to be Viro 1983 [@Viro_1983/84-new-prohibitions], and the survey Viro 1986 [@Viro_1986/86-Progress]. As explained there (Viro 1989/90, p.1085) this is a prohibition not coming from topology, but from Bézout. In fact this result is mentioned again in Viro 1989/90 [@Viro_1989/90-Construction p.1126, 5.3.E], with the exact cross-reference as being Viro 1983 [@Viro_1983/84-new-prohibitions]

[^104]: \[30.01.13\] This is another very interesting (inedited?) piece of information, as to my knowledge this never appeared under the (printed) pen of V.A. Rohlin. So here Viro tell us something very interesting not yet available in print (as far as I know).

[^105]: \[30.01.13\] This is probably true yet this requires a more highbrow dissipation theory, than in Gudkov’s second existence proof which apart from the trick with Cremona uses only dissipation of ordinary nodes à la Brusotti 1921 (so-called $A_1$-singularities).

[^106]: \[30.01.13\] This alas turned out to be quite foiled as the invisible part of the discriminant as real codimension 2.

[^107]: \[26.03.13\] It turned that this is wrong, cf. Fiedler’s letter dated \[21.03.13\].

[^108]: \[26.03.13\] This is again a misconception of mine, since the estimate $\chi\le k^2$ of Theorem \[Thom-Ragsdale:thm\] was based on the erroneous supposition that the Arnold surface is always orientable, cf. again Fiedler’s letter dated \[21.03.13\].

[^109]: For the same reason as the previous footnote this comment is not pertinent anymore.

[^110]: Presumably, the authors omit the rotational ambiguity.

[^111]: Of course Ahlfors’ statement is somewhat stronger giving $r \le n\le r+2p$, where $r$ is the number of contours and $p$ the genus.

[^112]: According to Havinson 2003/04 [@Havinson_2003/04-Erokhin1958], this terminology is due to Erokhin 1958: “In accordance with V.D. Erokhin’s proposal (1958), the quantity $\gamma(F)$ has been called the [*analytic capacity*]{} or the [*Ahlfors capacity*]{} since that time.”

[^113]: Who exactly? candidates: Golusin, Havinson, Havin, Vitushkin, etc., but see also Nehari (alias Willi Weisbach) as early as 1950. Indeed, “Ahlfors’ extremal function” occurs already in Nehari’s survey 1950 [@Nehari_1951-survey-BAMS p.357], and “Ahlfors mapping” alone occurs in Nehari 1950 [@Nehari_1950 p.267]. This probably beats any Russian contribution, for one of the first text is Golusin 1952/57 [@Golusin_1952/57], where actually the term “Ahlfors function” is not employed. However Havinson torrential list of publication on the topic starts as early as 1949 [@Havinson_1949].

[^114]: Existence is ensured under the mild condition that the domain supports nonconstant bounded analytic functions.

[^115]: A coinage of Carathéodory, cf. Carathéodory 1912 [@Caratheodory_1912].

[^116]: This seems to be a misprint, and should be “$n$ zeros” (\[27.09.12\]). Further it is tacitly assumed that the domain is bounded by Jordan curve, for pointlike punctures are removable singularities hence do not affect the Ahlfors function. To be concrete making $(n-1)$ punctures in the unit disc the domain reaches connectivity $n$ but its Ahlfors function is still the identity as if there were no punctures.

[^117]: Again “$n$ times covered disk” sounds more correct.

[^118]: This is indeed one of the fascinating difficulty also discussed in A. Mori 1951 [@Mori_1951] and Fedorov 1991 [@Fedorov_1991], who coins the lovely prose of a “rather opaque condition must be satisfied”.

[^119]: Here there is maybe a wrong cross-reference and Myrberg 1933 [@Myrberg_1933] was rather understood?

[^120]: Can one be more explicit? Hahn-Banach like in Read 1958 [@Read_1958_Acta] or Royden 1962 [@Royden_1962] or just something more in the realm of classical analysis.

[^121]: Of course for this purpose it would have been enough to cite Bieberbach 1925 [@Bieberbach_1925].

[^122]: This is true modulo the possibility of the planar case (i.e. Harnack-maximal Schottky double).

[^123]: Addition of Gabard, otherwise seems an abuse of notation.

[^124]: This argument looks all right, yet it seems to the writer than one can easily dispense of the concept of orientability, by just using the separation effected by the existence of the map induced on imaginary loci, i.e. $X({\Bbb C})-X({\Bbb R})\to {\Bbb P}^1({\Bbb C})-{\Bbb
    P}^1({\Bbb R})$.

[^125]: This is maybe a misprint and the “$K$” should be an $E$?

[^126]: Of course the notation $P$ instead of $N$ could have been more appealing, yet Forelli had obviously to reserve the letter $P$ for “probability measures”, to enter soon the arena! So imagine the “$N$” standing for non-negative real parts (which is incidentally more correct if we let penetrate the boundary behavior in the game).

[^127]: Of course behind both techniques there is the paradigm of compactness in suitable function spaces, first occurring as a such in the related Hilbert’s investigation on the Dirichlet principle (add maybe Arzelà-Vitali to be fair, cf. e.g. Zaremba 1910 [@Zaremba_1910]). So everything started to be solid after Hilbert 1900, and Montel 1907, etc.

[^128]: This is German for belt (=ceinture) in French.

[^129]: This is indeed quite trivial to see, if we know the Riemann(-Roch) inequality, cf. e.g. Gabard 2006 [@Gabard_2006].

[^130]: Of course any geometric topologist (or reasonable being) could find the writing $\partial \overline
    W$ semantically more precise, yet we follow Forelli’s alleged notation.

[^131]: “Les anglaises c’est comme le pudding, elles ne bougent pas quand on fait l’amour.” (Joke from Sherbrooke, learned from Gaston Boulé).

[^132]: Jargon of Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.42], implying that the map covers each point the same number of times (counting properly by multiplicity); but of course inspired by Stïlow’s book 1938 [@Stoilow_1938-Lecons].

[^133]: Some specialists from Grenoble (especially Emmanuel Ferrand) told me (ca. 1999/00) that the idea of filling the membrane by the insides of the ovals truly goes back to Arnold, which is probably essentially correct, yet Rohlin’s full credit for effecting the lovely perturbation and counting things properly is surely not at all affected.

[^134]: Gabard’s addition

[^135]: Severi perhaps? Try also Hurwitz??

[^136]: Try also Nevanlinna.

[^137]: \[26.03.13\] This is a misconception of Gabard, that was corrected in Fiedler’s letter dated \[21.03.13\].

[^138]: This prose is bracketed as it seems to be verbatim copied from Rüedy 1971 [@Ruedy_1971].

[^139]: It seems to be rather earlier!??

[^140]: No attempt to correct the English, since Gabard’s English is even more indigest than the everything what has been ever written.

[^141]: Means “mouche” in Polish

[^142]: Perhaps it would be better to say no accumulation point.

[^143]: Read perhaps multiconnected to be more faithful to Riemann’s original text.

[^144]: Easy to sharpen as Klein 1876 [@Klein_1876].

[^145]: This is especially true under the Russian perspective, yet in the West workers were a bit more universalist, e.g. Koebe 1907 [@Koebe_1907_UrAK], J. Douglas 1936 [@Douglas_1936-Some-new-results], Teichmüller 1939 [@Teichmueller_1939], Ahlfors 1950 [@Ahlfors_1950], Schiffer-Spencer 1954 [@Schiffer-Spencer_1954], etc.

[^146]: Read “finite” to be more conventional.

[^147]: Big challenge: find where? Possibly this is not to be found in Klein and Teichmüller (probably lacking a good library during the war time) sloppily extrapolated what he remembered from his Klein reading (namely reality of Riemann surfaces, yet as far as we know never the total reality of orthosymmetric curves).

[^148]: If the surface is open this the non-trivial result of Behnke-Stein 1947/49 [@Behnke-Stein_1947/49].

[^149]: This jargon goes back to Weierstrass (vgl. etwa Schottky 1877 [@Schottky_1877]).

[^150]: Maybe here one can pinpoint about a confusion with the uniformization of Klein-Poincaré-Koebe.

[^151]: This is essentially the theorem of Bieberbach-Grunsky (with antecedent by Riemann and Schottky).

[^152]: Compare maybe also Hilbert and Courant for similar works
