An old theme in geometry involves the study of constant curvature metrics on surfaces with isolated conic singularities and with prescribed cone angles. This has been studied from many points of view, ranging from synthetic geometry to geometric analysis to algebraic geometry to calculus of variations. I will describe some of this, with notable recent highlights by Mondello-Panov and Kapovich and Dey, but ultimately focus on some recent work with Xuwen Zhu, based on my earlier work with Weiss, concerning the complicated stratified structure of the moduli space ofthese metrics, and the analytic problem of obtaining an unobstructed deformation problem.
Traditional complex analysis focuses on a single space, like a domain in Euclidean space, or more generally a complex manifold, and studies holomorphic maps on that space, into some target space. The typical target space for a domain is the complex plane, but for complex manifolds there is sometimes a need to consider a different target space. The easiest target spaces are line bundles, and then one considers sections rather than functions; the case of the trivial bundle corresponds to studying the graphs of holomorphic functions.
Often the set of all holomorphic sections of a line bundle is simply too vast, so one considers natural subsets of sections. A fruitful approach is to study those sections that are square-integrable with respect to some Hermitian inner product structure and a volume form. The focus on Hilbert spaces was introduced by S. Bergman in the early 20th century, and came into maturity via PDE methods through the works of many distinguished mathematicians including Bochner, Kodaira, Spencer, Morrey, Kohn, H\ormander, Andreotti, Siu, Yau, Bombieri, Demailly and their many students and colleagues since.
With the vast knowledge about the case of a single manifold and Hilbert space in hand, people have begun to examine how these function spaces vary when certain parameters are changed. One could change the complex
structure of the underlying manifold (as was done for compact Riemann surfaces by Teichmuller, and more thoroughly and Ahlfors and his school), and one could also change the Hermitian structure used to define
the Hilbert spaces. Algebraic geometers have been exploring such deformations for a very long time, via the Hodge Theorem and its variants, to great effect, especially under certain strong curvature assumptions. However, as one relaxes these curvature assumptions somewhat, the black box given by the Hodge Theorem cannot be used any longer, and one has to look under the hood.
Perhaps the crowning achievement of these more refined Hilbert spaces methods is the L^2 extension theorem of Ohsawa-Takegoshi; a theorem that gives sufficient, almost necessary conditions for extending weighted square-integrable holomorphic functions from a submanifold. The theorem opened the door for arguments that use induction on dimension, and a major advance was Siu's deformation invariance of plurigenera, which eventually lead to a fundamental breakthrough in the minimal model program of birational geometry.
In 2009, Bo Berndtsson published a paper in the Annals of Mathematics, in which he established two theorems that measure, in a very precise way, the variation of Hilbert spaces of holomorphic sections of line bundles on a complete Kahler manifold. His first theorem, which is stated on the poster for this Atelier, deals with the case of pseudoconvex domains, while his second theorem deals with the case of compact Kahler manifolds. The two theorems have already seen an incredible number of applications, but surely this is just the beginning.
Ordinary differential equations in the complex plane is a classical topic that was related from the beginning with Hodge theory, i.e.the properties of holomorphic forms integrated over cycles on complex manifolds. These concepts can be considered also from a more dynamical perspective and in the talk I will discuss the relation with some invariants arising in dynamics. The thread that runs through the whole discussion is discrete subgroups of Lie groups, and I will provide an introduction to the relevant concepts in Hodge theory and dynamics.
The celebrated work of Eells and Sampson initiated a wide interest in the study of harmonic maps between Riemannian manifolds, and harmonic maps have proven to be a useful tool in geometry. A more recent development is the harmonic map theory for non-smooth spaces. The seminal works of Gromov-Schoen and Korevaar-Schoen consider harmonic maps from a Riemannian domain into a non-Riemannian target. Much of the work to date in the singular setting assumes non-positivity of curvature of the target. In this talk I will discuss joint work with Breiner, Huang, Mese, Sargent, Zhang on existence and regularity results for harmonic maps when the target curvature is bounded above by a constant that is not necessarily 0.
A recent work by Mazzeo-Swoboda-Weiss-Witt describes a stratum of the frontier of the space of SL(2,C) surface group representations in terms of 'limiting configurations' which solve a degenerate version of Hitchin's equations on a Riemann surface. We interpret these objects in (a mapping class group invariant way in) terms of the hyperbolic geometric objects of shearings of pleated surfaces. We study limits of opers in this perspective. This is joint with A. Ott, J. Swoboda, and R. Wentworth.