Uniqueness of the stereographic embedding
Michael Eastwood
The standard conformal compactification of Euclidean space is the round sphere. We use conformal geodesics to give an elementary proof that this is the only possible conformal compactification.
1412.0746.tex
\documentclass{amsart}
\setlength{\textwidth}{121mm}
\setlength{\textheight}{195mm}
% \addtolength{\headheight}{1pt}
\newtheorem{thm}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}
\newcommand{\im}{\operatorname{im}}
\newcommand{\Rho}{\mathrm{P}}
% \renewcommand{\thefootnote}{$\dagger$}
\begin{document}
\title{Uniqueness of the stereographic embedding}
\author{Michael Eastwood}
\address{\hskip-\parindent
Mathematical Sciences Institute\\
Australian National University\newline
ACT 0200, Australia}
\email{meastwoo@member.ams.org}
\subjclass{Primary 53A30; Secondary 53C22.}
% \footnotetext{Supported by the Australian Research Council.}
\begin{abstract}
The standard conformal compactification of Euclidean space is the round 
sphere. We use conformal geodesics to give an elementary proof that this is 
the only possible conformal compactification.
\end{abstract}
\renewcommand{\subjclassname}{\textup{2010} Mathematics Subject Classification}
\maketitle

\section{Introduction}\label{intro}
At the $34^{\mathrm{th}}$ Winter School on Geometry and Physics, held as always
in Srn\'{\i} in the Czech Republic, Charles Franc\`es gave a plenary series of
talks on `Conformal boundaries in pseudo-Riemannian geometry.' He discussed the
results contained in~\cite{F} and, in particular, the notions of {\em conformal
embedding\/}
$$(M,[g])\hookrightarrow^{\mathrm{open}}(N,[h])$$
and {\em conformal maximality\/} (where there is no non-trivial conformal 
embedding). The prototype conformal embedding is
$$({\mathbb{R}}^n,[\mbox{Euclidean metric}])\hookrightarrow
(S^n,[\mbox{round metric}])$$
where the image is the complement of a single point, which we shall write
as~$\infty$, and the mapping in the other direction
$S^n\setminus\{\infty\}\to{\mathbb{R}}^n$ is the standard stereographic
projection, which is well-known to be conformal. It is proved in \cite{F} that 
this is the only non-trivial conformal embedding of Euclidean space. In 
particular, it is the only possible {\em conformal compactification\/} 
of~${\mathbb{R}}^n$ as follows.
\begin{thm}\label{main_thm}
Suppose $n\geq 3$ and $(M,g)$ is a compact Riemannian $n$-manifold and
$\iota:{\mathbb{R}}^n\hookrightarrow M$ realises ${\mathbb{R}}^n$ as an open
subset of $M$ such that $\iota^*g$ is conformal to the standard Euclidean
metric on~${\mathbb{R}}^n$. Then $M$ must be the $n$-sphere~$S^n$, the metric
$g$ must be conformal to the standard round metric, and the embedding must be
inverse to the standard stereographic projection.
\end{thm}
Of course, this is the expected result and in \cite{F} two proofs are
presented. One proof \cite[Theorem~1.4]{F} is via a removable singularities
theorem based on the complement of ${\mathbb{R}}^n\hookrightarrow S^n$ having
small Hausdorff dimension. The other proof \cite[Proposition~2.3]{F} is based
on B.~Schmidt's notion of {\em conformal boundary\/} as further developed by
Franc\`es.

This article gives a proof of Theorem~\ref{main_thm} using {\em conformal
geodesics\/}. Apart from the elementary nature of this proof, it seems to be a
natural one and a good application of the theory conformal geodesics, which are
perhaps under-used in conformal differential geometry (unfortunately, as 
lamented in~\cite{T}, their behaviour is not so well understood in general).

Although Theorem~\ref{main_thm} also holds when $n=2$, the result is of a
different nature (global complex analysis rather than local to global
differential geometry). Also note that the corresponding theorem is false for
projective differential geometry: there are two distinct projective
compactifications of ${\mathbb{R}}^n$, namely
${\mathbb{R}}^n\hookrightarrow{\mathbb{RP}}_n$ as a standard affine
co\"ordinate patch and ${\mathbb{R}}^n\hookrightarrow S^n$ onto an open
hemisphere by inverse central projection.

This article starts with an exposition of conformal geodesics, which is more
than is strictly needed for the proof of Theorem~\ref{main_thm}. Hopefully,
this exposition will be useful for their further application in conformal
geometry.

I would like to thank Vladim\'{\i}r Sou\v{c}ek for his invitation to Charles
University and the Czech Winter School. I would also like to thank Charles
Franc\`es for an inspiring series of talks and for useful conversations.

\section{Conformal Geodesics}
Also known as {\em conformal circles\/}~\cite{BE}, these special curves may be
defined as follows. For any metric $g_{ab}$ in the conformal class, a curve
$\gamma\hookrightarrow M$ with parameterization $\tau:\gamma\to{\mathbb{R}}$ is
said to be a {\em conformal geodesic\/} if and only if
\begin{equation}\label{circles}
\partial A^a=3\frac{V.A}{V.V}A^a-\frac{3A.A}{2V.V}V^a+V.V\Rho_b{}^aV^b
-2\Rho_{bc}V^bV^cV^a,\end{equation}
where
\begin{itemize}
\item $V^a$ is tangent along $\gamma$ so that $V^a\nabla_a\tau=1$, the 
`velocity vector,'
\item $\partial=\partial/\partial\tau\equiv V^a\nabla_a$ acting on any tensor
field along $\gamma$,
\item $\nabla_a$ is the Levi-Civita connection for $g_{ab}$,
\item $A^a\equiv\partial V^a$, the `acceleration vector,'
\item $\Rho_{ab}$ is the conformal Schouten tensor 
$$\Rho_{ab}=\frac1{n-2}\left(R_{ab}-\frac{1}{2(n-1)}Rg_{ab}\right),$$
\item $R_{ab}$ is the Ricci tensor and $R$ the scalar curvature,
\item $X.Y$ is the inner product $X^aY_a$ of two fields $X^a$ and $Y^a$. 
\end{itemize}
The equation (\ref{circles}) is an ordinary differential equation for both the
curve $\gamma$ and its parameterization $\tau:\gamma\to{\mathbb{R}}$.
Alternatively, if one uses
$$B^a\equiv\frac1{V.V}A^a-2\frac{V.A}{(V.V)^2}V^a$$
instead of the na\"{\i}ve acceleration, then (\ref{circles}) becomes
\begin{equation}\label{Pauls_circles}
\textstyle\partial B^a=V.BB^a-\frac12B.BV^a+\Rho_b{}^aV^b\end{equation}
(which is equation (2) from~\cite{T}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Explicit verification of this! %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Paul implicitly advocates setting
% $$B^a=\frac1{V.V}A^a-2\frac{V.A}{(V.V)^2}V^a$$
% noting that
% $$V.B=-\frac1{V.V}V.A\quad\mbox{and}\quad 
% B.B=\frac{A.A}{(V.V)^2}$$
% so that instead of 
% $$\hat{A}^a=A^a-V.V\Upsilon^b+2(V.\Upsilon)V^b,$$
% noting that $V.\hat{A}=V.A+V.VV.\Upsilon$, we have 
% $$\hat{B}^a=\frac1{V.V}(A^a-V.V\Upsilon^b+2(V.\Upsilon)V^b)
% -2\frac{V.A+V.VV.\Upsilon}{(V.V)^2}V^a=B^a-\Upsilon^a$$
% and now note that
% $$\partial\frac1{V.V}=-2\frac{V.A}{(V.V)^2}\quad\mbox{and}\quad
% \partial\frac{V.A}{(V.V)^2}=\frac{A.A}{(V.V)^2}
% +\frac{V.\partial A}{(V.V)^2}-4\frac{(V.A)^2}{(V.V)^3},$$
% and also that
% $$V.\partial A=3\frac{(V.A)^2}{V.V}-\frac{3A.A}{2}-V.V\Rho_{bc}V^bV^c$$
% whence
% $$\begin{array}{rcl}\partial B^a&=&\displaystyle
% \frac1{V.V}\partial A^a-4\frac{V.A}{(V.V)^2}A^a
% -2\left(\frac{A.A}{(V.V)^2}
% +\frac{V.\partial A}{(V.V)^2}-4\frac{(V.A)^2}{(V.V)^3}\right)V^a\\
% &=&\displaystyle
% \frac1{V.V}\partial A^a-4\frac{V.A}{(V.V)^2}A^a
% +2\left(\frac{A.A}{2(V.V)^2}
% +\frac{(V.A)^2}{(V.V)^3}+\frac{\Rho_{bc}V^bV^c}{V.V}\right)V^a\\
% &=&\displaystyle
% -\frac{V.A}{(V.V)^2}A^a
% -\frac{A.A}{2(V.V)^2}V^a+\Rho_b{}^aV^b
% +2\frac{(V.A)^2}{(V.V)^3}V^a\\
% &=&\displaystyle
% -\frac{V.A}{V.V}B^a
% -\frac{A.A}{2(V.V)^2}V^a+\Rho_b{}^aV^b\\
% &=&V.BB^a-\frac12B.BV^a+\Rho_b{}^aV^b\\
% \end{array}$$
% which agrees with (2) in~\cite{T}.
Conformal geodesics were introduced in~\cite{Y} and one can verify, as is done
explicitly in~\cite{BE}, that (\ref{circles})
(equivalently~(\ref{Pauls_circles}) as is done in~\cite{T}), is conformally
invariant. Unlike ordinary geodesics, however, the parameter $\tau$ is
determined only up to projective freedom $\tau\mapsto(a\tau+b)/(c\tau+d)$.

Although dependent on a choice of metric in the conformal class, one can choose
instead to use ordinary arc-length $t:\gamma\to{\mathbb{R}}$ as a parameter.
Besides, as noted in~\cite{C} via the Cartan connection and directly verified
in~\cite{BE}, any smooth curve on a conformal manifold inherits a preferred
family of parameterizations defined up to projective freedom so the location of
conformal geodesics and how they are parameterized can be regarded as separate
issues. As noted in~\cite{T}, an advantage of using arc-length is that it
avoids the spurious feature that a projective parameterization $\tau$ might
blow up for no good reason. Thus, for a choice of metric in the conformal
class, one can insist on the unit vector field $U^a\equiv V^a/\sqrt{V.V}$ along
$\gamma$ with corresponding parameter $t:\gamma\to{\mathbb{R}}$ such that 
\begin{itemize}
\item $\delta t=1$ where $\delta=\partial/\partial t\equiv U^a\nabla_a$
acting on any tensor field,
\item $C^a=\delta U^a$ is the acceleration field along $\gamma$, 
\end{itemize}
and then it is readily verified that
\begin{equation}\label{Pauls_proper_circles}
\delta C^a=\Rho_b{}^aU^b-(C.C+\Rho_{bc}U^bU^c)U^a.\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Working is as follows. %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% $$C^a=\delta U^a=\frac{1}{\sqrt{V.V}}\partial\frac{V^a}{\sqrt{V.V}}
% =\frac{A^a}{V.V}-\frac{V.A}{(V.V)^2}V^a$$
% so
% $$C^a=B^a-U.BU^a$$
% and
% $$\delta C^a=\delta B^a-C.BU^a-U.\delta BU^a-U.BC^a.$$
% However,
% $$\begin{array}{rcl}U.\delta B=\frac{1}{V.V}V.\partial B
% &=&\frac{1}{V.V}((V.B)^2-\frac12B.BV.V+\Rho_{bc}V^bV^c)\\
% &=&(U.B)^2-\frac12B.B+\Rho_{bc}U^bU^c
% \end{array}$$
% and
% $$C.C=B.B-(U.B)^2=C.B$$
% and
% $$\textstyle\delta B^a=\frac1{\sqrt{V.V}}(V.BB^a-\frac12B.BV^a+\Rho_b{}^aV^b)
% =(U.B)B^a-\frac12B.BU^a+\Rho_b{}^aU^b$$
% so
% $$\begin{array}{rcl}\delta C^a
% &=&(U.B)B^a-\frac12B.BU^a+\Rho_b{}^aU^b
% -C.CU^a\\
% &&\quad{}-((U.B)^2-\frac12B.B+\Rho_{bc}U^bU^c)U^a-U.BC^a\\
% &=&(U.B)B^a+\Rho_b{}^aU^b
% -C.CU^a\\
% &&\quad{}-((U.B)^2+\Rho_{bc}U^bU^c)U^a-U.B(B^a-U.BU^a)\\
% &=&\Rho_b{}^aU^b-(C.C+\Rho_{bc}U^bU^c)U^a,
% \end{array}$$
% as advertised.
This is the form of the conformal geodesic equation used in \cite{T} to 
conclude that 
\begin{itemize}
\item conformal geodesics always exist for short time,
\item conformal geodesics are uniquely determined by initial conditions:
\begin{itemize}
\item an arbitrary unit velocity vector $U^a|_{t=0}$,
\item an arbitrary acceleration vector $C^a|_{t=0}$ orthogonal to $U^a|_{t=0}$,
\end{itemize}
\item a conformal geodesic can always be continued as long as $C^a$ is finite.
\end{itemize}
Indeed, as observed in~\cite[Theorem~1.1]{T}, the proof is immediate from 
Picard's Theorem and one can also add that, provided the acceleration remains 
bounded and that solutions exist for $0\leq t\leq 1$, 
\begin{itemize}
\item the end point $\gamma^{-1}(1)$ depends smoothly on the initial 
conditions.
\end{itemize}
These conclusions are also valid for conformal geodesics formulated according
to (\ref{circles}) provided that the acceleration but also the preferred
parameter $\tau$ both remain bounded. (Observe that if our chosen metric
$g_{ab}$ is conformally rescaled $\hat{g}_{ab}=\Omega^2g_{ab}$, then our
various notions of acceleration change:
\begin{itemize}
\item $\hat{A}^a=A^a-V.V\Upsilon^b+2(V.\Upsilon)V^b$,
\item $\hat{B}_a=B_a-\Upsilon_a$,
\item $\hat{C}_a=C_a-\Upsilon_a+(U.\Upsilon)U_a$,
\end{itemize}
where $\Upsilon_a=\nabla_a\log\Omega$ but whether any of these
notions remain finite is clearly independent of choice of metric in the
conformal class.)

Notice that if $\gamma\hookrightarrow M$ is a conformal geodesic passing
through $a,b,\in M$ with preferred parameterization
$\tau:\gamma\to{\mathbb{R}}$ so that $\tau(a)=0$ and $\tau(b)=1$, then the same
curve can be also be parameterized by $\hat\tau=1-\tau$. More specifically, the
defining equation (\ref{circles}) holds with $\tau$ replaced by 
$\hat\tau$ but now $\hat\tau(b)=0$ and~$\hat\tau(a)=1$. We shall refer to this 
man{\oe}uvre as {\em reversing the parameterization\/}. 

% \newpage
In Euclidean space, we can solve (\ref{circles}) explicitly. In fact, if one
restricts (\ref{circles}) to a Euclidean plane
${\mathbb{R}}^2\hookrightarrow{\mathbb{R}}^n$, then one may readily verify
% $$V^a(t)=\frac{4}{((2-\alpha t)^2+\beta^2t^2)^2}
% \Big((2-\alpha t)^2-\beta^2t^2,2(2-\alpha t)\beta t\Big)$$
% and acceleration
% $$A^a(t)=\frac{8}{((2-\alpha t)^2+\beta^2t^2)^3}
% \Big(\cdots,\cdots\Big)$$
that the curve
\begin{equation}\label{probe}\tau\mapsto\big(x(\tau),y(\tau)\big)=
\frac{2}{(2-\alpha\tau)^2+\beta^2\tau^2}
\Big((2-\alpha\tau)\tau,\beta\tau^2\Big)\end{equation}
satisfies 
$$\frac{d}{d\tau} A^a=3\frac{V.A}{V.V}A^a-\frac{3A.A}{2V.V}V^a,$$
% $$2\Big[\Big(\frac{dx}{dt}\Big)^2+\Big(\frac{dy}{dt}\Big)^2\Big]
% \Big(\frac{d^3x}{dt^3},\frac{d^3y}{dt^3}\Big)=
% 6\Big[\Big]\Big(\frac{d^2x}{dt^2},\frac{d^2y}{dt^2}\Big)
% -3\Big[\Big]\Big(\frac{dx}{dt},\frac{dy}{dt}\Big),$$ 
where
$$V^a=(dx/d\tau,dy/d\tau)\quad\mbox{and}\quad
A^a=(d^2x/d\tau^2,d^2y/d\tau^2).$$
% $$V^a=\Big(\frac{dx}{d\tau},\frac{dy}{d\tau}\Big)\quad\mbox{and}\quad
% A^a=\Big(\frac{d^2x}{d\tau^2},\frac{d^2y}{d\tau^2}\Big).$$
Hence, this is the unique conformal circle in ${\mathbb{R}}^n$ with initial 
conditions
$$\begin{array}{ll}(x,y,\ldots)|_{\tau=0}=(0,0,\ldots,0)\quad
&(dx/d\tau,dy/d\tau,\ldots)|_{\tau=0}=(1,0,\ldots,0)\\[4pt]
&(d^2x/d\tau^2,d^2y/d\tau^2,\ldots)|_{\tau=0}=(\alpha,\beta,0,\ldots,0)
\end{array}$$
(and one may also verify that for $\beta\not=0$, the trajectory is, indeed, a
round circle in~${\mathbb{R}}^2$ (centred at $(0,1/\beta)$) whilst if
$\beta=0$, the trajectory is the $x$-axis with projective parameterization
$x=2\tau/(2-\alpha\tau)$). Notice that the curve (\ref{probe}) is defined for
all $\tau$ provided $\beta\not=0$, has a limit at
$$\frac{2}{\alpha^2+\beta^2}\Big({-\alpha},\beta\Big)$$
as $\tau\to\pm\infty$, and passes through
$$\frac{2}{(2-\alpha)^2+\beta^2}\Big(2-\alpha,\beta\Big)$$
when $\tau=1$. 

\section{The Uniqueness Theorem}
In this section we prove Theorem~\ref{main_thm}. For convenience, let us denote
by $\Omega$ the open subset $\iota({\mathbb{R}}^n)\subset(M,g)$ that is
supposed conformal to~${\mathbb{R}}^n$. Initially, we know very little about
$\Omega\subset M$:
$$\begin{picture}(120,80)\linethickness{.6pt}
\qbezier(0,50) (0,80) (30,80)
\qbezier(30,80) (50,80) (60,70)
\qbezier(60,70) (70,60) (80,70)
\qbezier(80,70) (90,80) (110,40)
\qbezier(110,40) (120,20) (120,10)
\qbezier(120,10) (120,0) (80,20)
\qbezier(80,20) (70,25) (60,20)
\qbezier(60,20) (20,0) (10,10)
\qbezier(10,10) (0,20) (0,50)
\qbezier(90,50) (110,40) (100,25)
\qbezier(93,48) (90,38) (102,29)
\qbezier(65,66) (65,44) (75,22)
\qbezier(67,45) (60,47) (55,45)
\qbezier(60,55) (63,45) (60,35)
\qbezier(45,30) (60,50) (68,30)
\qbezier(50,20) (60,30) (50,40)
\put(10,45){$\Omega\cong{\mathbb{R}}^n$}
\put(110,55){$M$}
\end{picture}$$
In particular, the boundary $\partial\Omega\subset M$ may be very badly
behaved. Nevertheless, by surrounding each point of $\partial\Omega$ by a
co\"ordinate patch $U$ in which one considers Euclidean spheres centred on
points in $U\cap\Omega$ (cf.~\cite[Lemme~14]{Fr}), it follows that in 
$\partial\Omega$ there is a dense set of {\em highly accessible\/} points~$p$,
$$\begin{picture}(40,40)\linethickness{.6pt}
\put(20,20){\circle{40}}
\qbezier(40,20) (30,10) (20,10)
\qbezier(25,5) (25,20) (35,25)
\put(20,20){\circle{16}}
\put(24.5,14){$\bullet$}
\put(12,30){$\Omega$}
\put(20,18){$p$}
\end{picture}$$
namely those for which all smooth curves $\gamma(s)$ emanating from $p$ with
velocity on one side of a suitable hyperplane in $T_pM$ remain inside $\Omega$
for $0<s<\epsilon$ for some $\epsilon>0$. In particular, this property is valid
for conformal circles emanating from $p$ with velocity pointing into $\Omega$
in this sense. Let us fix a highly accessible point in $\partial\Omega$ and, at
the risk of putting the cart before the horse, denote it by~$\infty$. We aim to
show that $\partial\Omega=\{\infty\}$. Indeed, if we can show this, then
Theorem~\ref{main_thm} follows easily because then the conformal Weyl
curvature, in case $n\geq 4$, or the Cotton-York tensor, in case $n=3$,
vanishes on $M$ by continuity. This implies that $M$ is conformally flat and
Theorem~\ref{main_thm} follows from Liouville's theorem.

Let us fix a conformal geodesic $\gamma\hookrightarrow M$ with preferred
parameterization $\tau:\gamma\to{\mathbb{R}}$ emanating from $\infty$. By
rescaling the parameterization we may arrange that $\gamma$ is contained inside
$\Omega$ for $0<\tau\leq 1$. Finally, by reversing the parameterization, we
obtain a conformal geodesic starting at a point inside $\Omega$ when $\tau=0$
and ending up at a point $\infty\in\partial\Omega$ when $\tau=1$.

But recall that conformal geodesics and their preferred parameterizations
depend only on the conformal structure and that $\Omega$ is supposed to be
conformal to the standard Euclidean metric on~${\mathbb{R}}^n$. At the end of
the previous section we determined all the conformal circles
in~${\mathbb{R}}^n$ to be straight lines or round circles. A round circle is
bounded and so we are left with having constructed what, from the Euclidean
viewpoint, is simply a straight line. After a Euclidean motion, this straight
line may as well be the $x$-axis with parameterization starting at the origin
and to have the parameter $\tau$ run from $0$ to $1$ at $\infty$ completely
fixes the curve as
\begin{equation}\label{tauoveroneminustau}
\tau\mapsto\Big(\frac{\tau}{1-\tau},0,0,\ldots,0\Big).\end{equation}
In other words, it is the case $(\alpha,\beta)=(2,0)$ in the discussion above. 
Let us compare this curve with the conformal geodesic in 
${\mathbb{R}}^n$ having initial conditions
$$\begin{array}{c}\big(x,y,\ldots\big)|_{\tau=0}=\big(0,0,\ldots,0\big)\qquad
\big(dx/d\tau,dy/d\tau,\ldots\big)|_{\tau=0}=\big(1,0,\ldots,0\big)\\[4pt]
\big(d^2x/d\tau^2,d^2y/d\tau^2,\ldots\big)|_{\tau=0}=
\big(\alpha,\sigma(2-\alpha),0,\ldots,0\big)
\end{array}$$
for some fixed $\sigma$ and for some $0\leq\alpha<2$. According to our previous
discussion, it is the curve (\ref{probe}) with $\beta=\sigma(2-\alpha)$, namely
$$\tau\mapsto
\frac{2}{(2-\alpha\tau)^2+\sigma^2(2-\alpha)^2\tau^2}
\Big((2-\alpha\tau)\tau,\sigma(2-\alpha)\tau^2,0,\ldots,0\Big).$$
When $\tau=1$ this conformal geodesic passes through the point
\begin{equation}\label{endpoint}\frac{2}{2-\alpha}\frac1{1+\sigma^2}
\Big(1,\sigma,0,\ldots,0\Big).
\end{equation} 
Now let $\alpha\uparrow 2$. We see that the endpoint (\ref{endpoint}) moves
monotonely out along the straight line in the direction
$(1,\sigma,0,\ldots,0)$. At least, this is what we see in
$\Omega\cong{\mathbb{R}}^n$. Viewed in $M\supset\Omega$, there is no problem
with the limit curve with initial conditions $(\alpha,\beta)=(2,0)$. By
construction, it is the curve (\ref{tauoveroneminustau}) joining the origin
$0\in{\mathbb{R}}^n\cong\Omega\hookrightarrow M$ to $\infty\in
M\setminus\Omega$. We conclude that the endpoint (\ref{endpoint}) tends to
$\infty\in M$ along the straight line in the direction 
$(1,\sigma,0,\ldots,0)$. At this point, there are several ways to complete the 
proof, one of which is as follows.

Even if we restrict to $|\sigma|\leq 1$ to force uniform convergence
to~$\infty$, we are free to rotate around the $x$-axis to obtain an
entire cone 
% $$\begin{picture}(100,60)
% \qbezier (30., 0.), (30., 20.70), (21.21, 35.35)
% \qbezier (21.21, 35.35), (12.42, 50.), (0., 50.)
% \qbezier (0., 50.), (-12.42, 50.), (-21.21, 35.35)
% \qbezier (-21.21, 35.35), (-30., 20.70), (-30., 0.)
% \qbezier (-30., 0.), (-30., -20.7), (-21.21, -35.35)
% \qbezier (-21.21, -35.35), (-12.42, -50.), (0., -50.)
% \qbezier (0., -50.), (12.42, -50.), (21.21, -35.35)
% \qbezier (21.21, -35.35), (30., -20.70), (30., 0.)
% \end{picture}$$
$$\begin{picture}(100,80)(-50,-40)\linethickness{.6pt}
\qbezier (10., 0.), (10., 12.42), (7.070, 21.21)
\qbezier (7.070, 21.21), (4.14, 30.), (0., 30.)
\qbezier (0., 30.), (-4.14, 30.), (-7.070, 21.21)
\qbezier (-7.070, 21.21), (-10., 12.42), (-10., 1.5)
\qbezier (-10., -1.5), (-10., -12.42), (-7.070, -21.21)
\qbezier (-7.070, -21.21), (-4.14, -30.), (0., -30.)
\qbezier (0., -30.), (4.14, -30.), (7.070, -21.21)
\qbezier (7.070, -21.21), (10., -12.42), (10., 0.)
\thicklines
\put(-32,0){\line(1,0){40}}\put(12,0){\vector(1,0){40}}
\put(-32,0){\line(1,1){40}}
\put(-32,0){\line(1,-1){40}}
\put(20,3){$x$-axis}
\put(-40,0){$0$}
\end{picture}$$
all the rays of which end up at $\infty\in M$. But now we may repeat this
argument starting with any ray from this cone, rather than the $x$-axis. In
this way we obtain a finite collection of cones that cover all rays emanating
from the origin. We conclude that $\infty$ is the only boundary point
of~$\Omega\subset M$.\hfill Q.E.D.

% ROUGH IDEA AS FOLLOWS
% 
% Concentrate on ${\mathbb{R}}^2$ and explore the line through the origin in
% direction $(1,\sigma)$. The line is, of course, a conformal circle. If we 
% also fire out a conformal circle (\ref{probe}) from the origin for
% $\beta=\sigma(2-\alpha)$ then when $\tau=1$ we end up at
% $$\frac{2}{(2-\alpha)^2+\sigma^2(2-\alpha)^2}
% \Big(2-\alpha,\sigma(2-\alpha)\Big)=
% \frac{2}{(2-\alpha)}\frac{1}{1+\sigma^2}(1,\sigma)$$
% and now let $\alpha\uparrow 2$. On the one hand this point goes out along the
% straight line through $(1,\sigma)$. On the other hand, the intial conditions 
% converge to
% $$(\alpha,\beta)=(2,0)$$
% no matter what is $\sigma$ and the corresponding curve is the $x$-axis 
% parameterised by 
% $$\tau\mapsto\frac\tau{1-\tau}$$
% the end point of which is out at infinity and this conclusion is independent 
% of choice of $\sigma$. Now suppose there are two highly accessible points in
% $\partial\Omega$ and run conformal geodesics into $\Omega$ from to both of
% them. Joint endpoints by a straight line to obtain a contradiction unless 
% they are parallel with opposite orientation. Then they actually have to join
% up! Well, just two points in $\partial\Omega$ is also good enough for 
% conformal flatness and three points is just too much to bear.

\begin{thebibliography}{11}

\bibitem{BE} T.N. Bailey and M.G. Eastwood, 
{\em Conformal circles and parametrizations of curves in conformal manifolds},
Proc. Amer. Math. Soc. {\bf 108} (1990), 215--221.

\bibitem{C} \'E. Cartan,
{\em Les espaces \`a connexion conforme}, 
Ann. Soc. Polon. Math. {\bf 2} (1923), 171--221.

\bibitem{Fr} C. Franc\`es,
{\em Sur le groupe d'automorphismes des g\'eom\'etries paraboliques de rang 1},
Ann. Sci. \'Ecole Norm. Sup. {\bf 40} (2007), 741--764.

\bibitem{F} C. Franc\`es, 
{\em Rigidity at the boundary for conformal structures and other Cartan 
geometries}, arXiv:0806:1008.

\bibitem{T} K.P. Tod,
{\em Some examples of the behaviour of conformal geodesics}, 
Jour. Geom. Phys. {\bf 62} (2012), 1778--1792. 

\bibitem{Y} K. Yano,
{\em The Theory of Lie Derivatives and its Applications},
North-Holland 1957.

\end{thebibliography}
\end{document}

1412.0746.extracted_bib
{BE T.N. Bailey and M.G. Eastwood, {\em Conformal circles and parametrizations of curves in conformal manifolds, Proc. Amer. Math. Soc. {108 (1990), 215--221.
1412.0746.extracted_bib
{C \'E. Cartan, {\em Les espaces \`a connexion conforme, Ann. Soc. Polon. Math. {2 (1923), 171--221.
1412.0746.extracted_bib
{Fr C. Franc\`es, {\em Sur le groupe d'automorphismes des g\'eom\'etries paraboliques de rang 1, Ann. Sci. \'Ecole Norm. Sup. {40 (2007), 741--764.
1412.0746.extracted_bib
{F C. Franc\`es, {\em Rigidity at the boundary for conformal structures and other Cartan geometries, arXiv:0806:1008.
1412.0746.extracted_bib
{T K.P. Tod, {\em Some examples of the behaviour of conformal geodesics, Jour. Geom. Phys. {62 (2012), 1778--1792.
1412.0746.extracted_bib
{Y K. Yano, {\em The Theory of Lie Derivatives and its Applications, North-Holland 1957.