\abstracttitlefont\centre Hyperbolic Plateau problems\par
Plateau_050110.tex
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% Ici on definit les sous titres.

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% Ici on definit les sous-sous titres.

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% Commandes et Symboles
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% Ici on definit la famille mathroman...

\font\mathromanten=cmr10
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% Ici on definit la famille mathsf qui n'est pas definie dans Plain.

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% Ici on definit la famille mathbf qui n'est pas definie dans Plain.

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% Ici on definit \mathi et \mathj. En fait, cela n'est pas necessaire, car ces symboles existent deja sous les noms respectivement de \imath et \jmath. Mais,
% ca reste une exercice interessant.

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\mathchardef\mathi="7\hexa\mycmmifam7B
\mathchardef\mathj="7\hexa\mycmmifam7C


% Ici je definit \mybeth, \mygimmel et \mydaleth

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\mathchardef\mygimmel="7\hexa\mymsbmfam6A
\mathchardef\mydaleth="7\hexa\mymsbmfam6B

% Ici je definit la fonction qui insere des figures.

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\ht10=0pt%
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\def\placefigure#1#2#3{%
\medskip%
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\vbox{\line{\hfil#2\epsfbox{#3}#1\hfil}%
\vskip 0.3cm%
\line{\noindent\hfil\sl Figure \nextfigno\hfil}}%
\medskip%
\endinsert%
}

% Ici je definit les fonctions \proclaim et \endproclaim. Les premieres versions sont plus jolies mais je ne sais pas comment les couper verticalement, et il y
% a alors des problemes de mise en page. Les deuxiemes versions sont plus simples, mais il n'y a pas de problem de mise en page.

\newif\ifinproclaim%
\global\inproclaimfalse%
\def\proclaim#1{%
\medskip%
%
% J'insere ici un bgroup pour que les variables que je definis ici soit uniquement locales.
%
\bgroup%
\inproclaimtrue%
\setbox10=\vbox\bgroup\leftskip=0.8em\noindent{\bf #1}\sl%
}

\def\endproclaim{%
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\wd12=0pt%
\setbox13=\hbox{\noindent\box12\box10}%
\noindent\unhbox13%
\egroup%
\medskip\ignorespaces%
}

\def\proclaim#1{%
\medskip%
\bgroup%
\inproclaimtrue%
\noindent{\bf #1}%
\nobreak\medskip%
\sl%
}

\def\endproclaim{%
\mypar\egroup\penalty\proclaimpenalty\medskip\ignorespaces%
}

\def\noskipproclaim#1{%
\medskip%
\bgroup%
\inproclaimtrue%
\noindent{\bf #1}\nobreak\sl%
}

\def\endnoskipproclaim{%
\mypar\egroup\penalty\proclaimpenalty\medskip\ignorespaces%
}

% Ici on definit quelques autres commandes dont on aura besoin, en particulier, le colon francais.

\def\ninn{{n\in\Bbb{N}}}
\def\derivee#1{\frac{\partial}{\partial #1}}
\def\colon{\hphantom{0}:}
\def\proof{{\noindent\bf Proof:\ }}
\def\demonstration{{\noindent\sl D\'emonstration :\ }}
\def\remarque{{\noindent\sl Remarque :\ }}
\def\remark{{\noindent\sl Remark:\ }}
\def\suite#1#2{({#1}_{#2})_{#2\in\Bbb{N}}}
\def\mlim{\mathop{{\mathroman Lim}}}
\def\mlimsup{\mathop{{\mathroman LimSup}}}
\def\mliminf{\mathop{{\mathroman LimInf}}}
\def\msup{\mathop{{\mathroman Sup}}}
\def\minf{\mathop{{\mathroman Inf}}}
\def\msf#1{{\mathsf #1}}
\def\mbf#1{{\mathbf #1}}
\def\psl{\Bbb{P}SL}
\def\qed{~$\square$}
\def\munion{\mathop{\cup}}
\def\minter{\mathop{\cap}}
\def\myitem#1{%
\ifinproclaim%
    \item{#1}%
\else%
    \noindent\hbox to .5cm{\hfill#1\hss}%\kern 0.1em}%
\fi}%

\catcode`\@=11
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\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil%
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\catcode`\@=12

\def\myeqalign{\Eqalign}
\def\endmyeqalign{}

\def\makeop#1{%
\global\expandafter\def\csname op#1\endcsname{{\mathroman #1}}}%

\def\makeopsmall#1{%
\global\expandafter\def\csname op#1\endcsname{{\mathroman{\lowercase{#1}}}}}%

\makeopsmall{ArcTan}%
\makeopsmall{ArcCos}%
\makeop{Arg}%
\makeop{Det}%
\makeop{Log}%
\makeop{Re}%
\makeop{Im}%
\makeop{Dim}%
\makeopsmall{Tan}%
\makeop{Ker}%
\makeopsmall{Cos}%
\makeopsmall{Sin}%
\makeop{GL}%
\makeop{Exp}%
\makeopsmall{Tanh}%
\makeop{Tr}%
\def\nameSLfam{\Cal{SL}^+_\theta}%
\def\nameclosedSLfam{\overline{\Cal{SL}}^+_\theta}%
\makeop{End}%
\makeop{Long}%
\makeop{Ch}%
\makeop{Exp}%
\makeop{Int}%
\makeop{Ext}%
\makeop{Aire}%
\makeop{Im}%
\makeop{Conf}%
\makeop{Exp}%
\makeop{Mod}%
\makeop{Log}%
\makeop{Ext}%
\makeop{Int}%
\makeop{Dist}%
\makeop{Aut}%
\makeop{Id}%
\makeop{SO}%
\makeop{Homeo}%
\makeop{Fix}%
\makeop{Vol}%
\makeop{Ric}%
\makeop{Hess}%
\makeop{Euc}%
\makeop{Isom}%
\makeop{Max}%
\makeop{Long}%
\makeop{Fixe}%
\makeop{Wind}%
\makeop{Ad}%
\makeop{loc}%
\def\opTildeHomeo{{\mathroman H\widetilde{omeo}}}%

\let\emph=\bf

\hyphenation{quasi-con-formal}

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% Citations
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\ifmakebiblio%
    \openout\biblio=biblio.tex %
    {%
        \edef\fileoutput{\write\biblio{\bgroup\leftskip=2em}}%
        \fileoutput
    }%
\fi%

\def\BbbC{\Bbb{C}}
\def\BbbR{\Bbb{R}}
\def\BbbH{\Bbb{H}}

\newref{BallGromSch}{Ballman W., Gromov M., Schroeder V., {\sl Manifolds of nonpositive curvature}, Progress in Mathematics, {\bf 61}, Birkh\"auser, Boston, (1985)}
\newref{Bonk}{Bonk M., Private communication to Fran\c{c}ois Labourie}
\newref{Grom}{Gromov M., Foliated plateau problem, part I : Minimal varieties, {\sl GAFA} {\bf 1}, no. 1, (1991), 14--79}%
\newref{LabB}{Labourie F., Probl\`emes de Monge-Amp\`ere, courbes holomorphes et laminations, {\sl GAFA} {\bf 7}, no. 3, (1997), 496--534}
\newref{LabA}{Labourie F., Un lemme de Morse pour les surfaces convexes, {\sl Invent. Math.} {\bf 141} (2000), 239--297}
\newref{LV}{Lehto O., Virtanen K. I., {\sl Quasiconformal mappings in the plane}, Die Grund\-lehren der mathemathischen Wissenschaften, {\bf 126}, Springer-Verlag, New York-Heidelberg, (1973)}
\newref{Muller}{Muller M.P., Gromov's Schwarz lemma as an estimate of the gradient for holomorphic curves, In: {\sl Holomorphic curves in symplectic geometry},
Progress in Mathematics, {\bf 117}, Birkh\"auser, Basel, (1994)}
\newref{RosSpruck}{Rosenberg H., Spruck J. On the existence of convex hyperspheres of constant Gauss curvature in hyperbolic space, {\sl J. Diff. Geom.} {\bf 40} (1994), no. 2,
379--409} % Rosenberg, Spruck
\newref{SmiA}{Smith G., Special legendrian problems, In Preparation}%
\newref{SmiC}{Smith G., k-surfaces priv\'ees de points, In Preparation}%
\newref{SmiE}{Smith G., Th\`ese de doctorat, Paris (2004)}%

\ifmakebiblio%
    {\edef\fileoutput{\write\biblio{\egroup}}%
    \fileoutput}%
\fi%

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\document
\myfontdefault
\global\chapno=1
\global\showpagenumflag=1
\def\Pagetitle{}
\null
\vfill
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\font\abstracttitlefont=cmr10 at 14pt
{\abstracttitlefont\centre Hyperbolic Plateau problems\par}
\bigskip
{\centre Graham Smith\par}
\bigskip
{\centre \the\day\ \textmonth\month\ \the\year\par}
\bigskip
{\centre Equipe de topologie et dynamique,\par
Laboratoire des math\'ematiques,\par
B\^atiment 425,\par
UFR des sciences d'Orsay,\par
91405 Orsay CEDEX, FRANCE\par}
\bigskip
\noindent{\emph Abstract:\ } Using the notion of Plateau problems defined by Labourie in \cite{LabA}, we show that, for every $k\in(0,1)$, and for every locally conformal mapping from the Poincar\'e disc $\Bbb{D}$ into the ideal boundary $\partial_\infty\Bbb{H}^3$ of three dimensional hyperbolic space $\Bbb{H}^3$, viewed as a Plateau problem, there is a unique convex immersed surface $\Sigma$ in $\Bbb{H}^3$ of constant Gaussian curvature equal to $k$ which is a solution to this problem. We thus obtain a homeomorphism between the space of locally conformal mappings from $\Bbb{D}$ into $\hat{\Bbb{C}}\cong\partial_\infty\Bbb{H}^3$ and a family of holomorphic immersions from $\Bbb{D}$ into $U\Bbb{H}^3$.
\bigskip
\noindent{\emph Key Words:\ } pseudo-holomorphic curves, contact geometry, Plateau problem, Gaussian curvature, hyperbolic space.
\bigskip
\noindent{\emph AMS Subject Classification:\ } 32Q65, 51M10, 53C42, 53C45, 53D10, 58D10, 58J05.
\vfill
\nextoddpage
%
\def\Pagetitle{\sl Hyperbolic Plateau problems}
\newhead{Introduction}
%
\noindent In this paper, by establishing a form of the Schwarz lemma adapted to Cheeger/Gromov convergence, we shall prove the existence of constant Gaussian curvature solutions
in $\Bbb{H}^3$ of Plateau problems of hyperbolic type.
\medskip
\noindent Let $\Bbb{H}^3$ be three dimensional hyperbolic space and let $\partial_\infty\Bbb{H}^3$ be its ideal boundary (see, for example, \cite{BallGromSch}). The
ideal boundary of $\Bbb{H}^3$ may be identified canonically with the Riemann sphere $\hat{\Bbb{C}}$. In this context, we define a {\emph Plateau problem\/} to be a pair
$(S,\varphi)$ where $S$ is a Riemann surface and $\varphi:S\rightarrow\partial_\infty\Bbb{H}^3$ is a locally conformal mapping (i.e. a locally homeomorphic holomorphic
mapping). The Plateau problem $(S,\varphi)$ is said to be of hyperbolic, parabolic or elliptic type depending on whether $S$ is hyperbolic, parabolic or elliptic
respectively.
\medskip
\noindent Let $U\Bbb{H}^3$ be the unitary bundle over $\Bbb{H}^3$. For $i:S\rightarrow\Bbb{H}^3$ an immersion, using the canonical orientation of $S$, we may define the
unit normal exterior vector field $\msf{N}$ over $S$. This field is a section of $U\Bbb{H}^3$ over $i$. We define the {\emph Gauss lifting\/} $\hat{\mathi}$ of $i$ by
$\hat{\mathi}=\msf{N}$. For $k\in(0,1)$, a solution to the Plateau problem $(S,\varphi)$ is an immersion $i$ of constant Gaussian curvature $k$ such that, if we denote by
$\overrightarrow{n}$ the Gauss-Minkowski mapping of $\Bbb{H}^3$ which sends $U\Bbb{H}^3$ into $\partial_\infty\Bbb{H}^3$, then the Gauss lifting $\hat{\mathi}$ of
$i$ satisfies:
%
$$
\varphi = \overrightarrow{n}\circ\hat{\mathi}.
$$
%
\noindent In the case where $S$ is a Jordan domain in $\partial_\infty\Bbb{H}^3$ and $\varphi$ is the canonical embedding, the existence and uniqueness of solutions to
the Plateau problem was initially solved by Rosenberg \& Spruck in \cite{RosSpruck}. Rosenberg \& Spruck view the Plateau problem as a Dirichlet problem with asymptotic
boundary conditions in $\partial_\infty\Bbb{H}^3$ and using a classical approach to the treatment of Monge-Amp\`ere equations are able to find, for any topologically
embedded $(n-1)$-sphere $S$ in $\partial_\infty\Bbb{H}^{n+1}$, a constant Gaussian curvature hypersurface of $\Bbb{H}^{n+1}$ whose asymptotic boundary coincides with $S$.
The general Plateau problem in dimension $3$ was solved for a large class of Hadamard manifolds by Labourie in \cite{LabA}. Labourie shows that, for $M$ a three dimensional Hadamard manifold, the Gauss lifting of a hypersurface of constant Gaussian curvature in $M$ is a holomorphic curve in the unitary bundle $UM$ of $M$. Using techniques of holomorphic curve
theory, Labourie is able to establish various existence and uniqueness results for solutions to the Plateau problem. In this context, the results of this article may be viewed
as an extension of the results of \cite{LabA} for the Hadamard manifold $\Bbb{H}^3$. In \cite{SmiC} we will use the results of this paper to establish a link between
the family of constant Gaussian curvature hypersurfaces in $\Bbb{H}^3$ having cylindrical ends and the family of meromorphic mappings over a given compact Riemann surface
$S$. We believe that this equivalence will be of use in the study of the Teichm\"uller theory of compact surfaces with marked points.
\medskip
\noindent We obtain the following results concerning Plateau problems in $\Bbb{H}^3$:
%
\proclaim{Theorem \nextprocno\ {\sl Existence.}}
%\medskip
\noindent Let $\varphi:\Bbb{D}\rightarrow\hat{\Bbb{C}}$ be locally conformal. Then, for every $k\in (0,1)$, there
exists a unique solution  $i_k:\Bbb{D}\rightarrow\Bbb{H}^3$ to the Plateau problem $(\Bbb{D},\varphi)$ with constant Gaussian curvature $k$.
\endproclaim
\proclabel{PresentationChIIExistence}
%
\proclaim{Theorem \nextprocno\ {\sl Continuity.}}
%\medskip
\noindent Let $k\in (0,1)$ be a real number between $0$ and $1$. Let
$\suite{\varphi}{n},\varphi_0:\Bbb{D}\rightarrow\hat{\Bbb{C}}$ be locally conformal mappings. Let
$\suite{i}{n},i_0:\Bbb{D}\rightarrow\Bbb{H}^3$ be immersions such that, for all $n$, the mapping $i_n$ is the
unique solution to the Plateau problem $(\Bbb{D},\varphi_n)$ with constant Gaussian curvature $k$.
\medskip
\noindent If $\suite{\varphi}{n}$ converges to $\varphi_0$ locally uniformly, then
$\suite{i}{n}$ converges to $i_0$ in the $C^\infty_\oploc$ topology over $\Bbb{D}$.
\endproclaim
\proclabel{PresentationChIIContinuite}
%
\noindent In the first part of this paper, we provide a brief introduction to the theory of {\emph immersed submanifolds} and discuss {\emph Cheeger/Gromov} convergence. In the
second section, we study briefly the geometric structure of the unitary bundle $U\Bbb{H}^3$ over $\Bbb{H}^3$, we define {\emph $k$-surfaces} and show how they are related to
{\emph holomorphic curves} in $U\Bbb{H}^3$. In the third section, we establish the {\emph quasiconformal equivalence} of certain conformal structures that may be defined over $k$-surfaces. In the fourth section, we define the {\emph Plateau problem} and we recall existing results established by Labourie in \cite{LabB} and \cite{LabA}. In the fifth section, we establish the Cheeger/Gromov adapted {\emph Schwarz lemma}. In the sixth section, we show how this lemma may be elegantly used to obtain a {\emph compactness} result
for $k$-surfaces of {\emph hyperbolic} type. Finally in the seventh section, we show how this compactness result immediately yields theorems \procref{PresentationChIIExistence}
and \procref{PresentationChIIContinuite}, and how these results may be summarised in terms of a homeomorphism between two spaces.
\medskip
\noindent The results of this paper lead one to pose the following question:
%
\proclaim{Question \nextprocno}
%
\noindent let $D$ be the unit disc in $\Bbb{R}^2$. Let $\Cal{P}$ be the set of local homeomorphisms
$f:D\rightarrow\partial_\infty\Bbb{H}^3$ for which there exists a solution to the Plateau problem $(f,D)$. The group of homeomorphisms
of $S^2$ acts on each element of $\Cal{P}$ by left composition. Let $\opFix(\Cal{P})$ be the subgroup of the group of homeomorphisms of
$S^2$ which preserves $\Cal{P}$. What (if any) geometric information is contained in the group $\opFix(\Cal{P})$?
\endproclaim
%
\noindent This question is motivated by the following considerations: More generally, if $S^2$ denotes the unit sphere, then the group
$\opHomeo(D)$ of homeomorphisms of $D$ acts on the set of local homeomorphisms from $D$ into $S^2$ by right composition. Let $\Cal{F}$ be
a family of local homeomorphisms from $D$ into $S^2$ which is preserved by the action of this group. For example, the
family $\Cal{P}$ satisfies this property. Another such family is the family $\Cal{H}$ defined such that, if we denote by
$\hat{\Bbb{C}}$ the canonical conformal structure of $S^2$, then $f\in\Cal{H}$ if and only if the surface $D$ with the holomorphic
structure $f^*\hat{\Bbb{C}}$ is conformally equivalent to the Poincar\'e disc $\Bbb{D}$.
\medskip
\noindent We now consider, for any such family $\Cal{F}$, the subgroup $\opFix(\Cal{F})$ of homeomorphisms of $S^2$ which preserve $\Cal{F}$. For example, it is a non-trivial result \cite{Bonk} of M. Bonk, that $\opFix(\Cal{H})$ is a proper subgroup of the group of homeomorphisms of $S^2$ (in otherwords, there exist homeomorphisms of $S^2$ which do not preserve $\Cal{H}$). Consequently,
the group $\opFix(\Cal{H})$ (and thus also the family $\Cal{H}$) contains stronger geometric information than just the topological structure of $S^2$.
\medskip
\noindent The results of this paper show that $\Cal{H}$ is contained in $\Cal{P}$. If we were to show that $\Cal{H}$ coincides with
$\Cal{P}$, then we would find that $\opFix(\Cal{P})$ (and thus also the family $\Cal{P}$) contains geometric information beyond
the topological structure of $S^2$. Even if this were not the case, if we show that $\opFix(\Cal{P})$ is a proper subgroup
of $\opHomeo(S^2)$, then it follows that $\opFix(\Cal{P})$ nonetheless contains geometric information. In particular, since we may
define $\Cal{P}$ with $S^2$ replaced by the ideal boundary of any one of a large family of hyperbolic manifolds, we would thus obtain a
geometric structure over $S^2$ which may be generalised to these cases.
\medskip
\noindent I would like to thank Fran\c{c}ois Labourie for his guidance and his many helpful suggestions.
%
\newhead{Immersed Surfaces - Definitions and Notations}
%
\newsubhead{Definitions}
%
\noindent In this section we will review basic definitions from the theory of immersed submanifolds and establish the notations that will be used throughout this paper.
\medskip
\noindent Let $M$ be a smooth manifold. An {\emph immersed submanifold\/} is a pair $\Sigma=(S,i)$ where $S$ is a smooth manifold
and $i:S\rightarrow M$ is a smooth immersion. An {\emph immersed hypersurface\/} is an immersed submanifold of codimension $1$.
\medskip
\noindent Let $g$ be a Riemannian metric on $M$. We give $S$ the unique Riemannian metric $i^*g$ which makes $i$
into an isometry. We say that $\Sigma$ is {\emph complete\/} if and only if the Riemannian manifold $(S,i^*g)$ is.
%
\newsubhead{Normal Vector Fields, Second Fundamental Form, Convexity}
%
\noindent Let $\Sigma$ be a hypersurface immersed in the Riemannian manifold $M$. There exists a canonical embedding $i_*$ of the
tangent bundle $TS$ of $S$ into the pull-back $i^*TM$ of the tangent bundle of $M$. This embedding may be considered as a section
of $\opEnd(TS, i^*TM)$. We denote by $T\Sigma$ the image of $TS$ under the action of this embedding.
\medskip
\noindent Let us suppose that both $M$ and $S$ are oriented. We define $N\Sigma\subseteq i^*TM$, the normal bundle of $\Sigma$, by:
%
$$
N\Sigma = T\Sigma^\perp.
$$
%
\noindent $N\Sigma$ is a one dimensional subbundle of $i^*TM$ from which it inherits a canonical Riemannian metric. Using the orientations of $S$ and $M$, we define the
{\emph exterior unit normal vector field\/}, $\msf{N}_\Sigma\in\Gamma(S,N\Sigma)$, over $\Sigma$ in $M$. This is a global section of $N\Sigma$ which trivialises this bundle. We define the {\emph Weingarten operator\/}, $A_\Sigma$, which is a section of
$\opEnd(TS, T\Sigma)$, by:
%
$$
A_\Sigma(X) = (i^*\nabla)_X \msf{N}_\Sigma.
$$
%
\noindent Since there exists a canonical isomorphism (being $i_*$) between $T\Sigma$ and $TS$, we may equally well view $A_\Sigma$ as a section of $\opEnd(TS)$. This section is
self-adjoint with respect to the canonical Riemannian metric over $S$. We thus define the {\emph second fundamental form\/}, $II_\Sigma$, which is a symmetric bilinear form
over $TS$ by:
%
$$
II_\Sigma(X,Y) = \langle A_\Sigma X, Y \rangle.
$$
%
\noindent $\Sigma$ is said to be {\emph convex\/} at $p\in S$ if and only if the bilinear form $II_\Sigma$ is either positive or negative
definite at $p$. $\Sigma$ is then said to be {\emph locally convex\/} if and only if it is convex at every point. Through a slight
abuse of language, we will say that $\Sigma$ is {\emph convex\/} in this case. Bearing in mind that the sign of $II_\Sigma$ depends on the sign of $\msf{N}_\Sigma$, which in
turn depends on the choice of orientation of $S$, if $\Sigma$ is convex, then we may choose the orientation of $S$ such that $II_\Sigma$ is positive definite. Consequently, in
the sequel, if $\Sigma$ is convex, then we will assume that $II_\Sigma$ is positive definite.
%
\newsubhead{Curvature}
%
\noindent For $i\in\left\{0,...,n\right\}$, let us denote by $\xi_i:\opEnd(\Bbb{R}^n)\rightarrow\Bbb{R}$ the {\emph $i$'th symmetric function\/} over $\opEnd(\Bbb{R}^n)$. Thus,
for all matrices $A$ and for all real $t$:
%
$$
\opDet(I + tA) = \sum_{i=0}^n\xi_i(A)t^i.
$$
%
\noindent The group $\opGL(\Bbb{R}^n)$ acts on $\opEnd(\Bbb{R}^n)$ by the {\emph adjoint action\/} which we denote by $\opAd$. Thus, for all $A,B\in\opEnd(\Bbb{R}^n)$:
%
$$
\opAd(A)\cdot B = ABA^{-1}.
$$
%
\noindent For all $i$, the function $\xi_i$ is invariant under the adjoint action of $\opEnd(\Bbb{R}^n)$ on itself. For this reason, for any manifold $M$, and for all $i$, the
function $\xi_i$ defines a unique mapping $\tilde{\xi}_i:\Gamma(\opEnd(TM))\rightarrow C^\infty(M)$. For simplicity, we will denote $\tilde{\xi}_i$ also by $\xi_i$.
\medskip
\noindent Let $\Sigma$ be an oriented hypersurface immersed in an oriented Riemannian manifold $M$ of dimension $n+1$. For all $i\in\left\{0,...,n\right\}$, we refer to the
function $\xi_i(A_\Sigma)$ as the {\emph $i$'th higher principal curvature\/} of $\Sigma$. In particular, $\xi_1(A_\Sigma)$ is the {\emph mean curvature\/} of $\Sigma$, which is
also denoted by $H_\Sigma$, and $\xi_n(A_\Sigma)$ is the {\emph Gaussian curvature\/} of $\Sigma$, which is also denote by $k_\Sigma$.
\medskip
\noindent In this paper, we study oriented surfaces of constant Gaussian curvature immersed into three dimensional hyperbolic space.
%
\newsubhead{Pointed Manifolds, Convergence}
%
\noindent In this paper, we use the concept of Cheeger/Gromov convergence for complete pointed immersed submanifolds, which we explain in
this section.
\medskip
\noindent A {\emph pointed Riemannian manifold\/} is a pair $(M,p)$ where $M$ is a Riemannnian manifold and $p$ is a point in $M$. If $(M,p)$ and $(M',p')$ are pointed
manifolds then a {\emph morphism\/} (or {\emph mapping\/}) from $(M,p)$ to $(M',p')$ is a (not necessarily even continuous) function from $M$ to $M'$ which sends $p$ to $p'$ and
is $C^\infty$ in a neighbourhood of $p$. The family of pointed manifolds along with these morphisms forms a category. In this section, we will discuss a notion of convergence
for this family. It should be borne in mind that even though this family is not a set, we may consider it as such. Indeed, since every manifold may be plunged into an infinite
dimensional real vector space, we may discuss, instead, the equivalent family of pointed finite dimensional submanifolds of this vector space, and this is a set.
\medskip
\noindent Let $(M_n,p_n)_{n\in\Bbb{N}}$ be a sequence of complete pointed Riemannian manifolds. For all $n$, we denote by $g_n$ the Riemannian metric over $M_n$. We say that the
sequence $(M_n,p_n)_{n\in\Bbb{N}}$ {\emph converges\/} to the complete pointed manifold $(M_0,p_0)$ in the {\emph Cheeger/Gromov topology\/} if and only if:
\medskip
\noindent{(i)} for all $n$, there exists a mapping $\varphi_n:(M_0,p_0)\rightarrow (M_n,p_n)$,
\medskip
\noindent such that, for every compact subset $K$ of $M_0$, there exists $N\in\Bbb{N}$ such that for all $n\geqslant N$:
\medskip
\myitem{(i)} the restriction of $\varphi_n$ to $K$ is a $C^\infty$ diffeomorphism onto its image, and
\medskip
\myitem{(ii)} if we denote by $g_0$ the Riemannian metric over $M_0$, then the sequence of metrics $(\varphi_n^*g_n)_{n\geqslant N}$ converges to $g_0$ in the
$C^\infty$ topology over $K$.
\medskip
\noindent We refer to the sequence $(\varphi_n)_{n\in\Bbb{N}}$ as a sequence of {\emph convergence mappings\/} of the sequence $(M_n, p_n)_{n\in\Bbb{N}}$ with respect to the
limit $(M_0,p_0)$. The convergence mappings are trivially not unique. However, two sequences of convergence mappings $(\varphi_n)_{n\in\Bbb{N}}$ and $(\varphi'_n)_{n\in\Bbb{N}}$
are equivalent in the sense that there exists an isometry $\phi$ of $(M_0,p_0)$ such that, for every compact subset $K$ of $M_0$, there exists $N\in\Bbb{N}$ such that for
$n\geqslant N$:
\medskip
\myitem{(i)} the mapping $(\varphi_n^{-1}\circ\varphi_n')$ is well defined over $K$, and
\medskip
\myitem{(ii)} the sequence $(\varphi_n^{-1}\circ\varphi_n')_{n\geqslant N}$ converges to $\phi$ in the $C^\infty$ topology over $K$.
\medskip
\noindent One may verify that this mode of convergence does indeed arise from a topological structure over the space of complete pointed manifolds. Moreover, this topology
is Haussdorf (up to isometries).
\medskip
\noindent Most topological properties are unstable under this limiting process. For example, the limit of a sequence of simply connected manifolds is not necessarily
simply connected. On the other hand, the limit of a sequence of surfaces of genus $k$ is a surface of genus at most $k$ (but quite possibly with many holes).
\medskip
\noindent Let $M$ be a complete Riemannian manifold. A {\emph pointed immersed submanifold\/} in  $M$ is a pair $(\Sigma,p)$ where $\Sigma=(S,i)$ is an immersed submanifold in
$M$ and $p$ is a point in $S$.
\medskip
\noindent Let $(\Sigma_n,p_n)_{n\in\Bbb{N}}=(S_n,p_n,i_n)_{n\in\Bbb{N}}$ be a sequence of complete pointed immersed submanifolds in $M$. We say that
$(\Sigma_n,p_n)_{n\in\Bbb{N}}$ {\emph converges\/} to $(\Sigma_0,p_0)=(S_0,p_0,i_0)$ in the {\emph Cheeger/Gromov topology\/} if and only if $(S_n,p_n)_{n\in\Bbb{N}}$ converges
to $(S_0,p_0)$ in the Cheeger/Gromov topology, and, for every sequence $(\varphi_n)_{n\in\Bbb{N}}$ of convergence mappings of $(S_n,p_n)_\ninn$ with respect to this limit, and
for every compact subset $K$ of $S_0$, the sequence of functions $(i_n\circ\varphi_n)_{n\geqslant N}$ converges to the function $(i_0\circ\varphi_0)$ in the $C^\infty$ topology
over $K$.
\medskip
\noindent As before, this mode of convergence arises from a topological structure over the space of complete immersed submanifolds. Moreover, this topology is Haussdorf (up to isometries).
%
\newsubhead{``Common Sense'' Lemmata}
%
\noindent In order to make good use of the concept of Cheeger/Gromov convergence, it is helpful to recall some basic lemmata concerning the topological properties of functions
acting on open subset of $\Bbb{R}^n$. The results that follow are essentially formal expressions of ``common sense''. To begin with, we recall a result concerning the inverses
of a sequence of functions that converges:
%
\proclaim{Lemma \nextprocno}
%
\noindent Let $\Omega\subseteq\Bbb{R}^n$ be an open set. Let $\suite{f}{n},f_0:\Omega\rightarrow\Bbb{R}^n$ be such that,
for every $n$, the function $f_i$ is a homeomorphism onto its image. Let $\Omega'$ be the image of $\Omega$ under the action of $f_0$.
If $\suite{f}{n}$ converges towards $f_0$ locally uniformly in $\Omega$ as $n$ tends to infinity, then the sequence
$(f_n^{-1})_{n\in\Bbb{N}}$ converges towards $f_0^{-1}$ locally uniformly in $\Omega'$ as $n$ tends to infinity.
\medskip
\noindent To be precise, for every compact subset $K$ in $\Omega'$, there exists $N\in\Bbb{N}$ such that, for every $n\geqslant N$,
the set $K$ is contained within $f_n(\Omega)$ and $(f_n^{-1})_{n\geqslant N}$ converges towards $f_0^{-1}$ uniformly over $K$.
\medskip
\noindent Moreover, if every $f_n$ is of type $C^m$ and if $\suite{f}{n}$ converges to $f_0$ in the $C^m_{\oploc}$ topology, then $(f_n^{-1})_{n\in\Bbb{N}}$ also converges towards
$f_0^{-1}$ in the $C^m_{\oploc}$ topology.
\endproclaim
\proclabel{LemmeLimitesDInverses}
%
\noindent We recall a result concerning the injectivity of the limit:
%
\proclaim{Lemma \nextprocno}
%
\noindent Let $\Omega\subseteq\Bbb{R}^n$ be an open set. Let $\suite{f}{n},f_0:\Omega\rightarrow\Bbb{R}^n$ be such that for every $n>0$, the function $f_n$ is a homeomorphism
onto its image. If $(f_n)_{n\in\Bbb{N}}$ tends towards $f_0$ locally uniformly, and if, moreover, $f_0$ is a local homeomorphism, then $f_0$ is injective.
\endproclaim
\proclabel{LemmeInjectiviteDeLaLimite}
%
\noindent We recall a converse of this result for $C^2$ functions:
%
\proclaim{Lemma \nextprocno}
%
\noindent Let $\Omega\subseteq\Bbb{R}^n$ be an open set. Let $\suite{f}{n},f_0:\Omega\rightarrow\Bbb{R}^n$ be $C^2$ functions such that
$\suite{f}{n}$ converges towards $f_0$ in the $C^2_{\oploc}$ topology as $n$ tends to infinity. If $f_0$ is a diffeomorphism onto its image, then for every compact subset
$K\subseteq\Omega$ in $\Omega$, there exists $N\in\Bbb{N}$ such that, for $n\geqslant N$, the restriction of $f_n$ to $K$ is injective.
\endproclaim
\proclabel{LemmeInjectiviteLorsquOnSApprocheDuLimite}
%
\noindent Finally, we have a result concerning the surjectivity of the limit of a sequence of functions:
%
\proclaim{Lemma \nextprocno}
%
\noindent Let $\Omega\subseteq\Bbb{R}^n$ be an open set. Let $\suite{f}{n}:\Omega\rightarrow\Bbb{R}^n$ be such that for every $n$, the function $f_n$ is
a homeomorphism onto its image. If there exists a local homeomorphism $f_0:\Omega\rightarrow\Bbb{R}^n$ such that $\suite{f}{n}$ converges to $f_0$ locally uniformly, then, for
every compact subset $K\subseteq f_0(\Omega)$ in the image of $\Omega$ under $f_0$, there exists $N\in\Bbb{N}$ such that for $n\geqslant N$:
%
$$
K\subseteq f_n(\Omega).
$$
%
\endproclaim
\proclabel{LemmeCompacteDansLImageDeLaLimite}
%
\noindent The interested reader may find a discussion and proofs of these results in the appendix A of \cite{SmiE}.
%
\newhead{The Unitary Bundle of a Riemannian Manifold}
%
\newsubhead{Geometric Structures Over $TM$}
%
\noindent Let $M$ be a Riemannian manifold. We define $\pi:TM\rightarrow M$ to be the canonical projection of the tangent space of $M$ onto $M$. We denote by
$HTM\subseteq TTM$ the {\emph horizontal bundle\/} of the Levi-Civita covariant derivative of $M$. We denote by $VTM\subseteq TTM$ the {\emph vertical bundle\/}
over $TM$. To be precise, $VTM$ is defined to be the kernel in $TTM$ of the projection $\pi$. The tangent bundle to $TM$ is the direct sum of these two subbundles:
%
$$
TTM = HTM\oplus VTM.
$$
%
\noindent Each of $HTM$ and $VTM$ is canonically isomorphic to $\pi^*TM$. We denote by $i_H$ (resp. $i_V$), which is a section of $\opEnd(HTM,\pi^*TM)$ (resp. $\opEnd(VTM,\pi^*TM)$), the canonical isomorphism sending $HTM$ (resp. $VTM$) to $\pi^*TM$. We obtain the following isomorphism:
%
$$
i_H\oplus i_V:TTM\rightarrow\pi^*TM\oplus\pi^*TM.
$$
%
\noindent For every pair of vector fields $X,Y\in\Gamma(M,TM)$ over $M$ we define the vector field $\left\{X,Y\right\}$ over $TM$ by:
%
$$
(i_H\oplus i_V)(\left\{X,Y\right\}) = (\pi^* X, \pi^* Y).
$$
%
\noindent Trivially, every vector field over $TM$ may be expressed (at least locally) in terms of a linear combination of such vector fields. In the same way, for
a given point $p\in M$ in $M$ and for a given triplet of vectors $X,Y,q\in T_pM$ over $p$, we may define $\left\{X,Y\right\}_q\in T_qTM$ by:
%
$$
(i_H\oplus i_V)_q\left\{X,Y\right\}_q = (\pi^*_qX, \pi^*_q Y).
$$
%
\noindent Finally, for a given vector field $X$ over $M$, we may define $X^H$ and $X^V$ by:
%
$$\matrix
X^H \hfill&=\left\{X,0\right\}, \hfill\cr
X^V \hfill&=\left\{0,X\right\}. \hfill\cr
\endmatrix$$
%
\newsubhead{Geometric Structures Over $UM$}
%
\noindent For $M$ a Riemannian manifold, we define $UM$, the {\emph unitary bundle\/} over $M$ by:
\headlabel{HeadDifferentialGeometryInTTM}
%
$$
UM = \left\{X\in TM\ |\ \|X\| = 1\right\}.
$$
%
\noindent We define the {\emph tautological vector fields} $T^H$ and $T^V$ over the tangent space $TM$ to $M$ by:
%
$$\matrix
T^H(q) \hfill&=\left\{q,0\right\}_q, \hfill\cr
T^V(q) \hfill&=\left\{0,q\right\}_q. \hfill\cr
\endmatrix$$
%
\noindent Let $i:UM\rightarrow TM$ be the canonical embedding. Let $HUM$ (resp. $VUM$) be the restriction of $HTM$ (resp. $VTM$) to $UM$:
%
$$\matrix
HUM \hfill&=i^*HTM, \hfill\cr
VUM \hfill&=i^*VTM. \hfill\cr
\endmatrix$$
%
\noindent The section $i^*T^H$ (resp. $i^*T^V$) is nowhere vanishing. It consequently defines a one dimensional subbundle of $HUM$ (resp. $VUM$). In order
to simplify the notation we will also denote this section by $T^H$ (resp. $T^V$). We denote the one dimensional subbundle that it generates by $\langle q\rangle_H$
(resp. $\langle q\rangle_V$). We define the subbundles $\langle q\rangle_H^\perp$ and $\langle q\rangle_V^\perp$ of $HUM$ and $VUM$ respectively by:
%
$$\matrix
\langle q\rangle_H^\perp \hfill&=\langle T^H \rangle^\perp, \hfill\cr
\langle q\rangle_V^\perp \hfill&=\langle T^V \rangle^\perp. \hfill\cr
\endmatrix$$
%
\noindent Since parallel transport preserves the lengths of vectors and thus sends $UM$ onto itself, the immersion $i$ induces the following
isomorphism of vector bundles:
%
$$
i_*:TUM\rightarrow HUM\oplus \langle q\rangle_V^\perp.
$$
%
\noindent In order to simplify the notation, we consider $HUM$, $\langle q\rangle_H^\perp$ and $\langle q\rangle_V^\perp$ as subbundles of $TUM$. In particular, we define $WUM$
by:
%
$$
WUM = \langle q\rangle_H^\perp\oplus\langle q\rangle_V^\perp.
$$
%
\noindent The subbundle $WUM$ defines, in fact, a contact structure over $UM$, and we will consequently refer to it as the {\emph contact bundle\/} over $UM$.
In the sequel we will denote the bundles $HUM$, $VUM$, $WUM$ etc. by $H$, $V$, $W$ etc.
%
\noindent For $k>0$, we write $\nu=\sqrt{k}$ and we define the metric $g^\nu$ over $TTM$ such that, for every pair of vector fields $X,Y\in\Gamma(M,TM)$ over $M$ we have:
%
$$
g^\nu(\left\{X,Y\right\},\left\{X,Y\right\}) = \langle X, X\rangle + \nu^{-2}\langle Y, Y\rangle.
$$
%
\noindent We denote also by $g^\nu$ the metric induced over $UM$ by $g^\nu$ and the canonical embedding of $UM$ into $TM$.
\medskip
\noindent From now on, we will suppose that $M$ is oriented and three dimensional. This allows us to canonically identify $TM$ and $TM \wedge TM$ and consequently to define a
vector product $\wedge$ over $TM$. We then define the canonical {\emph complex structures\/} $J^H$ (resp. $J^V$) over $\langle q\rangle_H^\perp$ (resp.
$\langle q\rangle_V^\perp$) such that for every $q\in UM$ and for every vector $X$ orthogonal to $q$:
%
$$\matrix
J^H_q\left\{X,0\right\}_q \hfill&=\left\{q\wedge X, 0\right\}_q, \hfill\cr
J^V_q\left\{0,X\right\}_q \hfill&=\left\{0, q\wedge X\right\}_q. \hfill\cr
\endmatrix$$
%
\noindent In order to simplify notation we refer to both $J^H$ and $J^V$ by $J$. We define the isomorphism $j:HTM\rightarrow VTM$ by:
%
$$
j = i_V^{-1}\circ i_H.
$$
%
\noindent This isometry sends $\langle q\rangle_H$ onto $\langle q\rangle_V$ and consequently $\langle q\rangle_H^\perp$ onto $\langle q\rangle_V^\perp$. Moreover, we
trivially obtain the following commutative diagram:
%
$$
\commdiag{
\langle q\rangle_H^\perp &\mapright^j &\langle q\rangle_V^\perp \cr
\mapdown_J& &\mapdown_J\cr
\langle q\rangle_H^\perp &\mapright^j &\langle q\rangle_V^\perp}
$$
%
\noindent We define the {\emph complex structure\/} $J^\nu$ over $W=\langle q\rangle_H^\perp\oplus\langle q\rangle_V^\perp$ by:
%
$$
J^\nu = \pmatrix 0 & j^{-1}\nu^{-1}J \cr j\nu J & 0 \cr \endpmatrix.
$$
%
\noindent $J^\nu$ is compatible with the metric $g^\nu$ over $W$. In the sequel, in order to simplify the notation, we identify $\langle q\rangle_H^\perp$ and
$\langle q\rangle_V^\perp$ through the isomorphism $j$, and we thus write:
%
$$
J^\nu = \pmatrix 0 & \nu^{-1}J \cr \nu J & 0 \cr \endpmatrix.
$$
%
\noindent By composing this form with the orthogonal projection of $TUM$ onto $W$, we may extend it to a form defined on $TUM$.
\medskip
\noindent Let $q$ be a point in $UM$. Let $\Sigma\subseteq W_q$ be a plane in $W_q$. We say that $\Sigma$ is the graph of the matrix $A$ over $\langle q\rangle_H^\perp$
if and only if:
%
$$
\Sigma = \left\{ \left\{V,AV\right\} | V\in q^\perp \right\}.
$$
%
\noindent We say that the plane $\Sigma$ is {\emph k-complex\/} if and only if it is stable under the action of $J^\nu$. In this case, if it is the graph of a matrix $A$,
we may trivially show that $A$ is symmetric. The plane $\Sigma$ is then said to be {\emph positive\/} if and only if it is the graph of a positive definite matrix.
%
\newsubhead{Holomorphic Curves, k-Surfaces}
%
\noindent Let $M$ be a compact oriented three dimensional Riemannian manifold and let $\Sigma=(S,i)$ be a convex hypersurface immersed in $M$. Let $\mathsf{N}_\Sigma$ be
the normal exterior vector field to $\Sigma$. We define the {\emph Gauss lifting\/} $\hat{\Sigma}=(S,\hat{\mathi})$ of $\Sigma$ by:
%
$$
(S,\hat{\mathi}) = (S,\mathsf{N}_\Sigma).
$$
%
\noindent For $k>0$, we say that $\Sigma$ is a {\emph k-surface\/} if and only if $\hat{\Sigma}$ is complete and the Gaussian curvature of $\Sigma$ is always equal to $k$.
\medskip
\noindent We say that $\hat{\Sigma}$ is a {\emph k-holomorphic curve\/} if and only if all its tangent planes are k-complex planes, and we say that it is
{\emph positive\/} if and only if all its tangent planes are positive.
\medskip
\noindent These concepts are related by the following elementary result:
%
\proclaim{Lemma \nextprocno}
%
\noindent Let $M$ be an oriented three dimensional Riemannian manifold. Let $\Sigma=(S,i)$ be a convex hypersurface immersed in $M$. $\Sigma$ is a k-surface
if and only if $\hat{\Sigma}$ use a complete positive k-holomorphic curve.
\endproclaim
%
\proof See, for example \cite{LabB}.\qed
%
\newhead{Complex Structures of k-Surfaces in $\Bbb{H}^3$}
%
\newsubhead{Definitions}
%
\noindent Let $\Bbb{H}^3$ be three dimensional hyperbolic space of constant negative curvature equal to $-1$. Let $\partial_\infty\Bbb{H}^3$ be the
ideal boundary of $\Bbb{H}^3$. Let $\opExp:T\Bbb{H}^3\rightarrow\Bbb{H}^3$ be the exponential mapping. We define the {\emph Gauss-Minkowski\/} application
$\overrightarrow{n}:U\Bbb{H}^3\rightarrow\partial_\infty\Bbb{H}^3$ such that, for every $u\in U\Bbb{H}^3$:
%
$$
\overrightarrow{n}(u) = \mlim_{t\rightarrow +\infty}\opExp(tu).
$$
%
\noindent Let $p\in\Bbb{H}^3$ be an arbitrary point in $\Bbb{H}^3$. The restriction of $\overrightarrow{n}$ to the fibre over $p$ gives a bijection between this
fibre and $\partial_\infty\Bbb{H}^3$. This bijection permits us to give $\partial_\infty\Bbb{H}^3$ the topological structure of a $2$-sphere.
This topological structure is independant of the point $p$ chosen.
\medskip
\noindent We identify $\Bbb{H}^3$ with $\Bbb{C}\times\Bbb{R}^+$ and $U\Bbb{H}^3$ with $\Bbb{H}^3\times\Sigma^2$ where $\Sigma^2\subseteq\Bbb{R}^3$ denotes the
two dimensional sphere. We will denote these identifications by $i$ and $\hat{\mathi}$ respectively. In this model, the geodesics in $\Bbb{H}^3$ coincide with the
oriented vertical lines and the oriented vertical half circles centred on points contained in $\Bbb{C}=\Bbb{C}\times\left\{0\right\}$. Moreover, two geodesics are asymptotic
(i.e. they tend towards the same point at infinity) if and only if they terminate at the same point in $\hat{\Bbb{C}}=\Bbb{C}\munion\left\{\infty\right\}$ and we thus obtain an
identification $i_\infty$ between $\partial_\infty\Bbb{H}^3$ and $\hat{\Bbb{C}}$. If we denote by $p$ the stereographic projection of $\Sigma^2$ onto the extended complex
plane, a little Euclidian geometry permits us to show that $p$ coincides with $\overrightarrow{n}\circ\hat{\mathi}^{-1}$. More precisely, if we define
$p:\Bbb{H}^3\times\Sigma^2\rightarrow\hat{\Bbb{C}}$ by:
%
$$
p((z,\lambda),\underline{x}) = z + \frac{\lambda}{1-x_3}(x_1,x_2),
$$
%
\noindent then, for every $p\in\Bbb{H}^3$, the following diagram commutes:
%
$$\commdiag{
U\Bbb{H}^3 &\mapright^{\overrightarrow{n}} &\partial_\infty\Bbb{H}^3 \cr
\mapdown_{\hat{\mathi}} & &\mapdown_{i_\infty} \cr
(\Bbb{C}\times\Bbb{R}^+)\times\Sigma^2&\mapright^{p} &\hat{\Bbb{C}} \cr
}$$
%
\noindent It follows that, for every $p\in\Bbb{H}^3$, the complex structure induced over $\partial_\infty\Bbb{H}^3$ by $(\overrightarrow{n}_p)^{-1}$ and the
complex structure over $U_p\Bbb{H}^3$ coincides with the complex structure induced by $i_\infty$ and the conformal structure of $\hat{\Bbb{C}}$. Since this
latter structure does not depend on $p$, it follows that the bijection $\overrightarrow{n}_p$ allows us to canonically furnish $\partial_\infty\Bbb{H}^3$ with
the complex structure of the Riemann sphere in such a manner that this bijection is conformal.
%
\newsubhead{More Complex Structures}
%
\noindent In this section we will study in greater depth the relationship between the conformal structure that we have
constructed over $W$ and the canonical conformal structure over $\partial_\infty\Bbb{H}^3=\hat{\Bbb{C}}$. As in the
previous section, we will identify $\Bbb{H}^3$ with $\Bbb{C}\times\Bbb{R}^+$ and we will denote this identification by
$i:\Bbb{H}^3\rightarrow\Bbb{C}\times\Bbb{R}^+$. We define the point $P_0\in\Bbb{H}^3$ by:
%
$$
P_0 = (0,1).
$$
%
\noindent By definition, the application $Ti:T\Bbb{H}^3\rightarrow T(\Bbb{C}\times\Bbb{R}^+)$ is an isomorphism. Using $Ti$, we obtain an embedding of $U\Bbb{H}^3$ inside
$T(\Bbb{C}\times\Bbb{R}^+)\simeq (\Bbb{C}\times\Bbb{R}^+)\times(\Bbb{C}\times\Bbb{R})$, and we shall refer to the image of this embedding also by $U\Bbb{H}^3$.
\medskip
\noindent We define the vector field $N$ over $\Bbb{C}\times\Bbb{R}^+$ by:
%
$$
N(z,y) = (0,-y)_{(z,y)}.
$$
%
\noindent $N$ is a vector field contained in $U\Bbb{H}^3$. Indeed, if we denote respectively by $\|\cdot\|_{\Bbb{H}^3}$ and $\|\cdot\|_{\Bbb{E}}$ the hyperbolic and
Euclidian norms over $\Bbb{C}\times\Bbb{R}^+$, we obtain:
%
$$
\|N\|_{\Bbb{H}^3}^2 = \frac{1}{y^2}\|N\|_{\Bbb{E}}^2 = 1.
$$
%
\noindent We define the vector fields $\partial_1,\partial_2$ over $\Bbb{C}\times\Bbb{R}^+$ by:
%
$$\matrix
\partial_1(z,y) \hfill&=(1,0)_{(z,y)}, \hfill\cr
\partial_2(z,y) \hfill&=(i,0)_{(z,y)}. \hfill\cr
\endmatrix$$
%
\noindent By considering each of $\partial_1,\partial_2$ as mappings from $\Bbb{C}\times\Bbb{R}^+$ to
$T(\Bbb{C}\times\Bbb{R}^+)\simeq (\Bbb{C}\times\Bbb{R}^+)\times(\Bbb{C}\times\Bbb{R})$, we obtain:
%
$$\matrix
TN\cdot\partial_1 \hfill&=((1,0),(0,0))_{((z,y),(0,-y))}, \hfill\cr
TN\cdot\partial_2 \hfill&=((i,0),(0,0))_{((z,y),(0,-y))}. \hfill\cr
\endmatrix$$
%
\noindent Let $\nabla$ be the Levi-Civita covariant derivative over $\Bbb{H}^3$ and let $\omega$ be the connection
2-form of $\nabla$. For each $i\in\left\{1,2\right\}$ we obtain:
%
$$\matrix
&\omega(\partial_i,N)_{(z,y)} \hfill&=\partial_i(z,y), \hfill\cr
\Rightarrow &\nabla_{\partial_i}N \hfill&=\partial_i. \hfill\cr
\endmatrix$$
%
\noindent Let $\pi:T(\Bbb{C}\times\Bbb{R}^+)\rightarrow\Bbb{C}\times\Bbb{R}^+$ be the canonical projection. We have:
%
$$\matrix
(i_V^{-1}\pi^*\nabla_{\partial_1}N)_{N(z,y)} \hfill&=((0,0),(1,0))_{N(z,y)},\hfill\cr
(i_V^{-1}\pi^*\nabla_{\partial_2}N)_{N(z,y)} \hfill&=((0,0),(i,0))_{N(z,y)}.\hfill\cr
\endmatrix$$
%
\noindent Let $VT(\Bbb{C}\times\Bbb{R}^+)$ be the vertical subbundle of $TT(\Bbb{C}\times\Bbb{R}^+)$ associated to $\pi$. Let
$\pi_V:TT(\Bbb{C}\times\Bbb{R}^+)$ be the orthogonal projection of $TT(\Bbb{C}\times\Bbb{R}^+)$ into $VT(\Bbb{C}\times\Bbb{R}^+)$.
For each $i\in\left\{1,2\right\}$, we obtain:
%
$$
(\pi_V\cdot TN\cdot\partial_i)(z,y) = (i_V^{-1}\pi^*\nabla_{\partial_i}N)_{N(z,y)}.
$$
%
\noindent It follows that the horizontal bundle of $\nabla$ at $N(z,y)$ is given by:
%
$$
H_{N(z,y)} = \langle ((1,0),(-1,0))_{N(z,y)},((i,0),(-i,0))_{N(z,y)}\rangle.
$$
%
\noindent Consequently, for every vector $X_P$ over $P$ which is orthogonal to $N_P$, we obtain:
%
$$
\left\{X_P,0_P\right\}_{N(P)} = (X,-X)_{N(P)}.
$$
%
\noindent Let $\overrightarrow{n}:U\Bbb{H}^3\rightarrow\partial_\infty\Bbb{H}^3$ be the Gauss-Minkowski application. If we define $\hat{\mathi}$ as in the previous section,
we obtain:
%
$$
\hat{\mathi}(\underline{x}_{(z,\lambda)}) = \frac{1}{\lambda}Ti(\underline{x}_{(z,\lambda)}).
$$
%
\noindent Consequently:
%
$$
\overrightarrow{n}(\underline{x}_{(z,\lambda)}) = z + \frac{\lambda}{\lambda-x_3}(x_1,x_2).
$$
%
\noindent For every $X\in\Bbb{C}$, we define $X_P\in T_P\Bbb{H}^3$ by $X_P = (X,0)_P$ and we obtain:
%
$$\matrix
T\overrightarrow{n}_{N(P)}\cdot \left\{0,X_P\right\}_{N(P)} \hfill&=T\overrightarrow{n}_{((0,1),(0,-1))}(0,X)_{N(P)} \hfill\cr
&=(\frac{1}{2}X)_0. \hfill\cr
\endmatrix$$
%
\noindent At the same time, we obtain:
%
$$\matrix
T\overrightarrow{n}_{N(P)}\cdot \left\{X_P,0\right\}_{N(P)} \hfill&=T\overrightarrow{n}_{((0,1),(0,-1))}(X,-X)_{N(P)} \hfill\cr
&=(X-\frac{1}{2}X)_0 \hfill\cr
&=(\frac{1}{2}X)_0. \hfill\cr
\endmatrix$$
%
\noindent It follows that the restriction of $T\overrightarrow{n}$ to $W_{N(P)}$ is conformal relative the conformal
structure $J^1$ which we defined in the previous section. In otherwords, the following diagram commutes:
%
$$
\commdiag{
W_{N(P)} &\mapright^{J^1_{N(P)}} &W_{N(P)} \cr
\mapdown_{T\overrightarrow{n}_{N(P)}}& &\mapdown_{T\overrightarrow{n}_{N(P)}}\cr
T_0\Bbb{C} &\mapright^J &T_0\Bbb{C} \cr}
$$
%
\noindent By the homogeneity of $U\Bbb{H}^3$, it follows that $T\overrightarrow{n}$ is conformal over the whole
of $W$. We now obtain the following result for $J^\nu$ in general:
%
\proclaim{Lemma \nextprocno}
%\medskip
\noindent Let $\nu\in (0,1)$ be a positive real number less than $1$. Let us define $k$ by $k=\nu^2$. Let $\Sigma\subseteq W$ be a positive k-complex plane in $W$. The
restriction of $T\overrightarrow{n}$ to $\Sigma$ is k-quasiconformal, where $k=\nu^2$.
\endproclaim
\proclabel{LemmeLaMetriqueEstQC}
%
\proof Since $U\Bbb{H}^3$ is homogeneous, we may suppose that there exists $\lambda\in\Bbb{R}\setminus\left\{0\right\}$
such that $\Sigma$ is the subspace of $T_{N(P)}U\Bbb{H}^3$ generated by the vectors $\left\{1,\lambda\right\}$ and
$J^\nu\left\{1,\lambda\right\}$. Moreover, since $\Sigma$ is positive, we may suppose that $\lambda>0$. We obtain:
%
$$
T\overrightarrow{n}_{N(P)}\cdot\left\{1,\lambda\right\}_{N(P)} = \frac{1}{2}(1+\lambda)_0.
$$
%
\noindent At the same time, we have:
%
$$\matrix
T\overrightarrow{n}_{N(P)}\cdot J^{\nu}_{N(P)}\cdot\left\{1,\lambda\right\}_{N(P)} \hfill&=
    T\overrightarrow{n}_{N(P)}\cdot\left\{i\nu^{-1}\lambda, i\nu\right\}_{N(P)} \hfill\cr
&=\frac{i}{2}(\nu + \nu^{-1}\lambda)_0. \hfill\cr
\endmatrix$$
%
\noindent Since $\lambda\in(0,\infty)$ and $\nu\in(0,1)$, we have:
%
$$
\nu < \frac{\nu+\nu^{-1}\lambda}{1+\lambda} < \nu^{-1}.
$$
%
\noindent Consequently, there exists $q\in (\nu,\nu^{-1})$ such that if we define $Q_\Sigma$ by:
%
$$Q_\Sigma =
\pmatrix
q & \cr & q^{-1} \cr
\endpmatrix,$$
%
\noindent then the following diagram commutes:
%
$$
\commdiag{
\Sigma &\mapright^{J^\nu_{N(P)}} &\Sigma \cr
\mapdown_{T\overrightarrow{n}_{N(P)}}& &\mapdown_{T\overrightarrow{n}_{N(P)}}\cr
T_0\Bbb{C} &\mapright^{Q_\Sigma J} &T_0\Bbb{C} \cr}
$$
%
\noindent It thus follows that the restriction of $T\overrightarrow{n}$ to $\Sigma$ is k-quasiconformal,
where $k=\nu^2$, and the result follows.\qed
%
\newsubhead{Quasiconformal Applications}
%
\noindent In the sequel, we will require the following result which essentially says that, as far as we are concerned,
the space of quasiconformal mappings has the same compactness properties as the space of conformal mappings:
%
\proclaim{Lemma \nextprocno}
%
\noindent Let $(f_n)_{n\in\Bbb{N}}:(\Bbb{D},0)\rightarrow(\Bbb{D},0)$ be a sequence of k-quasiconformal mappings of the
unit disc which fix the origin. There exists a k-quasiconformal mapping $f_0:(\Bbb{D},0)\rightarrow(\Bbb{D},0)$
such that, after extraction of a subsequence, $\suite{f}{n}$ tends to $f_0$ as $n$ tends to infinity.
\endproclaim
\proclabel{LemmeCompaciteQuasiconforme}
%
\remark The proof is an elementary but technical application of classical results concerning quasiconformal mappings on the Riemann sphere. Consequently, the reader will not
find that it provides any enlightenment concerning the sequel.
\medskip
\proof We identify $\Bbb{D}$ with the hyperbolic half plane $\Bbb{H}^+$ and we recall that this identification
sends the origin and $1$ to $i$ and $0$ respectively. For $n\in\Bbb{N}$, let $\mu_n$ be the complex dilatation of
$f_n$. For all $n$, we extend $\mu_n$ to a measurable function $\tilde{\mu}_n$ defined on the whole of $\Bbb{C}$ by:
%
$$
\tilde{\mu}_n(\overline{z}) = \overline{\mu_n(z)}.
$$
%
\noindent For all $n$, let $g_n:\hat{\Bbb{C}}\rightarrow\hat{\Bbb{C}}$ be the unique quasiconformal mapping
with complex dilatation $\tilde{\mu}_n$ which preserves $0$, $1$ and $\infty$. We define
$h_n:\hat{\Bbb{C}}\rightarrow\hat{\Bbb{C}}$ by:
%
$$
h_n(\overline{z}) = \overline{g_n(z)}.
$$
%
\noindent The function $h_n$ has the same complex dilatation as $g_n$ and also fixes $0$, $1$ and $\infty$. It follows
by uniqueness that $h_n=g_n$, and consequently that $g_n$ preserves $\Bbb{R}$. Since $g_n$ is orientation
preserving, it follows that $g_n$ preserves $\Bbb{H}^+$. We may thus compose $g_n$ with a conformal mapping to obtain
a quasiconformal mapping $\tilde{g}_n$ whose complex dilatation is $\tilde{\mu}_n$ and which preserves $\Bbb{H}^+$ and the points $i$, $0$
and $-i$. By composing this mapping with the identification between $\Bbb{D}$ and $\Bbb{H}^+$, we obtain a
quasiconformal mapping $\tilde{f}_n:\hat{\Bbb{C}}\rightarrow\hat{\Bbb{C}}$ which sends $\Bbb{D}$ onto itself,
fixes $0$, $1$ and $\infty$ and has the same complex dilatation inside $\Bbb{D}$ as $f_n$. By the uniqueness of quasiconformal
mappings, if follows that there exists a rotation $\rho_n$ such that $\rho_n\circ\tilde{f}_n$ coincides with $f_n$
in the disc $\Bbb{D}$.
\medskip
\noindent By theorem $II\cdot 5\cdot 1$ of \cite{LV}, the sequence $\suite{\tilde{f}}{n}$ constitutes a normal
family. It follows that there exists a subsequence $(f_{k_n})_{n\in\Bbb{N}}$ that converges in $\hat{\Bbb{C}}$ to a limit function
$\tilde{f}_0$. Since the image of $\tilde{f}_0$ contains at least three different points (being $0$, $1$ and $\infty$),
theorem $II\cdot 5\cdot 3$ of \cite{LV} allows us to conclude that this function is a quasiconformal mapping and that the convergence is uniform over $\hat{\Bbb{C}}$.
\medskip
\noindent Since the group of rotations of $\Bbb{C}$ is compact, we may suppose that there exists a rotation
$\rho_0$ such that $(\rho_{k_n})_{n\in\Bbb{N}}$ converges towards $\rho_0$ as $n$ tends to infinity. Consequently,
if we define $f_0$ by:
%
$$
f_0 = \rho_0\circ\tilde{f}_0,
$$
%
\noindent then it follows that the sequence $(f_{k_n})_{k\in\Bbb{N}}$ converges towards $f_0$ locally uniformly in $\Bbb{D}$
as $n$ tends to infinity and the result follows.\qed
%
\newhead{The Plateau Problem}
%
\newsubhead{Definitions}
%
\noindent A {\emph Hadamard manifold} is a complete, connected and simply connected manifold of negative sectional curvature. The manifold $\Bbb{H}^3$ is an example of a
$3$-dimensional Hadamard manifold. In \cite{LabA}, Labourie studies the {\emph Plateau problem} for constant Gaussian curvature
hypersurfaces immersed in a Hadamard manifold $M$. In the language of this paper, a Plateau problem is a pair $(S,\varphi)$
where $S$ is a Riemann surface and $\varphi:S\rightarrow\partial_\infty\Bbb{H}^3$ is a locally conformal mapping. A solution to
this Plateau problem is an immersion $i:S\rightarrow\Bbb{H}^3$ such that the immersed hypersurface $(S,i)$ is a
k-surface and, if we denote by $\hat{\mathi}$ the Gauss lifting of $i$ and by $\overrightarrow{n}$ the Gauss-Minkowski mapping, then:
%
$$
\overrightarrow{n}\circ\hat{\mathi} = \varphi.
$$
%
\newsubhead{The Space of Solutions}
%
\noindent We define $\Cal{L}$ to be the set of Gauss liftings of pointed k-surfaces in $\Bbb{H}^3$:
%
$$
\Cal{L} = \left\{(\hat{\Sigma},p)|\Sigma\text{ is a k-surface in }\Bbb{H}^3, p\in\hat{\Sigma} \right\}.
$$
%
\noindent Let $\gamma$ be a geodesique in $\Bbb{H}^3$. We define $T_\gamma$, the tube about $\gamma$ in $U\Bbb{H}^3$
by:
%
$$
T_\gamma = \left\{ n_{\gamma(t)}\in U_{\gamma(t)}\Bbb{H}^3 |\langle n_{\gamma(t)},\partial_t\gamma(t)\rangle = 0 \right\}.
$$
%
\noindent We define a {\emph tube about $\gamma$\/} to be any complete surface immersed in $U\Bbb{H}^3$ whose image coincides
with $T_\gamma$. We define $\Cal{L}_\infty$ to be the set of pointed tubes in $U\Bbb{H}^3$:
%
$$
\Cal{L}_\infty = \left\{(T,p)| T\text{ is a tube about a geodesic }\gamma\text{ dans }\Bbb{H}^3,p\in T\right\}.
$$
%
\noindent We define $\overline{\Cal{L}}$ to be the union of these two sets:
%
$$
\overline{\Cal{L}} = \Cal{L}\munion\Cal{L}_\infty.
$$
%
\noindent The justification for this notation will become clear in the next section. Following \cite{LabA}, we will refer
to the surfaces contained in $\Cal{L}_\infty$ as {\emph curtain surfaces\/}.
%
\newsubhead{The Properties of the Space of Solutions}
%
\noindent In the sequel, we will require the following results of \cite{LabA} concerning Plateau problems. We quote the theorem
$7.3.2$ of this paper, which is an existence theorem for solutions to the Plateau problem:
%
\proclaim{Theorem \nextprocno\ {\bf [Labourie, 2000]}\ {\sl Existence of Solutions.}}
%\medskip
\noindent Let $S$ be a surface and let $\varphi:S\rightarrow\partial_\infty\Bbb{H}^3$ be a local homeomorphism. If $\Omega$ is
a relatively compact open subset of $S$, then there exists a solution to the Plateau problem $(\Omega,\varphi)$.
\endproclaim
\proclabel{ThmLabExistence}
%
\noindent The theorem $7.2.1$ of the same paper gives us uniqueness as well as some information concerning the structure of solutions to
the Plateau problem:
%
\proclaim{Theorem \nextprocno\ {\bf [Labourie, 2000]}\ {\sl Structure of Solutions.}}
%\medskip
\noindent There exists at most one solution to the Plateau problem. Moreover, let $\Sigma=(S,i)$ be a k-surface
in $\Bbb{H}^3$ and let $\hat{\Sigma}=(S,\hat{\mathi})$ be its Gauss lifting. Define $\varphi$ by
$\varphi=\overrightarrow{n}\circ\hat{\mathi}$ such that $i$ is the solution to the Plateau problem $(S,\varphi)$.
Let $\Omega$ by an open subset of $S$. There exists a solution to the Plateau problem $(\Omega,\varphi)$ which is
a graph over $\Omega$ in the extension of $\Sigma$. In otherwords, there exists $f:\Omega\rightarrow\Bbb{R}^+$ such
that the solution coincides (up to reparametrisation) with $(\Omega,\opExp(fN_\Sigma))$.
\endproclaim
\proclabel{LemmeLabNature}
%
\noindent Finally, as a compactness result, we use the principal result of \cite{LabB}, which translates into
our framework as follows:
%
\proclaim{Theorem \nextprocno\ {\bf [Labourie, 2000]}\ {\sl Compactness.}}
%\medskip
\noindent Let $(\Sigma_n)_{n\in\Bbb{N}}=(S_n,i_n)_{n\in\Bbb{N}}$ be a sequence of k-surfaces in $\Bbb{H}^3$ and, for every $n$, let
$\hat{\Sigma}_n=(S_n,\hat{\mathi}_n)$ be the Gauss lifting of $\Sigma_n$. For every $n$, let $p_n\in S_n$ be an
arbitrary point of $S_n$. If there exists a compact subset $K\subseteq U\Bbb{H}^3$ such that $\hat{\mathi}_n(p_n)\in K$ for every $n$, then there exists
$(\hat{\Sigma}_0,p_0)\in\overline{\Cal{L}}=\Cal{L}\munion\Cal{L}_\infty$ such that, after extraction of a
subsequence, $(\hat{\Sigma}_n,p_n)_{n\in\Bbb{N}}$ converges to $(\hat{\Sigma}_0,p_0)$ in the Cheeger/Gromov topology.
\endproclaim
\proclabel{ThmLabCompacite}
%
\remark We may also obtain this theorem as a special case of \cite{SmiA}.
%
\newhead{The Schwarz Lemma}
%
\newsubhead{A Multitude of Schwarz Lemmas}
%
\noindent The many different forms of the Schwarz lemma express the elliptic regularity of the generalised
Cauchy-Riemann equations. By enabling us, under certain conditions, to obtain bounds on the derivatives of
holomorphic functions, the Schwarz lemma provides us with important compactness results for certain families
of such functions.
\medskip
\noindent In this section, we prove a version of the Schwarz lemma which is adapted to the Cheeger/ Gromov topology:
%
\proclaim{Lemma \nextprocno}
%
\noindent Let $(S_n,g_n,p_n,\omega_n)_{n\in\Bbb{N}}$ be a sequence of complete oriented pointed surfaces. Suppose that this sequence converges towards a complete oriented pointed
surface $(S_0,g_0,p_0,\omega_0)$. For all $n$, let $J_n$ be the unique complex structure generated over $S_n$ by $g_n$ and $\omega_n$.
\medskip
\noindent For all $n$, let $\varphi_n:(S_n,J_n)\rightarrow\Bbb{D}$ be a holomorphic mapping. There exists a holomorphic mapping $\varphi_0:(S_0,J_0)\rightarrow\overline{\Bbb{D}}$
such that, after extraction of a subsequence, $(\varphi_n)_{n\in\Bbb{N}}$ converges to $\varphi_0$.
\medskip
\noindent In otherwords, if $(\psi_n)_{n\in\Bbb{N}}$ is a sequence of convergence mappings for $(S_n,g_n,p_n)_{n\in\Bbb{N}}$ with respect to $(S_0,g_0,p_0)$, then for every
compact subset $K$ of $S_0$ there exists $N\in\Bbb{N}$ such that:
\medskip
\item{(i)} for all $n\geqslant N$, the restriction of $\phi_n\circ\psi_n$ to $K$ is smooth, and
\medskip
\item{(ii)} $(\phi_n\circ\psi_n)_{n\geqslant N}$ converges to $\phi_0$ in the $C^\infty$ topology over $K$.
\endproclaim
\proclabel{LemmeCGAdaptedSchwarz}
%
\noindent In order to prove this result, we require the following lemma concerning conformal representations which is interesting in its own right:
%
\proclaim{Lemma \nextprocno}
%\medskip
\noindent Let $\Omega\subseteq\Bbb{R}^2$ be an open set in $\Bbb{R}^2$ containing the origin. Let $D\subseteq\Omega$
be a disc centred on the origin whose closure is contained within $\Omega$. Let $\suite{g}{n}$ be a sequence of metrics
over $\Omega$ which tends towards a metric $g_0$ in the $C^\infty_\oploc$ topology. For every $n$, let $J_n$ be
the almost complex structure over $\Omega$ generated by $g_n$ and the canonical orientation of $\Bbb{R}^2$.
\medskip
\noindent For every $n$, let $f_n:(\Bbb{D},0)\rightarrow (D,0,J_n)$ be the conformal representation of $(D,0,J_n)$.
There exists a sequence of rotations $\suite{R}{n}$ of $\Bbb{D}$ and a mapping $f_0$ such that
$(f_n\circ R_n)_{n\in\Bbb{N}}$ converges towards $f_0$ in the $C^\infty$ topology over $\Bbb{D}$
\endproclaim
\proclabel{LemmeConvergenceDesRepresentationsConformes}
%
\noindent We will use the following existing version of the Schwarz lemma, where the target manifold is of dimension $2$:
%
\proclaim{Lemma \nextprocno\ {\sl Schwarz Lemma.}}
%\medskip
\noindent Let $\Omega\subseteq\Bbb{R}^2$ be an open set in $\Bbb{R}^2$ and let $\suite{J}{n}$ and $J_0$ be (almost)
complex structures over $\Omega$ such that $(J_n)_\ninn$ tends towards $J_0$ in the $C^\infty_\oploc$ topology over $\Omega$.
\medskip
\noindent Let $K\subseteq\Omega$ be compact and let $\suite{f}{n}:\Bbb{D}\rightarrow\Omega$ be a sequence of
mappings such that:
\medskip
\item{(i)} for every $n$, the mapping $f_n$ is holomorphic with respect to the complex structure $J_n$, and
\medskip
\item{(ii)} for every $n$, the image of $f_n$ is contained within $K$.
\medskip
\noindent There exists a (possibly constant) mapping $f_0:\Bbb{D}\rightarrow\Omega$ such that, after extraction of a
subsequence, $\suite{f}{n}$ converges to $f_0$ locally uniformly $C^\infty_\oploc$ topology over $\Bbb{D}$.
\endproclaim
\proclabel{LemmeGromovSchwarz}
%
\remark The Schwarz lemma in dimension $2$ has simpler hypotheses than in the general case, since, firstly, every $2$-form over a two dimensional manifold is closed, and,
secondly, conformal metrics over open subsets of $\Bbb{R}^2$ are unique up to multiplication by positive functions.
\medskip
\remark A more general statement of this lemma, along with a proof, may be found in \cite{Muller}.
%
\newsubhead{Convergence of Conformal Representations}
%
\noindent We begin by recalling two elementary technical lemmas which are proven by Labourie in \cite{LabB}. For a function
$f:\Bbb{D}\rightarrow S$ defined on the Poincar\'e disc and taking values in an arbitrary surface, we define
$f(\partial\Bbb{D})$ to be the set of all points which may be obtained as limits of sequences $(f(x_n))_{n\in\Bbb{N}}$
where $\suite{x}{n}$ is a sequence of points in $\Bbb{D}$ which tends towards $\partial\Bbb{D}$.
\medskip
\noindent We obtain the following result:
%
\proclaim{Lemma \nextprocno}
%\medskip
\noindent With the hypotheses of lemma \procref{LemmeGromovSchwarz}, we suppose moreover that there exists $\epsilon>0$
such that for every $n$:
%
$$
d_n(f_n(\partial\Bbb{D}),f_n(0))\geqslant\epsilon.
$$
%
\noindent Then the mapping $f_0$ is not constant.
\endproclaim
\proclabel{LemmeLaLimiteNestPasConstante}
%
\remark Indeed, if $f_0$ is constant, then, for all $r\in (0,1)$, for sufficiently large $n$, the mapping $f_n$ sends the circle $\left\{\left|z\right|=r\right\}$ into the ball
of radius $\epsilon/2$ about $f_n(0)$. Consequently, for sufficently large n, $f_n(\left\{\left|z\right|=r\right\})$ is separated from $f_n(\partial\Bbb{D})$ by a distance of
at least $\epsilon/2$. Thus, it follows that for sufficiently large $n$, $f_n$ stretches the ring $\left\{r<\left|z\right|<1\right\}$ into a ring of ``width'' at least
$\epsilon/2$. We now obtain the result by considering modules of rings.
%
\proclaim{Lemma \nextprocno}
%\medskip
\noindent With the hypotheses of lemma \procref{LemmeGromovSchwarz}, the set $f_0(\partial\Bbb{D})$ is contained in the Haussdorf limit of $f_n(\partial\Bbb{D})$.
\endproclaim
\proclabel{LemmeEmplacementDuBordDeLaLimite}
%
\noindent We now prove lemma \procref{LemmeConvergenceDesRepresentationsConformes}\colon
\medskip
{\it\noindent Proof of lemma \procref{LemmeConvergenceDesRepresentationsConformes}:\ }
For every $n$, since $(D,J_n)$ is contained in a larger disc, it following that $(D,J_n)$ is of hyperbolic type, and
consequently that $f_n$ exists for every $n$. There exists a compact set $K\subseteq\Omega$ such that $D$ is always contained
in $K$. Thus, by lemma \procref{LemmeGromovSchwarz}, there exists a subsequence $(f_{k_n})_{n\in\Bbb{N}}$ and
a (possibly constant) pointed holomorphic application $f_0:(\Bbb{D},0)\rightarrow(D,0,J_0)$ such that
$(f_{k_n})_{n\in\Bbb{N}}$ converges towards $f_0$ in the $C^\infty_\oploc$ topology over $\Bbb{D}$.
\medskip
\noindent There exists $\epsilon>0$ such that, for every $n$:
%
$$
d_n(f_n(\partial\Bbb{D}),f_n(0))\geqslant\epsilon.
$$
%
\noindent Consequently, by lemma \procref{LemmeLaLimiteNestPasConstante}, the mapping $f_0$ is not constant.
\medskip
\noindent For every $p\in\Bbb{D}$, there exists $k\in\Bbb{N}$ and smooth mappings
$\varphi:(\Bbb{C},0)\rightarrow(\Bbb{C},p)$ and $\psi:(\Omega,f_0(p))\rightarrow(\Bbb{C},0)$ such that
$\psi\circ f_0\circ\varphi$ is defined in $B_2(0)$ and the following diagram commutes:
%
$$
\commdiag{
B_2(0) &\mapright^{z^k} &\Bbb{C} \cr
\mapdown_{\varphi}& &\mapup_{\psi}\cr
\Bbb{D} &\mapright^{f_0} &\Bbb{C} \cr}
$$
%
\noindent For every $n$, we define $\alpha_n$ by:
%
$$
\alpha_n = \psi\circ f_{k_n}\circ\varphi.
$$
%
\noindent Let $\|\cdot\|_0$ be the Euclidian norm over $\Bbb{C}$ and let $i:S^1\rightarrow\Bbb{C}$ be the canonical
embedding of the unit circle into $\Bbb{C}$. We see that $\|\alpha_0(z)\|_0=1$ for every $z$ in $S^1$. Since
$\alpha_n\rightarrow\alpha_0$ as $n$ tends to infinity, for all sufficiently large $n$ we obtain:
%
$$
\|\alpha_n(z)-\alpha_0(z)\| < 1\qquad\forall z\in S^1.
$$
%
\noindent It follows that, for all sufficiently large $n$, the curve $\alpha_n\circ i$ is homotopic to
$\alpha_0\circ i$ in $\Bbb{C}\setminus\left\{0\right\}$. Since every $\alpha_n$ is a diffeomorphism onto its image,
it follows that $k$ is equal to $1$. The function $f_0$ is thus everywhere a local diffeomorphism.
\medskip
\noindent By lemma \procref{LemmeInjectiviteDeLaLimite}, the mapping $f_0$ is injective.
\medskip
\noindent By lemma \procref{LemmeEmplacementDuBordDeLaLimite}, the set $f_0(\partial\Bbb{D})$ is contained in the
Haussdorf limit of $(f_{k_n}(\partial\Bbb{D}))_{n\in\Bbb{N}}$. However, since, for every $n$, the mapping $f_{k_n}$
is a diffeomorphism, it follows that $f_{k_n}(\partial\Bbb{D})$ is contained in $\partial D$. Consequently:
%
$$
f_0(\partial\Bbb{D})\subseteq\partial D.
$$
%
\noindent We may now show that $f_0$ is bijective. Indeed, $f_0(\Bbb{D})$ is a closed subset of $D$. However, since $f_0$ is a non-constant holomorphic mapping, by the open
mapping principal, $f_0(\Bbb{D})$ is an open subset of $D$. Since this subset is trivially non-empty, it follows that $f_0(\Bbb{D})$ coincides with $D$, and
the surjectivity follows.
\medskip
\noindent The mapping $f_0$ is thus a conformal representation of $(D,0,J_0)$. Let us suppose that $Df_0(0)\propto\opId$ and
let $\suite{R}{n}$ be a sequence of rotations of $\Bbb{D}$ such that, for every $n$:
%
$$
D(f_n\circ R_n)(0) \propto \opId.
$$
%
\noindent By uniqueness of conformal representations, we see that every subsequence of $(f_n\circ R_n)_{n\in\Bbb{N}}$
contains a subsubsequence which converges towards $f_0$. It thus follows that $(f_n\circ R_n)_{n\in\Bbb{N}}$ converges
itself towards $f_0$ and the result follows.\qed
%
\proclaim{Proposition \nextprocno}
%\medskip
\noindent Let $(\Sigma_n)_{n\in\Bbb{N}}=(S_n,g_n,\omega_n,p_n)_{n\in\Bbb{N}}$ be a sequence of pointed surfaces which converges to the pointed surface
$\Sigma_0=(S_0,g_0,\omega_0,p_0)$. Let $R>0$ be such that the injectivity radius of $\Sigma_0$ about $p_0$ is strictly greater than $R$. For every $n$, let $f_n:(\Bbb{D},0)\rightarrow (B_R(p_n),p_n)$ be the conformal representation of $(B_R(p_n),p_n)$. There exists a sequence of rotations $\suite{R}{n}$ of $\Bbb{D}$ such that $(f_n\circ R_n)_{n\in\Bbb{N}}$ converges towards $f_0$ in the $C^\infty_{\oploc}$ topology over $\Bbb{D}$.
\endproclaim
%
\proof Let $\epsilon>0$ be such that the injectivity radius of $\Sigma_0$ is greater than $R+2\epsilon$. We may suppose
that there exists $(\varphi_n)_{n\in\Bbb{N}}:\Sigma_0\rightarrow(\Sigma_n)_{n\in\Bbb{N}}$ such that:
\medskip
\myitem{(i)} for every $n$, the restriction of $\varphi_n$ to $B_{R+2\epsilon}(p_n)$ is a diffeomorphism, and
\medskip
\myitem{(ii)} $(\varphi_n^*g_n)_\ninn$ converges to $g_0$ in the $C^\infty_\oploc$ topology.
\medskip
\noindent For every $n$, let $A_n\in\opEnd(T_{p_0}S_0)$ be such that:
%
$$
A_n^*\varphi_n^*g_n(0) = g_0(0).
$$
%
\noindent We may suppose that $(A_n)_\ninn$ tends towards $\opId$. For all sufficiently large $n$,
we may define the exponential mapping $\opExp_n:A_n(B_{R+\epsilon}(0))\rightarrow S_0$ from $T_{p_0}S_0$ onto
$S_0$ with respect to the metric $\varphi_n^*g_n$. Since $(\varphi_n^*g_n)_\ninn$ converges to $g_0$ in the $C^\infty_\oploc$ topology over
$B_{R+2\epsilon}(p_0)$, it follows that $(\opExp_n\circ A_n)_\ninn$ converges to $\opExp_0$ in the $C^\infty_\oploc$ topology over
$B_{R+\epsilon}(0)$.
\medskip
\noindent If we define $\widetilde{\opExp}_n$ to be the exponential mapping of $(S_n,g_n)$ about $p_n$, we obtain the
following commutative diagram:
%
$$\commdiag{
B_{R+\epsilon}(0)\subseteq T_{p_n}S_n & \mapright^{\widetilde{\opExp}_n} & B_{R+\epsilon}(p_n)\subseteq S_n\cr
\mapdown_{A_n^{-1}\circ T_{p_n}\varphi_n^{-1}} & & \mapup_{\varphi_n} \cr
B_{R+\epsilon}(0)\subseteq T_{p_0}S_0 & \mapright^{\opExp_n\circ A_n} & \opIm(B_{R+\epsilon}(p_0))\subseteq S_0\cr
}$$
%
\noindent For every $n$, let us denote $\tilde{g}_n=(\opExp_n\circ A_n)^*\varphi_n^*g_n$ and
for $n=0$ we define $\tilde{g}_0=\opExp_0^*g_0$. The sequence $\suite{\tilde{g}}{n}$ converges towards $\tilde{g}_0$ in the
$C^\infty_\oploc$ topology over $B_{R+\epsilon}(0)$. In particular, for sufficiently large $n$,
the metric $\tilde{g}_n$ is non-degenerate in a neighbourhood of the closed ball $\overline{B}_R(0)$. Moreover,
by lemma \procref{LemmeInjectiviteLorsquOnSApprocheDuLimite}, for sufficiently large $n$, the injectivity radius of
$\Sigma_n$ is greater than $R$.
\medskip
\noindent For all $n$, let $\tilde{f}_n$ be the conformal representation of $(B_R(0),0,\tilde{g}_n)$. For sufficiently
large $n$, the restriction of $\varphi_n\circ\opExp_n\circ A_n$ to $B_R(0)$ is a well defined diffeomorphism onto its
image. By composing $\tilde{f}_n$ with a rotation if necessary, we obtain the following commutative diagram:
%
$$\commdiag{
\Bbb{D} &\mapright^{f_n} &B_R(p_n)\subseteq S_n \cr
\mapdown_{\tilde{f}_n} & & \mapup_{\varphi_n} \cr
B_R(0)\subseteq T_{p_0}S_0 &\mapright^{\opExp_n\circ A_n} &\opIm(B_R(p_0))\subseteq S_0 \cr
}$$
%
\noindent In other words, for every $n$:
%
$$
\varphi_n\circ(\opExp_n\circ A_n)\circ\tilde{f}_n = f_n.
$$
%
\noindent In the case where $n=0$, we obtain:
%
$$
\opExp_0\circ\tilde{f}_0 = f_0.
$$
%
\noindent By the preceeding lemma, there exists a sequence of rotations $\suite{R}{n}$ of $\Bbb{D}$ such that:
%
$$\matrix
&\tilde{f}_n\circ R_n \hfill&\rightarrow\tilde{f}_0\hfill&\text{ as }n\rightarrow\infty \hfill\cr
\Rightarrow\hfill&(\opExp_n\circ A_n)\circ\tilde{f}_n\circ R_n \hfill&\rightarrow\opExp_0\circ\tilde{f}_0\hfill&\text{ as }n\rightarrow\infty \hfill\cr
\Rightarrow\hfill&\varphi_n|_{B_{R+\epsilon}(p_0)}^{-1}\circ f_n\circ R_n \hfill&\rightarrow f_0\hfill&\text{ as }n\rightarrow\infty. \hfill\cr
\endmatrix$$
%
\noindent The result now follows.\qed
\medskip
\noindent We now obtain lemma \procref{LemmeCGAdaptedSchwarz} as a corollary:
\medskip
{\it\noindent Proof of lemma \procref{LemmeCGAdaptedSchwarz}:\ } Let $q_0$ be a point in $S_0$. Let $(q_n)_{n\in\Bbb{N}}\in(S_n)_{n\in\Bbb{N}}$ be a sequence of points that
converges to $q_0$. Let $\rho$ be strictly less than the injectivity radius of $S_0$ about $q_0$. By the common sense lemma \procref{LemmeInjectiviteLorsquOnSApprocheDuLimite},
for all sufficiently large $n$, we may assume that $\rho$ is less than the injectivity radius of $S_n$ about $q_n$.
\medskip
\noindent For all $n\in\Bbb{N}\cup\left\{0\right\}$, let $\alpha_n:(\Bbb{D},0)\rightarrow(B_\rho(q_n),q_n)$ be the conformal representation of $B_\rho(q_n)$. By the
preceeding proposition, there exists a sequence of rotations $(R_n)_{n\in\Bbb{N}}$ of $\Bbb{D}$ such that $(\alpha_n\circ R_n)_{n\in\Bbb{N}}$ converges to $\alpha_0$ in the
$C^\infty_\oploc$ topology over $\Bbb{D}$.
\medskip
\noindent Since (heuristically) the mapping which sends a function to its inverse is continuous in the $C^\infty_\oploc$ topology (common sense lemma
\procref{LemmeLimitesDInverses}), it follows that $(\alpha_n\circ R_n)^{-1}_{n\in\Bbb{N}}$ converges to $\alpha_0^{-1}$ in the $C^\infty_\oploc$ topology over $B_p(q_0)$. In
otherwords, if $(\psi_n)_{n\in\Bbb{N}}$ is a sequence of convergence mappings of $(S_n,p_n)_{n\in\Bbb{N}}$ with respect to $(S_0,p_0)$, then, for every compact subset $K$ of
$B_\rho(q_0)$ there exists a whole number $N$ such that:
\medskip
\myitem{(i)} for all $n\geqslant N$:
%
$$
\psi_n(K)\subseteq(\alpha_n\circ R_n)(\Bbb{D}),
$$
%
\myitem{(ii)} the restriction of $(\alpha_n\circ R_n)^{-1}\circ\psi_n$ is smooth, and
\medskip
\myitem{(iii)} the sequence $((\alpha_n\circ R_n)^{-1}\circ\psi_n)_{n\geqslant N}$ converges to $\alpha_0^{-1}$ in the $C^\infty_\oploc$ topology over $K$.
\medskip
\noindent By the classical Schwarz lemma, there exists a (possibly constant) function $f_0:\Bbb{D}\rightarrow\overline{\Bbb{D}}$ such that, after extraction of a subsequence:
%
$$
(\phi_n\circ(\alpha_n\circ R_n))_{n\in\Bbb{N}}\rightarrow f_0.
$$
%
\noindent We define $\psi_{0,q_0}$ by:
%
$$
\psi_{0,q_0} = f_0\circ\alpha_0^{-1}.
$$
%
\noindent If $K$ is a compact subset of $B_\rho(q_0)$ then, for all sufficiently large $n$:
%
$$
(\varphi_n\circ\psi_n)|_K = (\phi_n\circ(\alpha_n\circ R_n))\circ((\alpha_n\circ R_n^{-1})\circ\psi_n)|_K
$$
%
\noindent Thus, $(\varphi_n\circ\psi_n)_{n\geqslant N}$ converges to $f_0\circ\alpha_0^{-1}=\psi_{0,q_0}$ in the $C^\infty_\oploc$ topology over $K$.
\medskip
\noindent Since $S_0$ may be covered by countably many balls of the form $B_\rho(q_0)$, the result follows.\qed
%
\newhead{A Compactness Lemma}
%
\newsubhead{Our Framework}
%
\noindent Let $\Bbb{D}$ be the Poincar\'e disc and let $\suite{\varphi}{n},\varphi_0:\Bbb{D}\rightarrow\hat{\Bbb{C}}$
be locally conformal mappings such that $(\varphi_n)_{n\in\Bbb{N}}$ converges to $\varphi_0$ locally uniformly (and consequently in the
$C^\infty_\oploc$ topology) over $\Bbb{D}$.
\medskip
\noindent We assume that for all $n>0$, there exists a solution $j_n$ to the Plateau problem $(\Bbb{D},\varphi_n)$.
By theorem \procref{LemmeLabNature}, this solution is unique.
\medskip
\noindent For all $n$, let $\hat{\mathj}_n$ be the Gauss lifting of $j_n$. For all $n$, by lemma \procref{LemmeLaMetriqueEstQC}, the metric $\hat{\mathj}^*g^\nu$ is
$k$-quasi\-con\-formal over $\Bbb{D}$. For all $n$, let $\alpha_n:(\Bbb{D},0,g_{\opEuc})\rightarrow (\Bbb{D},0,\hat{\mathj}^*g^\nu)$ be the conformal representation
of $(\Bbb{D},0,\hat{\mathj}^*g^\nu)$. For all $n$, since $\hat{\mathj}_n^*g^\nu$ is a smooth metric and since any almost complex structure over any surface is
integrable, the application $\alpha_n$ is a smooth $k$-quasiconformal mapping of the Poincar\'e disc onto itself. By the relative compactness of the space of
$k$-quasiconformal mappings over $\Bbb{D}$ (lemma \procref{LemmeCompaciteQuasiconforme}), there exists a $k$-quasiconformal mapping $\alpha_0:(\Bbb{D},0)\rightarrow(\Bbb{D},0)$
such that, after extraction of a subsequence, $\alpha_n$ converges locally uniformly to $\alpha_0$ as $n$ tends to infinity.
\medskip
\noindent For every $n$, we denote $i_n=j_n\circ\alpha_n$. If we define
$\overrightarrow{n}:U\Bbb{H}^3\rightarrow\partial_\infty\Bbb{H}^3\cong\hat{\Bbb{C}}$
to be the Gauss-Minkowski mapping, we obtain the following commutative diagram:
%
$$\commdiag{
\Bbb{D} &\mapright^{\hat{\mathi}_n} &U\Bbb{H}^3 \cr
\mapdown_{\alpha_n} & & \mapdown_{\overrightarrow{n}} \cr
\Bbb{D} &\mapright^{\varphi_n} &\partial_\infty\Bbb{H}^3 \cr
}$$
%
\noindent In other words, $(\Bbb{D},i_n)$ is the solution to the Plateau problem $(\Bbb{D},\varphi_n\circ\alpha_n)$.
In the sequel, we shall denote $S_n=\Bbb{D}$ and $\Sigma_n=(S_n,i_n)$.
%
\newsubhead{The Compactness Result}
%
\noindent We shall prove the following compactness lemma:
%
\proclaim{Lemma \nextprocno}
%\medskip
\noindent Let $z\in\Bbb{D}$ be an arbitrary point in the Poincar\'e disc. The set $\left\{i_n(z)|n\in\Bbb{N}\right\}$
is relatively compact in $\Bbb{H}^3$.
\endproclaim
\proclabel{LemmeLImageDeZEstBornee}
%
\noindent We will prove this result by contradiction. This will require the following lemma:
%
\proclaim{Lemma \nextprocno}
%\medskip
\noindent Let $(p_n)_{n\in\Bbb{N}}\in\Bbb{H}^3$ be a sequence of points which tends towards
$p_0\in\partial_\infty\Bbb{H}^3$. Let $x_0\in\Bbb{H}^3$ be an arbitary point. Let $(A_n)_{n\in\Bbb{N}}$
be a sequence of automorphisms of $\Bbb{H}^3$. For every $n$, denote also by $A_n$ the action of
$A_n$ on $\partial_\infty\Bbb{H}^3$, and suppose that, for all $n$:
%
$$\matrix
A_n p_n \hfill&= x_0,\hfill\cr
A_n p_0 \hfill&= p_0.\hfill\cr
\endmatrix$$
%
\noindent There exists a subsequence $(A_{k_n})_{n\in\Bbb{N}}$ which converges locally uniformly in the complement
of $p$ in $\partial_\infty\Bbb{H}^3$ to a constant mapping.
\endproclaim
\proclabel{LemmeLimiteEstConstante}
%
{\noindent\sl Remark:\ }It is trivial that every $A_n$ is uniquely defined modulo post-composition by a rotation about
the unique geodesique linking $x_0$ to $p_0$.
\medskip
\proof We first identify $\Bbb{H}^3$ and $\partial_\infty\Bbb{H}^3$ with the positive half-space in $\Bbb{R}^3$
and the extended complex plane respectively:
%
$$\matrix
\Bbb{H}^3 \hfill&\simeq \left\{(x_1,x_2,x_3)\in\Bbb{R}^3|x_3>0\right\}, \hfill\cr
\partial\Bbb{H}^3 \hfill&\simeq \Bbb{R}^2\munion\left\{\infty\right\}. \hfill\cr
\endmatrix$$
%
\noindent Moreover, we may assume that $p_0=\infty$ and that $x_0=(0,0,1)$. For every $n$, let us define $\bold{x}_n=(x_{1,n},x_{2,n},x_{3,n})$ by:
%
$$
\bold{x}_n = p_n.
$$
%
\noindent Let $\|\cdot\|$ be the Euclidean metric over $\Bbb{R}^3$. Since $p_n\rightarrow p$ as $n$ tends towards infinity, it follows that $\|\bold{x}_n\|$ tends towards
infinity as $n$ tends to infinity. For every $n$, we denote $\lambda_n=\|\bold{x}_n\|$, and we define the mapping $M_n$ by:
%
$$
M_n\bold{x} = \frac{\bold{x}}{\lambda_n}.
$$
%
\noindent We denote also by $M_n$ the action of $M_n$ on $\partial_\infty\Bbb{H}^3$. Let $\Theta$ by an
automorphism of $\Bbb{H}^3$ which preserves to point $(0,0,1)$ and which sends the (Euclidien) hemisphere of
unit radius centred on the origin onto a vertical half plane. For every $n$, there exists a rotation $\Psi_n$
about the axis $(0,\infty)$ such that $(\Theta\cdot\Psi_n\cdot M_n)\bold{x}_n$ is a point directly above the
point $(0,0,1)$. For every $n$, we thus define $\mu_n\in (0,\infty)$ such that:
%
$$
(\Theta\cdot\Psi_n\cdot M_n)\bold{x}_n = (0,0,\mu_n),
$$
%
\noindent and we define $\Delta_n$ by:
%
$$
\Delta_n \bold{x}=\frac{1}{\mu_n}\bold{x}
$$
%
\noindent The mapping $\Delta_n$ sends $(\Theta\cdot\Psi_n\cdot A_n)\bold{x}_n$ onto $(0,0,1)$. For all $n\in\Bbb{N}$, we define $N_n$ by:
%
$$
N_n = (\Theta\cdot\Psi_n)^{-1}\Delta_n (\Theta\cdot\Psi_n).
$$
%
\noindent As for $\suite{M}{n}$, for every $n$, we denote also by $N_n$ the action of $N_n$ on
$\partial_\infty\Bbb{H}^3$. Let $\Bbb{D}$ be unit disc about the origin in the horizontal plane passing through the
origin (which we identify with $\partial_\infty\Bbb{H}^3$). $\Bbb{D}$ is preserved by $\Psi_n$ for every $n$. The
mapping $\Theta$ then sends it to a half plane which is in turn preserved by $\Delta_n$ for every $n$. Consequently, for
every $n$, the mapping $N_n$, being a composite of these mappings, preserves $\Bbb{D}$.
\medskip
\noindent Let $R>0$ be a positive real number. Let $N\in\Bbb{N}$ be such that for $n\geqslant N$:
%
$$
\lambda_n > R.
$$
%
\noindent For $n\geqslant N$, the set $M_n(B_R(0))$ is contained in $\Bbb{D}$, and consequently $N_n\circ M_n(B_R(0))$ is
also contained in $\Bbb{D}$. Since, for all $n$, the mapping $N_n\circ M_n$ is an isometry, it follows that the
action of $N_n\circ M_n$ on $\partial_\infty\Bbb{H}^3=\hat{\Bbb{C}}$ is conformal. By the classical Schwarz lemma, we
may suppose that the restriction of $(N_n\circ M_n)_{n\in\Bbb{N}}$ to $B_R(0)$ converges in the $C^\infty_\oploc$ topology over $B_R(0)$ to a
holomorphic mapping taking values in $\overline{\Bbb{D}}$. Since $R>0$ is arbitrary, we may assume that
$(N_n\circ M_n)_{n\in\Bbb{N}}$ converges in the $C^\infty_\oploc$ topology over $\Bbb{C}$ to a holomorphic mapping taking values in $\overline{\Bbb{D}}$,
which is consequently constant by Liouville's principal.
\medskip
\noindent For all $n$, there exists a rotation $R_n$, about $x_0$ such that:
%
$$
A_n = R_n\circ N_n\circ M_n.
$$
%
\noindent Since $SO(3)$ is compact, we may assume that there exists a rotation $R_0$ such that $\suite{R}{n}$ tends towards
$R_0$ as $n$ tends to infinity. Consequently, after extraction of a subsequence, the restriction of $\suite{A}{n}$ to $\Bbb{C}$ converges in the $C^\infty_\oploc$
topology over $\Bbb{C}$ to a constant function, and the result follows.\qed
\medskip
\noindent The following lemma places preliminary restrictions on the position of $i_n(z)$ for every
$n$:
%
\proclaim{Lemma \nextprocno}
%\medskip
\noindent Let $z\in\Bbb{D}$ be an arbitrary point. There exists a neighbourhood $\Omega$ of $\varphi_0\circ\alpha_0(z)$
in $\Bbb{H}^3\munion\partial_\infty\Bbb{H}^3$ such that, for all $n$:
%
$$
i_n(z) \notin \Omega.
$$
\endproclaim
\proclabel{LemmePNEstPasQ}
%
\proof Let $U\subseteq\Bbb{D}$ and $V_a\subseteq\partial_\infty\Bbb{H}^3\cong\hat{\Bbb{C}}$ be open sets such
that:
\medskip
\item{(i)} $z\in U$,
\medskip
\item{(ii)} $V_a$ is a disc about $\varphi_0(z)$ in $\hat{\Bbb{C}}$, and
\medskip
\item{(iii)} the mapping $\varphi_0$ sends $U$ diffeomorphically onto $V_a$.
\medskip
\noindent Let $V_b$ be another disc about $\varphi_0(z)$ in $\hat{\Bbb{C}}$ whose closure is strictly contained within
$V_a$. Since $\suite{\varphi}{n}$ converges towards $\varphi_0$ locally uniformly, by common sense lemma
\procref{LemmeCompacteDansLImageDeLaLimite}, for all sufficiently large $n$, we have:
%
$$
V_b \subseteq \varphi_n(U)
$$
%
\noindent Since $\suite{\varphi}{n}$ tends towards $\varphi_0$ in the $C^2_\oploc$ topology, by common sense
lemma \procref{LemmeInjectiviteLorsquOnSApprocheDuLimite}, restricting $U$ if necessary, we may assume that for all
sufficiently large $n$, the restriction of $\varphi_n$ to $U$ is a diffeomorphism onto its image. It follows that
for all sufficiently large $n$, there exists an open set $U_n$ contained in $U$ such that $\varphi_n$ sends $U_n$
diffeomorphically onto $V_b$.
\medskip
\noindent Since $\suite{\alpha}{n}$ tends towards $\alpha_0$ as $n$ tends to infinity, for all sufficiently large
$n$, the point $\alpha_n(z)$ is contained in $U$. Since $(\varphi_n\circ\alpha_n)_{n\in\Bbb{N}}$ converges towards
$\varphi_0\circ\alpha_0$ as $n$ tends to infinity, for all sufficiently large $n$, we have:
%
$$
(\varphi_n\circ\alpha_n)(z) \in V_b.
$$
%
\noindent Consequently, by injectivity, $\alpha_n(z)$ is in $U_n$.
\medskip
\noindent For all large $n$, the Plateau problem $(U_n,\varphi_n)$ is equivalent (up to reparametrisation) to
the problem $(V_b,i)$, where $i:V_b\rightarrow\hat{\Bbb{C}}$ is the canonical embedding of $V_b$ into $\Bbb{C}$. The
solution to this problem is a hypersurface equidistant to a complete totally geodesic surface contained in $\Bbb{H}^3$.
It follows that this solution is the boundary of a concave open set $\Omega$ in $\Bbb{H}^3\munion\partial_\infty\Bbb{H}^3$
which contains $(\varphi_0\circ\alpha_0)(z)$. Using the identification of $\Bbb{H}^3$ with the upper half space in $\Bbb{R}^3$, let $H$ be a (Euclidian) hemisphere centred on
$(\varphi_0\circ\alpha_0)(z)$ and contained in $\Omega$. Let $\Omega'$ be the interior of $H$. For all $p\in\Omega'$ and for all
$q\in\partial_\infty\Omega'$, the unique geodesique in $\Bbb{H}^3$ going from $p$ to $q$ stays in the interior of $H$, and
consequently does not intersect $\partial\Omega$.
\medskip
\noindent For all sufficiently large $n$, let $\gamma_n$ be the geodesique in $\Bbb{H}^3$ such that:
%
$$
(\partial_t\gamma_n)(0) = (\hat{\mathj}_n\circ\alpha_n)(z).
$$
%
\noindent By Labourie's uniqueness theorem (theorem \procref{LemmeLabNature}) the boundary of $\Omega$ is a graph over $U_n$ in the extension of
$\Sigma_n$. It follows that $\gamma_n$ intersects this boundary. However, for all sufficiently large $n$,
the point $(\varphi_n\circ\alpha_n)(z)$ is contained in $\partial_\infty\Omega'$ and consequently
$(j_n\circ\alpha_n)(z)$ is in the exterior of $H$.
\medskip
\noindent Consequently, for all sufficiently large $n$:
%
$$
(j_n\circ\alpha_n)(z)\notin\Omega',
$$
%
\noindent and the result follows.\qed
\medskip
\noindent We are now in a position to prove lemma \procref{LemmeLImageDeZEstBornee}:
\medskip
{\bf\noindent Proof of lemma \procref{LemmeLImageDeZEstBornee}:\ } We will assume the contrary in order to obtain a
contradiction.
\medskip
\noindent Let us denote $q=(\varphi_0\circ\alpha_0)(z)$. By compactness, we may suppose that there exists
$p\in\partial_\infty\Bbb{H}^3$ such that $i_n(z)\rightarrow p$ as $n$ tends to infinity. By the preceeding lemma
$p\neq q$.
\medskip
\noindent Let $x_0\in\Bbb{H}^3$ be an arbitrary point. Let $\suite{A}{n}$ be a sequence of isometries of $\Bbb{H}^3$
such that, for all $n$:
%
$$\matrix
A_ni_n(p_n) \hfill&=x_0,\hfill\cr
A_np_0 \hfill&=p_0.\hfill\cr
\endmatrix$$
%
\noindent For all $n$, we will note also by $A_n$ the actions of $A_n$ over $U\Bbb{H}^3$ and $\partial_\infty\Bbb{H}^3\cong\hat{\Bbb{C}}$.
\medskip
\noindent For all $n$, we define the application $k_n$ by $k_n=A_n\circ i_n$, and we define the immersed surface
$\Sigma_n'$ by $\Sigma_n'=(S_n,k_n)$. For all $n$, the immersed surface $\Sigma_n'$ is the unique solution to the Plateau
problem $(S_n,A_n\circ\varphi_n\circ\alpha_n)$. Let $\hat{\Sigma}'_n=(S_n,\hat{k}_n)$ be the Gauss lifting of $\Sigma'_n$.
For all $n$, we define $p_n$ by $p_n=z$.
\medskip
\noindent Since $\{\hat{k}_n(p_n) | n\in\Bbb{N}\}$ is relatively compact in $\Bbb{H}^3$, by Labourie's compactness theorem (theorem \procref{ThmLabCompacite})
we may assume that there exists a pointed immersed surface $\hat{\Sigma}'_0=(S_0,\hat{k}_0,p_0)$ in $U\Bbb{H}^3$ such that
$(\hat{\Sigma}'_n,p_n)_{n\in\Bbb{N}}$ converges to $(\hat{\Sigma}'_0,p_0)$ in the Cheeger/Gromov topology.
\medskip
\noindent For all $n$, let $\mu_n:(S_n,p_n)\rightarrow(\Bbb{D},z)$ be the tautological diffeomorphism. In particular, we have:
%
$$
\mu_n(p_n) = z.
$$
%
\noindent We obtain the following commutative diagram:
%
$$\commdiag{
(\Bbb{D},0) & \mapleft^{\mu_n} &(S_n,p_n) \cr
\mapdown_{\alpha_n} & & \mapdown_{\overrightarrow{n}\circ\hat{k}_n} \cr
(\Bbb{D},0) & \mapright^{A_n\circ\varphi_n} &\partial_\infty\Bbb{H}^3 \cr
}.$$
%
\noindent By the Cheeger/Gromov adapted Schwarz lemma (lemma \procref{LemmeCGAdaptedSchwarz}), we may assume that there exists a (possibly constant) holomorphic
mapping $\mu_0:S_0\rightarrow\Bbb{D}$ such that $\suite{\mu}{n}$ tends towards $\mu_0$ in the $C^\infty_\oploc$ topology.
\medskip
\noindent By lemma \procref{LemmeLimiteEstConstante} the sequence $\suite{A}{n}$ converges locally uniformly in the
complement of $p$ in $\partial_\infty\Bbb{H}^3$ to a constant mapping. Consequently $(A_n\circ\varphi_n)_{n\in\Bbb{N}}$
converges locally uniformly in the complement of $\varphi_0^{-1}(\left\{p\right\})$ in $\Bbb{D}$ to a constant mapping.
It then follows that $(A_n\circ\varphi_n\circ\alpha_n)_{n\in\Bbb{N}}$ converges in the complement of
$(\varphi_0\circ\alpha_0)^{-1}(\left\{p\right\})$ in $\Bbb{D}$ to a constant mapping. Let us define $\Omega$ by:
%
$$
\Omega = \Bbb{D}\setminus(\varphi_0\circ\alpha_0)^{-1}(\left\{p\right\}).
$$
%
\noindent Since $(\varphi_n\circ\alpha_n)(z)=q$ for all $n$, it follows that $(\varphi_0\circ\alpha_0)(z)=q$, and,
in particular that $z\in\Omega$. However, since $\mu_n(p_n)=z$ for all $n$, we may suppose that there exists $\epsilon>0$ such that
$\mu_n(B_\epsilon(p_n))\subseteq\Omega$ for all $n$. It follows that
$(A_n\circ\varphi_n\circ\alpha_n\circ\mu_n)_{n\in\Bbb{N}}$ converges uniformly over $B_\epsilon(p_0)$ towards a
constant mapping. However, for all $n$:
%
$$
A_n\circ\varphi_n\circ\alpha_n\circ\mu_n = \overrightarrow{n}\circ\hat{k}_n.
$$
%
\noindent By taking limits, we find that $\overrightarrow{n}\circ\hat{k}_0$ is constant over $B_\epsilon(0)$. We thus
obtain a contradiction, and the result follows.\qed
%
\newsubhead{Precompactness}
%
\noindent We now obtain a result concerning the precompactness of the family $\left\{\hat{\mathi}_n | n\in\Bbb{N}\right\}$
over $\Bbb{D}$:
%
\proclaim{Lemme \nextprocno}
%\medskip
\noindent There exists an immersion $\hat{\mathi}_0:\Bbb{D}\rightarrow U\Bbb{H}^3$ such that, after extraction of a
subsequence, $\suite{\hat{\mathi}}{n}$ converges towards $\hat{\mathi}_0$ in the $C^\infty_\oploc$ topology over $\Bbb{D}$.
Moreover, $\hat{\mathi}_0^*g^\nu$ defines a complete metric over $\Bbb{D}$.
\endproclaim
\proclabel{LemmeConvergenceDesImmersions}
%
\proof Let $z\in\Bbb{D}$ be an arbitrary point. For all $n$, let us denote $p_n=z$. For all $n$, let
$\hat{\Sigma}_n=(S_n, \hat{\mathi}_n)$ be the Gauss lifting of $\Sigma_n$. We furnish the surface $S_n$ with the
metric $\hat{\mathi}_n^*g^\nu$. By lemma \procref{LemmeLImageDeZEstBornee}, the set
$\left\{\hat{\mathi}(p_n) | n\in\Bbb{N}\right\}$ is precompact in $U\Bbb{H}^3$. By Labourie's compactness theorem (theorem \procref{ThmLabCompacite}),
we may consequently assume that there exists a pointed immersed surface $(\hat{\Sigma}_z,p_z) = (S_z,\hat{\mathi}'_z,p_z)$
such that $(\hat{\Sigma}_n,p_n)_{n\in\Bbb{N}}$ converges to $(\hat{\Sigma}_z,p_z)$ as $n$ tends to infinity.
\medskip
\noindent For all $n$, let $\mu_n:S_n\rightarrow\Bbb{D}$ be the tautological diffeomorphism. In particular, for every $n$:
%
$$
\mu_n(p_n)=z.
$$
%
\noindent Moreover, for every $n$, the mapping $\mu_n$ is injective and thus conformal.
\medskip
\noindent By the Cheeger/Gromov adapted Schwarz lemma (lemma \procref{LemmeCGAdaptedSchwarz}), there exists a (possibly constant) holomorphic mapping
$\mu_z:S_z\rightarrow\Bbb{D}$ such that, after extraction of a subsequence, $\suite{\mu}{n}$ converges towards $\mu_0$ in the $C^\infty_\oploc$ topology.
\medskip
\noindent Let $\overrightarrow{n}$ be the Gauss/Minkowski mapping which sends $U\Bbb{H}^3$ to $\partial_\infty\Bbb{H}^3$. For all $n$, we have:
%
$$
\varphi_n\circ\alpha_n\circ\mu_n = \overrightarrow{n}\circ\hat{\mathi}_n.
$$
%
\noindent Thus, by taking limits, we obtain:
%
$$
\varphi_0\circ\alpha_0\circ\mu_z = \overrightarrow{n}\circ\hat{\mathi}'_0.
$$
%
\noindent In particular $\mu_0$ is locally conformal (i.e., it is every a local homeomorphism). By common sense lemma
\procref{LemmeInjectiviteDeLaLimite}, the mapping $\mu_z$ is injective, and consequently conformal. For all $n$, let $d_n$ be the distance structure generated over $\Bbb{D}$ by
the Riemannian metric ${\mu_n}_*\hat{\mathi}_n^*g^\nu$.
\medskip
\noindent Let $w$ be another point in $\Bbb{D}$. We will show that $w$ is in the image of $\mu_z$ if and only if there exists $K\in\Bbb{R}^+$ such that for all $n$:
%
$$
d_n(z,w) \leqslant K.
$$
%
\noindent Indeed, if such a $K$ exists, then, by compactness, we may suppose that there exists a point $q\in S_0$ such that the sequence $(\mu_n^{-1}(w))_{n\in\Bbb{N}}$ converges to $q$. By
continuity:
%
$$
\mu_z(q) = \mlim_{n\rightarrow\infty}\mu_n(\mu_n^{-1}(w)) = w.
$$
%
\noindent On the other hand, suppose that there exists a point $q\in S_z$ such that $\mu_z(q_0)=w$. Since $S_z$ is complete, there exists a geodesic $\gamma:I\rightarrow S_z$
going from $p_z$ to $q$ which minimises the distance between these two points. Let us denote $\eta=\mu_z\circ\gamma$. By common sense lemma \procref{LemmeLimitesDInverses},
the sequence $(\mu_n^{-1})_{n\in\Bbb{N}}$ converges in the $C^\infty_\oploc$ topology over $\eta$ to $\mu_z^{-1}$. In otherwords, let $(\phi_n)_{n\in\Bbb{N}}$ be a sequence
of convergence mappings of the sequence $(S_n,p_n)_{n\in\Bbb{N}}$ with respect to $(S_0,p_0)$. There exists $R\in\Bbb{R}^+$ and $N\in\Bbb{N}$ such that:
\medskip
\myitem{(i)} for $n\geqslant N$, the restriction of the mapping $\mu_n\circ\phi_n$ to the ball of radius $R$ about $p_0$ is smoothly invertible,
\medskip
\myitem{(ii)} for $n\geqslant N$, the geodesic $\eta$ is contained in $(\mu_n\circ\phi_n)(B_R(p_0))$, and
\medskip
\myitem{(iii}) the sequence $((\mu_n\circ\phi_n)^{-1})_{n\in\Bbb{N}}$ converges to $\mu_0^{-1}$ in the $C^\infty$ topology over $\eta$.
\medskip
\noindent For every $n$, we have:
%
$$
{\mu_n}_*\hat{\mathi}_n^*g^\nu = ((\mu_n\circ\phi_n)^{-1})^*(\hat{\mathi}_n\circ\hat{\phi}_n)^*g^\nu.
$$
%
\noindent Consequently, $({\mu_n}_*\hat{\mathi}_n^*g^\nu)_{n\in\Bbb{N}}$ converges in the $C^\infty$ topology over $\eta$ to a limit $g_0$. Thus, if for all $n$ we denote by
$L_n$ the length of $\eta$ with respect to the metric ${\mu_n}_*\hat{\mathi}_n^*g^\nu$, we find that the sequence $(L_n)_{n\in\Bbb{N}}$ converges to $d_z(z,w)$. Thus, for all
$n$:
%
$$
d_n(z,w) \leqslant \msup_{n\in\Bbb{N}} L_n <\infty.
$$
%
\noindent The result now follows.
\medskip
\noindent Let $z'$ be another point in $\Bbb{D}$ and let us construct $\Sigma_{z'}=(S_{z'},i_{z'})$ and $\mu_{z'}:S_{z'}\rightarrow\Bbb{D}$ in the same way as we constructed
$\Sigma_z$ and $\mu_z$. Suppose that $\opIm(\mu_z)\cap\opIm(\mu_{z'})\neq\emptyset$. Let $w$ be a point in this intersection, and let $w'$ be another point in $\opIm(\mu_{z'})$.
There exists $K\in\Bbb{R}^+$ such that, for all $n$:
%
$$
d_n(z,w),d_n(w,z'),d_n(z',w')\leqslant K/3.
$$
%
\noindent Consequently, for all $n$:
%
$$
d_n(z,w') \leqslant K.
$$
%
\noindent It follows that $w'\in\opIm(\mu_z)$. Since $w'\in\opIm(\mu_{z'})$ is arbitrary, we obtain:
%
$$
\opIm(\mu_{z'}) \subseteq \opIm(\mu_z).
$$
%
\noindent Consequently, by symmetry:
%
$$
\opIm(\mu_{z'}) = \opIm(\mu_{z})
$$
%
\noindent Since, for all $z'\in\Bbb{D}$, we have:
%
$$
z'\in\opIm(\mu_{z'}),
$$
%
\noindent we obtain:
%
$$
\opIm(\mu_z)^c = \munion_{z'\in\opIm(\mu_z)^c}\opIm(\mu_{z'}).
$$
%
\noindent Since $\opIm(\mu_{z'})$ is an open set for all $z'\in\Bbb{D}$, it follows that the complement of $\opIm(\mu_z)$ in $\Bbb{D}$ is open. However, $\opIm(\mu_z)$ is itself
an open subset of $\Bbb{D}$, and thus, since it is non empty, it follows by connectedness of $\Bbb{D}$ that:
%
$$
\opIm(\mu_z) = \Bbb{D}.
$$
%
\noindent Consequently, $\mu_z$ is surjective, and is thus a diffeomorphism.
%
\noindent By the common sense lemma \procref{LemmeLimitesDInverses}, the sequence $(\mu_n^{-1})_{n\in\Bbb{N}}$ converges to $\mu_z^{-1}$ in the $C^\infty_\oploc$ topology
over $\Bbb{D}$. By composing with the sequence $(\hat{\mathi}_n)_{n\in\Bbb{N}}$, the obtain the desired result.\qed
\medskip
\noindent We define $S_0=S_z$, $\hat{\mathi}_0=\hat{\mathi}_z$ and $\hat{\Sigma}_0=(S_0,\hat{\mathi}_0)$. We obtain the following result:
%
\proclaim{Corollary \nextprocno}
%\medskip
\noindent There exists $i_0:\Bbb{D}\rightarrow\Bbb{H}^3$ such that $\hat{\mathi}_0$ is the Gauss lifting of
$i_0$. Moreover, $i_0$ is the solution to the Plateau problem $\varphi_0\circ\alpha_0$.
\endproclaim
%
\proof Since $\hat{\Sigma}_0$ is complete, it is in $\overline{\Cal{L}}=\Cal{L}\munion\Cal{L}_\infty$. Since it is
of hyperbolic type (i.e. it is conformally equivalent to $\Bbb{D}$), and since all the elements of $\Cal{L}_\infty$
are parabolic, it follows that $\hat{\Sigma}_0$ is the Gauss lifting of a k-surface $\Sigma_0=(S_0,i_0)$, and the
first result follows. Let $\overrightarrow{n}:U\Bbb{H}^3\rightarrow\partial_\infty\Bbb{H}^3$ be the Gauss-Minkowski mapping.
For all $n$, we have:
%
$$
\varphi_n\circ\alpha_n = \overrightarrow{n}\circ\hat{\mathi}_n.
$$
%
\noindent After taking the limits, we obtain:
%
$$
\varphi_0\circ\alpha_0 = \overrightarrow{n}\circ\hat{\mathi}_0,
$$
%
\noindent and the second result follows.\qed
\medskip
\noindent We define $j_0$ by $j_0=i_0\circ\alpha_0^{-1}$, and we obtain the following result:
%
\proclaim{Corollary \nextprocno}
%\medskip
\noindent The application $j_0$ is the solution to the Plateau problem $\varphi_0$, and the sequence $\suite{j}{n}$
converges towards $j_0$ in the $C^\infty_\oploc$ topology over $\Bbb{D}$.
\endproclaim
\proclabel{CorExistenceEtContinuite}
%
\proof We know that $\alpha_n$ tends to $\alpha_0$ locally $C^0$ as $n$ tends to infinity. As before, we
denote the Gauss-Minkowski mapping by $\overrightarrow{n}:U\Bbb{H}^3\rightarrow\partial_\infty\Bbb{H}^3$. For all $n$
(including $n=0$), we have:
%
$$
\varphi_n\circ\alpha_n = \overrightarrow{n}\circ\hat{\mathi}_n.
$$
%
\noindent For all $n$, the application $\varphi_n$ as everywhere locally invertible, and consequently, by abusing
slightly our notation, we obtain, for every $n$:
%
$$
\alpha_n = \varphi_n^{-1}\circ\overrightarrow{n}\circ\hat{\mathi}_n.
$$
%
\noindent In particular, $\alpha_0$ is smooth and $\suite{\alpha}{n}$ converges to $\alpha_0$ in the $C^\infty_\oploc$ topology.
By common sense lemma \procref{LemmeLimitesDInverses}, the sequence $(\alpha_n^{-1})_{n\in\Bbb{N}}$ converges
towards $\alpha_0^{-1}$ in the $C^\infty_\oploc$ topology over $\Bbb{D}$. However, for all $n$:
%
$$
j_n = i_n\circ\alpha_n^{-1}.
$$
%
\noindent It thus follows that $\suite{j}{n}$ converges towards $j_0$ in the $C^\infty_\oploc$ topology over $\Bbb{D}$.
Finally, let us denote by $\hat{\mathj}_0$ the Gauss lifting of $j_0$. We obtain:
%
$$
\hat{\mathj}_0 = \hat{\mathi}_0\circ\alpha_0^{-1}.
$$
%
\noindent In particular, the metric $\hat{\mathj}_0^*g^\nu$ is complete, and:
%
$$
\overrightarrow{n}\circ\hat{\mathj}_0 = \overrightarrow{n}\circ\hat{\mathi}_0\circ\alpha_0^{-1} = \varphi_0.
$$
%
\noindent It thus follows that $j_0$ is the solution to the Plateau problem $\varphi_0$ and the result follows.\qed
%
\newhead{The Main Results}
%
\newsubhead{Existence and Continuity}
%
\noindent Firstly, we obtain the existence result:
%
\proclaim{Theorem \procref{PresentationChIIExistence}}
%\medskip
\noindent Let $\varphi:\Bbb{D}\rightarrow\hat{\Bbb{C}}$ be locally conformal. Then, for every $k\in (0,1)$, there
exists a unique solution  $i_k:\Bbb{D}\rightarrow\Bbb{H}^3$ to the Plateau problem $(\Bbb{D},\varphi)$ with constant Gaussian curvature $k$.
\endproclaim
%
\proof Let $\suite{t}{n}\in (0,1)$ be such that $t_n\uparrow 1$ as $n$ tends to infinity. Let us denote:
%
$$\matrix
\varphi_n(z) \hfill&=\varphi(t_nz), \hfill\cr
\varphi_0 \hfill&=\varphi. \hfill\cr
\endmatrix$$
%
\noindent By Labourie's existence theorem (theorem \procref{ThmLabExistence}) and Labourie's uniqueness theorem (theorem \procref{LemmeLabNature}), for every $n\neq 0$ there
exists a unique solution $j_n$ to the Plateau problem $(\Bbb{D},\varphi_n)$. By the compactness lemma \procref{LemmeConvergenceDesImmersions} and the corollary
\procref{CorExistenceEtContinuite} there exists a solution $j_0$ to the Plateau problem $(\Bbb{D},\varphi_n)$. Finally, by Labourie's uniqueness theorem
(theorem \procref{LemmeLabNature}), this solution is unique.\qed
\medskip
\noindent Next we obtain continuity:
%
\proclaim{Theorem \procref{PresentationChIIContinuite}}
%\medskip
\noindent Let $k\in (0,1)$ be a real number between $0$ and $1$. Let
$\suite{\varphi}{n},\varphi_0:\Bbb{D}\rightarrow\hat{\Bbb{C}}$ be locally conformal mappings. Let
$\suite{i}{n},i_0:\Bbb{D}\rightarrow\Bbb{H}^3$ be immersions such that, for all $n$, the mapping $i_n$ is the
unique solution to the Plateau problem $(\Bbb{D},\varphi_n)$ with constant Gaussian curvature $k$.
\medskip
\noindent If $\suite{\varphi}{n}$ converges to $\varphi_0$ locally uniformly, then
$\suite{i}{n}$ converges to $i_0$ in the $C^\infty_\oploc$ topology over $\Bbb{D}$.
\endproclaim
%
\proof This follows directly from the compactness lemma \procref{LemmeConvergenceDesImmersions}, corollary \procref{CorExistenceEtContinuite} and the uniqueness of solutions to
the Plateau problem.\qed
%
\newsubhead{A Homeomorphism}
%
\noindent Let us denote by $\opConf(\Bbb{D},\hat{\Bbb{C}})$ the space of locally conformal mappings from $\Bbb{D}$
into $\hat{\Bbb{C}}$. We furnish this space with the topology of local uniform convergence. Let us denote by
$\overrightarrow{n}:U\Bbb{H}^3\rightarrow\partial_\infty\Bbb{H}^3\cong\hat{\Bbb{C}}$ the Gauss-Minkowski mapping.
We define $\Cal{M}_k(\Bbb{D})$ to be the space of immersions $i:\Bbb{D}\rightarrow\Bbb{H}^3$ such that:
\medskip
\myitem{(i)} the immersed surface $\Sigma = (\Bbb{D},i)$ is a k-surface, and
\medskip
\noindent{(ii)} the function $\overrightarrow{n}\circ\hat{\mathi}$ is locally holomorphic over $\Bbb{D}$, or, in other
words:
%
$$
\overrightarrow{n}\circ\hat{\mathi} \in \opConf(\Bbb{D},\hat{\Bbb{C}}).
$$
%
\noindent The second condition is not as restrictive as it appears. Indeed, let $\Sigma=(S,i)$ be a k-surface
such that $S$ is simply connected let $\hat{\Sigma}=(S,\hat{\mathi})$ be its Gauss lifting. The metric $g^\nu$ over
$UM$ induces a metric $\hat{\mathi}^*g^\nu$ over $S$. We say that this immersed surface if {\emph of hyperbolic type\/}
if and only if the Riemannian manifold $(S,\hat{\mathi}^*g^\nu)$ is conformally equivalent to $\Bbb{D}$. We obtain the
following result:
%
\proclaim{Lemme \nextprocno}
%
\noindent Let $\Sigma=(S,i)$ be a k-surface and let $\hat{\Sigma}=(S,\hat{\mathi})$ be its Gauss lifting. If $\hat{\Sigma}$
is of hyperbolic type, then there exists a diffeomorphism $\alpha:\Bbb{D}\rightarrow S$ such that:
%
$$
\overrightarrow{n}\circ\hat{\mathi}\circ\alpha \in\opConf(\Bbb{D},\hat{\Bbb{C}}).
$$
%
\endproclaim
%
\proof Let $\rho:\Bbb{D}\rightarrow S$ be the conformal representation of $(S,\hat{\mathi}^*g^\nu)$. By lemma
\procref{LemmeLaMetriqueEstQC}, $\varphi\circ\rho$ is a k-quasiconformal mapping. It follows that there exists a
quasiconformal mapping $\alpha':\Bbb{D}\rightarrow\Bbb{D}$ such that $\varphi'=\varphi\circ\rho\circ\alpha'$ is
conformal. In particular, $\varphi'$ is smooth and, since $\varphi$ is everywhere a local homeomorphism, by abusing
slightly our notation, we obtain:
%
$$
\rho\circ\alpha' = \varphi^{-1}\circ\varphi'.
$$
%
\noindent It follows that the mapping $\rho\circ\alpha'$ is smooth. We thus define $\alpha=\rho\circ\alpha'$ and the
result follows.\qed
\medskip
\noindent It follows that every simply connected k-surface surface is equivalent, modulo
reparametrisation to an element of $\Cal{M}_k(\Bbb{D})$.
\medskip
\noindent Next, we observe that if $i,i'\in\Cal{M}_k(\Bbb{D})$ and $\alpha:(\Bbb{D},0)\rightarrow(\Bbb{D},0)$ are
such that:
%
$$
i' = i\circ\alpha,
$$
%
\noindent then $\alpha$ is conformal and is thus a member of $\opSO(2)$. We may thus identify
the space of pointed simply connected k-surfaces in $\Bbb{H}^3$ with $\Cal{M}_k(\Bbb{D})/\opSO(2)$. Similarly, since
$\opAut(\Bbb{D})$ is isomorphic to $\psl(2,\Bbb{R})$, we may identify the space of simply connected k-surfaces
in $\Bbb{H}^3$ with $\Cal{M}_k(\Bbb{D})/\psl(2,\Bbb{R})$.
\medskip
\noindent We furnish $\Cal{M}_k(\Bbb{D})$ with the $C^\infty_\oploc$ topology. We define the function
$\Cal{P}:\opConf(\Bbb{D},\hat{\Bbb{C}})\rightarrow\Cal{M}_k(\Bbb{D})$ such that for every
$\varphi\in\opConf(\Bbb{D},\hat{\Bbb{C}})$, the immersion $\Cal{P}(\varphi)$ is the unique solution to the Plateau
problem $(\Bbb{D},\varphi)$.
\medskip
\noindent Our results may now be summarised by the following lemma:
%
\proclaim{Lemma \nextprocno}
%\medskip
\noindent The function $\Cal{P}$ is a homeomorphism.
\endproclaim
%
\demonstration By the existence theorem \procref{PresentationChIIExistence}, the function $\Cal{P}$ is well defined. By the
uniqueness of solutions to the Plateau problem, $\Cal{P}$ is injective. Let $i\in\Cal{M}_k(\Bbb{D})$ be an
arbitrary immersion in $\Cal{M}_k(\Bbb{D})$, and let $\hat{\mathi}$ be its Gauss lifting. Let $\overrightarrow{n}$
be the Gauss-Minkowski mapping. By definition of $\Cal{M}_k(\Bbb{D})$:
%
$$
\varphi = \overrightarrow{n}\circ\hat{\mathi} \in \opConf(\Bbb{D},\Bbb{C}).
$$
%
\noindent It follows that $i$ is the unique solution to the Plateau problem $(\Bbb{D},\varphi)$ and we thus obtain the
surjectivity of $\Cal{P}$.
\medskip
\noindent By the continuity theorem \procref{PresentationChIIContinuite} the function $\Cal{P}$ is continuous. Its inverse is trivially continuous, and
the result follows.\qed
\medskip
\noindent We have thus obtained a description of the moduli space of simply connected k-surfaces immersed into $\Bbb{H}^3$.
%
\newhead{Bibliography}
%
{\leftskip = 5ex \parindent = -5ex
\leavevmode\hbox to 4ex{\hfil \cite{BallGromSch}}\hskip 1ex{Ballman W., Gromov M., Schroeder V., {\sl Manifolds of nonpositive curvature}, Progress in Mathematics, {\bf 61}, Birkh\"auser, Boston, (1985)}
\medskip
\leavevmode\hbox to 4ex{\hfil \cite{Bonk}}\hskip 1ex{Bonk M., Private communication to Fran\c{c}ois Labourie}
\medskip
\leavevmode\hbox to 4ex{\hfil \cite{Grom}}\hskip 1ex{Gromov M., Foliated plateau problem, part I : Minimal varieties, {\sl GAFA} {\bf 1}, no. 1, (1991), 14--79}%
\medskip
\leavevmode\hbox to 4ex{\hfil \cite{LabB}}\hskip 1ex{Labourie F., Probl\`emes de Monge-Amp\`ere, courbes holomorphes et laminations, {\sl GAFA} {\bf 7}, no. 3, (1997), 496--534}
\medskip
\leavevmode\hbox to 4ex{\hfil \cite{LabA}}\hskip 1ex{Labourie F., Un lemme de Morse pour les surfaces convexes, {\sl Invent. Math.} {\bf 141} (2000), 239--297}
\medskip
\leavevmode\hbox to 4ex{\hfil \cite{LV}}\hskip 1ex{Lehto O., Virtanen K. I., {\sl Quasiconformal mappings in the plane}, Die Grund\-lehren der mathemathischen Wissenschaften, {\bf 126}, Springer-Verlag, New York-Heidelberg, (1973)}
\medskip
\leavevmode\hbox to 4ex{\hfil \cite{Muller}}\hskip 1ex{Muller M.P., Gromov's Schwarz lemma as an estimate of the gradient for holomorphic curves, In: {\sl Holomorphic curves in symplectic geometry},
Progress in Mathematics, {\bf 117}, Birkh\"auser, Basel, (1994)}
\medskip
\leavevmode\hbox to 4ex{\hfil \cite{RosSpruck}}\hskip 1ex{Rosenberg H., Spruck J. On the existence of convex hyperspheres of constant Gauss curvature in hyperbolic space, {\sl J. Diff. Geom.} {\bf 40} (1994), no. 2, 379--409} % Rosenberg, Spruck
\medskip
\leavevmode\hbox to 4ex{\hfil \cite{SmiA}}\hskip 1ex{Smith G., Positive special Legendrian structures and Weingarten problems, {\sl pr\'epub\-lication univ. Paris XI}, (2005)}%
\medskip
\leavevmode\hbox to 4ex{\hfil \cite{SmiC}}\hskip 1ex{Smith G., Pointed k-surfaces, {\sl pr\'epublication univ. Paris XI}, (2005)}%
\medskip
\leavevmode\hbox to 4ex{\hfil \cite{SmiE}}\hskip 1ex{Smith G., Th\`ese de doctorat, Paris (2004)}\par%
}%
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references.tex
\global\def\_@citation@BallGromSch{1}
\global\def\_@citation@Bonk{2}
\global\def\_@citation@Grom{3}
\global\def\_@citation@LabB{4}
\global\def\_@citation@LabA{5}
\global\def\_@citation@LV{6}
\global\def\_@citation@Muller{7}
\global\def\_@citation@RosSpruck{8}
\global\def\_@citation@SmiA{9}
\global\def\_@citation@SmiC{10}
\global\def\_@citation@SmiE{11}
\global\def\_@proc@PresentationChIIExistence{1.1}
\global\def\_@proc@PresentationChIIContinuite{1.2}
\global\def\_@proc@LemmeLimitesDInverses{2.1}
\global\def\_@proc@LemmeInjectiviteDeLaLimite{2.2}
\global\def\_@proc@LemmeInjectiviteLorsquOnSApprocheDuLimite{2.3}
\global\def\_@proc@LemmeCompacteDansLImageDeLaLimite{2.4}
\global\def\_@head@HeadDifferentialGeometryInTTM{3}
\global\def\_@proc@LemmeLaMetriqueEstQC{4.1}
\global\def\_@proc@LemmeCompaciteQuasiconforme{4.2}
\global\def\_@proc@ThmLabExistence{5.1}
\global\def\_@proc@LemmeLabNature{5.2}
\global\def\_@proc@ThmLabCompacite{5.3}
\global\def\_@proc@LemmeCGAdaptedSchwarz{6.1}
\global\def\_@proc@LemmeConvergenceDesRepresentationsConformes{6.2}
\global\def\_@proc@LemmeGromovSchwarz{6.3}
\global\def\_@proc@LemmeLaLimiteNestPasConstante{6.4}
\global\def\_@proc@LemmeEmplacementDuBordDeLaLimite{6.5}
\global\def\_@proc@LemmeLImageDeZEstBornee{7.1}
\global\def\_@proc@LemmeLimiteEstConstante{7.2}
\global\def\_@proc@LemmePNEstPasQ{7.3}
\global\def\_@proc@LemmeConvergenceDesImmersions{7.4}
\global\def\_@proc@CorExistenceEtContinuite{7.6}
