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% Author Package file for use with AMS-LaTeX 1.2
\controldates{5-APR-1999,5-APR-1999,5-APR-1999,5-APR-1999}
 
\documentclass{era-l}
\usepackage{graphicx}

%This is a plain TeX file.
%It uses the epsf macros for drawing figures.
%There are two figures which require files triangle.ps
%and triangulation.ps

\theoremstyle{remark}
\newtheorem{notation}{Notation}[section]
\newtheorem{remark}[notation]{Remark}
\newtheorem*{remarkn}{Remark}
\newtheorem*{notationn}{Notation}
\newtheorem*{finalremark}{2.10. Final Remark}

\swapnumbers
\theoremstyle{definition}
\newtheorem*{definitionn}{Definition}
\newtheorem{conj}{Conjecture}[section]

\newtheorem{thm}[conj]{Theorem}
\newtheorem{prop}[conj]{Proposition}


\begin{document}

\title{Metric minimizing surfaces}
\author{Anton Petrunin}
\address{Max-Planck-Institut f\"{u}r Mathematik in den Naturwissenschaften, 
Inselstrasse 22-26, D-04103 Leipzig, Germany}
\email{petrunin@mailhost.mis.mpg.de}


\issueinfo{5}{07}{}{1999}
\dateposted{April 8, 1999}
\pagespan{47}{54}
\PII{S 1079-6762(99)00059-1}
\def\copyrightyear{1999}
\copyrightinfo{1999}{American Mathematical Society}

\subjclass{Primary 53C21}
\date{September 14, 1998}

\commby{Dmitri Burago}

\thanks{The main part of this note was prepared when the author had a 
postdoctoral fellowship at MSRI (Berkeley).}

%\keywords{vanishing theorems; null spaces.}

\begin{abstract}
Consider a two-dimensional surface in an Alexandrov
space of curvature bounded above by $k$.
Assume that this surface does not admit contracting
deformations (a particular case of such surfaces is formed by area minimizing
surfaces). Then this surface inherits the upper curvature bound,
that is, this surface has also curvature bounded above by $k$, with respect to 
the intrinsic metric induced from its ambient space.
\end{abstract}

\maketitle

%\S0 
\section*{Introduction} Let $M$ be a Riemannian manifold
and $S$ a smooth surface in $M$. One can write a Gaussian
formula for the curvature of $S$,
\begin{equation*}K_S=K_M+K_G,\end{equation*}
where $K_S$ is the curvature of $S$, $K_M$ the curvature of $M$ in
the same direction, and $K_G$ the Gaussian curvature in the
same direction. Two important corollaries arise. First, if $M$
has positive curvature and $S$ is a convex hypersurface, then the
intrinsic metric on $S$ has positive curvature. Second, if $M$
has negative curvature and $S$ is a two-dimensional saddle
surface (i.e., the principal curvatures for any normal direction,
do not have the same sign at any point), then the intrinsic metric
on $S$ has negative curvature.

All notions in these corollaries have natural sense in
Alexandrov's geometry (one can define convex surface, saddle
surface (see below) and curvature bounded below and above
for general length-metric space). Therefore one can ask if
the generalization of such results is true for Alexandrov
spaces. Both of these problems are open so far.

Here we give a proof of a special case of the second problem.
Roughly speaking, we prove that upper bound on curvature is inherited
by a surface in an Alexandrov space
with  curvature $\le k$,
if this surface does not admit a contracting deformation
(this is stronger than just being saddle; see below).
The proof uses the well-known idea
of approximating our metric by a polyhedral one, but there is one
funny point below,
that is the place where we consider an approximation of a metric on a
surface by a metric on a graph
(it was a real surprise for me that this worked).

I thank the anonymous referee for invaluable help
in making this paper readable, as well as one of the editors
for bringing the thesis of C. Mese to my attention.


\section{Definitions and notation} Let $X$, $Y$ and $Z$ be
compact metric spaces.
%\setcounter{section}{1}
\subsection{Notation}
%Notation 1.1.} 
We will write $X-\epsilon\le Y$ or
$X\le Y+\epsilon$ if there is a mapping $f:X\to Y$ such that
$|xx'|_X\le|f(x)f(x')|_Y+\epsilon$.

It is easy to see that this is a transitive relation in the
following sense:

If $X-\epsilon\le Y$ and $Y-\epsilon'\le Z$ then
$X-\epsilon-\epsilon'\le Z$.
%\end{notation}
%{\bf Remark 1.2.} 
\subsection{Remark}\label{sub:1.2}
It is easy to see that in this notation if
$X$ is $\epsilon$-close to $Y$ in the Gro\-mov-Hausdorff sense
(see \cite{GLP}), then $X-2\epsilon\le Y$ and $Y-2\epsilon\le X$.
On the other hand, from the old folklore lemma which says
that if $X\ge Y$ and $Y\ge X$, then $X$ is isometric to $Y$ (a
careful proof can be found in \cite[1.2]{P}), it is easy to see that
if $X_n-\epsilon_n\le Y$ and $Y-\epsilon_n\le X_n$ and
$\epsilon_n\to 0$, then ${X_n} \overset{GH}{\to} Y$.
Therefore if one defines the metric
\begin{equation*}d'(X,Y)=\min\{\epsilon; X-\epsilon\le Y \hbox{ and }
Y-\epsilon\le X\},\end{equation*}
then this new metric will define the same topology on the
set of compact metric spaces as the standard
Gromov-Hausdorff metric.
%\end{remark}
%{\bf 1.3.} 
%\setcounter{subsection}{2}
\subsection{}
Now $M$ is a metric space.
%{\bf Definition.} 
\begin{definitionn}
A function $f:U\subset  M\to R$ is called \textit{convex}
if for any geodesic $\gamma\subset  U$ with length-parameter,
$f\circ\gamma$ is convex.
\end{definitionn}
%{\bf Definition.} 
\begin{definitionn}
We denote by $D$ a closed disk in $R^2$.
A continuous mapping $s:D\to M$ is called
\textit{saddle} if for any convex function $f$ on $M$ and any
subset $U\subset  D$
\begin{equation*}\sup_{x\in U}f(x)=\sup_{x\in \partial U}f(x).\end{equation*}
\end{definitionn}
%{\bf Remark.} 
\begin{remarkn}
The definition of saddle surfaces in $R^3$ used
by Shefel is that a plane cannot cut a cap from the surface.
If one applies the definition given
above to affine functions, it is easily seen that the two definitions
are equivalent for the case $M = R^3$.
\end{remarkn}
%{\bf Notation.} 
\begin{notationn}
For a mapping $s:D\to M$ define a
pseudometric on $D$ by $|xy|=|s(x)s(y)|_M$. Let $|xy|_s$,
$x,y\in D$, be the infimum of $|**|$-length of curves in
$D$ connecting $x$ and $y$. We will call $|**|_s$
the pull-back metric (generally
speaking, it is a pseudometric, i.e. it is possible to have $|xy|_s=0$
for $x\not=y$, but we call it a metric anyway).
\end{notationn}
%{\bf Definition.} 
\begin{definitionn}
A continuous mapping $s:D\to M$ is called
\textit{metric minimizing} if the pull-back metric is compact
and  there is no mapping $s':D\to M$
such that $s'|_{\partial D}\equiv
s|_{\partial D}$, 
$|xx'|_{s'}\le|xx'|_{s}$ for any two points $x,x'\in D$, 
and the inequality is strict for at least one pair.

In some sense it is the weakest ``minimal''-like property one could
express in terms of the intrinsic metric of a surface.
\end{definitionn}
\section{Results and proofs}
%{\bf 2.1. Conjecture.} 
%\setcounter{section}{2}
\subsection{Conjecture} (\textit{Probably due to Shefel}.) Let $M$ be an
Alexandrov space with curvature $\le 0$. Then any saddle
mapping $s:D\to M$ gives a pull-back metric on $D$
with curvature $\le 0$.
%\end{conj}

This conjecture was proved by S. Shefel (\cite{She1} and \cite{She2}) in
two special cases: first for $M=R^2$, and second for $M=R^3$
(the last one with the additional assumption that $s$ describe
a surface which is locally a graph of a function).
Aleksandrov \cite{A} proved that the upper bound on
curvature is inherited by ruled surfaces in
an Alexandrov space with curvature $\le k$.
C. Mese in her thesis proved this conjecture for minimal
surfaces in Alexandrov spaces (see \cite{M}).

Now look what I can do:

%{\bf 2.2. Theorem.}
%\begin{thm}
\subsection{Theorem} \textit{Let $M$ be a locally compact Alexandrov space with
curvature $\le k$ or a complete Alexandrov space with curvature $\le k\le0$.
Then any metric minimizing mapping
$s:D\to M$ gives a pull-back metric on $D$ with
curvature $\le k$.}
%\end{thm}

Let us prove first that this is really a special
case of the Conjecture.
%{\bf 2.3. Proposition.} 
\subsection{Proposition}
\textit{Any metric minimizing mapping in an Alexandrov
space with curvature $\le k$ is a saddle mapping.}
%\end{prop}
%{\bf Remark.} 
\begin{remarkn}
The author does not know a counterexample
to the conjecture that any saddle mapping in an Alexandrov
space with curvature $\le k$ and rectifiable
$s\circ \partial D$ is metric minimizing. Therefore one can
consider this Theorem as a way to approach the Conjecture.
\end{remarkn}
\begin{proof}[Proof of the Proposition] Let $M$ be an Alexandrov space
with curvature $\le k$ and $s:D\to M$ be metric minimizing.
Assume there is a convex function $f$ such that $f\circ s$
admits a strict local maximum at $C$, $C$ is
compact and $|C\partial D|_s>0$.

Direct application of Sharafutdinov retraction \cite{Sha}
for level surfaces of $f$ in a neighborhood of $s(C)$ (this is
an abstraction of deformation along gradient curves in the
smooth case) gives
us a $|**|_s$-nonexpanding deformation of $s$ in an arbitrary
small neighborhood of $C$. $\spadesuit$
\renewcommand{\qed}{}\end{proof}
%{\bf Remark.} 
\begin{remarkn}
The condition that our space be an Alexandrov
space with curvature $\le 0$ is not necessary. In order to
apply Sharafutdinov retraction it is enough that our space
have the first variation formula, i.e. one can define angle
between two geodesics which start from the same point
such that the standard first variation formula holds for
distance functions.
\end{remarkn}
%\setcounter{subsection}{3}
\subsection{ %2.4. 
Proof of the Theorem} \label{sub:2.4}
Sections \ref{sub:2.4}--\ref{sub:2.8} will deal with
the locally compact case; in Section \ref{sub:2.9} the necessary extra work is done
for complete negatively curved Alexandrov spaces.

For the convenience of the reader I will
consider only the case $k=0$. The general case requires only technical
modifications.

Without loss of generality one can assume that
$\partial D$ is rectifiable
in the pull-back metric and Im$(s)$ is inside some
$R_0$-domain (see \cite{N}).

Let us construct a ``formal'' triangulation $\Gamma$ in
$(D,|**|_s)$ whose $1$-skeleton with intrinsic metric is
$\epsilon$-GH-close to $(D,|**|_s)$ and no side of any
triangle is longer than $\epsilon$. ``Formal'' means that
the sides in the triangulation can have overlapping segments.
Therefore triangles could be degenerate like the one 
shown in Figure \ref{fig:1}.

\begin{figure}
\includegraphics[scale=.8]{era59el-fig-1}
\caption{}\label{fig:1}
\end{figure}

%\epsfxsize2.0truein
%\hfil\epsfbox{triangle.ps}\hfil
%\vskip .5cm


Let us denote by $\Gamma_0$, $\Gamma_1$ the $0$- and
$1$-skeleton of $\Gamma$.
\subsection{%2.5. 
Construction of $\Gamma$}\label{sub:2.5}
(If you do believe in this, skip to \ref{sub:2.6}.)

Let us consider a finite $\nu$-net $\{a_i\}$ in
$(D,|**|_s)$.
Consider the minimal closed curve  $\gamma\subset  (D,|**|_s)$ which surrounds
all $a_i$'s, and let us denote by $\mathrm{Conv}\{a_i\}$ the set of points
surrounded by $\gamma$.

First let us subdivide $\mathrm{Conv}\{a_i\}$ into polyhedra
with small diameters and perimeters. The steps of this
subdivision are illustrated in Figure \ref{fig:2}, in which
subfigure I shows the division into $C_i$ with a net of 5 points.

\begin{figure}
\includegraphics[scale=.90]{era59el-fig-2}
\caption{}\label{fig:2}
\end{figure}

For every $a_i$ consider the set
\begin{equation*}C_i=\{x\in \mathrm{Conv}
\{a_i\}; |xa_i|\le|xa_j|\}.\end{equation*}
We can add to $\{a_i\}$ finitely many points on
$\partial \operatorname{Conv}\{a_i\}$
to meet the following property:
if $C_j\cap\partial \operatorname{Conv}\{a_i\}\not=\emptyset$, then
$a_j\in\partial\operatorname{Conv}\{a_i\}$.

Let us consider a fundamental domain in the universal
covering $\tilde C_i$ of $C_i$,
\begin{equation*}D_i=\{x\in\widetilde C_i; |x\tilde a_i|\le
|x \gamma(\tilde a_i)| \hbox{ for any } \gamma\in\pi_1(C_i)\}.\end{equation*}
Mapping $D_i$ back to $D$, we obtain a map on
$\mathrm{Conv}\{a_i\}$ such that every ``country'' is a disk.
On every segment-border or point-border between
countries choose a point
(``customs point''), including borders with the same country.
(See II in Figure \ref{fig:2}.)
Connect every ``customs point'' by a minimal geodesic with the
``capital'' ($a_i$) of every adjacent country (if it is a border
between a country and itself, one has to consider two or
more connections depending on how many pieces of this
country meet at this customs point).
(See III in Figure \ref{fig:2}.) Therefore we have cut
$\mathrm{Conv}\{a_i\}$ into hexagons and quadrangles with
perimeter $\le 6\nu$ and diameter $\le 5\nu$.
(The number of vertices of a polygon is even because, when we are going
along its perimeter, ``capitals'' and ``customs points'' will alternate.
It is easy to see that the number of capitals in one polygon is at most $3$,
since
otherwise there must be some other customs inside the polygon;
therefore the total number of vertices is $\le 6$.)

It is easy
to see that the ``customs points'' can be chosen so that the
constructed hexagons and quadrangles cover all of
$\mathrm{Conv}\{a_i\}$.

Now let us remove the borders and connect each pair of the $a_i$'s by a minimal
geodesic;
we can easily do it to meet the rule: every two geodesics have either
empty intersection, or one-point intersection or one-segment
intersection (including the already constructed geodesics
between capitals and customs points). This insures that we obtain
a finite graph. (See IV in Figure \ref{fig:2}.)


We obtain a partition of $\mathrm{Conv}\{a_i\}$ into polyhedra
of perimeter $\le 6\nu$ and diameter $\le 5\nu$. Indeed,
every polygon of our graph is a result of finitely many
cuts of a hexagon or quadrangle along a geodesic, and
the perimeter as well as the diameter
do not increase at any such step.

Now let us subdivide every polygon into triangles with the same
vertices. (See V in Figure \ref{fig:2}.)

%\hfil\epsfbox{triangulation.ps}\hfil

We obtain a triangulation of Conv$\{a_i\}$ in
$(D,|**|_s)$. It is easy to show that the $1$-skeleton of
this triangulation is $8\nu$-close to $(D,|**|_s)$ in the sense of
the $d'$ metric (see \ref{sub:1.2}).

Indeed, by mapping each point to a closest $a_i$
we have $(D,|**|_s)-2\nu\le\Gamma_1$. Now the distance
between any point of $\Gamma_1$ (in its length-metric)
and a closest element of $\{a_i\}$ is less than $4\nu$
(for $3\nu/2$ we can get from any point to the perimeter
of a polyhedron, for another $3\nu/2$ we can traverse the
perimeter of the hexagon or quadrangle, and after another
$\nu$ we get a point from the $\nu$-net). Therefore by
mapping every point of $\Gamma_1$ to a closest (in the
intrinsic metric of $\Gamma_1$) point from the $\nu$-net
we obtain that $\Gamma_1-8\nu\le(D,|**|_s)$. Therefore
we obtain the needed triangulation $\Gamma$ for sufficiently small
$\nu=\nu(D,|**|_s,\epsilon)$ (see \ref{sub:1.2}). $\spadesuit$
\subsection{}\label{sub:2.6}%{2.6} 
One can naturally define the boundary of $\Gamma$
($\partial\Gamma$).

Let us consider the image $s(\Gamma_0)\subset  M$ and connect
vertices which are connected in $\Gamma$ by geodesics in
$M$. Call the new graph $\dot\Gamma$. It is easy to see that
$\dot\Gamma\le\Gamma_1$.

Now let us consider a graph $\ddot\Gamma$ with the same
combinatorial type, which minimizes the total length of edges in the
set of graphs such that every edge is less than or equal to the
corresponding edge of $\dot\Gamma$, and with the same
boundary as $\dot\Gamma$.


Applying Arzel\`a-Ascoli arguments we conclude that such a graph exists.
Obviously we have 
$\ddot\Gamma\le\dot\Gamma$.
%{\bf 2.7.} 
\subsection{}\label{sub:2.7}
Now let us consider a polyhedral metric on the triangulation,
with edge lengths given by those in $\ddot\Gamma$.
Simply consider for any triangle in $\ddot\Gamma$ (which
represent a triangle in the triangulation) a model triangle
in $R^2$ and glue a triangulation from them.
Let it be $C(\ddot\Gamma)$. It is easy to see that
$C(\ddot\Gamma)-2\epsilon\le\ddot\Gamma$ because from
the construction every side of every triangle is not more than
$\epsilon$.

It is easy to construct a nonexpanding mapping
of $m:C(\ddot\Gamma)\to M$ (by \cite{R}) such that
$m|_{\ddot\Gamma}=$id. We can consider the pull-back
metric on $C(\ddot\Gamma)$.

Therefore,
\begin{equation*} (C(\ddot\Gamma),|**|_m)-2\epsilon\le
C(\ddot\Gamma)-2\epsilon\le\ddot\Gamma\le
\dot\Gamma\le\Gamma_1\le (D,|**|_s) + 2\epsilon.\end{equation*}
Hence
\begin{equation*}(C(\ddot\Gamma),|**|_m)\le C(\ddot\Gamma)\le
(D,|**|_s) + 4\epsilon.\end{equation*}
As $\epsilon\to 0$ (again applying Arzel\`a-Ascoli arguments)
we obtain a nonexpanding mapping
$(D,|**|_s)\to M$ with the same boundary values as $s$.
From the metric minimizing property of $s$, this new
mapping has exactly the same pull-back metric, i.e.,
\begin{equation*}\lim_{\epsilon\to0}(C(\ddot\Gamma),|**|_m)=(D,|**|_s).\end{equation*}
From the inequality above,
\begin{equation*}\lim_{\epsilon\to0}C(\ddot\Gamma)=(D,|**|_s).\end{equation*}
Therefore, the following Proposition finishes the proof.
%\setcounter{conj}{7}
\subsection{Proposition}\label{sub:2.8} 
\textit{$C(\ddot\Gamma)$ has nonpositive curvature.}

\begin{proof}
Indeed, assume the contrary; then there is a
vertex in $C(\ddot\Gamma)$ with sum of angles $<2\pi$.
Let us consider the space of directions $\overline{\Omega_p(M)}$
at this point. It follows from \cite[Theorem 1]{N} that it is a CAT(1) space, and
from the comparison inequality we obtain that the directions of the edges
at $p$ form a broken geodesic with total length $l<2\pi$. By
using Reshetniak's theorem (see \cite{R} or \cite[Corollary 2]{N}) we
obtain that there is a convex domain in $C\subset  S^2$ with perimeter
$l$ admitting a nonexpanding mapping
$r:C\to\overline{\Omega_p(M)}$ such that $\operatorname{Im}\partial C$ is
our broken geodesic. Therefore, there is a direction
$\omega\in\overline{\Omega_p(M)}$ which has angle $<\pi/2$
with the direction of every edge at $p$ (it is the image of the
``center'' of $C$). By moving $p$ along a curve in a direction
close to $\omega$ we reduce every edge of $\ddot\Gamma$,
a contradiction. $\spadesuit$
\renewcommand{\qed}{}\end{proof}
%2.9 What to do when $M$ is only complete, but $k\le 0$.}
%\setcounter{subsection}{8}
\subsection{What to do when $M$ is only complete, but $k\le 0$}\label{sub:2.9}
For spaces which are not locally compact the very same proof works, but
we cannot apply Arzel\`a-Ascoli arguments.
There are two places where we did it: the first is the
existence of $\ddot\Gamma$ (\ref{sub:2.6}),
and the second is the
existence of the limit $\ddot\Gamma$ as $\epsilon\to0$ (\ref{sub:2.7}).

The existence of $\ddot\Gamma$ can be proved in the following way:

Let $p_i(\dot\Gamma)$ be a sequence of graphs
with the same boundary as $\dot\Gamma$, such that the
total length of edges of $p_i(\dot\Gamma)$ goes to infimum.
Passing to a subsequence if necessary one can
insure that the length of each edge converges as $i\to \infty$.
Assume it is not a Cauchy sequence. Then there
are two subsequences, $p_{n_i}(\dot\Gamma)$ and
$p_{m_i}(\dot\Gamma)$ such that for some $\epsilon>0$,
\begin{equation*}
\max_{x\in\Gamma_0}|p_{n_i}(x)p_{m_i}(x)|>\epsilon.\end{equation*}
Let us take  $q_i(x),
x \in \Gamma_0$, to be the
midpoint of $p_{n_i}(x)p_{m_i}(x)$.
Then for any two points $x,y\in \dot\Gamma_0$ the Bruhat-Tits inequality gives
\begin{equation*}
2|q_i(x)q_i(y)|\le|p_{n_i}(x)p_{n_i}(y)|+|p_{m_i}(x)p_{m_i}(y)|\end{equation*}
and for some edge the difference between the right and left sides of the 
inequality is greater than some positive number which does not depend on $i$.

Indeed, let $d(x)= |p_{n_i}(x)p_{m_i}(x)|$. Then since the length
of each edge converges,
the last statement is
a trivial corollary of the following
inequality for $x,y\in \dot\Gamma_0$:
\begin{equation*}2|q_i(x)q_i(y)|^2+\frac{1}{2}|d(x)-d(y)|^2\le
|p_{n_i}(x)p_{n_i}(y)|^2+|p_{m_i}(x)p_{m_i}(y)|^2.\end{equation*}
This inequality (as well as the one above) is obvious for a
plane quadrangle, and from triangle comparison
it is true for Alexandrov spaces with nonpositive curvature,
a contradiction. $\spadesuit$


The fact that $\lim_{\epsilon\to 0}(C(\ddot\Gamma),|**|_m)= (D,|**|_s)$,
needed in \ref{sub:2.7},
can be obtained in a similar way:

First note that for each $\ddot\Gamma=\ddot\Gamma(\epsilon)$ one can construct
a mapping $f_\epsilon:D\to M$ such that:

(i) $|f_\epsilon(x)f_\epsilon(y)|\le |xy|_s+\epsilon$;

(ii) $f_\epsilon|_{\partial D}\equiv s|_{\partial D}$.

(One simply needs to take $f_\epsilon|_{\partial D}\equiv s|_{\partial D}$
and then send each point in $D\backslash\partial D$
to the
vertex of $\ddot\Gamma$ corresponding to the closest
point $a_i$ of the $\nu$-net;
see \ref{sub:2.5}.)

Now, for $\epsilon\to0$, consider the set of all possible GH-limits of $D$
with the extrinsic pseudometric
induced from $f_\epsilon$ (i.e., $|xy|=|f_\epsilon(x)f_\epsilon(y)|$;
here $f_\epsilon$ has the above properties).

Obviously each such limit $X\le(D,|**|_s)$. Consider a minimal element $X_0$
in this set, and take a sequence
$f_{\epsilon_n}\ (\epsilon_n\to0)$ which corresponds
to $X_0$.
Then the same arguments as above show that $f_{\epsilon_n}$ is a Cauchy
sequence. Therefore as a limit we have a mapping $f:D\to M$ such that
$|f(x)f(y)|\le |xy|_s$, but from the definition of pull-back metric we
then have
 $|xy|_f\le|xy|_s$, and from the metric minimizing property we have
$|xy|_f=|xy|_s$. Since for appropriately chosen $X_0$ we can assume
\begin{equation*}
(D,|**|_f) \le
\lim_{\epsilon\to 0}(C(\ddot\Gamma),|**|_m)\le (D,|**|_s),\end{equation*}
we have
\begin{equation*}\lim_{\epsilon\to 0}(C(\ddot\Gamma),|**|_m)= (D,|**|_s).\quad
\spadesuit\end{equation*}
%{\bf 2.10. Final Remark.} 
%\begin{finalremark}
\subsection{Final remark} It is really strange that the sign of $k$ comes
in the game in such a place.
If one tried to repeat \ref{sub:2.9} for $k>0$, then some positive
term would appear in the Bruhat-Tits inequality, which is very small but
which will poison all the fun.
Direct application of the above proof would still give convergence if our surface
were in addition
\textit{hodograph minimizing}, in the sense of the following definition:

%{\bf Definition.} 
\begin{definitionn}
A metric minimizing mapping $s:D\to M$ is called
\textit{hodograph minimizing} if in addition there is no mapping $s':D\to M$,
such that $s'|_{\partial D}\equiv
s|_{\partial D}$, $|xx'|_{s'}\le|xx'|_{s}$ 
for any two points $x,x'\in \partial D$,
and if the inequality is strict for at least one pair.
\end{definitionn}
\begin{thebibliography}{She1}

\bibitem[A]{A} A.D. Aleksandrov, \textit{ Ruled surfaces in metric spaces}, 
Vestnik LGU
Ser. Mat., Mech., Astr. 1957, vyp. 1, 5-26. (Russian)
 \MR{19:59a}
\bibitem[GLP]{GLP} M. Gromov, \textit{ 
Structures m\'etriques pour les vari\'et\'es riemanniennes},
J. Lafontaine and  P. Pansu, eds., CEDIC, Paris, 1981.
\MR{85e:53051}
\bibitem[M] {M} C. Mese, \textit{The curvature of 
minimal surfaces in singular spaces},
to appear in Comm. Anal. Geom.

\bibitem[N]{N} I. Nikolaev,
\textit{ The tangent cone of an Aleksandrov space of curvature $\leq K$},
Manuscr. Math. 86 (1995), No. 2, 137-147. 
\MR{95m:53062}
\bibitem[P]{P} A. Petrunin, \textit{ Parallel transportation for Alexandrov space with curvature
bounded below},
Geom. Funct. Anal. 8 (1998), No. 1, 123-148. 
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\end{thebibliography}
\end{document}
<\PRE>