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% Author Package file for use with AMS-LaTeX 1.2
%% Translation via Omnimark script a2l, November 11, 2003 (all in one day!)
\controldates{16-DEC-2003,16-DEC-2003,16-DEC-2003,16-DEC-2003}
 
\RequirePackage[warning,log]{snapshot}
\documentclass{era-l}
\issueinfo{9}{17}{}{2003}
\dateposted{December 17, 2003}
\pagespan{135}{141}
\PII{S 1079-6762(03)00120-3}
\usepackage{graphicx}

\theoremstyle{plain}
\newtheorem*{theorem1}{Theorem A}
\newtheorem*{theorem2}{Theorem B}
\newtheorem*{theorem3}{Corollary}
\newtheorem*{theorem4}{Theorem C}
\newtheorem*{theorem5}{Theorem D}
\newtheorem*{theorem6}{Lemma}
\newtheorem*{theorem7}{Key Lemma}

\newcommand{\RR}{\mathbb{R}}

\begin{document}

\title{Harmonic functions on Alexandrov spaces \linebreak[1] 
and their applications}
\author{Anton Petrunin}
\address{Department of Mathematics, The Pennsylvania State University,
University Park, PA 16802}
\email{petrunin@math.psu.edu}
\thanks{The main part of this paper was written while I had postdoctoral 
fellowship at MSRI in
1995--1996. I would like to thank  this institute for providing 
excellent conditions 
to conduct this research. I was also supported by NSF DMS-0103957.}
\subjclass[2000]{Primary 51K10; Secondary 31B99}
\copyrightinfo{2003}{American Mathematical Society}
\date{March 4, 2003}
\commby{Dmitri Burago}
\begin{abstract}
The main result can be stated roughly as follows:
Let $M$ be an Alexandrov space, $\Omega \subset M$ 
an open domain and $f:\Omega \to \mathbb{R}$ a
harmonic function. Then $f$ is Lipschitz on any compact subset of $\Omega $.

Using this result I extend proofs of some classical theorems in Riemannian
geometry to Alexandrov spaces.
\end{abstract}
\maketitle

\section*{Introduction}
Most of the work presented in this paper was done seven years ago; here,
I publish only the announcement of the results, since the proofs are very 
technical.
For complete proofs I refer the reader to \cite{Pet1}.

During the years since most of the work was done, there have been some 
developments in the
field, mostly due to Kuwae,  Machigashira and Shioya \cite{KMS1}, 
\cite{KMS2}. The reader is referred to
their papers for the updates.

In this paper, I am trying to establish
basic properties of harmonic functions defined on an Alexandrov space.
As an application I obtain generalizations of the
Isoperimetric Inequality of Gromov-Levy type
and extend the standard estimates of eigenvalues to Alexandrov spaces.

The main technical tool relies on the fact that a
harmonic function on a space with a lower curvature bound
is Lipschitz on any compact subdomain.
More precisely, if $M$ is an Alexandrov space with a lower curvature bound,
 $F:U\subset M\to \RR ^{n}$ is a harmonic map, and $C\subset U$ is compact, then 
$F|_{C}$ is Lipschitz.
In a certain sense this statement is an addition to the Gromov-Shoen Theorem,
which says that any harmonic map $\RR ^{n}\to N$, where
$N$ is a negatively curved (singular) space, is Lipschitz.
But this addition is not so general as it should be.
There is a strong feeling that the statement should be true in the case of
a lower Ricci curvature
bound, not just for spaces with a lower sectional curvature bound.
Unfortunately, before proving this generalization, one has to define
generalized spaces with a lower Ricci bound.
The technique presented below could provide help in
making the right definition of such spaces.

The original motivation for investigating harmonic functions was to
 get a local analysis, which is 
desperately needed for Alexandrov spaces
(for example, to prove that the boundary of an Alexandrov space is an 
Alexandrov space).
This motivation seemed to be reasonable, since harmonic functions already 
give a way
to establish a good analysis on both Alexandrov spaces
with two-sided bounded curvature (see \cite{N}) and on $2$-dimensional 
Alexandrov spaces with
bounded curvature
(in particular, for curvature bounded below) (see \cite{R}).
Unfortunately this idea encountered an apparently small obstacle:
I could not construct a harmonic chart in a neighborhood
of any point of an Alexandrov space (even a ``very good'' point).

\section*{1. List of results and some discussion}

\subsection*{1.A. Main Theorem}
In an Alexandrov $n$-space, the space of Sobolev functions can be understood as 
the  closure of
Lipschitz functions under the following norm:
\begin{equation*}|\!|f|\!|^{2}_{1,2}=\int _{\Omega }(f^{2} + 
|\text{grad}f|^{2}) d h_{n},\end{equation*}
where $h_{n}$ is Hausdorff measure and $|\text{grad}f|$ is the maximal 
rate of growth of $f$
at a point.  This norm makes it possible to define a $\lambda $-harmonic 
function
as the energy minimizing function having fixed boundary values, where the
 energy of a function $f$ is given by
\begin{equation*}E_{\lambda }(f)=\int _{\Omega }(\lambda f + 
|\text{grad}f|^{2}) d h_{n}.\end{equation*}
The $\lambda $-subharmonic ($\lambda $-superharmonic)
functions could be defined in the same way, but $f$ is supposed to
minimize the energy of all functions $\le f$  ($\ge f$) with the same
boundary values.

The main result of this paper can now be formulated:

\begin{theorem1} Let $\Omega $ be an open domain in an Alexandrov space $M$
with curvature $\ge k$,
and let $C\subset \Omega $ be a compact subset of diameter $\le D$ and volume 
$\ge v$.
Then there is a constant $L= L(D,v,k,d i s t(\partial \Omega ,C))$ such that
if $f:\Omega \to \RR $ is a harmonic function,
then
$f|_{C}$ is a Lipschitz function with Lipschitz constant $L|\!|f|\!|_{1,2}$.
\end{theorem1}


\subsection*{1.B. Properties of harmonic functions}
The next theorem says, roughly, that the limit of harmonic functions on an 
Alexandrov space is
again a harmonic function.

\begin{theorem2} Let $\{M_{i}\}$ be a noncollapsing sequence of Alexandrov
spaces, and let $M$ be its Gromov-Hausdorff limit (dim\,$M=$ dim\,$M_{i}$).
Let $f_{i}:B_{R}(p_{i})\subset M_{i}\to \RR $ be a sequence of harmonic
functions with a uniformly bounded norm and
$f:B_{R}(p)\subset M\to R$ its limit.
Then $f$ is a harmonic function and for any $R'0$,
the volume
$vol(\Sigma _{\pm }\cap B_{\epsilon }(x))>0$.

Consider the functions $f_{+}:\Sigma _{+}\to \RR _{+}$ and $f_{-}:\Sigma _{-}\to 
\RR _{+}$,
$f_{\pm }=d i s t_{\sigma _{\alpha }}$.

Now let $s^{*}$ be a sphere in $S^{n}$. One defines $S^{n}_{+}$, $S^{n}_{-}$,
$f^{*}_{+}:S^{n}_{+}\to \RR $ and $f^{*}_{-}:S^{n}_{-}\to \RR $
in the same way as above.

The following lemma should be regarded as an analog of the fact that a 
minimal
surface has constant mean curvature.
\renewcommand{\qed}{}
\end{proof}
\begin{theorem6}
There is a sphere $s^{*}$ in $S^{n}$ such that if
$f^{*}_{\pm }(y)=f_{\pm }(x)$, then $\Delta f^{*}_{\pm }(y)\ge \Delta f_{\pm 
}(x)$.
\end{theorem6}

The last inequality should be understood as an estimate for $\Delta 
f_{\pm }$,
i.e., for any $\epsilon >0$ and any $x\in \Sigma _{\pm }$, there is a 
neighborhood
$U\ni x$
such that $f_{\pm }$ is a $(\Delta f^{*}_{\pm }(y)-\epsilon )$-superharmonic 
function in $U$.

It is easy to construct two functions $g_{\pm }$  defined 
in a
neighborhood of $x_{\pm }$, which support $f_{\pm }$ at these points
(i.e. $g_{\pm }(x)\le f_{\pm }(x)$ and the inequality is exact only if 
$x=x_{\pm }$),
and such that $\Delta g_{\pm }\ge \Delta f_{\pm }(y_{\pm })+\epsilon $.
One can assume that the foot points of $x_{+}$ and $x_{-}$ (say $z_{+}$ and 
$z_{-}$)
on $\sigma _{\alpha }$ are distinct.

Consider the functions $\tilde g_{\pm }(y)=\max g_{\pm }(x)-|x y|$. These 
functions $\tilde g_{\pm }(y)$ are well defined in a neighborhood of
the minimal geodesic $x_{\pm }z_{\pm }$, i.e. the minimum is admitted for 
interior points
 of the domain of $g_{\pm }$.

 As a corollary of the Key Lemma (for Theorem A), it is easy to see that
 $\tilde g_{\pm }$ are $(\Delta f_{\pm }(y_{\pm })+\epsilon )$-subharmonic.
Moreover, direct calculations
(analogous to the Rauch comparison for Riemannian manifolds) show that in 
some neighborhoods of
$z_{\pm }$
 these functions are
$\lambda _{\pm }$-subharmonic with $\lambda _{+} +\lambda _{-}>0$.

The ``direct calculations'' above for the general case use
the second variation formula for Alexandrov spaces \cite{Pet2} and Perelman's
representation of a semiconvex function.  The latter says that for any 
semiconvex function $f$ one has the following representation for
almost all $x$:
\begin{equation*}f(y)=f(x)+\langle \text{grad}f,\log _{x} y\rangle + H(\log 
_{x} y)+o(|x y|^{2}),\end{equation*}
 where $H$ is a quadratic form, $\log _{x} y$ is an element of the 
tangent space $C_{x}$
 at $x$ such that $|\log _{x} y|=|x y|$, and the direction of $\log _{x} y$ 
coincides with the
 direction from $x$ to $y$.

The second variation gives an estimate of $H$.  The regular part
of  $\Delta f$ is simply $\text{trace}(H)$,
and the singular part is positive since
the function is semiconvex.

Let me cut two small caps with the same volume along level
sets of these functions (Figure 1).
\begin{figure}[b]
\includegraphics[scale=.45]{era120el-fig-1}
\caption{}
\end{figure}
Let $A_{+old}$, $A_{-old}$ be the areas of these caps and $A_{+new}$, 
$A_{-new}$ the areas of
the level sets which cut these caps.
Let $v=v_{+}=v_{-}$ be the volume being cut. Then from the Stokes theorem,
\begin{equation*}A_{\pm new}-A_{\pm old}\le \lambda _{\pm }v.\end{equation*}
Therefore
\begin{equation*}A_{+new}+A_{-new}