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\controldates{19-NOV-2003,19-NOV-2003,19-NOV-2003,19-NOV-2003}
 
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\documentclass{era-l}
\issueinfo{9}{15}{}{2003}
\dateposted{November 26, 2003}
\pagespan{121}{124}
\PII{S 1079-6762(03)00119-7}
\copyrightinfo{2003}{American Mathematical Society}

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\begin{document}


\title[Singularity structure in mean curvature flow]{Singularity structure in mean curvature flow \linebreak[1]
of mean-convex sets}
\author{Tobias H. Colding}
\address{Courant Institute of Mathematical Sciences,  251 Mercer Street, New
York, New York 10012}
\email{colding@cims.nyu.edu}
\thanks{The first author was supported by NSF grant DMS-0104453}
\author{Bruce Kleiner}
\address{Department of Mathematics, University of Michigan, 2072 East Hall, 525
E University Avenue, Ann Arbor, Michigan 48109-1109}
\email{bkleiner@umich.edu}
\thanks{The second author was supported by NSF grant DMS-0204506}
\date{September 24, 2003}
\subjclass[2000]{Primary 53C44}
\keywords{Mean-convex, mean curvature, singularities}
\commby{Svetlana Katok}

\begin{abstract}
In this note we announce results on the mean curvature flow of
mean-convex sets in three dimensions.  Loosely speaking,  our results justify the
naive picture of mean curvature flow where the only singularities are neck
pinches, and components which collapse to asymptotically round spheres.  
\end{abstract}

\maketitle

In this note we announce results on the mean curvature flow of mean-convex
sets; all the statements below have natural generalizations to the setting of
Riemannian $3$-manifolds, but for the sake of simplicity we will primarily
discuss subsets of $\R^3$ here. Loosely speaking,  our results justify the
naive picture of mean curvature flow where the only singularities are neck
pinches, and components which collapse to asymptotically round spheres.  
Recall that a one-parameter family of smooth hypersurfaces $\{ M_t \} \subset
\R^{n+1}$ {\it flows by mean curvature} if \begin{equation}\label{e:meanflow}
z_t = {\bf{H}} (z) = \Delta_{M_t} z \, , \end{equation} where
$z=(z_1,\ldots,z_{n+1})$ are coordinates on $\R^{n+1}$ and  ${\bf{H}} = - H
{\bf{n}}$ is the mean curvature vector. The papers  \cite{evansspruck} and
\cite{chengigagoto} defined a level set flow for any closed subset $K$ of
$\R^n$.  This is a $1$-parameter family of closed sets $K_t\subset \R^n$ with
$K_0=K$ (if $K$ is a domain bounded by a smooth compact hypersurface, 
then the
evolution of $\D K$ for a short time interval coincides with the classical mean
curvature evolution).  Following \cite{white1}, we say that a compact subset
$K\subset \R^n$ is {\em mean-convex} if $K_t\subset\Int(K)$ for all $t>0$. In
this case there is also an associated Brakke flow $\M:t\mapsto M_t$ of
rectifiable varifolds \cite{brakke,ilmanen,white1}, and the pair $(\M,\K)$,
where
\[\K:=\bigcup_{t\geq 0}\,K_t\times\{t\}\subset\R^n\times\R\]
is called a {\em mean-convex flow}, \cite{white2}. The fundamental papers
\cite{white1,white2} developed a far-reaching partial regularity theory for
mean curvature flow of mean-convex subsets of $\R^n$.  Our results build on
\cite{white1,white2}, giving finer understanding of the singularities  in the
$3$-dimensional case.  Recall that the main result of \cite{white1}  asserts
that the space-time singular set of the region swept out by a  mean--convex set
in $\R^{n+1}$ has parabolic Hausdorff dimension at most  $(n-1)$, and 
\cite{white2} proved a structure theorem for  blowups of mean-convex flows;
cf. also   \cite{huiskensinestrari1,huiskensinestrari0}.   We expect that the
more refined description of singularities given here  will open the way for
applications of mean-convex flow to geometric and/or topological problems
involving mean-convex surfaces.


If $(\M,\K)$ is a mean-convex flow in $\R^3$, then for almost every time $t$
the time slice $K_t$ is a domain with smooth boundary, \cite[Corollary to
Theorem 1.1]{white1}.  Our first result  shows that the high curvature portion
of such smooth time slices has standard local geometry:   
\begin{theorem} For
all $\eps > 0$ there is a number $h_0=h_0(\eps)$ with the following property. 
If $(\M,\K)$ is a mean-convex flow in $\R^3$ and  $K_t$ is a regular time slice
of $\K$ for some $t>0$,  then there is a decomposition $K_t=G_t\cup B_t$ such
that

\begin{itemize}
\item For all $x\in G_t$,
and after rescaling by the factor $\frac{h_0}{d(x,\D K)}$ 
the pointed subset $(K_t,x)$ is  $\eps$-close
to some pointed half-space $(P,p)$ in the pointed
$C^{\frac{1}{\eps}}$-topology.

\medskip
\item Each component of $B_t$ is diffeomorphic to the $3$-ball or a solid torus, and for all
$x\in \D K_t\cap B_t$, the pointed subset $(K_t,x)$ becomes, after rescaling
by the factor $H(x)$, $\eps$-close to a pointed convex model subset $(V,v)$
in the pointed $C^{\frac{1}{\eps}}$-topology.  Here $V\subset \R^3$ is a 
convex set whose
tangent cone at infinity is either a point, or a line, or a ray, and $V$
looks like a round cylinder near infinity, in the following sense: 
for every $\de>0$ there is a compact set $K\subset V$ such that
for every  $v'\in V$ lying 
 outside $K$, if we rescale $V$ by $H(v')$,
the resulting pointed subset $(V,v')$ is $\de$-close to a round
cylinder in the pointed $C^{\frac{1}{\de}}$-topology.
\end{itemize}
\end{theorem}
\noindent
 Note that the bounds on the geometry deteriorate
as one approaches $\D K$; this is by necessity, since no regularity
condition has been imposed on $K$.  If $K$ happens to be smooth,
then standard estimates for smooth mean curvature flow 
control the geometry of $K_t$ when $t\lesssim \sqrt r$, where
$r$ is the normal injectivity radius of $\D K$.
Theorem 1 may be compared with the recent work of 
Huisken-Sinestrari \cite{huiskensinestrari},
in which a similar geometric description was obtained for mean curvature 
flow of smooth hypersurfaces in $\R^n$
where the sum of the first two 
principal curvatures is positive.  
The results in 
\cite[Sections 11, 12]{perelman1} are also in a similar spirit.  Note that
their results only apply to the evolution prior to the formation of the first
singularity, whereas our results, like those in \cite{white1,white2},
apply even after the formation of
a singularity.  (In fact,
the methods yield a decomposition of arbitrary time slices, which we
omit for the sake of simplicity.)

 
It follows from the strong maximum principle and compactness that the
sets $\D K_t$ for $t\geq 0$ are disjoint, and define a ``singular foliation'' of the
original set $K$.   Our next theorem proves H\"older regularity of the singular
set of the foliation $\partial K_t$.  

\begin{theorem}

The foliation defined by the sets $\D K_t$
is smooth on the complement of a closed subset $S\subset K$
which satisfies the following Reifenberg-type condition: for all $\eps>0$  
there is an $r_0=r_0(\eps)$ such that if $r