% 
% Lectures on mean curvature flow without singularities
% Oliver Schn\"urer
% Uni Konstanz, Winter 2018
%
\documentclass[a4paper,draft]{amsart}
%\usepackage{fullpage}
\usepackage{citesort}
\usepackage{amssymb,amsmath}
%\usepackage{ngerman}
\usepackage[arrow,curve,matrix,tips]{xy}
% Hilfe: xypic-xygraph-tutorial.pdf
%\usepackage[notref,notcite]{showkeys} 
\usepackage{enumerate}
\usepackage{titleref}
\usepackage{ifthen}
%\usepackage[T1]{fontenc}
%\usepackage[latin1]{inputenc}
%\usepackage{a4}
%\CompileMatrices
%\xymatrixnocompile schaltet dies fuer eine einzelne Matrix aus.
\setcounter{secnumdepth}{1}

%\usepackage[a4paper,twoside,includeheadfoot,%
%inner=1.0cm,outer=1.0cm,top=1.0cm,bottom=1.0cm]{geometry}

\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\diam}{diam}
\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\Image}{Im}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\graph}{graph}
\DeclareMathOperator*{\fueralle}{\forall}
\DeclareMathOperator*{\existiert}{\exists}
\DeclareMathOperator{\interior}{int}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\rang}{rang}
\DeclareMathOperator{\ind}{ind}
\DeclareMathOperator{\Null}{null}
\DeclareMathOperator{\divergenz}{div}
\DeclareMathOperator{\Iso}{Iso}
\def\ol#1{\overline{#1}}
\def\ul#1{\underline{#1}}
\def\R{{\mathbb{R}}}
\def\N{{\mathbb{N}}}
\def\C{{\mathbb{C}}}
\def\S{{\mathbb{S}}}
\def\T{{\mathbb{T}}}
\def\Q{{\mathbb{Q}}}
\def\Z{{\mathbb{Z}}}
\def\O{{\mathcal{O}}}
\def\P{{\mathcal{P}}}
\def\B{{\mathcal{B}}}
\def\U{{\mathcal{U}}}
\def\F{{\mathcal{F}}}
\def\M{{\mathcal{M}}}
\def\theta{{\vartheta}}
\def\phi{{\varphi}}
\def\epsilon{{\varepsilon}}
\def\dt{{\frac{d}{dt}}}
\def\fracd#1#2{{\frac{\displaystyle #1}{\displaystyle #2}}}
\def\fracp#1#2{{\frac{\partial #1}{\partial #2}}}
\def\fracpd#1#2{{\frac{\displaystyle\partial #1}
    {\displaystyle\partial #2}}}
\def\tfracp#1#2{{\tfrac{\partial #1}{\partial #2}}}
\long\def\umbruch{{\displaybreak[1]}}
\long\def\neueZeile{{\rule{0mm}{1mm}\\[-3.25ex]\rule{0mm}{1mm}}}
\long\def\neueZeilealt{{\rule{0mm}{1mm}\\\rule{0mm}{1mm}}}
\long\def\Zeileweg{{\rule{0mm}{1mm}\\[-7.1ex]\rule{0mm}{1mm}}}
\def\dh{d.\,h.\ }
\def\impliziert{\text{ }\ensuremath{\Longrightarrow}\text{ }}
\def\folgtaus{\text{ }\ensuremath{\Longleftarrow}\text{ }}
\def\aequivalent{\text{ }\ensuremath{\Longleftrightarrow}\text{ }}
\def\Hinrichtung{\glqq\ensuremath{\Longrightarrow}\grqq:\text{ }}
\def\Rueckrichtung{\glqq\ensuremath{\Longleftarrow}\grqq:\text{ }}
\def\eins{\mbox{\ensuremath{1\hspace*{-0.9ex}1}}}
\def\stern{\ensuremath{\star} }
\def\interior#1{\overset{{}_\circ}{#1}}
%\newcommand{\margincomment}[1]{\marginpar{\footnotesize{#1}}}
\def\gap{\rule{2cm}{1ex}}
\def\bzw{bzw.{} }
\def\zB{z.\,B.{} }
\def\delt{{\frac{\partial}{\partial t}}}
\def\lx{\lambda_1}
\def\ly{\lambda_2}
\def\g{\mathcal G}
\newcommand{\A}[1]{\ifthenelse{#1 = 2}{\lvert A\rvert^{#1}}{\tr A^{#1}}}
\DeclareMathOperator*{\osc}{osc}

\def\di{\tfracp{}{x^i}}
\def\dj{\tfracp{}{x^j}}
\def\dk{\tfracp{}{x^k}}
\def\dl{\tfracp{}{x^l}}

\def\nablai{\nabla_\di}
\def\nablaj{\nabla_\dj}
\def\nablak{\nabla_\dk}
\def\nablal{\nabla_\dl}

\def\heat{\left(\tfrac d{dt}-\Delta\right)}

\DeclareMathOperator{\Ric}{Ric}

\def\linksanfuegen#1{\ar@{}[r]|{\displaystyle #1}}
\def\rechtsanfuegen#1{\ar@{}[l]|{\displaystyle #1}}
\def\anfuegen#1#2{\ar@{}[#1]|{\displaystyle #2}}
 
\mathchardef\ordinarycolon\mathcode`\:
  \mathcode`\:=\string"8000
  \begingroup \catcode`\:=\active
    \gdef:{\mathrel{\mathop\ordinarycolon}}
  \endgroup

% Hier ggf. vor der Veroeffentlichung noch etwas aendern.
\long\def\loesung#1{\begin{proof}[L\"osung]#1\end{proof}}
%\long\def\loesung#1{}

%\newtheorem{theorem}{Theorem}[chapter]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{exercise}[theorem]{Exercise}

\newtheorem{aufgabe}{Aufgabe}[section]
%\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

%\numberwithin{section}{chapter}
%\numberwithin{equation}{chapter}
\numberwithin{equation}{section}
\setcounter{tocdepth}{1}

%    Absolute value notation
\newcommand{\abs}[1]{\left\lvert#1\right\rvert}

%    Blank box placeholder for figures (to avoid requiring any
%    particular graphics capabilities for printing this document).
\newcommand{\blankbox}[2]{%
   \parbox{\columnwidth}{\centering
%    Set fboxsep to 0 so that the actual size of the box will match the
%    given measurements more closely.
     \setlength{\fboxsep}{0pt}%
     \fbox{\raisebox{0pt}[#2]{\hspace{#1}}}%
   }%
}

\hyphenation{}

\begin{document}

\title[Lectures on mean curvature flow without singularities]{Lectures
  on mean curvature flow\\ without singularities} 

%    Information for first author
\author{Oliver C. Schn\"urer}
%    Address of record for the research reported here
\address{Oliver C. Schn\"urer, FB Mathematik und Statistik,
  Universit\"at Konstanz, Germany}
%    Current address
\curraddr{}
\def\AmSeeHome{@uni-konstanz.de}
\email{Oliver.Schnuerer\AmSeeHome}
%    \thanks will become a 1st page footnote.
\thanks{Lecture notes for the Winter School ``Geometric
  Evolution Equation'', Konstanz, 05.-09.02.2018,
  organised by K. Kr\"oncke, L. Simon, B. Vertman and the author.}

%    General info
\subjclass[2000]{53C44}
% 53C44 Geometric evolution equations (mean curvature flow)

\date{\today.}

\dedicatory{}

\keywords{}

%\setlength{\parindent}{0pt}
%\setlength{\parskip}{0.5em}

\begin{abstract}
  In these lectures, we study hypersurfaces that solve geometric
  evolution equations. More precisely, we investigate hypersurfaces
  that evolve with a normal velocity depending on a curvature function
  like the mean curvature. In two lectures, we will address
  \begin{itemize}
  \item hypersurfaces, principal curvatures and evolution equations
    for geometric quantities like the metric and the second
    fundamental form.
  \item the evolution of graphical hypersurfaces under mean curvature
    flow.
  \end{itemize}
\end{abstract}

\maketitle

\tableofcontents

%\mainmatter
%\addtocounter{section}{-1}

%\markboth{\footnotesize{\textsc{\thesection.{} \currenttitle}}}
%{\footnotesize{\textsc{\thesection.{} \currenttitle}}}

\section{Overview and plan for the winter school}
We consider flow equations that deform hypersurfaces according to
their curvature. \par
If $X_0:M^n\to\R^{n+1}$ is an embedding of an $n$-dimensional
manifold, we can define principal curvatures $(\lambda_i)_{1\le i\le
  n}$ and a normal vector $\nu$. We deform the embedding vector $X$
according to
\[\begin{cases}
  \dt X=-F\nu,&\\
  X(\cdot,0)=X_0,&
\end{cases}\] where $F$ is a symmetric function of the principal
curvatures, e.\,g.\ the mean curvature $H=\lambda_1+\cdots+\lambda_n$.
In this way, we obtain a family $X(\cdot,t)$ of embeddings. Graphical
solutions are shown to exist for all times or to disappear to
infinity. \par
Nowadays classical results in this direction were obtained by
G. Huisken \cite{HuiskenRoundSphere} and K. Ecker and G. Huisken
\cite{EckerHuiskenInvent} for mean curvature flow.

\begin{remark}\neueZeile
  \begin{enumerate}[(i)]
  \item We will use geometric flow equations as a tool to deform a
    manifold.
  \item The flow equations considered are parabolic equations like the
    heat equation.
  \item In order to control the behaviour of the flow, we will look for
    properties of the manifold that are preserved under the flow. For
    that purpose, we will also look for quantities that are monotone
    and have geometric significance, i.\,e.\ their boundedness implies
    geometric properties of the evolving manifold.
  \end{enumerate}
\end{remark}

\subsection{Plan for the winter school}
These notes contain some background material that will be covered as
necessary. Then we will derive evolution equations for geometric
quantities and study geometric problems. More precisely, our plan
is to study the following: 
\begin{itemize}
\item Geometric prerequisites and evolution equations of geometric
  quantities. 
\item Mean curvature flow of complete graphs.
% and local $C^1$-bounds, Theorem~\ref{C1 est thm}.
\end{itemize}

\section{Differential geometry of submanifolds}
\label{sec:diff-geom-subm}
We will only consider hypersurfaces in Euclidean space. 

We use $X=X(x,\,t)=\left(X^\alpha\right)_{1\le\alpha\le n+1}$ to denote
the time-dependent embedding vector of a manifold $M^n$ into
$\R^{n+1}$ and $\dt X=\dot X $ for its total time derivative.  Set
$M_t:=X(M,\,t)\subset\R^{n+1}$. We will often identify an embedded
manifold with its image. We will assume that $X$ is smooth.  Assume
furthermore that $M^n$ is smooth, orientable, connected, complete and
$\partial M^n=\emptyset$.  We choose
$\nu=\nu(x)=\left(\nu^\alpha\right)_{1\le\alpha\le n+1}$ to be the
outer (or downward pointing) unit normal vector to $M_t$ at $x\in
M_t$.  The embedding $X(\cdot,\,t)$ induces at each point on $M_t$ a
metric $(g_{ij})_{1\le i,\,j\le n}$ and a second fundamental form
$(h_{ij})_{1\le i,\,j\le n}$. Let $(g^{ij})$ denote the inverse of
$(g_{ij})$. These tensors are symmetric. The principal curvatures
$(\lambda_i)_{1\le i\le n}$ are the eigenvalues of the second
fundamental form with respect to that metric. That is, at $p\in M$,
for each principal curvature $\lambda_i$, there exists $0\neq\xi\in
T_pM\cong\R^n$ such that
\[\lambda_i\sum\limits_{l=1}^n g_{kl}\xi^l=\sum\limits_{l=1}^n
h_{kl}\xi^l\text{ or, equivalently, }
\lambda_i\xi^l=\sum\limits_{k,r=1}^n g^{lk}h_{kr}\xi^r.\] As usual,
eigenvalues are listed according to their multiplicity. A hypersurface
is called strictly convex, if all principal curvatures are strictly
positive.  

Latin indices range from $1$ to $n$ and refer to geometric quantities
on the hypersurface, Greek indices range from $1$ to $n+1$ and refer
to components in the ambient space $\R^{n+1}$.  In $\R^{n+1}$, we will
always choose Euclidean coordinates.  We use the Einstein summation
convention for repeated upper and lower indices. Latin indices are
raised and lowered with respect to the induced metric or its inverse
$\left(g^{ij}\right)$, for Greek indices we use the flat metric $(\ol
g_{\alpha\beta})_{1\le\alpha,\beta\le n+1}
=(\delta_{\alpha\beta})_{1\le\alpha,\beta\le n+1}$ of $\R^{n+1}$. So
the defining equation for the principal curvatures becomes $\lambda_i
g_{kl}\xi^l =h_{kl}\xi^l$.

Denoting by $\langle\cdot,\,\cdot\rangle$ the Euclidean scalar
product in $\R^{n+1}$, we have
$$g_{ij}=\left\langle X_{,\,i},\,X_{,\,j}\right\rangle
=X^\alpha_{,\,i}\delta_{\alpha\beta}X^\beta_{,\,j},$$ where we used
indices, preceded by commas, to denote partial derivatives. We write
indices, preceded by semi-colons, e.\,g.\ $h_{ij;\,k}$ or $v_{;k}$, to
indicate covariant differentiation with respect to the induced metric.
Later, we will also drop the commas and semi-colons, if the meaning is
clear from the context. We set $X^\alpha_{;i}\equiv X^\alpha_{,i}$ and
\begin{equation}
  \label{eq:Xij}
  X^\alpha_{;\,ij}=X^\alpha_{,\,ij}-\Gamma^k_{ij}X^\alpha_{,\,k},
\end{equation}
where 
$$\Gamma^k_{ij}=\tfrac12g^{kl}(g_{il,\,j}+g_{jl,\,i}-g_{ij,\,l})$$
are the Christoffel symbols of the metric $(g_{ij})$. So
$X^\alpha_{;ij}$ becomes a tensor. 

The Gau{\ss} formula relates covariant derivatives of the
position vector to the second fundamental form and 
the normal vector 
\begin{equation}\label{Gauss formula}
  X^\alpha_{;\,ij}=-h_{ij}\nu^\alpha.
\end{equation}

The Weingarten equation allows to compute derivatives of the 
normal vector 
\begin{equation}\label{Weingarten equation}
  \nu^\alpha_{;\,i}=h^k_iX^\alpha_{;\,k}.
\end{equation}

We can use the Gau{\ss} formula \eqref{Gauss formula} or
the Weingarten equation \eqref{Weingarten equation} to compute
the second fundamental form. 

Symmetric functions of the principal curvatures are well-defined, we
will use the mean curvature $H=\lambda_1+\ldots+\lambda_n$, the square
of the norm of the second fundamental form
$|A|^2=\lambda_1^2+\ldots+\lambda_n^2$, $\tr A^k= \lambda_1^k
+\ldots+\lambda_n^k$, and the Gau{\ss} curvature $K=\lambda_1 \cdot
\ldots \cdot \lambda_n$. It is
often convenient to choose coordinate systems such that, at a fixed
point, the metric tensor equals the Kronecker delta,
$g_{ij}=\delta_{ij}$, and $(h_{ij})$ is diagonal,
$(h_{ij})=\diag(\lambda_1,\ldots,\lambda_n)$, e.\,g.
$$\sum\lambda_k h_{ij;k}^2=\sum\limits_{i,\,j,\,k=1}^n \lambda_k h_{ij;k}^2
=h^{kl}h^i_{j;\,k}h^j_{i;\,l}
=h_{rs}h_{ij;\,k}h_{ab;\,l}g^{ia}g^{jb}g^{rk}g^{sl}.$$ Whenever we use
this notation, we will also assume that we have fixed such a
coordinate system.

A normal velocity $F$ can be considered as a function of
$(\lambda_1,\ldots,\lambda_n)$ or $(h_{ij},\,g_{ij})$.  If
$F(\lambda_i)$ is symmetric and smooth, then $F(h_{ij},g_{ij})$ is
also smooth \cite[Theorem 2.1.20]{CGCPBook}. We set
$F^{ij}=\fracp{F}{h_{ij}}$, $F^{ij,\,kl}=\fracp{^2F}{h_{ij}\partial
  h_{kl}}$.  Note that in coordinate systems with diagonal $h_{ij}$
and $g_{ij}=\delta_{ij}$ as mentioned above, $F^{ij}$ is diagonal.
For $F=|A|^2$, we have $F^{ij}=2h^{ij}=2\lambda_i g^{ij}$, and for
$F=K^{\alpha}$, $\alpha>0$, we have $F^{ij}=\alpha K^{\alpha}\tilde
h^{ij}= \alpha K^{\alpha}\lambda_i^{-1}g^{ij}$.

The Gau{\ss} equation expresses the Riemannian curvature tensor of the
hypersurface in terms of the second fundamental form
\begin{equation}\label{Riem}
  R_{ijkl}=h_{ik}h_{jl}-h_{il}h_{jk}.
\end{equation}

As we use only Euclidean coordinate systems in $\R^{n+1}$,
$h_{ij;\,k}$ is symmetric in all three indices according to the
Codazzi equations.

The Ricci identity allows to interchange covariant derivatives. We
will use it for the second fundamental form
\begin{equation}\label{Ricci}
  h_{ik;\,lj}=h_{ik;\,jl}+h^a_kR_{ailj}+h^a_iR_{aklj}.
\end{equation}

For tensors $A$ and $B$, $A_{ij}\ge B_{ij}$ means that 
$(A_{ij}-B_{ij})$ is positive definite.

Finally, we use $c$ to denote universal, estimated constants. 

\subsection{Graphical submanifolds}
\begin{lemma}\label{graph geom lem}
  Let $u:\R^n\to\R$ be smooth. Then $\graph u$ is a submanifold in
  $\R^{n+1}$. The metric $g_{ij}$, the lower unit normal vector $\nu$,
  the second fundamental form $h_{ij}$, the mean curvature $H$, and the
  Gau\ss{} curvature $K$ are given by
  \begin{align*}
    g_{ij}=&\,\delta_{ij}+u_iu_j,\\
    g^{ij}=&\,\delta^{ij}-\frac{u^iu^j}{1+|Du|^2},\\
    \nu=&\,\frac{((u_i),-1)}{\sqrt{1+|Du|^2}}\equiv\frac{((u_i),-1)}v,\\
    h_{ij}=&\,\frac{u_{ij}}{\sqrt{1+|Du|^2}}\equiv\frac{u_{ij}}v,\\
    H=&\,\divergenz\left(\frac{Du}{\sqrt{1+|Du|^2}}\right),\\
    \intertext{and}\\
    K=&\,\frac{\det D^2u}{\left(1+|Du|^2\right)^{\frac{n+2}2}},
  \end{align*}
  where $u_i\equiv\fracp u{x^i}$, $u^i=u_j\delta^{ji}$ and
  $u_{ij}=\fracp{^2u}{x^i\partial x^j}$. Note that in Euclidean space,
  we do not distinguish between $Du$ and $\nabla u$.
\end{lemma}
This result also holds, if $u$ is defined on an open subset of $\R^n$.
\begin{proof}\stern
  \begin{enumerate}[(i)]
  \item We use the embedding vector $X(x):=(x,u(x))$,
    $X:\R^n\to\R^{n+1}$. The induced metric is the pull-back of the
    Euclidean metric in $\R^{n+1}$,
    $g:=X^*g_{\R^{n+1}_{\text{Eucl.}}}$. We have $X_{,i}=(e_i,u_i)$.
    Hence \[g_{ij}=X_{,i}^\alpha\delta_{\alpha\beta}X^\beta_{,j}
    =\langle X_{,i},X_{,j}\rangle =\langle(e_i,u_i),(e_j,u_j)\rangle
    =\delta_{ij}+u_iu_j.\]
  \item It is easy to check, that $g^{ij}$ is the inverse of $g_{ij}$.
    Note that $u^i:=\delta^{ij}u_j$, i.\,e., we lift the index with
    respect to the flat metric. It is convenient to choose a
    coordinate system such that $u_i=0$ for $i<n$.
  \item The vectors $X_{,i}=(e_i,u_i)$ are tangent to $\graph u$. The
    vector $((-u_i),1)\equiv(-Du,1)$ is orthogonal to these vectors,
    hence, up to normalization, a unit normal vector. 
  \item We combine \eqref{eq:Xij}, \eqref{Gauss formula} and compute
    the scalar product with $\nu$ to get
    \begin{align*}
      h_{ij} =&\,-\langle X_{;ij},\nu\rangle =-\langle
      X_{,ij}-\Gamma^k_{ij}X_{,k},\nu\rangle =-\langle
      X_{,ij},\nu\rangle\\
      =&\,-\left\langle(0,u_{ij}),\frac{((u_i),-1)}v\right\rangle
      =\frac{u_{ij}}v.
    \end{align*}
  \item We obtain
    \begin{align*}
      H =&\,\sum\limits_{i=1}^n\lambda_i=g^{ij}h_{ij}
      =\left(\delta^{ij}-\frac{u^iu^j}{1+|Du|^2}\right)
      \frac{u_{ij}}{\sqrt{1+|Du|^2}}\\
      =&\,\frac{\delta^{ij}u_{ij}}{\sqrt{1+|Du|^2}}
      -\frac{u^iu^ju_{ij}}{\left(1+|Du|^2\right)^{3/2}}\\
      =&\,\frac{\Delta u}{\sqrt{1+|Du|^2}}
      -\frac{u^iu^ju_{ij}}{\left(1+|Du|^2\right)^{3/2}}
      \intertext{and, on the other hand,}
      \divergenz\left(\frac{Du}{\sqrt{1+|Du|^2}}\right)
      =&\,\sum\limits_{i=1}^n\fracp{}{x^i}\frac{u_i}{\sqrt{1+|Du|^2}}\\
      =&\,\sum\limits_{i=1}^n\frac{u_{ii}}{\sqrt{1+|Du|^2}}
      -\sum\limits_{i,j=1}^n
      \frac{u_iu_ju_{ji}}{\left(1+|Du|^2\right)^{3/2}}\\ 
      =&\,H.
    \end{align*}
  \item From the defining equation for the principal curvatures, we
    obtain
    \begin{align*}
      K=&\,\prod\limits_{i=1}^n\lambda_i
      =\det\,\left(g^{ij}h_{jk}\right) =\det g^{ij}\cdot\det h_{ij}
      =\frac{\det h_{ij}}{\det g_{ij}}\\
      =&\,\frac{v^{-n}\det u_{ij}}{v^2} =\frac{\det
        D^2u}{\left(1+|Du|^2\right)^{\frac{n+2}2}}.
    \end{align*}
  \end{enumerate}
\end{proof}

\begin{exercise}[Spheres]
  \stern The lower part of a sphere of radius $R$ is locally given as
  $\graph u$ with $u:B_R(0)\to\R$ defined by
  $u(x):=-\sqrt{R^2-|x|^2}$.  Compute explicitly for that example all
  the quantities mentioned in Lemma \ref{graph geom lem} and the
  principal curvatures.
\end{exercise}

\section{Evolving submanifolds}
\label{sec:evolv-subm}
\subsection{General assumption}
We will only consider the evolution of manifolds of dimension $n$
embedded into $\R^{n+1}$, i.\,e.\ the evolution of hypersurfaces in
Euclidean space. (Mean curvature flow is also considered for manifolds
of arbitrary codimension. Another generalization is to study flow
equations of hypersurfaces immersed into a (Riemannian or Lorentzian)
manifold.)
\begin{definition}
  Let $M^n$ denote an orientable manifold of dimension $n$. Let
  $X(\cdot,t):M^n\to\R^{n+1}$, $0\le t\le T\le\infty$, be a smooth
  family of smooth embeddings. Let $\nu$ denote one choice of the
  normal vector field along $X(M^n,t)$. Then $M_t:=X(M^n,t)$ is said
  to move with normal velocity $F$, if
  \[\dt X=-F\nu\quad\text{in }M^n\times[0,T).\]
\end{definition}

In the following we will often identify an embedded submanifold and
its image under the embedding.

\subsection{Evolution of graphs}
\begin{lemma}
  Let $u:\R^n\times[0,\infty)\to\R$ be a smooth function such that
  $\graph u$ evolves according to $\dt X=-F\nu$. Then
  \[\dot u=\sqrt{1+|Du|^2}\cdot F.\]
\end{lemma}
This result also holds, if $u$ is defined on an open subset of                
$\R^n\times[0,\infty)$.
\begin{proof} \stern
  Beware of assuming that considering the $(n+1)$-st component in the
  evolution equation $\dt X=-F\nu$ were equal to $\dot u$ as a
  hypersurface evolving according to $\dt X=-F\nu$ does not only move
  in vertical direction but also in horizontal direction. \par
  Let $p$ denote a point on the abstract manifold embedded via $X$
  into $\R^{n+1}$. As our embeddings are graphical, we see that
  \[X(p,t)=(x(p,t),u(x(p,t),t)).\] We consider the scalar product of
  both sides of the evolution equation with $\nu$ and obtain
  \[F=\langle F\nu,\nu\rangle=\left\langle-\dt X,\nu\right\rangle
  =-\left\langle\left(\left(\dot x^k\right),u_i\dot x^i+\dot
      u\right),\frac{((u_i),-1)}{\sqrt{1+|Du|^2}}\right\rangle
  =\frac{\dot u}{\sqrt{1+|Du|^2}}.\]
\end{proof}

\begin{corollary}
  \stern Let $u:\R^n\times[0,\infty)\to\R$ be a smooth function such
  that $\graph u$ solves mean curvature flow $\dt X=-H\nu$. Then
  \[\dot u=\sqrt{1+|Du|^2}\cdot
  \divergenz\left(\frac{Du}{\sqrt{1+|Du|^2}}\right).\]
\end{corollary}

\subsection{Examples}
\begin{lemma}\label{sphere lem}
  Consider mean curvature flow, i.\,e.\ the evolution equation $\dt
  X=-H\nu$, with $M_0=\partial B_R(0)$. Then a smooth solution exists
  for $0\le t<T:=\frac1{2n}R^2$ and is given by $M_t=\partial
  B_{r(t)}(0)$ with $r(t)=\sqrt{2n(T-t)}=\sqrt{R^2-2nt}$.
\end{lemma}
\begin{proof}
  The mean curvature of a sphere of radius $r(t)$ is given by $H=\frac
  n{r(t)}$. Hence we obtain a solution to mean curvature flow, if
  $r(t)$ fulfills
  \[\dot r(t)=\frac{-n}{r(t)}.\]
  A solution to this ordinary differential equation is given by
  $r(t)=\sqrt{2n(T-t)}$.\par
  (The theory of partial differential equations implies that this
  solution is actually unique and hence no solutions exist that are
  not spherical.)
\end{proof}

\begin{exercise}
  Find a solution to mean curvature flow with $M_0=\partial
  B_{R}(0)\times\R^k\subset\R^l\times\R^k$.  This includes in
  particular cylinders. Note that for $k>1$, it is not obvious,
  whether these solutions are unique.
\end{exercise}


\begin{remark}[Level-set flow] \stern
  \label{lsf rem}
  If a hypersurface moves with velocity $F$, we use a function
  $u\colon\R^n\times[0,\infty)\to\R$ such that for each $c\in\R$, the
  set $M_t:=\{x\in\R^n\colon u(x,t)=c\}$ (if it is a smooth
  hypersurface) is an embedded hypersurface that moves with velocity
  $F$. \par
  We fix the unit normal $\nu=\frac{Du}{|Du|}$. Recall that $\dot
  X=-F\nu$. If $u$ is as described above, we have $u(X(p,t),t)=0$
  along the flow. Differentiating this equation yields $0=\dot
  u+Du\cdot\dot X =\dot u+Du\cdot(-\nu)\cdot F=\dot u-|Du|\cdot
  F$. \par For mean curvature flow, we obtain \[\dot
  u=|Du|\cdot\divergenz\left(\frac{Du}{|Du|}\right) =\left(\delta^{ij}
    -\frac{u^iu^j}{|Du|^2}\right)u_{ij}.\] We leave it as an exercise
  that the converse implication is also true if the level sets are
  regular in the sense that $Du\neq0$, i.\,e.{} that $\{x\colon
  u(x,t)=c\}$ evolves with normal velocity $F$ if $\dot u=|Du|\cdot F$
  and $Du\neq0$ along $\{x\colon u(x,t)=c\}$.
\end{remark}

\subsection{Short-time existence and avoidance principle}
In the case of closed initial hypersurfaces, short-time existence is
guaranteed by the following
\begin{theorem}[Short-time existence]\label{short time ex thm}
  Let $X_0:M^n\to\R^{n+1}$ be an embedding describing a smooth closed
  hypersurface. Let $F=F(\lambda_i)$ be smooth, symmetric, and
  $\fracp{F}{\lambda_i}>0$ everywhere on $X(M^n)$ for all $i$. Then
  the initial value problem
  \[\begin{cases}
    \dt X=-F\nu,\\
    X(\cdot,0)=X_0
  \end{cases}\]
  has a smooth solution on some (short) time interval $[0,T)$, $T>0$. 
\end{theorem}
\begin{proof}[Idea of Proof]
  Represent potential solutions locally as graphs in a tubular
  neighbourhood of $X_0(M^n)$. Then $\fracp F{\lambda_i}>0$ ensures
  that the evolution equation for the height function in this
  coordinate system is strictly parabolic. Linear theory and the
  implicit function theorem guarantee that there exists a solution on
  a short time interval.\par
  For details see \cite[Theorem 3.1]{HuiskenPolden}.
\end{proof}

On the other hand, starting with a closed hypersurface gives rise to
solutions that exist at most on a finite time interval. This is a
consequence of the avoidance principle. 
We will only consider the avoidance principle for mean curvature flow: 
\begin{theorem}[Avoidance principle]\label{avoidance thm}
  Let $M^1_t$ and $M^2_t\subset\R^{n+1}$ be two embedded closed
  hypersurfaces and smooth solutions to $\dt X=-H\nu$ on a common time
  interval $[0,T)$. If $M^1_0$ and $M^2_0$ are disjoint, then $M^1_t$
  and $M^2_t$ are also disjoint.\par In particular, if $M^1_0$ is
  contained in a bounded component of $\R^{n+1}\setminus M^2_0$, then
  $M^1_t$ is contained in a bounded component of $\R^{n+1}\setminus
  M^2_t$.
\end{theorem}
\begin{proof}
  Otherwise there would be some minimal $t_0>0$ such that $M^2_{t_0}$
  touches $M^1_{t_0}$ at some point $p\in\R^{n+1}$. We get for the
  normals $\nu^1=\pm\nu^2$ at $p$. Observe that if we change $\nu$ to
  $-\nu$, $H$ also changes sign and $H\nu$ remains
  unchanged. Therefore it does not matter for mean curvature flow,
  which normal we choose and we may assume without loss of generality
  that $\nu^1=\nu^2$ at $p$. Writing $M^i_t$ locally as $\graph u^i$
  over the common tangent hyperplane $T_pM^i_{t_0}\subset\R^{n+1}$, we
  see that the functions $u^i$ fulfill \[\dot
  u^i=\sqrt{1+\left|Du^i\right|^2}
  \cdot\divergenz\left(\fracd{Du^i}{\sqrt{1+\left|Du^i\right|^2}}\right)
  \equiv F\left(D^2u^i,Du^i\right).\] We may assume that $u^1>u^2$ for
  $t<t_0$. The evolution equation for the difference $w:=u_1-u_2$
  fulfills $w>0$ for $t<t_0$ locally in space-time and $w(0,t_0)$, if
  we have $p=(0,0)$ in our coordinate system. The evolution equation
  for $w$ can be computed as follows
  \begin{align*}
    \dot w=&\,\dot u^1-\dot u^2 =F\left(D^2u^1,Du^1\right)
    -F\left(D^2u^2,Du^2\right)\\
    =&\,\int\limits_0^1\frac d{d\tau} F\left(\tau
      D^2u^1+(1-\tau)D^2u^2,
      \tau Du^1+(1-\tau)Du^2\right)\,d\tau\\
    =&\,\int\limits_0^1\fracp F{r_{ij}}(\ldots)\,d\tau\cdot
    \left(u^1-u^2\right)_{ij}
    +\int\limits_0^1\fracp F{p_i}(\ldots)\,d\tau\cdot
    \left(u^1-u^2\right)_i\\ 
    \equiv&\,a^{ij}w_{ij}+b^iw_i.
  \end{align*}
  Hence we can apply the parabolic Harnack inequality or the strong
  parabolic maximum principle and see that it is impossible that
  $w(x,t)>0$ for small $|x|$ and $t<t_0$, but $w(0,t_0)=0$. Hence
  $M^1_t$ cannot touch $M^2_t$ in a point, where $\nu^1=\nu^2$. The
  theorem follows.
\end{proof}

\begin{corollary}[Finite existence time]\label{finite ex time cor}
  Let $M_0$ be a smooth closed embedded hypersurface in $\R^{n+1}$.
  Then a smooth solution $M_t$ to $\dt X=-H\nu$ can only exist on some
  finite time interval $[0,T)$, $T<\infty$.
\end{corollary}
\begin{proof}
  Choose a large sphere that encloses $M_0$. According to Lemma
  \ref{sphere lem}, that sphere shrinks to a point in finite
  time. Thus the solution $M_t$ can exist smoothly at most up to that
  time. 
\end{proof}

\begin{remark}[Maximal existence time]
  Consider $T$ maximal such that a smooth solution $M_t$ as in
  Corollary \ref{finite ex time cor} exists on $[0,T)$. Then the
  embedding vector $X$ is uniformly bounded according to Theorem
  \ref{avoidance thm}.  Then some spatial derivative of the embedding
  $X(\cdot,t)$ has to become unbounded as $t\nearrow T$. For otherwise
  we could apply Arzel\`a-Ascoli and obtain a smooth limiting
  hypersurface $M_T$ such that $M_t$ converges smoothly to $M_T$ as
  $t\nearrow T$. This, however, is impossibly, as Theorem \ref{short
    time ex thm} would allow to restart the flow from $M_T$. In this
  way, we could extend the flow smoothly all the way up to
  $T+\epsilon$ for some $\epsilon>0$, contradicting the maximality of
  $T$. \par
  It can often be shown that extending a solution beyond $T$ is
  possible provided that $\Vert X(\cdot,t)\Vert_{C^2}$ is uniformly
  bounded. For mean curvature flow, this follows from explicit
  estimates. For other normal velocities, additional assumptions (the
  principal curvatures stay in a region, where $F$ has nice
  properties) and Krylov-Safonov-estimates can imply such a result.
\end{remark}

\section{Evolution equations for submanifolds}
\label{sec:evol-equat-subm}
In this chapter, we will compute evolution equations of geometric
quantities, see e.\,g.\
\cite{HuiskenPolden,HuiskenRoundSphere,OSA2}.

For a family $M_t$ of hypersurfaces solving the evolution equation
\begin{equation}\label{X dot}
  \dt X=-F\nu
\end{equation}
with $F=F(\lambda_i)$, where $F$ is a smooth symmetric function, we
have the following evolution equations.

\begin{lemma}
  The metric $g_{ij}$ evolves according to 
  \begin{equation}
    \label{g evol}
    \dt g_{ij}=-2Fh_{ij}.
  \end{equation}
\end{lemma}
\begin{proof}
  By definition, $g_{ij}=\langle X_{,i}, X_{,j}\rangle =X^\alpha_{,i}
  \delta_{\alpha\beta}X^\beta_{,j}$. We differentiate with respect to
  time. Derivatives of $\delta_{\alpha\beta}$ vanish. The term
  $X^\alpha_{,i}$ involves only partial derivatives. We obtain
  \begin{align*}
    \dt g_{ij}=&\,\left(\dot X^\alpha\right)_{,i}
    \delta_{\alpha\beta}X^\beta_{,j} 
    +X^\alpha_{,i}\delta_{\alpha\beta} \left(\dot X^\beta\right)_{,j} 
    \intertext{(we may exchange partial spatial and time derivatives)}
    =&\,(-F\nu^\alpha)_{,i}\delta_{\alpha\beta}X^\beta_{,j}
    +X^\alpha_{,i}\delta_{\alpha\beta}(-F\nu\beta)_{,j} \intertext{(in
      view of the evolution equation $\dt X=-F\nu$)}
    =&\,-F\nu^\alpha_{;i}\delta_{\alpha\beta}X^\beta_{,j}
    -X^\alpha_{,i}\delta_{\alpha\beta}F\nu_{;j} \intertext{(terms
      involving derivatives of $F$ vanish as $\nu$ and $X^\alpha_{,i}$
      are orthogonal to each other; as the background metric $\ol
      g_{\alpha\beta}=\delta_{\alpha\beta}$ is flat, covariant and
      partial derivatives of $\nu$ coincide)}
    =&\,-Fh_i^kX^\alpha_{,k}\delta_{\alpha\beta}X^\beta_{,j}
    -FX^\alpha_{,i}\delta_{\alpha\beta}h_j^kX^\beta_{,k} \intertext{(in
      view of the Weingarten equation \eqref{Weingarten equation})}
    =&\,-Fh_i^kg_{kj}-Fg_{ik}h_j^k \intertext{(by the definition of
      the metric)} =&\,-2Fh_{ij} \intertext{(by the definition of
      $h^i_j:=h_{jk}g^{ki}$).}
  \end{align*}
  The lemma follows.
\end{proof}

\begin{corollary}
  \stern
  The evolution equation of the volume element $d\mu:=\sqrt{\det
    g_{ij}}\,dx$ is given by
  \begin{equation}
    \label{eq:dmu.evol}
    \dt d\mu=-FH\,d\mu.
  \end{equation}
\end{corollary}
\begin{proof}
  Exercise. Recall the formulae for differentiating the determinant.
\end{proof}

\begin{lemma}
  The unit normal $\nu$ evolves according to 
  \begin{equation}
    \label{nu evol}
    \dt\nu^\alpha=g^{ij}F_{;\,i}X^\alpha_{;\,j}.
  \end{equation}
\end{lemma}
\begin{proof}
  By definition, the unit normal vector $\nu$ has length one,
  $\langle\nu,\nu\rangle=1=\nu^\alpha\delta_{\alpha\beta}\nu^\beta$.
  Differentiating yields
  \[0=\dot\nu^\alpha \delta_{\alpha\beta} \nu^\beta.\] Hence it
  suffices to show that the claimed equation is true if we take on
  both sides the scalar product with an arbitrary tangent vector. The
  vectors $X_{,i}$ (which we will also denote henceforth by $X_i$ as
  there is no danger of confusion; we will also use this convention
  if partial and covariant derivatives of some quantity coincide) form
  a basis of the tangent plane at a fixed point. We differentiate the
  relation
  \[0=\langle\nu,X_i\rangle=\nu^\alpha\delta_{\alpha\beta}X^\beta_i\]
  and obtain
  \begin{align*}
    0=&\,\dt\nu^\alpha\delta_{\alpha\beta}X^\beta_i
    +\nu^\alpha\delta_{\alpha\beta}\dt X^\beta_i\\
    =&\,\dt\nu^\alpha\delta_{\alpha\beta}X^\beta_i
    +\nu^\alpha\delta_{\alpha\beta}\left(\dt X^\beta\right)_i\\
    =&\,\dt\nu^\alpha\delta_{\alpha\beta}X^\beta_i
    -\nu^\alpha\delta_{\alpha\beta}\left(F\nu^\beta\right)_i.
  \end{align*}
  Hence
  \begin{align*}
    \dt\nu^\alpha\delta_{\alpha\beta}X^\beta_i
    =&\,\nu^\alpha\delta_{\alpha\beta}\nu^\beta F_i
    +F\nu^\alpha\delta_{\alpha\beta}\nu^\beta_i\\ 
    =&\,F_i+F\tfrac12\langle\nu,\nu\rangle_i=F_i
  \end{align*}
  and the lemma follows as taking the scalar product of the claimed
  evolution equation with $X_k$, i.\,e.\ multiplying it with
  $\delta_{\alpha\beta}X^\beta_k$, yields
  \[\dt\nu^\alpha\delta_{\alpha\beta}X^\beta_k
  =g^{ij}F_iX^\alpha_j\delta_{\alpha\beta}X^\beta_k =g^{ij}F_ig_{jk}
  =\delta^i_kF_i=F_k.\qedhere\]
\end{proof}

\begin{lemma}
  The second fundamental form $h_{ij}$ evolves according to 
  \begin{equation}
    \label{h evol}
    \dt h_{ij}=F_{;\,ij}-Fh_i^kh_{kj}.
  \end{equation}
\end{lemma}
\begin{proof}
  The Gau\ss{} formula \eqref{Gauss formula} implies that
  $h_{ij}=-X^\alpha_{;ij}\nu_\alpha$. Differentiating yields
  \begin{align*}
    \dt h_{ij}=&\,-\dt\langle X_{;ij},\nu\rangle\umbruch\\
    =&\,-\left\langle\dt X_{;ij},\nu\right\rangle
    -\left\langle-h_{ij}\nu,\dt\nu\right\rangle\umbruch\\
    =&\,-\left\langle\dt X_{;ij},\nu\right\rangle
    +h_{ij}\left\langle\nu,\dt\nu\right\rangle\umbruch\\
    =&\,-\left\langle\dt X_{;ij},\nu\right\rangle\umbruch\\
    =&\,-\dt\left(X^\alpha_{,ij}-\Gamma^k_{ij}X^\alpha_k\right)
    \nu_\alpha\umbruch\\
    =&\,-\left(\dt X^\alpha\right)_{,ij}\nu_\alpha
    +\Gamma^k_{ij}\left(\dt X^\alpha\right)_{,k}\nu_\alpha
    \intertext{(where no time derivatives of $\Gamma^k_{ij}$ show up
      as $X^\alpha_i\nu_\alpha=0$)}\umbruch\\
    =&\,(F\nu^\alpha)_{,ij}\nu_\alpha
    -\Gamma^k_{ij}(F\nu^\alpha)_{,k}\nu_\alpha \intertext{(in view of
      the evolution equation)}\umbruch\\
    =&\,F_{,ij}\nu^\alpha\nu_\alpha +F_{,i}\nu^\alpha_{,j}\nu_\alpha
    +F_{,j}\nu^\alpha_{,i}\nu_\alpha +F\nu_{,ij}^\alpha\nu_\alpha
    -\Gamma^k_{ij}F_{,k}\nu^\alpha\nu_\alpha
    -\Gamma^k_{ij}F\nu^\alpha_{,k}\nu_\alpha\umbruch\\
    =&\,F_{;ij}+F\nu^\alpha_{,ij}\nu_\alpha
  \end{align*}
  as $F_{;ij}=F_{,ij}-\Gamma^k_{ij}F_{,k}$ and
  $\nu^\alpha_{,j}\nu_\alpha =\frac12(\nu^\alpha\nu_\alpha)_j=0$. It
  remains to show that $\nu^\alpha_{,ij}\nu_\alpha=-h_i^kh_{kj}$. We
  obtain
  \begin{align*}
    \nu^\alpha_{,ij}\nu_\alpha=&\,\nu^\alpha_{;i,j}\nu_\alpha
    \intertext{(as $\nu^\alpha_i=\nu^\alpha_{;i}$)}\\
    =&\,\nu^a_{;ij}\nu_\alpha
    \intertext{($\nu^\alpha_{;ij}=(\nu^\alpha_{;i})_{,j}
      -\Gamma^k_{ij}\nu^\alpha_k$ and $0=\nu^\alpha_k\nu_\alpha$)}\\
    =&\,\left(h^k_iX^\alpha_k\right)_{;j}\nu_\alpha
    \intertext{(according to the Weingarten equation \eqref{Weingarten
        equation})}\\
    =&\,h^k_i(-h_{kj}\nu^\alpha)\nu_\alpha \intertext{(due to the
      Gau\ss{} equation \eqref{Gauss formula} and the orthogonality
      $X^\alpha_k\nu_\alpha=0$)}\\
    =&\,-h^k_ih_{kj}
  \end{align*}
  as claimed. The Lemma follows.
\end{proof}

\begin{lemma}
  The normal velocity $F$ evolves according to
  \begin{equation}
    \label{F evol}
    \dt F-F^{ij}F_{;ij}=FF^{ij}h_i^kh_{kj}.
  \end{equation}
\end{lemma}

% \begin{remark}
%   For mean curvature flow, we have $F=H=g^{ij}h_{ij}$ and deduce that 
%   \begin{align*}
%     F^{ij}&\,\fracp{H}{h_{ij}} =g^{ij},\umbruch\\
%     \fracp{H}{g_{ij}}=&\,-g^{ik}h_{kl}g^{lj} =-g^{ik}h_k^j
%     \intertext{so that}
%     \dt H-g^{ij}H_{;ij}=&\,\dt H-\Delta H=H|A|^2. 
%   \end{align*}
% \end{remark}

\begin{proof}%[Proof of Lemma \ref{F evol}]
  We have, see \cite[Lemma 5.4]{OSMathZ}, the proof of \cite[Theorem
  2.1.20]{CGCPBook}, or check this explicitly for the normal velocity
  considered,
  $$\fracp{F}{g_{kl}}=-F^{il}h^k_i$$
  and compute the evolution equation of the normal velocity $F$
  \begin{align*}
    \dt F-F^{ij}F_{;ij}=&-F^{il}h^k_i\dt g_{kl}
    +F^{ij}\dt h_{ij}-F^{ij}F_{;\,ij}\umbruch\\
    =&FF^{ij}h^k_ih_{kj},
  \end{align*}
  where we used \eqref{g evol} and \eqref{h evol}.
\end{proof}

We will need more explicit evolution equations for geometric
quantities $\boxplus$ involving $\dt\boxplus-F^{ij}\boxplus_{;ij}$.
\begin{lemma}
  The second fundamental form $h_{ij}$ evolves according to
  \begin{equation}
    \label{2ff evol}
    \begin{split}
      \dt h_{ij}-F^{kl}h_{ij;\,kl}=&\,F^{kl}h^a_kh_{al}\cdot h_{ij}
      -F^{kl}h_{kl}\cdot h^a_ih_{aj}\\
      &\,-Fh^k_ih_{kj}+F^{kl,\,rs}h_{kl;\,i}h_{rs;\,j}.
    \end{split}
  \end{equation}
\end{lemma}
\begin{proof}
  Direct calculations yield
  \begin{align*}
    \dt h_{ij}-F^{ij}h_{ij;\,kl}
    =&\,F_{;ij}-Fh^k_ih_{kj}-F^{ij}h_{ij;kl}&&
    \text{by \eqref{h evol}}\umbruch\\
    =&\,F^{kl}h_{kl;\,ij}
    +F^{kl,\,rs}h_{kl;\,i}h_{rs;\,j}&&\\
    &\,-Fh^k_ih_{kj}-F^{ij}h_{ij;\,kl}&&\umbruch\\
    =&\,F^{kl}h_{ik;\,lj}
    +F^{kl,\,rs}h_{kl;\,i}h_{rs;\,j}&&\\
    &\,-Fh^k_ih_{kj}-F^{ij}h_{ik;\,jl}&&\text{by Codazzi}\umbruch\\
    =&\,F^{kl}\left(h^a_kR_{ailj}+h^a_iR_{aklj}\right)
    -Fh^k_ih_{kj}\\
    &\,+F^{kl,\,rs}h_{kl;\,i}h_{rs;\,j}
    &&\text{by \eqref{Ricci}}\umbruch\\
    =&\,F^{kl}h^a_kh_{al}h_{ij}-F^{kl}h^a_kh_{aj}h_{il}\\
    &\,+F^{kl}h^a_ih_{al}h_{kj}-F^{kl}h^a_ih_{aj}h_{kl}\\
    &\,-Fh^k_ih_{kj}+F^{kl,\,rs}h_{kl;\,i}h_{rs;\,j}&&
    \text{by \eqref{Riem}}\umbruch\\
    =&\,F^{kl}h^a_kh_{al}h_{ij}-F^{kl}h^a_ih_{aj}h_{kl}\\
    &\,-Fh^k_ih_{kj}+F^{kl,\,rs}h_{kl;\,i}h_{rs;\,j}.&&
    \qedhere
  \end{align*}
\end{proof}

\begin{remark}
  A direct consequence of \eqref{X dot} and \eqref{Gauss formula} is
  \begin{align}
    \dt X^\alpha-F^{ij}X^\alpha_{;\,ij}
    =&\left(F^{ij}h_{ij}-F\right)\nu^\alpha.\umbruch
    \label{X evol}\\
    \intertext{Hence}
    \dt\abs{X}^2-F^{ij}\left(\abs{X}^2\right)_{;ij}
    =&2\left(F^{ij}h_{ij}-F\right)\langle X,\,\nu\rangle
    -2F^{ij}g_{ij}.\nonumber
  \end{align}
\end{remark}
\begin{proof}
  \stern We have
  \begin{align*}
    \dt|X|^2-F^{ij}\left(|X|^2\right)_{;ij}=&\, 2\left\langle X,\dt
      X\right\rangle-2F^{ij}\langle X_i,X_j\rangle
    -2F^{ij}\langle X,X_{;ij}\rangle\\
    =&\,2\langle X,-F\nu\rangle-2F^{ij}g_{ij} -2F^{ij}\langle
    X,-h_{ij}\nu\rangle.\qedhere
  \end{align*}
\end{proof}

\begin{lemma}
  The evolution equation for the unit normal $\nu$ is
  \begin{equation}\label{nu evol ganz}
    \dt\nu^\alpha-F^{ij}\nu^\alpha_{;ij}
    =F^{ij}h^k_ih_{kj}\cdot\nu^\alpha. 
  \end{equation}
\end{lemma}
\begin{proof}
  We compute
  \begin{align*}
    \dt\nu^\alpha-F^{ij}\nu^\alpha_{;ij} =&g^{ij}F_{;\,i}X^\alpha_{;\,j}
    -F^{ij}\left(h^k_iX^\alpha_{;\,k}\right)_{;\,j}
    &&\text{by \eqref{nu evol} and \eqref{Weingarten equation}}\umbruch\\
    =&g^{ij}F^{kl}h_{kl;\,i}X^\alpha_{;\,j}
    -F^{ij}h^k_{i;\,j}X^\alpha_{;\,k}
    -F^{ij}h^k_iX^\alpha_{;\,kj}&&\umbruch\\ 
    =&F^{ij}h^k_ih_{kj}\nu^\alpha&&\text{by \eqref{Gauss formula}.}
    \qedhere
  \end{align*}
\end{proof}

\begin{lemma}
  \stern The evolution equation for the scalar product $\langle
  X,\nu\rangle$ is
  \begin{equation}\label{X nu evol ganz}
    \dt\langle X,\nu\rangle-F^{ij}\langle X,\nu\rangle_{;ij}
    =-F^{ij}h_{ij}-F+F^{ij}h^k_ih_{kj}
    \langle X,\,\nu\rangle.
  \end{equation}
\end{lemma}
\begin{proof}
  Exercise.
%   We obtain
%   \begin{align*}
%     \dt\langle X,\,\nu\rangle-F^{ij}\langle X,\,\nu\rangle_{;ij}=&
%     X^\alpha\delta_{\alpha\beta}
%     \left(\dt\nu^\beta-F^{ij}\nu^\alpha_{;ij}\right)\\
%     &+\left(\dt
%       X^\alpha-F^{ij}X^\alpha_{;\,ij}\right)\delta_{\alpha\beta}
%     \nu^\beta\\
%     &-2F^{ij}X^\alpha_{;\,i}\delta_{\alpha\beta}
%     \nu^{\beta}_{;\,j}\umbruch\\ 
%     =&F^{ij}h^k_ih_{kj}\langle X,\,\nu\rangle
%     +\left(F^{ij}h_{ij}-F\right)\langle\nu,\,\nu\rangle\\
%     &-2F^{ij}X^\alpha_{;\,i}\delta_{\alpha\beta}h^k_jX^\beta_{;\,k}\\
%     \intertext{by \eqref{Weingarten equation}, \eqref{X evol},
%       and \eqref{nu evol ganz}}\\
%     =&F^{ij}h^k_ih_{kj}\langle X,\,\nu\rangle-F^{ij}h_{ij}-F.
%     \qedhere
%   \end{align*}
\end{proof}

\begin{lemma}
  Let $\eta_\alpha=(-e_{n+1})_\alpha=(0,\ldots,0,-1)$. Then $\tilde
  v:=\langle\eta,\nu\rangle\equiv\eta_\alpha\nu^\alpha$ fulfills
  \begin{align}
    \dt\tilde v-F^{ij}\tilde v_{;ij}=&\,F^{ij}h^k_ih_{kj}\tilde v
    \label{v tilde evol}
    \intertext{and $v:=\tilde v^{-1}$ fulfills}
    \dt v-F^{ij}v_{;ij}=&\,-vF^{ij}h^k_ih_{kj}-2\frac1vF^{ij}v_iv_j.
    \label{v evol}
  \end{align}
\end{lemma}
\begin{proof}
  The evolution equation for $\tilde v$ is a direct consequence of
  \eqref{nu evol ganz}. For the proof of the evolution equation of $v$
  observe that 
  \begin{align*}
    v_i=&\,-\tilde v^{-2}\tilde v_i=-v^2\tilde v_i\\
    \intertext{and} v_{;ij}=&\,-\tilde v^{-2}\tilde v_{;ij}+2\tilde
    v^{-3}\tilde v_i\tilde v_j=-v^2\tilde v_{;ij} +2v^{-1}v_iv_j.
    \qedhere
  \end{align*}
\end{proof}

\section{Mean curvature flow of entire graphs} 
For mean curvature flow of entire graphs, K. Ecker and G. Huisken
proved the following existence theorem \cite[Theorem
5.1]{EckerHuiskenInvent}
\begin{theorem}\label{thm:entire-mcf}
  Let $u_0:\R^n\to\R$ be locally Lipschitz continuous. Then there
  exists a function $u\in
  C^\infty\left(\R^n\times(0,\infty)\right)\cap
  C^0\left(\R^n\times[0,\infty)\right)$ solving
  \[\begin{cases}
    \dot u=\sqrt{1+|Du|^2}\divergenz
    \left(\frac{Du}{\sqrt{1+|Du|^2}}\right) &\text{in
    }\R^n\times(0,\infty),\\
    u(\cdot,0)= u_0 &\text{in }\R^n.
  \end{cases}\]
\end{theorem}

The key ingredient in the existence proof is the following localised
gradient estimate.
\begin{theorem}\label{mcf gradient bound}
  Let $u:B_R(0)\times[0,T]\to\R$ be a smooth solution to graphical
  mean curvature flow. Then 
  \[\sqrt{1+|Du|^2(0,t)}\le c(n)\sup\limits_{B_R(0)}
  \sqrt{1+|Du|^2(\cdot,0)}\cdot
  \exp\left(c(n)\,R^{-2}\left(\osc\limits_{B_R(0)\times[0,T]}
      u\right)^2\right).\]
\end{theorem}
We do not prove this Theorem in this course. However, if we
additionally assume that $u(x,0)\to\infty$ as $|x|\to\infty$, Theorem
\ref{C1 est thm}, that is much easier to prove, can be used instead of
Theorem \ref{mcf gradient bound}.

Theorem \ref{thm:entire-mcf} has been extended to continuous initial
data by J. Clutterbuck \cite{JulieParabolicContinuous} and T. Colding
and W. Minicozzi \cite{ColdingMinicozzi}.

If $u$ is initially close to a cone in an appropriate sense, graphical
mean curvature flow converges, as $t\to\infty$, after appropriate
rescaling, to a self-similarly expanding solution ``coming out of a
cone'', see the papers by K. Ecker and G. Huisken
\cite{EckerHuiskenInvent} and N. Stavrou \cite{StavrouSelfSim}.

Stability of translating solutions to graphical mean curvature flow
without rescaling is considered in \cite{JCOSFSMCFStability}.

\section{Mean curvature flow of complete graphs}
The material in this section is based on joint work with M. S\'aez,
see \cite{OSMarielMCFwithoutSing}. Have a look at the article for
illustrations.

\subsection{Intuition}

\begin{remark}
  \neueZeile
  \begin{enumerate}[(i)]
  \item Long time existence for entire graphs was shown before by
    K. Ecker and G. Huisken \cite{EckerHuiskenInvent}, see Theorem
    \ref{thm:entire-mcf}. 
  \item We wish to study the evolution of complete graphs defined on
    subsets of Euclidean space $\R^{n+1}$. The additional dimension is
    related to Theorem \ref{weak flow thm intro}. 
  \item We assume for the moment that such initial data have smooth
    solutions. Then the following figures should give some intuition
    about the behaviour of these solutions.
    \begin{enumerate}[a)]
    \item A rotationally symmetric solution defined on a ball: Figure
      \fbox{1} shows a rotationally symmetric graph in $\R^{n+2}$
      defined on a ball in $\R^{n+1}$. A cylinder over the boundary of
      the ball encloses this graph. Asymptotically, these two
      hypersurfaces coincide as $x^{n+2}\to\infty$. Under mean
      curvature flow, the cylinder in $\R^{n+2}$ collapses to a line
      in finite time. The sphere in $\R^{n+1}$ collapses to a point in
      finite time. As the principal curvatures of any cylinder
      $M^n_t\times\R$ are $\lambda_1,\ldots,\lambda_n,0$, where
      $\lambda_1,\ldots,\lambda_n$ are the principal curvatures of
      $M^n_t$, the projection of the evolving cylinder coincides at
      all times with the evolving sphere. \par The evolution of the
      graph stays graphical and asymptotic to the evolving cylinder as
      $x^{n+2}\to\infty$. As the curvature near the tip is larger than
      that of the cylinder, the tip moves faster and moves up to
      infinity at precisely the time when the cylinder collapses to a
      line. Thus for all times, the boundary of the projections of the
      graphs coincides with the evolving spheres and hence fulfills
      mean curvature flow.
    \item A solution initially defined on a domain that will form a
      neckpinch under mean curvature flow: In Figure \fbox{2}, the
      graph is initially defined over a domain whose boundary will
      develop a neckpinch in finite time, i.\,e.{} the thin neck will
      collapse. There are methods to continue the flow past this
      neckpinch singularity. After this singularity, the hypersurface
      splits into two topologically spherical components. Once again,
      the evolution of the graph above is such that the boundary of
      its projection or, equivalently, of the domain of definition of
      the graph, fulfills mean curvature flow. This happens as
      follows: As the neckpinch singularity forms downstairs, the mean
      curvature in $\R^{n+1}$ blows up. Meanwhile, above the neck
      region in $\R^{n+2}$, the mean curvature becomes even larger so
      that the graph over the neck region moves to infinity while the
      rest of the graph remains finite.  Then the graph separates into
      two disjoint components.
    \item A solution initially defined on an annulus: In Figure
      \fbox{3}, the domain of definition is an annulus. Its boundary
      consists of two disjoint spheres that disappear at different
      times. The graph above is asymptotic to two cylinders as
      $x^{n+2}\to\infty$. When the inner cylinder collapses, a ``cap
      at infinity'' is added to the graph and its topology
      changes. Similarly to the example of a contracting sphere, this
      cap can travel in finite time from infinity downwards and become
      visible. Later, the situation is similar to that of Figure
      \fbox{1}. 
    \item A solution defined on a domain in the plane bounded by
      possibly countably many disjoint curves: For a planar domain
      with finitely many holes, see Figure \fbox{3}, there are
      finitely many times, where boundary components shrink to points
      and vanish. At those times, caps at infinity are added to the
      graphical solution similarly to the annulus situation
      above. \par Finally, if a planar domain has countably many
      holes, we can arrange so that the holes disappear on a dense set
      of times. We get a smoothly evolving graph whose mean curvature
      is unbounded at all times.
    \end{enumerate}
  \end{enumerate}
\end{remark}

\subsection{Results}
Let us consider mean curvature flow for graphs defined on a relatively
open set
\begin{equation}
  \label{set eq}
  \Omega\equiv\bigcup\limits_{t\ge0}\Omega_t\times\{t\}
  \subset\R^{n+1}\times[0,\infty).
\end{equation}
Our existence result for bounded domains is
\begin{theorem}[Existence]
  \label{exist thm intro}
  Let $A\subset\R^{n+1}$ be a bounded open set and $u_0\colon A\to\R$
  a locally Lipschitz continuous function with $u_0(x)\to\infty$ for
  $x\to x_0\in\partial A$.\par
  Then there exists $(\Omega,u)$, where
  $\Omega\subset\R^{n+1}\times[0,\infty)$ is relatively open, such
  that $u$ solves graphical mean curvature flow
  \[\dot u=\sqrt{1+|Du|^2}\cdot \divergenz\left(\frac{Du}
    {\sqrt{1+|Du|^2}}\right)\quad\text{in }\Omega\cap\{t>0\},\] $u$ is
  smooth for $t>0$ and continuous up to $t=0$, $\Omega_0=A$,
  $u(\cdot,0)=u_0$ in $A$ and $u(x,t)\to\infty$ as
  $(x,t)\to(x_0,t_0)\in\partial\Omega$, where $\partial\Omega$ is the
  relative
  boundary of $\Omega$ in $\R^{n+1}\times[0,\infty)$.
\end{theorem}

Such smooth solutions yield weak solutions to mean curvature flow. We
have 
\begin{theorem}[Weak flow]
  \stern
  \label{weak flow thm intro}
  Let $(A,u_0)$ and $(\Omega,u)$ be as in Theorem \ref{exist thm
    intro}. Let $\partial\mathcal D_t$ be the level set evolution of
  $\partial\Omega_0$ with $\mathcal D_0=\Omega_0$. If $\partial
  \mathcal D_t$ does not fatten, the measure theoretic boundaries
  of $\Omega_t$ and $\mathcal D_t$ coincide for every $t\ge0$.
\end{theorem}
Here, $\mathcal D_t=\left\{x\in\R^{n+1}\colon w(x,t)<0\right\}$ and
$w$ solves $\dot w=|Dw|\cdot\divergenz\left(\frac{Dw}{|Dw|}\right)$ as
in Remark \ref{lsf rem}. The equation is solved in the viscosity
sense, see e.\,g.{} \cite{ChenGigaGotolevelset,evanssprucklevelset1}
for more details.
  
\subsection{Strategy of proof}

\begin{proof}[Strategy of the proof of Theorem \ref{exist thm intro}] 
  \neueZeile
  \begin{enumerate}[(i)]
  \item Fix $L>0$. Then there exists a solution with initial value
    $\min\{u_0,L\}$ for all $t\in[0,\infty]$, see
    \cite{EckerHuiskenInvent}. 
  \item If $L_1<L$, we prove a priori estimates for the part of the
    evolving graphs which is below $L_1$. This is done in Theorem
    \ref{C1 est thm} for the (spatial) first order derivatives of
    $u$. See Theorem \ref{C2 est thm} for the second derivative
    bounds. Similar techniques imply bounds for all higher
    derivatives.
  \item We let $L\to\infty$ and use a variant of the Theorem of
    Arzel\`a-Ascoli to pass to a subsequence which is our
    solution. \qedhere
  \end{enumerate}
\end{proof}

\begin{proof}[Sketch of the strategy of the proof of Theorem \ref{weak
    flow thm intro}]
  \neueZeilealt
  In the following sketch of a proof we try to give an idea of the
  argument without mentioning technical details, e.\,g.{}
  approximations or fattening. None of the steps
  works exactly as described below. 
  \begin{enumerate}[(i)]
  \item The constructed solution corresponds to a level-set solution.
  \item The level-set solution starting from $\partial A\times\R$ is
    an outer barrier to the graphical solution $\graph
    u(\cdot,t)$. Observe that $\Omega_t$ is the projection of the
    evolving graph at time $t$ to $\R^{n+1}$. Hence $\Omega_t$ is
    contained in the level-set evolution of $A$.
  \item By shifting downwards the level set solution, we obtain
    convergence to the level set solution starting with the cylinder
    $\partial A\times\R$. This prevents $\graph u(\cdot,t)$ from
    detaching near infinity from the evolution of the cylinder.    
    \qedhere
  \end{enumerate}
\end{proof}

\subsection{The a priori estimates}

Recall the definition $v=\sqrt{1+|Du|^2}$, where we consider $u$ as a
function defined on some subset of $\R^{n+1}\times[0,\infty)$.
                                                                              
Let $\eta:=(\eta_\alpha)=(0,\ldots,0,1)$. In the following, whenever
quantities like $v$ or $|A|^2$ are involved, we consider $u$ and $v$
as functions on the evolving hypersurfaces rather than as functions
depending on $(x,t)\in\R^{n+1}\times[0,\infty)$, i.\,e.{} we consider
$u:=X^\alpha\eta_\alpha$ and $v:=-\langle\nu,\eta\rangle^{-1}$.

\begin{theorem}\label{evol eq thm}
  Let $X$ be a solution to mean curvature flow. Then we have the
  following evolution equations. 
  \begin{align*}
    \heat u=&\,0,\umbruch\\
    \heat v=&\,-|A|^2v-\tfrac2v|\nabla v|^2,\umbruch\\
    \heat |A|^2=&\,-2|\nabla A|^2+2|A|^4,\umbruch\\
%    \heat \left|\nabla^mA\right|^2
%    \le&\,-2\left|\nabla^{m+1}A\right|^2\\
%    &\,\quad +c(m,n)\cdot\sum\limits_{i+j+k=m}
%    \left|\nabla^mA\right|\cdot \left|\nabla^iA\right|\cdot
%    \left|\nabla^jA\right|\cdot \left|\nabla^kA\right|,\umbruch\\
    \heat \g\le&\,-2k\cdot\g^2 -2\phi v^{-3}\langle\nabla
    v,\nabla \g\rangle,
  \end{align*}
  where $\g =\phi|A|^2 \equiv\frac{v^2}{1-kv^2}|A|^2$ and $k>0$ is
  chosen so that $kv^2\le \frac12$ in the domain considered.
\end{theorem}
\begin{proof}
  We leave it to the reader to prove the evolution equations for $u$,
  $v$ and $|A|^2$.  For the evolution equation of $\g$, see
  \cite{EckerHuiskenInvent,KEMCFBook}. 
\end{proof}

\begin{assumption}\label{est ass}
  For the proof of the a priori estimates, we will assume that
  $u\colon\R^{n+1}\times[0,\infty)$ is a smooth solution to mean
  curvature flow such that \[\{x\colon u(x)\le 0\}\subset B_R(0)\] for 
  some $R>0$. In order to be able to consider smooth solutions, a few
  extra constructions are necessary. 
\end{assumption}

\begin{theorem}[$C^1$-estimates]
  \label{C1 est thm}
  Let $u$ be as in Assumption \ref{est ass}. Then
  \[vu^2\le\max\limits_{\genfrac{}{}{0pt}{}{t=0}{\{u<0\}}}vu^2\]
  at points where $u<0$.
\end{theorem}
Here and in the following, it is often possible to increase the
exponent of $u$. 
\begin{proof}
  Exercise. Consider also $v(-u)$. 
  % According to Theorem \ref{evol eq thm}, $w:=vu^2$ fulfills
  % \begin{align*}
  %   \dot w=&\,\dot vu^2+2vu\dot u,\umbruch\\
  %   w_i=&\,v_iu^2+2vuu_i,\umbruch\\
  %   w_{ij}=&\,v_{ij}u^2+2vuu_{ij} +2vu_iu_j
  %   +2u(v_iu_j+v_ju_i),\umbruch\\ 
  %   \heat w=&\,u^2\heat v
  %   -2v|\nabla u|^2-4u\langle\nabla v,\nabla u\rangle\\
  %   =&\,u^2\left(-v|A|^2-\tfrac2v|\nabla v|^2\right)
  %   -2v|\nabla u|^2 -4\left\langle\tfrac u{\sqrt v}\nabla v,\sqrt
  %     v\nabla u\right\rangle\umbruch\\
  %   \le&\,-u^2v|A|^2\le0.
  % \end{align*}
  % The estimate follows from the maximum principle applied to $w$ in
  % the domain where $u<0$. 
\end{proof}

\begin{remark}
  We recommend to consider Theorem \ref{C1 est thm} as an estimate for
  $v(-u)^2$. 
\end{remark}

\begin{corollary}\label{C1 cor}
  Let $u$ be as in Assumption \ref{est ass}. Then
  \[v\le\max\limits_{\genfrac{}{}{0pt}{}{t=0}{\{u<0\}}}vu^2\] 
  at points where $u\le-1$. 
\end{corollary}

\begin{remark}
  Corollaries similar to Corollary \ref{C1 cor} also hold for the
  following a priori estimates for points with $u\le-\epsilon<0$ or
  $t\ge\epsilon>0$. We do not write them down explicitly. \par
  In Theorem \ref{C1 est thm} and later, we may replace every $u$ by
  $u-h$ for any constant $h$.
\end{remark}

\begin{remark}\label{nabla u v est rem}
  \stern For later use, we estimate derivatives of $u$ and $v$,
  \begin{align*}
    |\nabla u|^2=&\,\eta_\alpha X^\alpha_ig^{ij}X^\beta_j\eta_\beta
    =\eta_\alpha\left(\delta^{\alpha\beta}
      -\nu^\alpha\nu^\beta\right)\eta_\beta =1-v^{-2}\le 1
    \intertext{and, according to \eqref{Weingarten equation},} |\nabla
    v|^2=&\,\left(\left(-\eta_\alpha\nu^\alpha\right)^{-1}\right)_i
    g^{ij}\left(\left(-\eta_\beta\nu^\beta\right)^{-1}\right)_j
    =v^4\eta_\alpha
    X^\alpha_kh^k_ig^{ij} h^l_jX^\beta_l\eta_\beta \le v^4|A|^2\\
    \le&\,v^2\phi|A|^2 =v^2\g.  \intertext{So we get} |\langle\nabla
    u,\nabla v\rangle| \le&\,|\nabla u|\cdot|\nabla v| \le v^2|A|\le
    v\sqrt{\g}.
  \end{align*}
\end{remark}

\begin{theorem}[$C^2$-estimates]
  \label{C2 est thm}
  \stern Let $u$ be as in Assumption \ref{est ass}.
  \begin{enumerate}[(i)]
  \item Then there exist $\lambda>0$, $c>0$ and $k>0$ (the constant in
    $\phi$ and implicitly in $g$), depending on the $C^1$-estimates,
    such that
    \[tu^4\g+\lambda
    u^2v^2\le\sup\limits_{\genfrac{}{}{0pt}{}{t=0}{\{u<0\}}} \lambda
    u^2v^2 +ct\] at points where $u<0$ and $0<t\le 1$. 
  \item Moreover, if $u$ is in $C^2$ initially, we get $C^2$-estimates
    up to $t=0$: Then there exists $c>0$, depending only on the
    $C^1$-estimates, such
    that \[u^4\g\le\sup\limits_{\genfrac{}{}{0pt}{}{t=0}{\{u<0\}}}u^4\g
    +ct\] at points where $u<0$.
  \end{enumerate}
\end{theorem}
\begin{proof}
  In order to prove both parts simultaneously, we set \[w:=(\mu
  t+(1-\mu))u^4\g+\lambda u^2v^2 \equiv\mu_tu^4\g+\lambda u^2v^2.\] If
  we set $\mu=1$, we obtain $\mu_t=t$ and later the first claim, if
  $\mu=\lambda=0$, we get $\mu_t=1$ and deduce in the following the
  second claim. We calculate
  \begin{align*}
    \dot w=&\,\mu u^4\g +4\mu_tu^3\g\dot
    u+\mu_tu^4\dot \g +2\lambda
    v^2u\dot u +2\lambda u^2v\dot v,\umbruch\\
    w_i=&\,4\mu_tu^3\g u_i +\mu_tu^4\g_i +2\lambda v^2uu_i
    +2\lambda
    u^2vv_i,\umbruch\\
    w_{ij}=&\,4\mu_tu^3\g u_{ij} +\mu_tu^4\g_{ij} +2\lambda
    v^2uu_{ij} +2\lambda u^2vv_{ij} +12\mu_tu^2\g u_iu_j\\
    &\,+4\mu_tu^3(\g_iu_j+\g_ju_i) +2\lambda v^2u_iu_j +2\lambda
    u^2v_iv_j +4\lambda
    vu(u_iv_j+u_jv_i),\umbruch\\
    \mu_tu^3\nabla\g=&\,\frac1u\nabla w -4\mu_tu^2\g\nabla u
    -2\lambda
    v^2\nabla u -2\lambda uv\nabla v,\umbruch\\
    \heat w\le&\,\mu{u^4\g} +\mu_tu^4\left(-2k\g^2-2\phi
      v^{-3}\langle\nabla v,\nabla\g\rangle\right) +2\lambda
    u^2v\left(-|A|^2v -\tfrac2v|\nabla v|^2\right)\\
    &\,-12\mu_tu^2\g|\nabla u|^2
    -8\mu_tu^3\langle\nabla\g,\nabla u\rangle -2\lambda v^2|\nabla
    u|^2 -2\lambda u^2|\nabla v|^2\\ &\,-8\lambda uv\langle\nabla
    u,\nabla v\rangle.
  \end{align*}
  In the following, we will use the notation $\langle\nabla
  w,b\rangle$ with a generic vector $b$. The constants $c$ are allowed
  to depend on $\sup\{|u|\colon u<0\}$ (which does not exceed its
  initial value) and the $C^1$-estimates.  It may also depend on an
  upper bound for $t$, but we assume that $0<t\le 1$ whenever $t$
  appears explicitly. I.\,e.,{} we suppress dependence on already
  estimated quantities. \par We estimate the terms involving
  $\nabla\g$ separately. Let $\epsilon>0$ be a constant. We fix its
  value below. Using Remark \ref{nabla u v est rem} for estimating
  terms, we get
  \begin{align*}
    -2\phi\mu_tu^4v^{-3}\langle\nabla v,\nabla\g\rangle
    =&\,-2\frac{\phi u}{v^3}\left\langle\nabla v,\frac1u\nabla
      w-4\mu_tu^2\g\nabla u-2\lambda v^2\nabla u-2\lambda uv\nabla
      v\right\rangle\umbruch\\
    \le&\,\langle\nabla w,b\rangle +8\mu_t\frac{\phi |u|^3}v\g|A|
    +4\lambda\phi v|u||A| +4\frac{\lambda\phi u^2}{v^2} |\nabla
    v|^2\umbruch\\
    =&\,\langle\nabla w,b\rangle
    +8\mu_t\phi^2\frac{|u|^3\g^{3/2}}{\phi^{3/2}} \frac1v
    +4\lambda\phi v|u||A| +\lambda
    u^2|\nabla v|^2\cdot4\frac\phi{v^2}\umbruch\\
    \le&\,\langle\nabla w,b\rangle +\epsilon\mu_tu^4\g^2
    +\epsilon\lambda u^2v^2|A|^2 +\lambda u^2|\nabla v|^2\cdot
    4\frac{\phi}{v^2}\\ &\,+c(\epsilon,\lambda),\umbruch\\
    -8\mu_tu^3\langle\nabla\g,\nabla u\rangle =&\,-8\left\langle\nabla
      u,\frac1u\nabla w -4\mu_tu^2\g\nabla u -2\lambda v^2\nabla u
      -2\lambda uv\nabla v\right\rangle\umbruch\\
    \le&\,\langle\nabla w,b\rangle +32\mu_tu^2\g+16\lambda
    v^2+16\lambda |u|v^3|A|\umbruch\\
    \le&\,\langle\nabla w,b\rangle +\epsilon\mu_tu^4\g^2
    +\epsilon\lambda u^2v^2|A|^2 +c(\epsilon,\lambda).
  \end{align*}
  We obtain 
  \begin{align*}
    \heat w\le&\,\mu{u^4\g} +\mu_tu^4\g^2(-2k+2\epsilon)
    +\langle\nabla w,b\rangle\\ &\,+\lambda u^2v^2|A|^2(-2+3\epsilon)
    +\lambda u^2|\nabla v|^2\left(4\frac\phi{v^2}-6\right)
    +c(\epsilon,\lambda).
  \end{align*}
  Let us assume that $k>0$ is chosen so small that $kv^2\le\frac13$ in
  $\{u<0\}$. This implies $\phi\le2v^2$.  We may assume that
  $\lambda\ge 2u^2$ in $\{u<0\}$ and get $\mu{u^4\g}\le\frac12\lambda
  u^2\phi|A|^2 \le\lambda u^2v^2|A|^2$. We get \[4\frac\phi{v^2}-6
  =\frac4{1-kv^2} -6\le 0.\] Finally, fixing $\epsilon>0$ sufficiently
  small, we obtain \[\heat w\le\langle\nabla w,b\rangle +c.\] Now,
  both claims follow from the maximum principle.
\end{proof}

\begin{appendix}
  \section{Parabolic maximum principles}
  \label{sec:parab-maxim-princ}
  The following maximum principle is fairly standard. For non-compact,
  strict or other maximum principles, we refer to
  \cite{EckerHuiskenInvent} or \cite{ProtterWeinberger}, respectively.

  We will use $C^{2;1}$ for the space of functions that are two times
  continuously differentiable with respect to the space variables and
  once continuously differentiable with respect to the time variable.
  \begin{theorem}[Weak parabolic maximum principle]
    \label{thm:parab-maxim-princ}
    Let $\Omega\subset\R^n$ be open and bounded and $T>0$. Let
    $a^{ij}$, $b^i\in L^\infty(\Omega\times[0,T])$. Let $a^{ij}$ be
    strictly elliptic, i.\,e.\ $a^{ij}(x,t)>0$ in the sense of
    matrices. Let $u\in C^{2;1}(\Omega\times[0,T))\times
    C^0\left(\ol\Omega\times[0,T]\right)$ fulfill
    \[\dot u\le a^{ij}u_{ij}+b^iu_i\quad\text{in
    }\Omega\times(0,T).\] Then we get for $(x,t)\in\Omega\times(0,T)$
    \[u(x,t)\le\sup\limits_{\mathcal P\left(\Omega\times(0,T)\right)}
    u,\] where $\mathcal
    P\left(\Omega\times(0,T)\right):=(\Omega\times\{0\})
    \cup(\partial\Omega\times(0,T))$.
  \end{theorem}
  \begin{proof}\neueZeile
    \begin{enumerate}[(i)]
    \item Let us assume first that $\dot u<a^{ij}u_{ij}+b^iu_i$ in
      $\Omega\times(0,T)$.  If there exists a point
      $(x_0,t_0)\in\Omega\times(0,T)$ such that
      $u(x_0,t_0)>\sup\limits_{\mathcal
        P\left(\Omega\times(0,T)\right)} u$, we find
      $(x_1,t_1)\in\Omega\times(0,T)$ and $t_1$ minimal such that
      $u(x_1,t_1)=u(x_0,t_0)$. At $(x_1,t_1)$, we have $\dot u\ge0$,
      $u_i=0$ for all $1\le i\le n$, and $u_{ij}\le 0$ (in the sense
      of matrices). This, however, is impossible in view of the
      evolution equation.
    \item Define for $0<\epsilon$ the function $v:=u-\epsilon t$. It
      fulfills the differential inequality
      \[\dot v=\dot u-\epsilon<\dot u\le a^{ij}u_{ij}+b^iu_i
      =a^{ij}v_{ij}+b^iv_i.\]
      Hence, by the previous considerations,
      \[u(x,t)-\epsilon t=v(x,t)\le\sup\limits_{\mathcal
        P(\Omega\times(0,T))} v =\sup\limits_{\mathcal
        P(\Omega\times(0,T))} u-\epsilon t\] and the result follows
      as $\epsilon\searrow0$.\qedhere
    \end{enumerate}
  \end{proof}

  \section{Some linear algebra}
  \label{sec:some-linear-algebra}
  \begin{lemma}
    We have
    $$\frac\partial{\partial a_{ij}}\det(a_{rs})
    =\det(a_{rs})a^{ji},$$ if $a_{ij}$ is invertible with inverse
    $a^{ij}$, i.\,e. if $a^{ij}a_{jk}=\delta^i_k$.
  \end{lemma}
  We skip the proof and point out that in two dimensions, this follows
  directly from \[\det
  \begin{pmatrix}
    a&b\\ c&d
  \end{pmatrix}
  =ad-bc.\]

  \begin{lemma}
    Let $a_{ij}(t)$ be differentiable in $t$ with inverse $a^{ij}(t)$.
    Then
    $$\dt a^{ij}=-a^{ik}a^{lj}\dt a_{kl}.$$
  \end{lemma}
  \begin{proof}
    We have
    $$a^{ik}a_{kj}=\delta^i_j.$$
    Assume that there exists $\tilde a^{ij}$ such that
    $$a_{ik}\tilde a^{kj}=\delta^j_i.$$
    Then $a^{ij}=\tilde a^{ij}$, as
    \[a^{ij}=a^{ik}\delta^j_k=a^{ik}\left(a_{kl}\tilde a^{lj}\right)
    =\left(a^{ik}a_{kl}\right)\tilde a^{lj}=\tilde a^{ij}.\] We
    differentiate and obtain
    \[0=\dt\delta^i_j=\dt\left(a^{ik}a_{kj}\right) =\dt
    a^{ik}a_{kj}+a^{ik}\dt a_{kj}.\] Hence
    \[\dt a^{il}=\dt a^{ik}\delta^l_k
    =\dt a^{ik}a_{kj}a^{jl} =-a^{ik}\dt a_{kj}a^{jl}.\]
  \end{proof}

\section{Exercises}
\begin{exercise}
  Consider a solution $(M_t)_{0\le t<T)}$ to mean curvature flow in
  $\R^{n+1}$.
  \begin{enumerate}
  \item Recall that $H=\lambda_1+\lambda_2+\ldots+\lambda_n$ and
    $|A|^2=\lambda_1^2+\lambda_2^2+\ldots+\lambda_n^2$.\\ Show that
    $H=g^{ij}h_{ij}$ and $|A|^2=h_{ij}h_{kl}g^{ik}g^{jl}$.
  \item Show that $F^{ij}=g^{ij}$. 
  \item Show that $\heat X=0$ and $\heat u=0$, where $u=\langle X,
    e_{n+1}\rangle$.
  \item Show that $\heat v=-|A|^2v-\frac2v|\nabla v|^2$. 
  \item Show that $\heat h_{ij}=|A|^2h_{ij}-2Hh_i^kh_{kj}$. 
  \item Compute the evolution equation for $|X|^2$.
  \item Compute the evolution equation for $H$. You may use $H=F$ or
    $H=g^{ij}h_{ij}$. 
  \item Show that \[\heat|A|^2=-2|\nabla A|^2+2|A|^4.\]
  \item Show that $H>0$ is preserved for closed hypersurfaces.
  \item Show that \[t\mapsto\max_{M_t}\frac{|A|^2}{H^2}\] is
    non-increasing for closed hypersurfaces with $H>0$.
  \item Consider the case $n=2$ and $\lambda_1\geq \lambda_2
    >0$. Deduce from the monotonicity of $|A|^2/H^2$ that
    $\lambda_1/\lambda_2$ stays bounded. Hint: Rewrite
    $\frac{(\lambda_1-\lambda_2)^2}{(\lambda_1+\lambda_2)^2}$ in terms
    of $|A|^2$ and $H$ and consider the function
    $x\mapsto\frac{x-1}{x+1}$.
  \item Prove Theorem 6.6 or a version with $v(-u)$ instead of $v(-u)^2$.
  \item Derive an estimate of the form $H^2\leq c\, |A|^2$. Use the
    evolution equation of $H$ to derive an upper bound for $T$ for
    closed hypersurfaces with $H>0$.
  \item Show that $H\ge0$ is preserved for closed hypersurfaces. 
  \item Read and understand the proof of Theorem 6.11.
  \end{enumerate}
  Consider a solution $(M_t)_{0\le t<T)}$ to mean curvature flow in
  $\R^{n+1}$.
  \begin{enumerate}
  \item Show that $F^{ij}=g^{ij}$. 
  \item Show that $\heat X=0$ and $\heat u=0$. 
  \item Show that $\heat v=-|A|^2v-\frac2v|\nabla v|^2$. 
  \item Show that $\heat h_{ij}=|A|^2h_{ij}-2Hh_i^kh_{kj}$. 
  \item Compute the evolution equation for $|X|^2$.
  \item Recall that $H=\lambda_1+\lambda_2+\ldots+\lambda_n$ and
    $|A|^2=\lambda_1^2+\lambda_2^2+\ldots+\lambda_n^2$.\\ Show that
    $H=g^{ij}h_{ij}$ and $|A|^2=h_{ij}h_{kl}g^{ik}g^{jl}$.
  \item Compute the evolution equation for $H$. You may use $F=H$ or
    $H=g^{ij}h_{ij}$. 
  \item Show that \[\heat|A|^2=-2|\nabla A|^2+2|A|^4.\]
  \item Show that \[t\mapsto\max_{M_t}\frac{|A|^2}{H^2}\] is
    non-increasing for closed hypersurfaces. \\ Hint: Use Kato's
    inequality $|\nabla|A||^2\le|\nabla A|^2$. 
  \item Consider the case $n=2$. Deduce from the monotonicity of
    $|A|^2/H^2$ that $\lambda_1/\lambda_2$ stays bounded. Hint:
    Rewrite $\frac{(\lambda_1-\lambda_2)^2}{(\lambda_1+\lambda_2)^2}$
    in terms of $|A|^2$ and $H$ and consider the function
    $x\mapsto\frac{1-x}{1+x}$.
  \item Prove Theorem \ref{C1 est thm}. Consider $v(-u)$ and
    $v(-u)^2$. 
  \item Show that $H>0$ is preserved for closed hypersurfaces.
  \item Derive an estimate between $H^2$ and $|A|^2$. Use the
    evolution equation of $H$ to derive an upper bound for $T$ for
    closed hypersurfaces with $H>0$.
  \item Show that $H\ge0$ is preserved for closed hypersurfaces. 
  \item Read and understand the proof of Theorem \ref{C2 est thm}.
  \end{enumerate}
\end{exercise}

\end{appendix}

\bibliographystyle{amsplain} 
%\bibliography{/Users/os/uni/math/bib/biblio.bib}
\bibliography{/Users/admin/uni/math/bib/biblio.bib}
\end{document}
