% 
% Geometric Flow Equations
% Oliver Schn\"urer
% Uni Konstanz, Sommer 2016
%
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\hyphenation{Du-al-raum Nor-men Gleich-heit Kom-pakt-heits-satz 
  Viel-fach-heit Funk-tion Funk-tion-en}

\begin{document}

\title{Geometric flow equations}

%    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 Summer School and Workshop ``Geometric
  Flows and the Geometry of Space-Time'', Hamburg, 19.-23.09.2016,
  organised by V. Cort\'es, K. Kr\"oncke and J. Louis.}

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

\date{\today.}

\dedicatory{}

\keywords{}

%\setlength{\parindent}{0pt}
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\begin{abstract}
  In this minicourse, 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 or Gau\ss{} curvature. In three 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 convergence of convex hypersurfaces to round points. Here,
    we will also show some computer algebra calculations.
  \item the evolution of graphical hypersurfaces under mean curvature
    flow.
  \end{itemize}
  % In these lectures, we study hypersurfaces that solve geometric
  % evolution equations. The normal velocity of the evolving
  % hypersurfaces depends on their Gau\ss{} or mean curvature. We will
  % discuss the asymptotic behaviour of those solutions for large
  % times. Depending on the initial conditions there are many known
  % results that describe different ways solutions might evolve. We will
  % focus on two aspects related to the work of the lecturer:
  % \begin{itemize}
  % \item Hypersurfaces that become round as they shrink to points.
  % \item The behaviour of complete non-compact solutions. 
  % \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 summer 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 and study
their behavior especially near singularities and for large times. We
consider hypersurfaces that contract to a point in finite time and,
after appropriate rescaling, to a round sphere. 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 behavior 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 summer school}
These notes contain some background material that will be covered as
necessary. Then we will derive evolution equations for geometric
quantities and study two geometric problems. More precisely, our plan
is to study the following: 
\begin{itemize}
\item Geometric prerequisites and evolution equations of geometric
  quantities. 
\item Convex surfaces contracting to a round point and an estimate for
  Gau\ss{} curvature flow, Theorem \ref{thm:andrews.stones.mon},
  measuring the deviation from being umbilic. 
%\item Gau\ss{} curvature flows of entire graphs and the model case for
%  a local $C^2$-bound, Theorem \ref{mod C2 bound thm}.
\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.  The inverse of the second fundamental form is denoted by
$\big(\tilde h^{ij}\big)_{1\le i,\,j\le n}$.

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 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 $u_i\equiv\fracp u{x^i}$ and $u_{ij}=\fracp{^2u}{x^i\partial
    x^j}$. Note that in Euclidean space, we don't need to distinguish
  between $Du$ and $\nabla u$.
\end{lemma}
\begin{proof}\neueZeile
  \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
    metric in Euclidean $\R^{n+1}$,
    $g:=X^*g_{\R^{n+1}_{\text{Eucl.}}}$. 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]
  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}

\begin{exercise}
  Give a geometric definition of the (principal) curvature of a curve
  in $\R^2$ in terms of a circle approximating that curve in an
  optimal way.\par
  \stern Use the min-max characterization of eigenvalues to extend
  that geometric definition to $n$-dimensional hypersurfaces in
  $\R^{n+1}$.
\end{exercise}

\begin{exercise}[Rotationally symmetric graphs]\label{rot graph ex} 
  \neueZeilealt Assume that the function $u:\R^n\to\R$ is smooth and
  $u(x)=u(y)$, if $|x|=|y|$.  Then $u(x)=f(|x|)$ for some
  $f:\R_+\to\R$. Compute once again all the geometric quantities
  mentioned in Lemma \ref{graph geom lem}.
\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}

\begin{remark}
  \stern In codimension $1$, we often don't need to assume that $M^n$
  is orientable: Let $X:M^n\to N^{n+1}$ be a $C^2$-immersion and
  $H_1(N;\Z/2\Z)=0$.  Assume that $X$ is proper, $X^{-1}(\partial
  N)=\partial M$, and $X$ is transverse to $\partial N$. Then
  $N\setminus f(M)$ is not connected \cite{CodimOneOrientable}. Hence,
  if $M^n$ is closed and embedded in $\R^{n+1}$, $M^n$ is orientable.
\end{remark}
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}
\begin{proof}
  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}
  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}

\begin{exercise}[Rotationally symmetric translating solutions]
  Let $u:=\R^n\times\R\to\R$ be rotationally symmetric. Assume that
  $\graph u$ is a translating solution to mean curvature flow $\dt
  X=-H\nu$, i.\,e.\ a solution such that $\dot u$ is constant. \par
  Why does it suffice to consider the case $\dot u=1$? \par
  Similar to Exercise \ref{rot graph ex}, derive an ordinary
  differential equation for translating rotationally symmetric
  solutions to mean curvature flow.
\end{exercise}

\begin{remark}
  \stern Consider a physical system consisting of a domain
  $\Omega\subset\R^3$. Assume that the energy of the system is
  proportional to the surface area of $\partial\Omega$. Then the
  $L^2$-gradient flow for the area is mean curvature flow. We check
  that in a model case for graphical solutions in Lemma \ref{mcf is l2
    gradient flow lem}.
\end{remark}

\begin{lemma}\label{mcf is l2 gradient flow lem}
  \stern Let $u:\R^n\times[0,\infty)\to\R$ be smooth. Assume that
  $u(x,t)\equiv0$ for $|x|\ge R$. Then the surface area is maximally
  reduced among all normal velocities $F$ with given $L^2$-norm, if
  the normal velocity of $\graph u$ is given by $H$, i.\,e.\ if $\dot
  u=\sqrt{1+|Du|^2}\,H$.
\end{lemma}
\begin{proof}
  The area of $\graph u(\cdot,t)|_{B_R}$ is given by
  \[A(t)=\int\limits_{B_R}\sqrt{1+|Du|^2}\,dx.\] Define the induced area
  element $d\mu$ by $d\mu:=\sqrt{1+|Du|^2}\,dx$. We obtain using
  integration by parts
  \begin{align*}
    \left.\dt A(t)\right|_{t=0}=&\,\left.\int\limits_{B_R}\dt
      \sqrt{1+|Du|^2}\,dx\,\right|_{t=0}
    =\left.\int\limits_{B_R(0)}\frac1{\sqrt{1+|Du|^2}} \langle Du,D\dot
      u\rangle\,\right|_{t=0}\\
    =&\,-\left.\int\limits_{B_R} \divergenz\left(\frac{Du}
        {\sqrt{1+|Du|^2}}\right)\frac{\dot u}v\cdot v\,dx\,\right|_{t=0}
    =-\left.\int\limits_{B_R}H\,F\,d\mu\,\right|_{t=0}\\
    \ge&\,-\left.\left(\;\;\int\limits_{B_R}H^2\,d\mu\right)^{1/2}
      \left(\;\;\int\limits_{B_R}F^2\,d\mu\right)^{1/2}\right|_{t=0}.
  \end{align*}
  Here, we have used H\"older's inequality $\|ab\|_{L^1}
  \le\|a\|_{L^2} \cdot\|b\|_{L^2}$. There, we get equality precisely
  if $a$ and $b$ differ only by a multiplicative constant. Hence the
  surface area is reduced most efficiently among all normal velocities
  $F$ with $\|F\|_{L^2}=\|H\|_{L^2}$, if we choose $F=H$. In this
  sense, mean curvature flow is the $L^2$-gradient flow for the area
  integral.
\end{proof}

\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{exercise}\label{sphere evol ex}
  Find solutions for $\dt X=-|A|^2\nu$, $\dt X=-K\nu$, $\dt
  X=\frac1H\nu$, and $\dt X=\frac1K\nu$ if $M_0=\partial
  B_R(0)\subset\R^{n+1}$, especially for $n=2$.
\end{exercise}

\begin{remark}[Level-set flow for $F>0$] 
  \stern Let $M_t$ be a family of smooth embedded hypersurfaces in
  $\R^{n+1}$ that move according to $\dt X=-F\nu$ with $F>0$. Impose
  the global assumption that each point $x\in\R^{n+1}$ belongs to at
  most one hypersurface $M_t$. Then we can (at least locally) define a
  function $u:\R^{n+1}\to\R$ by setting $u(x)=t$, if $x\in M_t$. That
  is $u(x)$ is the time, at which the hypersurface passes the point
  $x$. We obtain the equation $F\cdot|Du|=1$.\par
  If $F<0$, we get $F\cdot|Du|=-1$.\par
  This formulation is used to describe weak solutions, where
  singularities in the classical formulation occur. See for example
  \cite{HuiskenIlmanenPenrose}, where the inverse mean curvature flow
  $F=-\frac1H$ is considered to prove the Riemannian Penrose
  inequality. Note that $H=\divergenz \left(\frac{Du}{|Du|}\right)$ as
  the outer unit normal vector to a closed expanding hypersurface
  $M_t=\{u=t\}$ is given by $\frac{Du}{|Du|}$. According to
  \eqref{Weingarten equation}, the divergence of the unit normal
  yields the mean curvature as the derivative of the unit normal in
  the direction of the unit normal vanishes. Hence the evolution
  equation $\dt X=\frac1H\nu$ can be rewritten as
  \[\divergenz\left(\frac{Du}{|Du|}\right)=|Du|.\]\par
  Mean curvature flow can be rewritten as
  $|Du|\divergenz\left(\frac{Du}{|Du|}\right)=-1$.
\end{remark}

\begin{exercise}
  Verify the formula for the mean curvature in the level-set
  formulation. Compute level-set solutions to the flow equations $\dt
  X=-H\nu$ and $\dt X=\frac1H\nu$, where $u$ depends only on $|x|$,
  i.\,e.\ the hypersurfaces $M_t$ are spheres centered at the
  origin. Compare the result to your earlier computations.
\end{exercise}

We will use the level-set formulation to study a less trivial solution
to mean curvature flow which can be written down in closed form.
\begin{exercise}[Paper-clip solution]
  Let $v\neq0$. Consider the set 
  \[M_t:=\left\{(x,y)\in\R^2:e^{v^2t}\cosh(vy)=\cos(vx)\right\}.\]
  Show that $M_t$ solves mean curvature flow. Describe the shape of
  $M_t$ for $t\to-\infty$ and for $t\nearrow0$ (after appropriate
  rescaling).\par
  Compare this to Theorem \ref{mcf round thm}.\par
  Note that you may also rewrite solutions equivalently (on an
  appropriate domain) as
  \[y_{\pm} := \frac1v \log\left(\cos(vx)
    \pm\sqrt{\cos^2(vx)-e^{2v^2t}} \right)-vt.\] Hint: You should
  obtain $t_x=u_x=-\frac{\sin(vx)}{v\cos(vx)}$ and
  $u_y=-\frac{\sinh(vy)}{v\cosh(vy)}$.
\end{exercise}

\begin{remark}[Level-set flow]
  \stern If a hypersurface moves with velocity $F$, where $F$ is not
  necessarily positive, we cannot use the level-set formulation from
  above. Instead, we can 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}

\begin{exercise}\label{introduce speeds}
  \neueZeile
  \begin{enumerate}[(i)]
  \item Check, for which initial data the conditions in Theorem
    \ref{short time ex thm} are fulfilled if $F=H, \,K, \,|A|^2,
    \,-1/H, \,-1/K$.\par
  \item Find examples of closed hypersurfaces such that
    \begin{enumerate}[a)]
    \item $H>0$,
    \item $K>0$,
    \item $H$ is not positive everywhere,
    \item $H>0$, but $K$ changes sign.
    \end{enumerate}
  \item Show that on every smooth closed hypersurface
    $M^n\subset\R^{n+1}$, there is a point, where $M^n$ is strictly
    convex, i.\,e.\ $\lambda_i>0$ is fulfilled for every $i$.
  \end{enumerate}
\end{exercise}

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 following
\begin{theorem}[Avoidance principle]\label{avoidance thm}
  \stern Let $F=F(\lambda_i)$ be smooth and symmetric. Let $M^1_t$ and
  $M^2_t\subset\R^{n+1}$ be two embedded closed hypersurfaces and
  smooth solutions to a strictly parabolic flow equation $\dt
  X=-F\nu$, i.\,e.\ $\fracp F {\lambda_i}>0$ during the flow. Assume
  that $F$, considered as a function of $(D^2u,Du)$ for graphs, is
  elliptic on a set, which is convex and independent of $Du$. If
  $M^1_0$ and $M^2_0$ are disjoint, $M^1_t$ and $M^2_t$ can only touch
  if the respective normal vectors fulfill $\nu^1=-\nu^2$ there.
  Hence, 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 $\R^{n+1}\setminus M^2_t$ unless the hypersurfaces
  touch each other in a point with opposite normals.
\end{theorem}
The technical condition on the convexity of the domain, where $F$,
considered as a function of $(D^2u,Du)$, is convex, is technical and
always fulfilled for the evolution equations considered here (besides
for the inverse mean curvature flow). It can be relaxed, but makes the
proof less transparent. Only for $F=-\frac1H$, a separate argument is
needed. (It suffices to choose coordinates such that $|Du|\ll1$. Then
interpolation does not destroy positivity of the denominator. The
technical details are left as an exercise.)\par
The normal velocity $F$ is a symmetric function of the principal
curvatures. Thus it is well-defined, as the principal curvatures are
defined only up to permutations.\par
We have considered $F$ as a function of the principal curvatures.
Writing an evolving hypersurface locally as $\graph u$, we also wish
to express $F$ in terms of $\left(D^2u,Du\right)$. We continue to call
this function $F$. In the cases considered here, it is clear from the
explicit expressions, that $F$ is also a smooth function of
$\left(D^2u,Du\right)$. In general, this is a theorem \cite[Theorem
2.1.20]{CGCPBook}.\par
A similar statement is true for the condition $\fracp F{\lambda_i}>0$
and the ellipticity of $F\left(D^2,Du\right)$, i.\,e.\
$0<\frac{F(r,p)}{r_{ij}}<\infty$ in the sense of matrices. Once again,
it can be checked by direct computations for the normal velocities
considered here, that these two statements are equivalent.

We will only consider the avoidance principle for mean curvature flow: 
\begin{theorem}[Avoidance principle]
  Let $M^1_t$ and $M^2_t\subset\R^{n+1}$ be two embedded closed
  hypersurfaces and smooth solutions to $\dt X=-H\nu$. 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}[Proof of Theorem \ref{avoidance thm}]
  Otherwise there would be some $t_0>0$ such that $M^2_{t_0}$ touches
  $M^1_{t_0}$ at some point $p\in\R^{n+1}$ with normal vectors
  $\nu^1=\nu^2$ 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 $\dot u^i=F\left(D^2u^i,Du^i\right)$ for some strictly
  elliptic differential operator $F$ corresponding to the normal
  velocity $F$. 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
  $M^1_t$ can't touch $M^2_t$ in a point, where $\nu^1=\nu^2$. The
  theorem follows.
\end{proof}

\begin{exercise}
  Show that the normal velocities as considered in Exercise
  \ref{introduce speeds} can be represented (in an appropriate domain)
  as smooth functions of $\left(D^2u,Du\right)$ for hypersurfaces that
  are locally represented as $\graph u$. 
\end{exercise}

\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{exercise}
  Deduce similar corollaries for the normal velocities in Exercise
  \ref{introduce speeds}. You may use Exercise \ref{sphere evol ex}.   
\end{exercise}

\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}
  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 adopt 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{proof}
  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}
  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}
  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{Convex hypersurfaces}
\subsection{Mean curvature flow}
\label{sec:convex-hypersurfaces}
G. Huisken obtained the following theorem \cite{HuiskenRoundSphere}
for $n\ge2$. The corresponding result for curves by M. Gage,
R. Hamilton, and M. Grayson is even better, see
\cite{GageHamilton,GraysonRoundPoints}. It is only required that
$M\subset\R^2$ is a closed embedded curve.
\begin{theorem}\label{mcf round thm}
  Let $M\subset\R^{n+1}$ be a smooth closed convex hypersurface. Then
  there exists a smooth family $M_t$ of hypersurfaces solving
  \[\begin{cases}
    \dt X=-H\nu&\text{for }0\le t<T,\\
    M_0=M
  \end{cases}\] for some $T>0$.\par As $t\nearrow T$,
  \begin{itemize}
  \item $M_t\to Q$ in Hausdorff distance for some $Q\in\R^{n+1}$
    (convergence to a point),
  \item $(M_t-Q)\cdot(2n(T-t))^{-1/2}\to\S^n$ smoothly (convergence to
    a ``round point'').
  \end{itemize}
\end{theorem}

The key step in the proof of Theorem \ref{mcf round thm} (in the case
$n\ge2$) is the following
\begin{theorem}\label{mcf monotone}
  Let $M_t\subset\R^{n+1}$ be a family of convex closed hypersurfaces
  flowing according to mean curvature flow. Then there exists some
  $\delta>0$ such that
  \[\max\limits_{M_t}\frac{n|A|^2-H^2}{H^{2-\delta}}\] is
  bounded above.
\end{theorem}
The proof involves complicated integral estimates.
\begin{exercise}
  Prove Theorem \ref{mcf monotone} for $\delta=0$.\par
  \textbf{Hint:} Use Kato's inequality.
\end{exercise}

\begin{theorem}[Kato's inequality]
  We have \[|\nabla|A||^2\le|\nabla A|^2.\]
\end{theorem}
\begin{proof}
  We prove this inequality if $|A|\neq0$. In the exercise above, we
  only need that case. As $\nabla|A|^2=2|A|\nabla|A|$, the claim is
  equivalent to $\frac14\left|\nabla|A|^2\right|^2\le|A|^2\cdot|\nabla
  A|^2$. We choose a coordinate system such that $g_{ij}=\delta_{ij}$
  and $h_{ij}$ is diagonal with eigenvalues $\lambda_i$. We obtain there
  \begin{align*}
    \frac14\left|\nabla|A|^2\right|^2
    =&\,\frac14\sum\limits_k\left(\nabla_k|A|^2\right)^2
    =\sum\limits_{i,j,k} \lambda_ih_{ii;k}\lambda_jh_{jj;k}\umbruch\\
    \le&\, \sum\limits_{i,j,k}\left(\frac12\lambda_i^2h_{jj;k}^2
      +\frac12\lambda_j^2h_{ii;k}^2\right) =\sum\limits_{i,j,k}
    \lambda_i^2h_{jj;k}^2 \le\sum\limits_{i,j,k,l}
    h_{ij;k}^2\lambda_l^2\umbruch\\ =&\,|A|^2\cdot|\nabla
    A|^2. \qedhere 
  \end{align*}
\end{proof}

\begin{remark}
  For simplicity, we will illustrate the significance of the quantity
  considered in Theorem \ref{mcf monotone} only in the case $n=2$.
  These considerations extend to higher dimensions.\par
  As
  \begin{align*}
    2|A|^2-H^2=&\,2(\lambda_1^2+\lambda_2^2)-(\lambda_1+\lambda_2)^2\\
    =&\,2\lambda_1^2+2\lambda_2^2
    -\lambda_1^2-2\lambda_1\lambda_2-\lambda_2^2\\
    =&\,\lambda_1^2-2\lambda_1\lambda_2+\lambda_2^2\\
    =&\,(\lambda_1-\lambda_2)^2,
  \end{align*}
  it measures the difference from being umbilic
  $(\lambda_1=\lambda_2)$ and vanishes precisely if $M_t$ is a sphere.
  Recall from differential geometry that, according to Codazzi,
  $\lambda_1=\lambda_2$ everywhere implies that $M_t$ is locally part
  of a sphere or hyperplane.\par
  Assume that $\min\limits_{M_t}H\to\infty$ as $t\nearrow T$. Assume
  also that $\lambda_1\le\lambda_2$ and that the surfaces stay
  strictly convex, i.\,e.\ $\min\limits_{M_t}\lambda_1>0$. Then
  Theorem \ref{mcf monotone} implies for any $\epsilon$ there exists
  $t_\epsilon$, such that for $t_\epsilon\le t<T$
  \begin{align*}
    \epsilon\ge&\,\frac{2|A|^2-H^2}{H^2}
    =\frac{(\lambda_1-\lambda_2)^2}{(\lambda_1+\lambda_2)^2}
    \ge\frac{(\lambda_1-\lambda_2)^2}{4\lambda_2^2}
    =\frac14\left(\frac{\lambda_1}{\lambda_2}-1\right)^2.
  \end{align*}
  Hence $\frac{\lambda_1}{\lambda_2}\approx1$ and thus this implies
  that $M_t$ is, in terms of the principal curvatures $\lambda_i$,
  close to a sphere.
\end{remark}

\subsection{Gau\ss{} curvature flow and other normal velocities}
There are many results showing that convex hypersurfaces converge to
round points under certain flow equations, see e.\,g.\
\cite{AndrewsContractCalc,AndrewsStones,ChowNthRoot,FireyStones,%
  GageHamilton,CGFlowSpheres,JamesMcCoyArea,UrbasExpandJDG,OSA2,OSIGCF}.

Let us consider normal velocities of homogeneity bigger than one. In
this case, the calculations, that lead to a theorem corresponding to
Theorem \ref{mcf monotone} for mean curvature flow, are much simpler
and rely only on the maximum principle.

\begin{theorem}\label{thm:andrews.stones.mon}[\cite[Proposition
  3]{AndrewsStones}]
  Let $M_t$ be a smooth family of closed strictly convex solutions to
  Gau\ss{} curvature flow $\dt X=-K\nu$. Then
  \[t\mapsto\max\limits_{M_t}\,(\lambda_1-\lambda_2)^2\] is
  non-increasing.
\end{theorem}
\begin{proof}
  Recall that $H^2-4K =(\lambda_1+\lambda_2)^2-4\lambda_1\lambda_2
  =(\lambda_1-\lambda_2)^2=:w$. For Gau\ss{} curvature flow, we have,
  according to Appendix \ref{sec:some-linear-algebra},
  \begin{align*}
    F^{ij}=&\,K^{ij} =\fracp{}{h_{ij}}\frac{\det h_{kl}}{\det g_{kl}}
    =\frac{\det
      h_{kl}}{\det g_{kl}}\tilde h^{ij} =K\tilde h^{ij},\umbruch\\
    F^{ij,kl}=&\,K\tilde h^{ij}\tilde h^{kl}-K\tilde h^{ik}h^{lj},
  \end{align*}
  where $\tilde h^{ij}$ is the inverse of $h_{ij}$. Recall the
  evolution equations \eqref{g evol}, \eqref{F evol}, and \eqref{2ff
    evol} which become for Gau\ss{} curvature flow
  \begin{align*}
    \dt g_{ij}=&\,-2Kh_{ij},\umbruch\\
    \dt K-K\tilde h^{kl}K_{kl}=&\,KK\tilde h^{ij}h^k_ih_{kj}
    =K^2H,\\
    \intertext{and}\\
    \dt h_{ij}-K\tilde h^{kl}h_{ij;kl}=&\,K\tilde
    h^{kl}h^a_kh_{al}h_{ij}
    -K\tilde h^{kl}h_{kl}h^a_ih_{aj} -Kh^k_ih_{kj}\\
    &\,+K\left(\tilde h^{kl}\tilde
      h^{rs} -\tilde h^{kr}\tilde h^{sl}\right)h_{kl;i}h_{rs;j}\umbruch\\
    =&\,KHh_{ij}-(n+1)Kh_i^ah_{aj} +K\left(\tilde h^{kl}\tilde
      h^{rs} -\tilde h^{kr}\tilde h^{sl}\right)h_{kl;i}h_{rs;j},\umbruch\\
  \end{align*}
  where $n=2$. We have
  \begin{align*}
    \dt H-K\tilde h^{ij}H_{;ij}=&\,-h_{ij}g^{ik}g^{jl}\dt g_{kl}
    +g^{ij}\left(\dt h_{ij}-K\tilde h^{kl}h_{ij;kl}\right)\umbruch\\
    =&\,2K|A|^2+KH^2-3K|A|^2+Kg^{ij}\left(\tilde h^{kl}\tilde h^{rs}
      -\tilde h^{kr}\tilde
      h^{sl}\right)h_{kl;i}h_{rs;j}\umbruch\\
    =&\,K\left(H^2-|A|^2\right)+Kg^{ij}\left(\tilde h^{kl}\tilde
      h^{rs} -\tilde h^{kr}\tilde
      h^{sl}\right)h_{kl;i}h_{rs;j}\umbruch\\
    =&\,2K^2+Kg^{ij}\left(\tilde h^{kl}\tilde h^{rs} -\tilde
      h^{kr}\tilde
      h^{sl}\right)h_{kl;i}h_{rs;j},\umbruch\\
    \intertext{hence}\\
    \dt w-K\tilde h^{ij}w_{;ij}=&\, 2H\left(\dt H-K\tilde
      h^{ij}H_{;ij}\right)
    -2K\tilde h^{ij}H_iH_j\\
    &\,-4\left(\dt K-K\tilde h^{ij}K_{;ij}\right)\umbruch\\
    =&\,2H\left(2K^2+Kg^{ij}\left(\tilde h^{kl}\tilde h^{rs} -\tilde
        h^{kr}\tilde
        h^{sl}\right)h_{kl;i}h_{rs;j}\right)\\
    &\,-2K\tilde h^{ij}H_iH_j -4K^2H\umbruch\\
    =&\,2HKg^{ij}\left(\tilde h^{kl}\tilde h^{rs} -\tilde h^{kr}\tilde
      h^{sl}\right)h_{kl;i}h_{rs;j} -2K\tilde h^{ij}H_iH_j.
  \end{align*}
  In a coordinate system, such that $g_{ij}=\delta_{ij}$ and
  $h_{ij}=\diag\,(\lambda_1,\lambda_2)$, we obtain
  \begin{align*}
    &\dt w-K\tilde h^{ij}w_{;ij}=
    2KH\sum\limits_{i,j,k=1}^2\frac1{\lambda_i\lambda_j}h_{ii;k}h_{jj;k}
    -2KH\sum\limits_{i,j,k=1}^2\frac1{\lambda_i\lambda_j}h_{ij;k}^2\\
    &\qquad\qquad\qquad\qquad\qquad
    -2K\sum\limits_{i,j,k=1}^2\frac1{\lambda_k}h_{ii;k}h_{jj;k}
    \umbruch\\
    =&\,2KH\sum\limits_{\genfrac{}{}{0pt}{}{i,j,k=1}{i\neq j}}^2
    \frac1{\lambda_i\lambda_j}h_{ii;k}h_{jj;k}
    -2KH\sum\limits_{\genfrac{}{}{0pt}{}{i,j,k=1}{i\neq j}}^2
    \frac1{\lambda_i\lambda_j}h_{ij;k}^2
    -2K\sum\limits_{i,j,k=1}^2\frac1{\lambda_k}h_{ii;k}h_{jj;k}
    \umbruch\\
    =&\,\frac{4KH}{\lx\ly}\left(h_{11;1}h_{22;1}-h_{12;1}^2
      +h_{11;2}h_{22;2}-h_{12;2}^2\right)\\
    &\,-\frac{2K}\lx(h_{11;1}+h_{22;1})^2
    -\frac{2K}\ly(h_{11;2}+h_{22;2})^2.
  \end{align*}
  From now on, we consider a positive spatial maximum of $H^2-4K$.
  There, we get $2Hg^{ij}h_{ij;k}-4K\tilde h^{ij}h_{ij;k}=0$ for
  $k=1,2$. In a coordinate system as above, this (divided by 2)
  becomes
  \begin{align*}
    0=&\,Hh_{11;k}+Hh_{22;k}-2\frac K\lx h_{11;k}-2\frac K\ly
    h_{22;k}\umbruch\\
    =&\,(\lx+\ly-2\ly)h_{11;k}+(\lx+\ly-2\lx)h_{22;k}\umbruch\\
    =&\,(\lx-\ly)(h_{11;k}-h_{22;k}).
  \end{align*}
  This enables us to replace $h_{11;2}$ in the evolution equation in a
  positive critical point by $h_{22;2}$: $h_{11;2}=h_{22;2}$ and
  $h_{22;1}=h_{11;1}$. Using also the Codazzi equations, we can
  rewrite the evolution equation in a positive critical point as
  \begin{align*}
    \dt w-K\tilde h^{ij}w_{;ij}=&\,
    4(\lx+\ly)\left(h_{11;1}^2-h_{22;2}^2+h_{22;2}^2-h_{11;1}^2\right)\\
    &\,-\frac{2K}\lx(h_{11;1}+h_{22;1})^2
    -\frac{2K}\ly(h_{11;2}+h_{22;2})^2\umbruch\\
    \le&\,0.
  \end{align*}
  Hence, by the parabolic maximum principle, Theorem
  \ref{thm:parab-maxim-princ}, the claim follows.
\end{proof}

A consequence of Theorem \ref{thm:andrews.stones.mon} is the following
result, see \cite[Theorem 1]{AndrewsStones}.
\begin{theorem}\label{thm:andrews.stones}
  Let $M\subset\R^3$ be a smooth closed strictly convex surface.  Then
  there exists a smooth family of closed strictly convex hypersurfaces
  solving Gau\ss{} curvature flow $\dt X=-K\nu$ for $0\le t<T$. As
  $t\nearrow T$, $M_t$ converges to a round point.
\end{theorem}
\begin{proof}[Sketch of proof]
  The main steps are
  \begin{enumerate}[(i)]
  \item The convergence to a point is due to K. Tso \cite{TsoPoint}.
    There, the problem is rewritten in terms of the support function
    and considered in all dimensions. It is shown that a positive
    lower bound on the Gau\ss{} curvature is preserved during the
    evolution.  This ensures that the surfaces stay convex.  The
    evolution equation of $\frac K{\langle X,\nu\rangle-\frac12R}$ is
    used to bound the principal curvatures as long as the surface
    encloses $B_R(0)$. Thus a positive lower bound on the principal
    curvatures follows. Parabolic Krylov-Safonov estimates imply
    bounds on higher derivatives.
  \item Theorem \ref{thm:andrews.stones.mon},
  \item Show that $M_t$ is between spheres of radius $r_+(t)$ and
    $r_-(t)$ and center $q(t)$ with $\frac{r_+(t)}{r_-(t)}\to 1$ as
    $t\nearrow T$.
  \item Show that the quotient $\frac{K(p,t)}{K_{r(t)}}$ converges to
    $1$ as $t\nearrow T$. Here $r(t)=(3(T-t))^{1/3}$ is the radius of
    a sphere flowing according to Gau\ss{} curvature flow that becomes
    singular at $t=T$ and $K_{r(t)}=(3(T-t))^{-2/3}$ its Gau\ss{}
    curvature. This involves a Harnack inequality for the normal
    velocity.
  \item Show that $\frac{\lambda_i}{(3(T-t))^{-1/3}}\to1$ as
    $t\nearrow T$.
  \item Obtain uniform a priori estimates for a rescaled version of
    the flow and hence smooth convergence to a round sphere.\qedhere
  \end{enumerate}
\end{proof}

\subsection{\stern The tensor maximum principle} 
Often, the tensor maximum principle can be used to deduce a priori
bounds. 

We see directly from the parabolic maximum principle for tensors that
a positive lower bound on the principal curvatures is preserved for
surfaces moving with normal velocity $|A|^2$.
\begin{lemma}\label{A2 conv pre}
  For a smooth closed strictly convex surface $M$ in $\R^3$, 
  flowing according to $\dt X=-|A|^2\nu$, the minimum of the
  principal curvatures is non-decreasing.
\end{lemma}
\begin{proof}
  We have $F=|A|^2=h_{ij}g^{jk}h_{kl}g^{li}$,
  $F^{ij}=2g^{ia}h_{ab}g^{bj}$, and
  $F^{ij,kl}=2g^{ik}g^{jl}$. 
  Consider $M_{ij}=h_{ij}-\epsilon g_{ij}$ with $\epsilon>0$ so small
  that $M_{ij}$ is positive semi-definite for some time $t_0$. We wish
  to show that $M_{ij}$ is positive semi-definite for $t>t_0$. Using
  \eqref{2ff evol}, we obtain
  $$\dt h_{ij}-F^{kl}h_{ij;\,kl}=2\A3 h_{ij}-3\A2h^k_ih_{kj}
  +2g^{kr}g^{ls}h_{kl;\,i}h_{rs;\,j}.$$
  In the evolution equation for $M_{ij}$, we drop the positive definite 
  terms involving derivatives of the second fundamental form
  $$\dt M_{ij}-F^{kl}M_{ij;\,kl}\ge 2\A3 h_{ij}-3\A2h^k_ih_{kj}
  +2\epsilon\A2 h_{ij}.$$
  Let $\xi$ be a zero eigenvalue of $M_{ij}$ with $\abs{\xi}=1$,
  $M_{ij}\xi^j=h_{ij}\xi^j-\epsilon g_{ij}\xi^j=0$.
  So we obtain in a point with $M_{ij}\ge0$
  \begin{align*}
    \left(2\A3h_{ij}-3\A2 h^k_ih_{kj}+2\epsilon\A2 h_{ij}\right)
    \xi^i\xi^j=&2\epsilon\A3-3\epsilon^2\A2+2\epsilon^2\A2\\
    =&2\epsilon\A3-\epsilon^2\A2\\
    \ge&2\epsilon^2\A2-\epsilon^2\A2>0
  \end{align*}
  and the maximum principle for tensors, Theorem
  \ref{thm:weak.parab.max.princ.tensor}, which extends to the case $\dt
  M_{ij}\ge\ldots$, gives the result.
\end{proof}

\begin{exercise}
  Show that under mean curvature flow of closed hypersurfaces, the
  following inequalities are preserved during the flow.
  \begin{enumerate}[(i)]
  \item $0\le H$, $0<H$,
  \item $h_{ij}\ge0$,
  \item $\epsilon Hg_{ij}\le h_{ij}\le \beta Hg_{ij}$ for
    $0<\epsilon\le\frac1n<\beta<1$.
  \end{enumerate}
\end{exercise}
Such estimates exist also for other normal velocities.

\subsection{Two dimensional surfaces}
\begin{theorem}[\cite{OSA2}]\label{thm:a2.mon}
  Let $M_t$ be a family of closed strictly convex hypersurfaces
  evolving according to $\dt X=-|A|^2\nu$. Then
  \[t\mapsto \max\limits_{M_t} \frac{(\lambda_1+\lambda_2)
    (\lambda_1-\lambda_2)^2} {\lambda_1\lambda_2}\] is non-increasing.
\end{theorem}
\begin{exercise}\neueZeile\label{ex:a2}
  \begin{enumerate}[(i)]
  \item Prove Theorem \ref{thm:a2.mon}.\par
    Hint: In a positive critical point of
    $w:=\frac{(\lambda_1+\lambda_2) (\lambda_1-\lambda_2)^2}
    {\lambda_1\lambda_2}$, for $F=|A|^2$, the evolution equation of
    $w$ is given by
    \begin{align*}
      \dt w-F^{ij}w_{;ij}=
      &\,-4(\lx-\ly)^2\lx\ly\umbruch\\
      &\,-2\frac{5\lx^8-4\lx^7\ly +46\lx^6\ly^2 +48\lx^5\ly^3
        +72\lx^4\ly^4}
      {\left(\lx^2 +\lx\ly +\ly^2\right)^2\lx^4}h_{11;1}^2\umbruch\\
      &\,-2\frac{44\lx^3\ly^5 +34\lx^2\ly^6 +8\lx\ly^7
        +3\ly^8}{\left(\lx^2
          +\lx\ly +\ly^2\right)^2\lx^4}h_{11;1}^2\umbruch\\
      &\,-2\frac{5\ly^8-4\ly^7\lx +46\ly^6\lx^2 +48\ly^5\lx^3
        +72\ly^4\lx^4}
      {\left(\ly^2 +\ly\lx +\lx^2\right)^2\ly^4}h_{22;2}^2\umbruch\\
      &\,-2\frac{44\ly^3\lx^5 +34\ly^2\lx^6 +8\ly\lx^7
        +3\lx^8}{\left(\ly^2 +\ly\lx +\lx^2\right)^2\ly^4}h_{22;2}^2.
    \end{align*}
    (This is a longer calculation.)
  \item Show that the only closed strictly convex surfaces contracting
    self-similarly (by homotheties) under $\dt X=-|A|^2\nu$, are round
    spheres.  A surface $M_t$ is said to evolve by homotheties, if for
    every $t_1,t_2$, there exists $\lambda\in\R$ such that
    $M_{t_1}=\lambda M_{t_2}$.
  \item Show that for closed strictly convex initial data $M$, there
    exists some $c>0$ such that $\frac1c\le\frac\lx\ly +\frac\ly\lx\le
    c$ for surfaces evolving according to $\dt X=-|A|^2\nu$ for all
    $0\le t<T$, where $T$ is, as usual, the maximal existence time.
  \end{enumerate}
\end{exercise}

Similar results also exist for expanding surfaces
\begin{theorem}[\cite{OSIGCF}]\label{thm:IGCF}
  Let $M_t$ be a family of closed strictly convex hypersurfaces
  evolving according to $\dt X=\frac1K\nu$. Then
  \[t\mapsto \max\limits_{M_t} \frac{(\lambda_1-\lambda_2)^2}
  {\lambda_1^2\lambda_2^2}\] is non-increasing.
\end{theorem}
\begin{exercise}
  Prove Theorem \ref{thm:IGCF} and deduce consequences similar to
  those in Exercise \ref{ex:a2}.\par
  Hint: In a critical point of $w:=\frac{(\lx-\ly)^2}{\lx^2\ly^2}$,
  the evolution equation of $w$ reads
  \[\dt w-F^{ij}w_{;ij}=-2\frac{(\lx+\ly)(\lx-\ly)^2}{\lx^3\ly^3}
  -\frac8{\lx^6\ly}h_{11;\,1}^2 -\frac8{\lx\ly^6}h_{22;\,2}^2.\]
\end{exercise}

\iffalse
\subsection{\stern Isoperimetric inequalities}
In a situation, where we know, that mean curvature flow exists until
the enclosed volume shrinks to zero, it can be used to prove
isoperimetric inequalities. Flow equations and isoperimetric
inequalities are studied by G. Huisken, F. Schulze \cite{FelixIso},
and P. Topping \cite{ToppingMCFInequ}. We want to describe such an
approach in a model situation. \par
Consider a family $M^2_t$ of smooth closed surfaces that moves by mean
curvature flow until the volume of the enclosed area $\Omega_t$
shrinks to zero. In this case, the isoperimetric inequality reads
\[\frac1{3\sigma}\left(\mathcal H^2(M_t)\right)^{3/2}-\mathcal
H^3(\Omega_t) \equiv \frac1{3\sigma}|M_t|^{3/2}-|\Omega_t|\ge0\] for
$\sigma=\sqrt{4\pi}$. (Recall that $|\partial B_1^2|=4\pi$ and
$|B_1^2|=\frac{4\pi}3$.) We want to prove this inequality using mean
curvature flow for surfaces $M_t$. Note first that by H\"older's
inequality
\[\int\limits_{M_t} H\le\left(\,\,\,\int\limits_{M_t}1\right)^{1/2}
\left(\,\,\,\int\limits_{M_t}H^2\right)^{1/2}.\] Secondly, by
Gau\ss{}-Bonnet, 
\[\int\limits_{M_t}H^2=\int\limits_{M_t}(\lambda_1-\lambda_2)^2
+\int\limits_{M_t}4K\ge 4\int\limits_{M_t}K=4\mathcal H^2(\partial
B_1(0)) \equiv 4|\partial B_1(0)| =16\pi.\] If $M_t$ is not a
topological sphere, the integration has to be restricted to a set
$\tilde M_t\subset M_t$ such that $\nu\colon\tilde M_t\to\S^2$ is
bijective.  Hence, under the evolution by mean curvature flow, we get
according to \eqref{eq:dmu.evol} the following estimate for the
isoperimetric difference (we may assume that $\int_{M_t}H>0$ for
otherwise the inequality derived in the following follows already from
the first line)
\begin{align*}
  \dt\left(\frac1{3\sigma}|M_t|^{3/2}-|\Omega_t|\right) =&\,
  \frac1{2\sigma}|M_t|^{1/2}\int\limits_{M_t}-H^2\,d\mu
  -\int\limits_{M_t}-H\,d\mu\umbruch\\
  \le&\,-\frac1{2\sigma} \left(\,\,\,\int\limits_{M_t}H^2\right)^{1/2}
  \int\limits_{M_t}H+\int\limits_{M_t}H\umbruch\\
  =&\,\left(1-\frac1{2\sigma}
    \left(\,\,\,\int\limits_{M_t}H^2\right)^{1/2}\right) 
  \int\limits_{M_t}H \umbruch\\
  \le&\,\left(1-\frac1{2\sigma}\sqrt{16\pi}\right)\int H\umbruch\\
  =&\,0.
\end{align*}
Setting $f(t):=\frac1{3\sigma}|M_t|^{3/2}-|\Omega_t|$, we have
$f(T)\ge0$ (considered as a limit as $t\nearrow T$) as we have assumed
that $|\Omega_T|=0$ (in the sense of a limit). Integrating backwards
in time yields $f(t)\ge0$ for $t<T$, which is the isoperimetric
inequality claimed above.
\fi

\subsection{Calculations on a computer algebra system}
For checking the monotonicity of \[t\mapsto\max\limits_{M_t}
\frac{(\lx+\ly)(\lx-\ly)^2}{\lx\ly},\] see Theorem \ref{thm:a2.mon},
the calculations become quite long. In the following we describe how
the calculations leading to this theorem can be done by a computer
provided that you trust these machines. 
\begin{enumerate}[(i)]
\item Rewrite $w=\frac{(\lx+\ly)(\lx-\ly)^2}{\lx\ly}$ in terms of $H$
  and $K$, $H$ and $K$ in terms of $g_{ij}$ and $h_{ij}$ and finally
  $g_{ij}$ and $h_{ij}$ as a function of $Du$ and $D^2u$, provided
  that the surface is locally described as $\graph u$. 
\item Proceed similarly with the normal velocity
  $|A|^2=F\left(Du,D^2u\right)$. Then $u$ fulfills the partial
  differential equation \[\dot u=\sqrt{1+|Du|^2}\cdot F(Du,D^2u)
  \equiv vF.\] 
\item Differentiating this equation \[\dot u_k=vF_{r_{ij}}u_{ijk}
  +vF_{p_i}u_{ik} +\frac{u^i}vu_{ik}\] and dropping lower order terms
  suggests to consider the linearised operator \[LW:=\dot
  W-vF_{r_{ij}}W_{ij},\] where $v$ and $F$ are evaluated at
  $\left(Du,D^2u\right)$.
\item We would like to show that $w$ is non-increasing. This follows
  from the maximum principle if we can show that $\dt
  w-F^{ij}w_{;ij}\equiv\dt w-\fracp F{h_{ij}}w_{;ij}\le0$ in a
  positive maximum of $w$. By the chain rule, we get \[\fracp
  F{r_{ij}} =\fracp F{h_{kl}}\cdot\fracp{h_{kl}}{r_{ij}} =\fracp
  F{h_{ij}}\cdot\frac1v.\]
\item The considerations in the last paragraph do not depend on the
  coordinate system. We choose a coordinate system such that a
  positive maximum is attained at the origin and $Du(0)=0$. We may
  assume in addition that $D^2u(0)$ is diagonal. At the origin, both
  factors that distinguish covariant and partial derivatives in
  $w_{;ij}= w_{,ij}-\Gamma^k_{ij}w_{,k}$ vanish. Hence it suffices to
  show that $Lw|_{x=0}\le0$. This can be carried out with the help of
  a computer. 
\end{enumerate}
The algorithm in words:
\begin{enumerate}
\item Write $w=w\left(Du,D^2u\right)$ and $F=F\left(Du,D^2u\right)$.
\item Compute the following derivatives in terms of derivatives of
  $u$: $F_{r_{ij}}$, $\dot w$, $w_i$, $w_{ij}$.
\item Combine those derivatives and get $Lw=:N_1$ in terms of
  derivatives of $u$.
\item Use the relations obtained from differentiating $\dot u=vF$,
  $\dot u_k=(vF)_k$ and $\dot u_{kl}=(vF)_{kl}$ to remove any time
  derivative from $N_1$: Call the result $N_2$. 
\item As $w$ is positive and maximal at the point we want to consider,
  we can solve $w_k=0$ for $u_{11k}$ and $u_{22k}$. We use this to
  replace the terms $u_{112}$ and $u_{221}$ in $N_2$ and get $N_3$. 
\item Assume that $Du(0)=0$ and $D^2u(0)=\left(
  \begin{smallmatrix}
    a&0\\ 0&b
  \end{smallmatrix}\right)$ in $N_3$ to get $N_4$. 
\item $N_4$ consists of three terms: \[N_4=A+Bu_{111}^2+Cu_{222}^2,\]
  no terms involving $u_{111}u_{222}$ show up. Observe that $A$, $B$
  and $C$ do only depend on $a$ and $b$ and that $B$ and $C$ are equal
  up to interchanging $a$ and $b$. 
\item It is easy to see that $A\le0$ and $B\le0$ for $a,b\ge0$ in the
  situation of Theorem \ref{thm:a2.mon}.\par If it is not obvious,
  whether these inequalities hold, Sturm's algorithm \cite{SturmThm}
  can be used to check the underlying polynomials for positivity.
\item Applying the steps above for different choices of $w$ can be
  used to find monotone quantities, see \cite{OSA2,OSIGCF}. 
\end{enumerate}

Two warnings:
\begin{itemize}
\item Do not use the simplifications valid at a single point,
  especially $Du=0$, before differentiating.
\item The computer might identify $u_{12}$ and $u_{21}$. Take this
  into account when computing $F_{r_{12}}$. 
\end{itemize}

\begin{exercise}
  Prove Theorem \ref{thm:a2.mon} based on computer algebra
  calculations.  
\end{exercise}

\iffalse
\section{Entire solutions to Gau\ss{} curvature flows}

\subsection{Mean curvature flow}
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
  \[\begin{cases}
    \dot u=\sqrt{1+|Du|^2}\divergenz
    \left(\frac{Du}{\sqrt{1+|Du|^2}}\right) &\text{in
    }\R^n\times(0,\infty),\\
    u(\cdot,t)\to u_0 &\text{as }t\searrow0 \text{ in
    }C^0_{loc}\left(\R^n\right).
  \end{cases}\]
\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}

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}.

\subsection{Gau\ss{} curvature flows}
The results of this section are joint work with J. Urbas and not yet
published elsewhere. 

Let $\alpha>0$. An entire graphical solution moving with normal
velocity $K^\alpha$ fulfills
\begin{equation}\label{Ka flow}
\dt X=-K^\alpha\nu
\end{equation}
or, equivalently, with an initial condition 
\begin{equation}\label{Kau flow}
\begin{cases}
\dot u=\sqrt{1+|Du|^2} \left(\fracd{\det D^2u}
{\left(1+|Du|^2\right)^{\frac{n+2}2}} \right)^\alpha &\text{in
}\R^n\times(0,\infty),\\ u(\cdot,0)=u_0&\text{in }\R^n.
\end{cases}
\end{equation}
It is a parabolic equation if $u$ is strictly convex.

In order to formulate our existence theorem, we need the following
definition. 
\begin{definition}[$\nu$-condition]
  Let $u\colon\R^n\to\R$ be differentiable. Then $u$ (or $\graph u$)
  fulfills the $\nu$-condition if for every $\epsilon>0$ there exists
  $r>0$ such that for points $p,q\in\graph u$ with $|p-q|\le1$ and
  $|p|,|q|\ge r$ \[|\nu(p)-\nu(q)|\le\epsilon\] holds.
\end{definition}

For general functions $u$, this condition is restrictive. This is
different for convex functions. 
\begin{example}
  \neueZeile
  \begin{enumerate}[(i)]
  \item $u(x)=|x|^k$ fulfills the condition for every $k>1$ and for
    every $k>0$ outside the origin. 
  \item We say that $k\colon\R^n\to\R$ (or $\graph k$) is a cone if
    $k$ is positive homogeneous of degree one. We call it a smooth
    cone if $k$ is smooth outside the origin. \par
    Let $u$ be convex, differentiable and asymptotic to a smooth
    convex cone $k$ in the sense that \[\lim\limits_{r\to\infty}
    \sup\limits_{\R^n\setminus B_r(0)} |u-k|=0.\] Then $u$ fulfills
    the $\nu$-condition. \par This is also true if $u_0-k$ growth at
    most sublinearly at infinity. 
  \item A function $u$ close to the cone $k\colon\R^2\to\R$ with
    $k(x,y):=\max\{\pm x,\pm y\}$ does not fulfill the
    $\nu$-condition.
  \end{enumerate}
\end{example}

We prove an existence result for initial data that fulfill the
$\nu$-condition.
\begin{theorem}\label{existence thm}
  Let $0<\alpha<\frac1{n-1}$. Let $u_0\in
  C^{2,\beta}_{loc}\left(\R^n\right)$ be strictly convex for some
  $0<\beta<1$. Assume that $u_0$ fulfills the $\nu$-condition. Then
  there exists
  $$u\in C^{2;1}_{loc}\left(\R^n\times(0,\infty)\right) \cap
  C^0_{loc}\left(\R^n\times[0,\infty)\right)$$ solving \eqref{Kau
    flow}.
  \par If $\sup\limits_{\R^n}|Du_0|<\infty$, $u$ is unique. 
\end{theorem}

\begin{remark}
  Corresponding to the convergence results for mean curvature flow
  \cite{StavrouSelfSim,JCOSFSMCFStability}, we get:
  \begin{enumerate}[(i)]
  \item If $u_0$ is asymptotic to a smooth convex cone $k$ at
    infinity, then \[\sup\limits_{\R^n}|u(\cdot,t)-U(\cdot,t)|\to0
    \quad\text{as }t\to\infty,\] where $U$ is the homothetically
    expanding solution with $U(\cdot,0)=k$, see
    \cite{UrbasGaussSelfSim1}.
  \item If $u_0$ deviates at most sublinearly from $k$ near infinity,
    then the graph of $u(\cdot,t)$ converges to the graph of
    $U(\cdot,t)$ after suitable rescaling. 
  \end{enumerate}
  The proofs follow along the lines of the corresponding results for
  mean curvature flow. Observe that $u_0\ge k$ if $u$ is asymptotic to
  $k$.
\end{remark}

The crucial local $C^2$ a priori estimates in the proof of Theorem
\ref{existence thm} are contained in 
\begin{theorem}\label{loc C2 thm}
  Let $0<\alpha<\frac1{n-1}$, $T>0$ and $(M_t)_{t\in[0,T]}$ be a
  family of complete convex $C^2$-hypersurfaces solving \eqref{Ka
    flow}. Pick a coordinate system such that each
  $M_t^-:=M_t\cap\{x^{n+1}\le 0\}$ can be written as $\graph
  u(\cdot,t)$ in some domain with $|Du(\cdot,t)|\le G$ in $M_t^-$. If
  $M_0^-$ is bounded, $\beta=\beta(n,\alpha)\ge1$ is large and
  $G=G(n,\alpha,\beta)>0$ is small enough, then
  $$\left(-\eta_\alpha X^\alpha\right)\cdot \max\limits_{|\xi|=1}
  \frac{h_{ij}\xi^i\xi^j}{g_{ij}\xi^i\xi^j}\cdot e^{\beta v}$$ is
  bounded in $M_t^-$ by the maximum of
  $c=c\left(n,\alpha,\beta,G,\max-\eta_\alpha X^\alpha\right)$ and its
  value at $t=0$.
\end{theorem}
We will prove this theorem below in a model situation. 

\begin{proof}[Main steps of the proof of Theorem \ref{existence thm}]
  \neueZeile
  \begin{enumerate}[(i)]
  \item We approximate our initial data with closed strictly convex
    surfaces and prove a priori estimates that allow to pass to a
    limit. Using spheres as barriers, we see that we can derive the a
    priori estimates for entire prospective solutions.
  \item Spheres, used as barriers, and the convexity of $u$ imply
    local $C^0$- and $C^1$-bounds. 
  \item The $\nu$ condition is preserved during the flow: Use spheres
    as barriers and the convexity of $u$. 
  \item There exist $(R,T,G,H)$-coordinate systems, where $R,T,G>0$
    are given and $H>0$ depends on $u_0$ and on these parameters:
    \begin{itemize}
    \item The coordinate systems with coordinates $(\tilde x^\gamma)$
      differ by a translation and rotation from the original
      coordinate systems.
    \item $\graph u\cap\{\tilde x^{n+1}<0\}$ can be written as a
      convex graph of a function $\tilde u$ with $|\nabla\tilde u|\le
      G$. 
    \item Those coordinate systems exist for a set of points $C$ of
      distance at least $R$ from the origin such that $\tilde u\le -H$
      for at least one of these coordinate systems for each point in
      $C$. $C$ surrounds the origin, i.\,e.{} a bounded component of
      $M_t\setminus C$ contains $M_t\cap B_R(0)$. 
    \item Those coordinate systems exist for $t\in[0,T]$.       
    \end{itemize}
  \item Theorem \ref{loc C2 thm} implies local upper $C^2$ a priori
    estimates. 
  \item The $\nu$-condition allows to control the normal image near
    infinity. Hence a Harnack inequality \cite{AndrewsHarnack} and an
    explicit barrier imply local lower bounds on $K$ for positive
    times.
  \item Those estimates suffice to pass to a limit of the evolving
    approximates solutions and to obtain a solution for all $t>0$.
    \qedhere
  \end{enumerate}
\end{proof}

Instead of Theorem \ref{loc C2 thm}, we consider a model
situation. This is a partial differential equation for which we will
prove similar estimates. It is not motivated geometrically. 
\begin{theorem}
  \label{mod C2 bound thm}
  Let $u\colon\R^n\times[0,\infty)\to\R$ be convex (in space). Define
  $\Omega_t:=\{x\colon u(x,t)<0\}$. Assume that $\bigcup\limits_{t\in
    I}\Omega_t$ is bounded for any compact time interval and that $u$
  is of class $C^{4;2}$ in a neighborhood of these sets. Let
  $0<\alpha<\frac1{n-1}$.  If $|Du|\le G$ in each $\Omega_t$ for $G$
  sufficiently small, then for any such $u$ solving
  \begin{align*}
    \dot u=&\,\tfrac1\alpha \cdot(\det D^2u)^\alpha,\umbruch\\ 
    W:=&\, (-u)\cdot \max_{|\xi|=1}u_{\xi\xi}\cdot
    e^{\frac12|Du|^2},
  \end{align*}
  is bounded above in $\Omega_t$ in terms of $n,\alpha,G$ and its
  value at $t=0$.
\end{theorem}
\begin{proof}
  \def\D{\mathcal D}
  We differentiate the evolution equation and obtain
  \begin{align*}
    \dot u=&\,\tfrac1\alpha\cdot (\det D^2u)^\alpha
    \equiv\tfrac1\alpha\cdot\D,\umbruch\\
    \dot u_i=&\, \D u^{kl}u_{kli},\umbruch\\
    \dot u_{11}=&\, \D u^{ij}u_{ij11}-\D
    u^{ik}u^{jl}u_{ij1}u_{kl1} +\alpha\D
    u^{ij}u_{ij1}u^{kl}u_{kl1}.
  \end{align*}
  Instead of bounding 
  \begin{align*}
    W=&\, (-u)\cdot \max_{|\xi|=1}u_{\xi\xi}\cdot
    e^{\frac12|Du|^2},
    \intertext{we will consider the test function}
    w=&\,\log(-u)+\log u_{11}+\tfrac12|Du|^2.
  \end{align*}
  It suffices to bound this function: Without loss of generality we
  may assume that $\max\limits_{|\xi|=1} u_{\xi\xi} =u_{11}$ at the
  point, where $W$ attains its maximum. Then $W=e^w$ there and $W\le
  e^w$ elsewhere.  Hence it suffices too bound $w$. \par
  In an increasing maximum of $w$, we obtain
  \begin{align*}
    0\le \dot w=&\, \frac{-\dot
      u}{-u} +\frac{\dot u_{11}}{u_{11}} +u^k\dot u_k,\umbruch\\
    0=w_i=&\, \frac{-u_i}{-u} +\frac{u_{11i}}{u_{11}}
    +u^ku_{ki},\umbruch\\ 
    0\ge w_{ij}=&\, \frac{-u_{ij}}{-u} +\frac{u_{11ij}}{u_{11}}
    +u^ku_{kij}\\ &\, -\frac{u_iu_j}{(-u)^2} 
    -\frac{u_{11i}u_{11j}}{u_{11}^2} +u^k_ju_{ki}.
  \end{align*}
  We get there
  \begin{align*}
    0\le&\,\dot w-\D u^{ij}w_{ij}\umbruch\\
    =&\,\frac{-\dot u}{-u}+\frac{\dot u_{11}}{u_{11}}
    +u^k\dot u_k\\
    &\,-\D u^{ij}\left(\frac{-u_{ij}}{-u} +\frac{u_{11ij}}{u_{11}}
      +u^ku_{kij} -\frac{u_iu_j}{(-u)^2}
      -\frac{u_{11i}u_{11j}}{u_{11}^2} +u^k_ju_{ki}\right)\umbruch\\
    =&\,\frac1{u_{11}}\left(\dot u_{11}-\D u^{ij}u_{ij11}\right)
    +u^k\left(\dot u_k-\D u^{ij}u_{ijk}\right)\\
    &\,+\D\frac{u^{ij}u_iu_j}{(-u)^2}
    +\D\frac1{u_{11}^2}u^{ij}u_{11i}u_{11j} -\frac{\D}{\alpha(-u)}
    +\D\frac{n}{(-u)} -\D\Delta u.
  \end{align*}
  We use the differentiated evolution equation and obtain in the
  maximum considered
  \begin{align*}
    0\le&\,\frac{\D}{u_{11}}\left(-u^{ik}u^{jl}u_{ij1}u_{kl1}
      +\alpha u^{ij}u_{ij1} u^{kl}u_{kl1}\right) +0\\
    &\, +\D\frac{u^{ij}u_iu_j}{(-u)^2}
    +\frac{\D}{u_{11}^2}u^{ij}u_{11i}u_{11j}
    +\D\frac{n}{-u}
    -\D\Delta u.
  \end{align*}
  Let us assume that $u_{ij}$ is diagonal at the point considered. We
  obtain there
  \begin{align*}
    0=&\,w_i=\frac{-u_i}{-u} +\frac{u_{11i}}{u_{11}}
    +u_iu_{ii},\umbruch\\  
    \frac{u^{ij}u_iu_j}{(-u)^2}=&\, \sum\limits_{i=1}^n
    u^{ii}\left(\frac{u_{11i}}{u_{11}} +u_iu_{ii}\right)^2\\
    =&\,\sum\limits_{i=1}^n\left(u^{ii}\frac{u_{11i}^2}{u_{11}^2}
      +2u_i\frac{u_{11i}}{u_{11}} +u_i^2u_{ii}\right)\\
    =&\,\sum\limits_{i=1}^n\left(u^{ii}\frac{u_{11i}^2}{u_{11}^2}
      +2u_i\left(\frac{u_i}{-u}-u_iu_{ii}\right)
      +u_i^2u_{ii}\right)\\
    \le&\,\sum\limits_{i=1}^n u^{ii}\frac{u_{11i}^2}{u_{11}^2}
       +2\frac{|Du|^2}{-u}.
  \end{align*}
  We use this estimate in our previous inequality, divide by $\D$ and
  obtain 
  \begin{align*}
    0\le&\,\frac{1}{u_{11}}\left(-u^{ik}u^{jl}u_{ij1}u_{kl1} +\alpha
      u^{ij}u_{ij1} u^{kl}u_{kl1}\right)\\
    &\,+\sum\limits_{i=1}^n
    u^{ii}\frac{u_{11i}^2}{u_{11}^2}
    +\frac{1}{u_{11}^2}u^{ij}u_{11i}u_{11j}
    +2\frac{|Du|^2}{-u} +\frac{n}{-u}-\Delta u.
  \end{align*}
  Let us assume that $u_{ij}$ (and hence $u^{ij}$) is diagonal at the
  point considered.  We obtain for the $3^{rd}$-order terms
  \begin{align*}
    u_{11}\cdot 3^{rd}\text{o.t.}\le&\, \sum\limits_{i,j=1}^n
    \left(-u^{ii}u^{jj}u_{ij1}^2 +\alpha
      u^{ii}u_{ii1}u^{jj}u_{jj1}\right)
    +2\frac1{u_{11}}\sum\limits_{i=1}^nu^{ii}u_{11i}^2\umbruch\\
    =&\,-\frac{u_{111}^2}{u_{11}^2} -2\sum\limits_{r=2}^n
    \frac{u^{rr}}{u_{11}} u_{11r}^2 -\sum\limits_{r,s=2}^n
    u^{rr}u^{ss}u_{rs1}^2\\ &\, +\alpha\frac{u_{111}^2}{u_{11}^2}
    +2\alpha\frac{u_{111}}{u_{11}}\sum\limits_{r=2}^n u^{rr}u_{rr1}
    +\alpha\sum\limits_{r,s=2}^n u^{rr}u_{rr1} u^{ss}u_{ss1}\\
    &\,+2\frac{u_{111}^2}{u_{11}^2}
    +2\sum\limits_{r=2}^n\frac{u^{rr}}{u_{11}} u_{11r}^2\umbruch\\
    \le&\,(1+\alpha)\frac{u_{111}^2}{u_{11}^2} -\sum\limits_{r=2}^n
    \left(u^{rr}u_{rr1}\right)^2
    +\alpha(n-1)\sum\limits_{r=2}^n\left(u^{rr}u_{rr1}\right)^2\\
    &\,+\epsilon\sum\limits_{r=2}^n \left(u^{rr}u_{rr1}\right)^2
    +\frac{\alpha^2(n-1)}
    {\epsilon}\frac{u_{111}^2}{u_{11}^2}\\
    \intertext{where $\epsilon>0$}
    \le&\,\left(1+\alpha+\frac{\alpha^2(n-1)}{\epsilon}\right) \cdot
    \frac{u_{111}^2}{u_{11}^2} +\underbrace{\left(-1 +\alpha(n-1)
        +\epsilon \right)}_{=0\text{ if
      }\epsilon=1-\alpha(n-1)>0}\cdot\sum\limits_{r=2}^n
    \left(u^{rr}u_{rr1}\right)^2\umbruch\\
    \equiv&\,c(n,\alpha)\cdot\frac{u_{111}^2}{u_{11}^2} \umbruch\\ 
    =&\,c(n,\alpha)\cdot\left(\frac{-u_1}{-u} +u_1u_{11}\right)^2
    \umbruch\\ 
    =&\,c(n,\alpha)\cdot\left(\frac{u_1^2}{(-u)^2}
      -2\frac{u_1^2u_{11}}{-u}
      +u_1^2u_{11}^2\right) \umbruch\\
    \le&\,u_{11}\cdot c(n,\alpha)\cdot
    \left(\frac{u_1^2}{u_{11}(-u)^2} 
      +u_1^2u_{11}\right).
  \end{align*}
  This implies at the increasing maximum considered
  \begin{align*}
    0\le&\,c(n,\alpha)\cdot\left(\frac{u_1^2}{u_{11}(-u)^2}
      +u_1^2u_{11}\right)
    +2\frac{|Du|^2}{-u} +\frac{n}{-u} -u_{11}.
  \end{align*}
  We multiply this bound by $(-u)$, add $(-u)\cdot u_{11}$ and obtain
  \begin{align*}
    (-u)\cdot u_{11}\le&\,c(n,\alpha)\left(\frac{|Du|^2}{u_{11}(-u)}
      +|Du|^2(-u)u_{11}\right) +2|Du|^2 +n,\\
    e^w=&\,(-u)\cdot u_{11}\cdot e^{\frac12|Du|^2} \le c(\max
    w(\cdot,0), n,\alpha,\sup|Du|).
  \end{align*}
  The claim follows. 
\end{proof}
\fi

\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
  \[\begin{cases}
    \dot u=\sqrt{1+|Du|^2}\divergenz
    \left(\frac{Du}{\sqrt{1+|Du|^2}}\right) &\text{in
    }\R^n\times(0,\infty),\\
    u(\cdot,t)\to u_0 &\text{as }t\searrow0 \text{ in
    }C^0_{loc}\left(\R^n\right).
  \end{cases}\]
\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 pictures (only on the blackboard,
    not in these notes) should give an intuition about the behavior of
    these solutions.
    \begin{enumerate}[a)]
    \item A rotationally symmetric solution defined on a ball.
    \item A solution initially defined on a domain that will form a
      neck-pinch under mean curvature flow.
    \item A solution initially defined on an annulus.
    \item A solution defined on a domain in the plane bounded by
      up to countably many disjoint curves.  
    \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]
  \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}

\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}

Let $\eta:=(\eta_\alpha)=(0,\ldots,0,1)$ and define
$u:=X^\alpha\eta_\alpha$. Here, we consider $u$ as a function on the
evolving hypersurfaces rather than as a function depending on
$(x,t)\in\R^{n+1}\times[0,\infty)$.  Whenever we use quantities like
$v$ or $|A|^2$, we also use this meaning of $u$.

\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}
  The first equation follows from $\heat X=0$. For the remaining
  claims see \cite{EckerHuiskenInvent,KEMCFBook}. \par
  More details: For mean curvature flow, we have
  $F^{ij}=g^{ij}$. Hence $F^{ij}h_{ij}=H$ and \eqref{X evol} implies
  the evolution equation for $u$. \par
  For the evolution equation of $w:=|A|^2$, we calculate
  \begin{align*}
    \heat g_{ij} =&\,-2Hh_{ij},\qquad\text{see }\eqref{g
      evol},\umbruch\\
    \heat h_{ij} =&\,|A|^2h_{ij} -2Hh_i^ah_{aj},\qquad\text{see
    }\eqref{2ff evol},\umbruch\\
    w=&\,g^{ik}h_{ij}g^{jl}h_{kl},\umbruch\\
    \dot w=&\,2g^{ik}h_{ij}g^{jl}h_{kl}
    -2g^{ir}g^{sk}h_{ij}g^{jl}h_{kl}\dot g_{rs},\umbruch\\
    w_r=&\,2g^{ik}h_{ij;r}g^{jl}h_{kl},\umbruch\\
    w_{rs}=&\,2g^{ik}h_{ij;rs}g^{jl}h_{kl}
    +2g^{ik}h_{ij;r}g^{jl}h_{kl;s},\umbruch\\ 
    \heat |A|^2 =&\, 2g^{ik}(|A|^2h_{ij}-2Hh^a_ih_{aj})g^{jl}h_{kl}
    +4H\tr A^3 -2|\nabla A|^2\umbruch\\
    =&\,2|A|^4 -2|\nabla A|^2.\qedhere
  \end{align*}
\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}
  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{exercise}
  Consider $v(-u)$ to obtain similar $C^1$-estimates. 
\end{exercise}

\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 blow. 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\\
    \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, where time derivatives
  are counted twice. 
  \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 $\epsilon\searrow0$.
    \end{enumerate}
  \end{proof}

  There is also a parabolic maximum principle for tensors, see
  \cite[Theorem 9.1]{HamiltonThree}. (See the AMS-Review for a
  small correction of the proof.)\par
  A tensor $N_{ij}$ depending smoothly on $M_{ij}$ and $g_{ij}$, 
  involving contractions with the metric, is said to fulfill the
  null-eigenvector condition, if $N_{ij}v^iv^j\ge0$ for all
  null-eigenvectors of $M_{ij}$.
  \begin{theorem}\label{thm:weak.parab.max.princ.tensor}
    Let $M_{ij}$ be a tensor, defined on a closed Riemannian manifold
    $(M,g(t))$, fulfilling
    \[\delt M_{ij}=\Delta M_{ij}+b^k\nabla_kM_{ij}+N_{ij}\]
    on a time interval $[0,T)$, where $b$ is a smooth vector field and
    $N_{ij}$ fulfills the null-eigenvector condition. If $M_{ij}\ge0$
    at $t=0$, then $M_{ij}\ge0$ for $0\le t<T$.
  \end{theorem}

  \section{Some linear algebra}
  \label{sec:some-linear-algebra}
  \begin{lemma}
    We have
    $$\frac\partial{\partial a_{ij}}\det(a_{kl})
    =\det(a_{kl})a^{ji},$$ if $a_{ij}$ is invertible with inverse
    $a^{ij}$, i.\,e. if $a^{ij}a_{jk}=\delta^i_k$.
  \end{lemma}
  \begin{proof}
    It suffices to prove that the claimed inequality holds when we
    multiply it with $a_{ik}$ and sum over $i$. Hence, we have to show
    that
    $$\frac\partial{\partial a_{ij}}
    \det(a_{kl})a_{ik}=\det(a_{kl})\delta^j_k.$$ We get
    $$\frac\partial{\partial a_{ij}}\det(a_{kl})=
    \det\begin{pmatrix}
      a_{1\,1} & \hdots & a_{1\,j-1} & 0      & a_{1\,j+1} & \hdots &
      a_{1\,n}\\ 
      \vdots &        & \vdots    & \vdots & \vdots    &        &
      \vdots\\ 
      a_{i-1\,1} & \hdots & a_{i-1\,j-1} & 0 & a_{i-1\,j+1} & \hdots &
      a_{i-1\,n}\\ 
      0 & \hdots & 0 & 1 & 0 & \hdots & 0\\
      a_{i+1\,1} & \hdots & a_{i+1\,j-1} & 0 & a_{i+1\,j+1} & \hdots &
      a_{i+1\,n}\\ 
      \vdots &        & \vdots    & \vdots & \vdots    &        &
      \vdots\\ 
      a_{n\,1} & \hdots & a_{n\,j-1} & 0 & a_{n\,j+1} & \hdots &
      a_{n\,n}\\ 
    \end{pmatrix}.$$ and thus
    \begin{align*}
      \frac\partial{\partial a_{ij}}\det(a_{kl})\cdot a_{ik}=&
      \det\begin{pmatrix}
        0&\hdots&0&a_{1\,k}&0&\hdots&0\\
        a_{2\,1}&\hdots&a_{2\,j-1}&0&a_{2\,j+1}&\hdots&a_{2\,n}\\
        \vdots&&\vdots&\vdots&\vdots&&\vdots\\
        a_{n\,1}&\hdots&a_{n\,j-1}&0&a_{n\,j+1}&\hdots&a_{n\,n}
      \end{pmatrix}\umbruch\\
      &+\det\begin{pmatrix}
        a_{1\,1}&\hdots&a_{1\,j-1}&0&a_{1\,j+1}&\hdots&a_{1\,n}\\
        0&\hdots&0&a_{2\,k}&0&\hdots&0\\
        a_{3\,1}&\hdots&a_{3\,j-1}&0&a_{3\,j+1}&\hdots&a_{3\,n}\\
        \vdots&&\vdots&\vdots&\vdots&&\vdots\\
        a_{n\,1}&\hdots&a_{n\,j-1}&0&a_{n\,j+1}&\hdots&a_{n\,n}
      \end{pmatrix}\umbruch\\
      &+\hdots\\
      =&\det\begin{pmatrix}
        a_{1\,1}&\hdots&a_{1\,j-1}&a_{1\,k}&a_{1\,j+1}&\hdots&a_{1\,n}\\
        a_{2\,1}&\hdots&a_{2\,j-1}&0&a_{2\,j+1}&\hdots&a_{2\,n}\\
        \vdots&&\vdots&\vdots&\vdots&&\vdots\\
        a_{n\,1}&\hdots&a_{n\,j-1}&0&a_{n\,j+1}&\hdots&a_{n\,n}
      \end{pmatrix}\umbruch\\
      &+\det\begin{pmatrix}
        a_{1\,1}&\hdots&a_{1\,j-1}&0&a_{1\,j+1}&\hdots&a_{1\,n}\\
        a_{2\,1}&\hdots&a_{2\,j-1}&a_{1\,k}&a_{2\,j+1}&\hdots&a_{2\,n}\\
        a_{3\,1}&\hdots&a_{3\,j-1}&0&a_{3\,j+1}&\hdots&a_{3\,n}\\
        \vdots&&\vdots&\vdots&\vdots&&\vdots\\
        a_{n\,1}&\hdots&a_{n\,j-1}&0&a_{n\,j+1}&\hdots&a_{n\,n}
      \end{pmatrix}\umbruch\\
      &+\hdots\\
      =&\det\begin{pmatrix}
        a_{1\,1}&\hdots&a_{1\,j-1}&a_{1\,k}&a_{1\,j+1}&\hdots&a_{1\,n}\\
        \vdots&&\vdots&\vdots&\vdots&&\vdots\\
        a_{n\,1}&\hdots&a_{n\,j-1}&a_{n\,k}&a_{n\,j+1}&\hdots&a_{n\,n}
      \end{pmatrix}\umbruch\\
      =&\delta^j_k\det(a_{rs}).
    \end{align*}
  \end{proof}

  \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}

\end{appendix}

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\end{document}
