% 
% Geometric evolution equations 
% Oliver Schn\"urer
% FU Berlin, Sommer 2007
% Sommerschule Graduiertenkolleg 02.07.-07.07.2007, Alpbach (Tirol), 
%
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\begin{document}
\title{Geometric evolution equations}

%    Information for first author
\author{Oliver C. Schn\"urer}
%    Address of record for the research reported here
\address{Oliver C. Schn\"urer, Freie Universit\"at Berlin,
   Arnimallee 6, 14195 Berlin, Germany}
%    Current address
\curraddr{}
\def\fuaddress{@math.fu-berlin.de}
\email{Oliver.Schnuerer\fuaddress}
%    \thanks will become a 1st page footnote.
\thanks{}

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

\date{September 2007.}

\dedicatory{}

\keywords{}

\begin{abstract}
  Lecture notes for the Summer School of the International Research
  Training Group Arithmetic and Geometry, 02.09.--07.09.2007 in
  Alpbach (Tirol).
\end{abstract}

\maketitle

\tableofcontents

%\mainmatter
\addtocounter{section}{-1}
\section{Overview}
We consider flow equations that deform manifolds according to their
curvature. In the lectures, I will focus on two types of such flow
equations. \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
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 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. \par
We will also consider Ricci flow, where the metric $g(t)$ of a
Riemannian manifold $M$ evolves according to
\[\begin{cases}
  \delt g(t)=-2\Ric\,(g(t)),&\\
  g(0)=g_0.&\\
\end{cases}\] Manifolds that are initially positively curved in an
appropriate sense converge to space-forms after rescaling. For other
manifolds, solutions exist for all times. \par
Nowadays classical results were obtained by G. Huisken
\cite{HuiskenRoundSphere} for mean curvature flow and by R. Hamilton
\cite{HamiltonThree} for Ricci flow. \par
During the last years, geometric evolution equations have been used to
study geometric questions like isoperimetric inequalities, the
Sch\"onfliess conjecture, the Poincar\'e conjecture, Thurston's
geometrization conjecture, the $1/4$-pinching theorem, or Yau's
uniformization conjecture.
\par
In this course, I'll give an introduction to both types of flow
equations. We will learn to compute evolution equations, study
manifolds that become round in finite time and those that evolve
smoothly for all times. At the end, I'll indicate briefly, how
evolution equations and surgery can be used to address geometric
problems.

\begin{remark}\neueZeile
  \begin{enumerate}[(i)]
  \item We will use geometric flow equations as a tool to canonically
    deform a manifold into a manifold with nicer properties.
  \item The flow equations considered share many properties with the
    heat equation. In particular, they tend to balance differences,
    e.\,g.\ of the curvature, on the manifold. 
  \item In order to control the behavior of the flow, we will look for
    properties of the manifold that are preserved under the flow.
  \item For precise control on the behavior of the evolving manifold,
    we will look for quantities that are monotone and have geometric
    significance, i.\,e.\ their boundedness implies geometric
    properties of the evolving manifold. Often these quantities are
    not scaling invariant. Hence bounding them ``brakes the
    scaling invariance''. 
  \end{enumerate}
\end{remark}

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

We use $X=X(x,\,t)=\left(X^\alpha\right)_{1\le\alpha\le n}$ 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 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 surface is
called strictly convex, if all principal curvatures are strictly
positive.  The inverse of the second fundamental form is denoted by
$\left(\tilde h^{ij}\right)_{1\le i,\,j\le n}$.

Latin indices range from $1$ to $n$ and refer to geometric quantities
on the surface, 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 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
$F=-K^{-1}$, we have $F^{ij}=K^{-1}\tilde h^{ij}=
K^{-1}\lambda_i^{-1}g^{ij}$.

The Gau{\ss} equation expresses the Riemannian curvature tensor
of the surface 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 $\R^3$, $h_{ij;\,k}$ 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
  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 Definition}
We will only consider the evolution of manifolds of dimension $n$
embedded into $\R^{n+1}$, i.\,e.\ the evolution of hypersurfaces in
Euclidean space. (Mean curvature flow is also considered for manifolds
of arbitrary codimension. Another generalization is to study flow
equations of hypersurfaces immersed into a (Riemannian or Lorentzian)
manifold.)
\begin{definition}
  Let $M^n$ denote an orientable manifold of dimension $n$. Let
  $X(\cdot,t):M^n\to\R^{n+1}$, $0\le t\le T\le\infty$, be a smooth
  family of smooth embeddings. Let $\nu$ denote one choice of the
  normal vector field along $X(M^n,t)$. Then $M_t:=X(M^n,t)$ is said
  to move with normal velocity $F$, if
  \[\dt X=-F\nu\quad\text{in }M^n\times[0,T).\]
\end{definition}

In codimension $1$, we often don't need to assume that $M^n$ is
orientable.
\begin{remark}
  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
  \[\dot u=\sqrt{1+|Du|^2}
  \divergenz\left(\frac{Du}{\sqrt{1+|Du|^2}}\right).\]
\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}
  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}
  Let $u:\R^n\times[0,\infty)\to\R$ be smooth. Assume that
  $u(x,0)\equiv0$ 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]
  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 $t\uparrow0$ (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}

\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 solutions locally as graphs in a tubular neighborhood 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}
  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 
  Find examples of closed hypersurfaces such that 
  \begin{enumerate}[(i)]
  \item $H>0$,
  \item $K>0$,
  \item $H$ is not positive everywhere,
  \item $H>0$, but $K$ changes sign.
  \end{enumerate}
  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{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}
  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.
\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 $\graph u$. \par
  Denote by $\Gamma_F$ the set of $(\lambda_i)\subset\R^n$ such that
  $\fracp F{\lambda_i}>0$. Show that this set is a convex cone. Prove
  that $F$ as a function $\left(D^2u,Du\right)$ is strictly elliptic
  precisely if the principal curvatures corresponding to $(D^2u,Du)$
  lie in $\Gamma_F$.
\end{exercise}

\begin{corollary}\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}

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 $t\uparrow T$. 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 $t\uparrow T$. 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. 

\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
  and the inverse of a matrix. 
\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
\[\dt\nu^\alpha\delta_{\alpha\beta}X^\beta_k
=g^{ij}F_iX^\alpha_j\delta_{\alpha\beta}X^\beta_k =g^{ij}F_ig_{jk}
=\delta^i_kF_i=F_k.\]
\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}.&&
\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\umbruch\\
\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.
  \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_{;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}.}
  \end{align*}
\end{proof}

\begin{lemma}
  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.
  \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.
  \end{align*}
\end{proof}

\section{Graphical Solutions to Mean Curvature Flow}
\label{sec:graph-solut-mean}
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}
\begin{proof}[Strategy of proof]\neueZeile
  \begin{enumerate}[(i)]
  \item Approximate $u_0$ by smooth functions. Hence we will assume in
    the following that $u_0$ is smooth.
  \item Consider $R>0$. Eventually, we will let $R\to\infty$ in order
    to obtain a solution. Consider graphical mean curvature flow with
    initial condition $u_R$, 
    \[\begin{cases}
      \dot u=\sqrt{1+|Du|^2}\divergenz
      \left(\frac{Du}{\sqrt{1+|Du|^2}}\right) &\text{in
      }B_{3R}(0)\times(0,\infty),\\
      u(\cdot,0)=u_R &\text{in }B_{3R}(0),\\
      u(\cdot,t)=0&\text{on }\partial B_{3R}(0),
    \end{cases}\]
    where $u_R$ is smooth, $u_R=u_0$ in $B_R(0)$, and $u_R=0$ in
    $B_{3R}(0)\setminus B_{2R}(0)$. 
    Solutions to that initial value problem are known to exist for all
    times $t\ge0$ and they are smooth. We call the solution to that
    initial value problem $u^R$. 
  \item In order to be able to let $R\to\infty$ and to obtain a 
    subsequence that converges to a solution, we need interior
    estimates. For the first spatial derivatives, we will prove such
    an estimate in Theorem \ref{mcf gradient bound}. For higher order
    spatial derivatives, there are a priori estimates for the second
    fundamental form $A$ of the form
    \[\sup\limits_{B_{\frac12R}(x_0)}|\nabla^mA|^2(\cdot,t)\le
    c\left(m,n,\sup\limits_{B_R(x_0)\times[0,t]}|Du|\right)
    \cdot\left(\frac1{R^2}+\frac1t\right)^{m+1},\] if a solution to
    mean curvature flow can be written as a graph over a set
    $B_R(x_0)$, see \cite[Corollary 3.5]{EckerHuiskenInvent} for
    details.
  \item Apply Arzel\`a-Ascoli and obtain a subsequence that converges
    to a solution $u$ as desired.
  \end{enumerate}
\end{proof}

The following result is Theorem 2.3 in \cite{EckerHuiskenInvent},
obtained by K. Ecker and G. Huisken,
\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}
\begin{proof}
  Remember that $\osc u:=\sup u-\inf u$. Assume first that $u\ge0$ in
  $B_R(0)\times[0,T]$ by considering $u-\inf
  \limits_{B_R(0)\times[0,T]} u$ instead of $u$. (In the following
  definition of $\phi$, $u>0$ corresponds to $\langle X,\eta\rangle<0$
  for $\eta=(0,\ldots,0,-1)$. We will see, however, that $\dt\langle
  X,\eta\rangle-\Delta\langle X,\eta\rangle=0$, whereas $\dot u=vH$.
  The difference comes from the fact, that $\dot u$ describes the
  speed in vertical direction, whereas mean curvature flow prescribes
  the normal velocity.) Scaling the solution according to $\tilde
  u(x,t):=\frac1Ru\left(Rx,R^2t\right)$, we obtain another solution to
  mean curvature flow, defined on $B_1(0)\times\left[0,\frac
    T{R^2}\right]$. Proving the theorem for $\tilde u$ is equivalent
  to proving the theorem for $u$. Let us therefore assume that $R=1$.
  \par
  Consider the function $v\psi$ with $\psi:=-1+\exp(\lambda\phi)$ and
  \[\phi:=\left(\frac1{2\beta}\langle X,\eta\rangle
    +1-\left(|X|^2-\langle X,\eta\rangle^2\right)\right)_+,\] where
  $\lambda,\beta>0$ are constants to be chosen and
  $\eta=(0,\ldots,0,-1)$. Assume that for the first time $t_0$,
  $v\psi$ reaches a new positive maximum. Then $\psi$ is positive and
  hence smooth near the maximum and we get there
  \[(v\psi)_{;i}=0, \quad\dt(v\psi)-\Delta(v\psi)
  \equiv\dt(v\psi)-g^{ij}(v\psi)_{;ij}\ge0.\] The following
  calculations are also valid in this new maximum. Using the
  evolution equation for $v$, \eqref{v evol}, $F^{ij}=H^{ij}=g^{ij}$,
  and the extremal condition, we obtain
\begin{align*}
  0\le&\,\left(\dt v-\Delta v\right)\psi
  +v\left(\dt\psi-\Delta\psi\right) -2g^{ij}v_{;i}\psi_{;j}\\
  =&\,-vg^{ij}h^k_ih_{kj}\psi-2\frac\psi vg^{ij}v_iv_j
  +v\left(\dt\psi-\Delta\psi\right) -2g^{ij}v_{;i}\psi_{;j}\\
  \equiv&\,-v|A|^2\psi-2\frac\psi v|\nabla v|^2
  +v\left(\dt\psi-\Delta\psi\right)-2\langle\nabla
  v,\nabla\psi\rangle\\ 
  =&\,-v|A|^2\psi-2\frac\psi v|\nabla v|^2
  +v\left(\dt\psi-\Delta\psi\right)+2\left\langle\nabla
  v,\frac{\psi\nabla v}v\right\rangle\\ 
  =&\,-v|A|^2\psi+v\left(\dt\psi-\Delta\psi\right). 
\end{align*}
As $v>0$, $\psi\ge0$, we get
\[\dt\psi-\Delta\psi\ge0.\]
According to the definition of $\psi$, 
\begin{align}
  \psi_{;i}=&\,e^{\lambda\phi}\lambda\phi_{;i},\nonumber\umbruch\\
  \psi_{;ij}=&\,e^{\lambda\phi}\left(\lambda\phi_{;ij}
    +\lambda^2\phi_{;i}\phi_{;j}\right)\nonumber\umbruch\\
  \intertext{and hence}
  \dt\phi-\Delta\phi\ge&\,\lambda|\nabla\phi|^2.
  \label{heat phi}
\end{align}
Due to the definition of $\phi$, we see that
\begin{align*}
  \phi_{;i}=&\,\frac1{2\beta}X^\alpha_i\eta_\alpha
  -\left(|X|^2-\langle X,\eta\rangle^2\right)_i,\umbruch\\
  |\nabla\phi|^2=&\, \frac1{4\beta^2} \eta_\alpha
  X^\alpha_ig^{ij}X^\beta_j\eta_\beta +\left|\nabla\left(|X|^2-\langle
      X,\eta\rangle^2\right)\right|^2\\
  &\,-\frac1\beta\eta_\alpha X^\alpha_ig^{ij} \left(|X|^2-\langle
    X,\eta\rangle^2\right)_j\umbruch\\
  \ge&\, \frac1{4\beta^2} \eta_\alpha
  X^\alpha_ig^{ij}X^\beta_j\eta_\beta -\frac1\beta\eta_\alpha
  X^\alpha_ig^{ij} \left(|X|^2-\langle X,\eta\rangle^2\right)_j,
  \umbruch\\
  \dt\phi-\Delta\phi=&\,\frac1{2\beta}\left(\dt\langle X,\eta\rangle
    -\Delta\langle X,\eta\rangle\right)\\
  &\,-\left\{\dt\left(|X|^2-\langle X,\eta\rangle^2\right)
    -\Delta\left(|X|^2-\langle
      X,\eta\rangle^2\right)\right\}\umbruch\\
  =&\,\frac1{2\beta}\left\langle\dt X-\Delta X,\eta\right\rangle
  -2\left\langle X,\dt X-\Delta X\right\rangle\\
  &\,+2\langle
  X,\eta\rangle\left\langle\dt X-\Delta X,\eta\right\rangle
  +2g^{ij}X^\alpha_iX^\beta_j\ol g_{\alpha\beta}
  -2g^{ij}X^\alpha_iX^\beta_j\eta_\alpha\eta_\beta\\
  \le&\,2g^{ij}X^\alpha_iX^\beta_j\ol g_{\alpha\beta}\umbruch\\
  =&\,2g^{ij}g_{ij}\umbruch\\
  =&\,2n,\\
  \intertext{where we have used that}
  \dt X-\Delta X=&\,-H\nu-g^{ij}X_{;ij}
  =-H\nu+g^{ij}h_{ij}\nu =0.
\end{align*}
Using \eqref{heat phi} and the estimate for $|\nabla\phi|^2$, we
deduce that 
\begin{equation}
  \label{lambda beta est}
  \lambda\left(\frac1{4\beta^2} \eta_\alpha
    X^\alpha_ig^{ij}X^\beta_j\eta_\beta -\frac1\beta\eta_\alpha
    X^\alpha_ig^{ij} \left(|X|^2-\langle
      X,\eta\rangle^2\right)_j\right)\le 2n.
\end{equation}
Note that $\ol g^{\alpha\beta}\equiv\delta^{\alpha\beta}
=X^\alpha_ig^{ij}X^\beta_j +\nu^\alpha\nu^\beta$. This can be verified
by testing the equation with the unit normal $\nu$ and tangent vectors
$X_i$. Hence
\begin{align*}
  \eta_\alpha X^\alpha_ig^{ij}X^\beta_j\eta_\beta=&\, \eta_\alpha
  \ol g^{\alpha\beta}\eta_\beta
  -\eta_\alpha\nu^\alpha\nu^\beta\eta_\beta\umbruch\\
  =&\,1-\langle\eta,\nu\rangle^2\umbruch\\
  =&\,1-v^{-2},\umbruch\\
  \eta_\alpha X^\alpha_ig^{ij}\left(|X|^2-\langle
    X,\eta\rangle^2\right)_j =&\,\eta_\alpha X^\alpha_ig^{ij} \cdot
  X^\gamma_j \left(2X^\beta\ol g_{\beta\gamma} -2\langle X,\eta\rangle
    \eta_\gamma\right)\umbruch\\
  =&\,2\eta_\alpha\left(\ol g^{\alpha\gamma}-\nu^\alpha\nu^\gamma\right)
  \left(X^\beta\ol g_{\beta\gamma} -\langle X,\eta\rangle
    \eta_\gamma\right)\umbruch\\
  =&\,2\langle\eta,X\rangle
  -2\langle\eta,\nu\rangle\langle\nu,X\rangle\\
  &\,-2|\eta|^2\langle
  X,\eta\rangle +2\langle
  X,\eta\rangle\langle\eta,\nu\rangle^2\umbruch\\ 
  =&\,-2\langle\eta,\nu\rangle\langle\nu,X\rangle +2\langle
  X,\eta\rangle\langle\eta,\nu\rangle^2.
\end{align*}
In coordinates, we have $\eta=(0,-1)$, $\nu=v^{-1}(Du,-1)$, and
$X=(x,u)$. According to the definition of $\phi$ and in view of
$\langle X,\eta\rangle<0$, $v\psi\equiv0$ if $|x|^2+u^2\ge1$. Hence we
may assume that $|x|<1$ and $|u|<1$. We compute
\begin{align*}
  -2\langle\eta,\nu\rangle\langle\nu,X\rangle +2\langle
  X,\eta\rangle\langle\eta,\nu\rangle^2=&\,
  -2v^{-1}v^{-1}(\langle Du,x\rangle-u)-2uv^{-2}\umbruch\\
  \le&\,2v^{-2}|Du|\cdot1\umbruch\\
  \le&\,2v^{-1}.
\end{align*}
Combining these estimates with \eqref{lambda beta est}, we get
\[\lambda\left(\frac1{4\beta^2}(1-v^{-2})-\frac2\beta v^{-1}\right)
\le2n.\] Now we define $\lambda:=64n\beta^2$ and obtain (by a direct
calculation) $7v^2-64\beta v-8\le0$ and $v\le4+16\beta$. Note that
$\phi\le1$. We denote $\graph u(\cdot,t)|_{B_1(0)}\cap\{\psi>0\}$ by
$\tilde M_t$. Hence
\[\max\limits_{\tilde M_{t_0}}v\psi\le(4+16\beta)\sup\limits_{\tilde
  M_{t_0}}\psi \le(4+16\beta)e^\lambda=(4+16\beta)e^{64n\beta^2}\] and
thus, as we have considered an arbitrary new increasing maximum of
$v\psi$,
\[v\psi\le\sup\limits_{\tilde M_0}v\psi+(4+16\beta)e^{64n\beta^2} \le
e^{64n\beta^2}\left(4+16\beta+\sup\limits_{B_1(0)}\sqrt{1+|Du|^2(\cdot,0)}
\right)\] everywhere on $\tilde M_t$, $0\le t<T$.  We set
$\beta:=1+\sup\limits_{t\in[0,T]}u(0,t)$. Therefore, we get for
$t\in[0,T]$ and $x=0$
\begin{align*}
  v\psi=&\,\sqrt{1+|Du|^2(0,t)}
  \left(-1+e^{64n\beta^2\left(\frac{-u(0,t)}{2+2\sup\limits_{t\in[0,T]}
          u(0,t)} +1\right)_+}\right)\umbruch\\
  \ge&\,\sqrt{1+|Du|^2(0,t)}
  \left(-1+e^{64n\left(-\frac12 +1\right)}\right)\umbruch\\
  \ge&\,\sqrt{1+|Du|^2(0,t)}.
\end{align*}
We conclude that 
\[\sqrt{1+|Du|^2(0,t)}\le c(n)\sup\limits_{B_1(0)\times\{0\}}
\sqrt{1+|Du|^2}\cdot \exp\left(c(n) \sup\limits_{t\in[0,T]}
  u(0,t)^2\right).\] The theorem follows.
\end{proof}

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{Convex Hypersurfaces}
\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 $\delta=0$.
\end{exercise}

\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{n|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}

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}\umbruch\\
    =&\,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 $h_{22;2}$. 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 es 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.
  \end{enumerate}
\end{proof}

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.
\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}

\subsection{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 $M_t$ which are topologically spheres.
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, as we are on a topological sphere, 
\[\int\limits_{M_t}H^2=\int\limits_{M_t}(\lambda_1-\lambda_2)^2
+\int\limits_{M_t}4K\ge 4\int\limits_{M_t}K=4\mathcal H^2(\partial
B_1(0)) \equiv 4|\partial B_1(0)| =16\pi.\] 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.

\section{Mean Curvature Flow with Surgery}
\label{sec:mean-curvature-flow}
We consider mean curvature flow. Even for a smooth initial surface
which is topologically a sphere, singularities will occur before the
surface can shrink to a point.\par
The following example can be found in greater generality in
\cite{KEMCFBook}.
\begin{example}
In order to see this, we consider a family of
hypersurfaces in $\R^3$ given by
\[M_t=\left\{(x,y,z)\in\R^3:\tfrac12z^2=x^2+y^2-1+t\right\}\] for
$0\le t<1$. For $t\nearrow 1$, these hypersurfaces converge to a cone
with singularity at the origin. We want to see that this hypersurface
is a barrier for mean curvature flow. Locally, we can write it near
$y=0$ as a graph $x=\sqrt{\frac12z^2-y^2+1-t}$. Due to the symmetry,
it suffices to understand the evolution near $y=0$. Observe that the
function $u(x^1,x^2):= \sqrt{\frac12(x^1)^2-(x^2)^2+1-t}$ fulfills
\[\dot u\ge\sqrt{1+|Du|^2} \divergenz
\left(\frac{Du}{\sqrt{1+|Du|^2}}\right)\] for $0\le t<1$ and
$x^2\approx0$. This is a direct calculation (exercise). That is, the
surface $M_t$ shrinks slower in the direction of the axis $z=0$ than a
surface flowing by mean curvature flow. Hence we can apply a
comparison principle principle (proved exactly as the avoidance
principle, Theorem \ref{avoidance thm}, using the inequality in the
maximum principle) to see that any compact surface flowing according
to mean curvature flow which is initially in the set $\{\frac12z^2\ge
x^2+y^2-1+0\}$, will be contained in the set $\{\frac12z^2\ge
x^2+y^2-1+t\}$ for all $0\le t<1$, if it evolves smoothly. Note that
the surfaces $M_t$ are not compact, but the avoidance principle still
applies as it only needs that one of the surfaces is compact, as in
such a case there has to be a point of first contact. \par
We now pick a surface $N$ that encloses two sufficiently large balls
(which evolve smoothly up to $t=2$) on different sides of the neck of
$M_t$. If the surface $N_t$ evolves smoothly, it continues to enclose
those balls and to be contained in the set $\{\frac12z^2\ge
x^2+y^2-1+t\}$ for $0\le t<1$. Hence it has to pass through the neck.
This however, is only possible, if the embedding becomes singular
before $t=1$. Therefore, singularities may occur before a surface
shrinks to a point.
\end{example}

We have seen that in general, it is not to be expected that a smooth
solution to mean curvature flow exists for smooth initial data. It is
possible to study mean curvature flow in a weak setting, e.\,g.\ using
a level-set flow. Another possibility is to do surgery before
singularities occur. This is done by G. Huisken and C. Sinestrari in
\cite{HuiskenSinestrari3}. A hypersurface with principal curvatures
$\lambda_1\le\ldots\le\lambda_n$ is called \emph{$2$-convex,} if,
everywhere on the hypersurface, $\lambda_1+\lambda_2\ge0$. The main
theorem in \cite{HuiskenSinestrari3} implies
\begin{theorem}
  Let $M\subset\R^{n+1}$, $n\ge3$, be a smooth closed embedded
  $2$-convex hypersurface. Then there exists a mean curvature flow
  starting from $M$ with a finite number of surgeries.
\end{theorem}
\begin{proof}[Strategy of proof]\neueZeile
  \begin{enumerate}[(i)]
  \item Show that $2$-convexity is preserved during mean curvature
    flow.
  \item Let the surfaces evolve until the curvature becomes big
    somewhere.
  \item Prove a priori estimates that allow to control the behavior of
    hypersurfaces near a point, where the curvature is big.
  \item Convex parts shrink to round points and are understood. Other
    regions of high curvature look like a neck $\S^{n-1}\times[a,b]$
    or $\S^{n-1}\times\S^1$, or like a degenerate neck-pinch. Necks of
    the form $\S^{n-1}\times\S^1$ are shown to be diffeomorphic to the
    model space $\S^{n-1}\times\S^1$. Necks of the form
    $\S^{n-1}\times[a,b]$ or degenerate neck-pinches are cut out so
    that the curvature drops. Therefore, it has to be shown that a
    piece (=connected component) of $M_t$ with a point of high
    curvature that is not convex or diffeomorphic to
    $\S^{n-1}\times\S^1$ has a neck-like region.
  \item Each time a surgery is performed, a certain amount of area is
    cut out. Hence there can be at most finitely many surgeries. In
    order to cut out a certain amount of area, the authors follow a
    neck until it opens up. This ensures the lower area bound on the
    differences of the areas in the surgery process.
  \item The hard part is to control the behavior of the flow
    analytically. We list a few key steps to do that
    \begin{enumerate}[(a)]
    \item In regions, where the curvature is big, the hypersurface is
      almost convex. For every $\delta>0$, there exists
      $c(\delta,M_0)>0$, such that 
      \[\lambda_1\ge-\delta H-c(\delta,M_0).\]
    \item If $|\lambda_1|$ is small, then the other principal
      curvatures are close to each other. (Note that this corresponds
      to a cylinder.) It has to be shown that such a ``curvature
      cylinder'' is also a ``geometric cylinder''. The (pointwise)
      cylindrical estimate reads: For every $\eta>0$,
      \[|\lambda_1|\le\eta H\quad\Longrightarrow\quad
      |\lambda_i-\lambda_j|^2\le\eta H^2+c(\eta,M_0,n)\quad
      \text{ for }i,j\ge2.\]
    \item A pointwise gradient estimate. It states that the second
      fundamental form does not change much between points nearby.
      \[|\nabla A|^2\le c(n)|A|^4+c(M_0).\]
    \item All this is done with few parameters that control the
      geometry of the hypersurfaces. Up to the choice of these
      parameters, the surgeries are carried out in a canonical way. It
      is important that the control on the geometry is uniformly
      maintained during the surgeries.
    \end{enumerate}
  \end{enumerate}
\end{proof}

\begin{remark}\neueZeile
  \begin{enumerate}[(i)]
  \item The surgery is a procedure that allows to control the geometry
    of the hypersurfaces. Such is in general not known for other weak
    formulations. See, however, \cite{HuiskenIlmanenPenrose}, where
    the Euler characteristic is controlled for a weak formulation of
    inverse mean curvature flow. Hence mean curvature flow with
    surgeries allows to reconstruct the topology of the original
    manifold. It turns out that all $2$-convex hypersurfaces are
    topologically $\S^n$ or a finite connected sum of
    $\S^{n-1}\times\S^1$.
  \item The idea to combine a geometric flow equation with a surgery
    procedure appears also in the work of R. Hamilton and G. Perelman
    on the Ricci flow.
  \item Mean curvature flow with surgery can be considered a
    ``canonical'' way of deforming a manifold into standard pieces.
    Keep in mind that surgery and forming a connected sum are
    operations inverse to each other. Hence such procedure is a way to
    ``geometrize'' embedded $2$-convex manifolds.
  \item In Section \ref{sec:convex-hypersurfaces}, we had seen such a
    geometrization program for convex hypersurfaces. Without imposing
    curvature conditions, the possible singularities seem not to allow
    something like surgery. This is different in $3$ dimensions if we
    consider abstract manifolds, i.\,e.\ manifolds which are not
    embedded as hypersurfaces. 
  \item There is a strong similarity between extrinsic flows, i.\,e.\
    flows of manifolds embedded in some other manifold like
    $\R^{n+1}$, and intrinsic flows, i.\,e.\ flows, where only the
    (Riemannian) metric changes in time. 
  \end{enumerate}
\end{remark}

\section{Ricci Flow}
\label{sec:ricci-flow}

\begin{definition}
  A Riemannian metric $g(t)$ is evolving under Ricci flow with initial
  condition $g(0)$, a Riemannian metric, if
  \begin{equation}
    \label{eq:Ricci-flow}
    \begin{cases}
      \delt g(t)=-2\Ric(g(t)),\\
      g(0)=g_0.
    \end{cases}
  \end{equation}
\end{definition}

\begin{remark}[Existence]
For smooth initial data on a compact manifold, short-time existence is
proved in \cite{HamiltonThree} using the Nash-Moser implicit function
theorem. A problem in the existence proof is that the evolution
equation is degenerate parabolic, corresponding to the invariance of
the Ricci flow under diffeomorphisms. In a more direct approach one
considers an equivalent flow, the Ricci-DeTurck or Ricci harmonic map
heat flow, which is strictly parabolic, to prove existence. This is
known as the DeTurck trick \cite{DeTurckTrick}.
\end{remark}

\begin{definition}[Solitons]
Special solutions are those that evolve only by diffeomorphisms and
scaling. They are called Ricci solitons. For these solitons the metric
evolves as follows
\[g(t)=\sigma(t)\phi^*(t)g(0),\] where $\sigma(t)>0$ and $\phi(t)$ is
a family of diffeomorphisms.  This can be rewritten in terms of a
constant $\epsilon$ corresponding to $\dot\sigma$ and a vector field
$X$ induced by $\phi(t)$ as
\begin{align*}
-2\Ric(g_0)=&\,\epsilon g_0+\mathcal L_{X_0}g_0\umbruch\\
\intertext{or, equivalently, as}
-2R_{ij}=&\,\nabla_iX_j+\nabla_jX_i+\epsilon (g_0)_{ij}.
\end{align*}
For $\epsilon>0$, the soliton is called, expanding; solitons with
$\epsilon=0$ are called steady; if $\epsilon<0$, the soliton is
shrinking. (Note that $\R^n$ with its standard Euclidean metric is at
the same time an expanding, a steady, and a shrinking soliton.)\par
If $X$ is the gradient of a function $f$, the soliton equation becomes
\[R_{ij}+\nabla_i\nabla_jf+\frac\epsilon2g_{ij}=0.\]
\end{definition}

\begin{example}[Cigar soliton]
  Consider $\left(\R^2,u(x,t)\delta\right)$, where $\delta$ denotes
  the standard Euclidean metric, $\lambda>0$, and
  $u(x,t)=\frac1{\lambda|x|^2+e^{4\lambda t}}$. Then
  $\left(\R^2,u(x,t)\delta\right)$ fulfills Ricci flow.
\end{example}
\begin{exercise}
  Prove that in this situation, Ricci flow is equivalent to $\dot
  u=\Delta\log u$. It might be helpful to prove first that
  $R=-1/u\cdot\Delta\log u$. Show that the cigar soliton is a gradient
  soliton to Ricci flow. 
\end{exercise}

\section{Evolution Equations} 
\label{sec:evolution-equations}
We use the following definitions, see e.\,g.\ \cite{HamiltonThree}.
\begin{definition}
We define\\
the Christoffel symbols
\[\Gamma^h_{ij}=\tfrac12g^{hk}(g_{jk,i}+g_{ik,j}-g_{ij,k}),\]
the Riemannian curvature tensor $R^k_{ijl}=R_{ij}{}^k{}_l$ 
\[R_{ij}{}^k{}_l:=\Gamma^k_{jl,i}-\Gamma^k_{il,j}
+\Gamma^k_{ip}\Gamma^p_{jl} -\Gamma^k_{jp}\Gamma^p_{il},\]
and in covariant form
\[R_{ijkl}=g_{kh}R_{ij}{}^h{}_l,\]
the Ricci tensor
\[R_{ik}=g^{jl}R_{ijkl},\]
and the scalar curvature
\[R=g^{ij}R_{ij}.\]  
\end{definition}

In order to compute the evolution equation for the Riemannian
curvature tensor, it is convenient to consider a normal coordinate
system such that $\Gamma^k_{ij}=0$ at the point considered. To avoid
confusion about the order of differentiation, we write $\partial_i$
and $\partial_t$. We obtain
\begin{align*}
  \partial_t\Gamma^h_{jl}=&\,\tfrac12g^{hm}(\partial_j\partial_t
  g_{lm}
  +\partial_l\partial_tg_{jm}-\partial_m\partial_tg_{jl}),\umbruch\\
  \partial_tR_{ij}{}^h{}_l=&\,\partial_i\partial_t\Gamma^h_{jl}
  -\partial_j\partial_t\Gamma^h_{il},\umbruch\\ 
  \partial_tR_{ijkl}=&\,g_{hk}\partial_tR_{ij}{}^h{}_l
  +\partial_tg_{hk}R_{ij}{}^h{}_l,\umbruch\\
\intertext{and hence}
\partial_tR_{ijkl}=&\,\partial_i\partial_kR_{jl}-\partial_i\partial_lR_{jk}
-\partial_j\partial_kR_{il} +\partial_j\partial_lR_{ik}\\
&\,-g^{pq}(R_{ijkp}R_{ql}+R_{ijpl}R_{qk}). 
\end{align*}



The following Lemma \cite[Lemma 7.2]{HamiltonThree} is a consequence
of the Bianchi identities
\begin{align*}
  0=&\,R_{ijkl}+R_{iklj}+R_{iljk},\umbruch\\
  0=&\,\partial_iR_{jklm}+\partial_jR_{kilm}+\partial_kR_{ijlm},
\end{align*}
of differentiating these identities, exchanging derivatives, permuting
indices and contracting. 
\begin{lemma}
  We have
  \begin{align*}
    \Delta R_{ijkl}+&\,2(B_{ijkl}-B_{ijlk}-B_{iljk}+B_{ikjl})\\
=&\,\partial_i\partial_kR_{jl} -\partial_i\partial_lR_{jk}
-\partial_j\partial_kR_{il} +\partial_j\partial_lR_{ik}\\
&\,+g^{pq}(R_{pjkl}R_{qi}+R_{ipkl}R_{qj}),\\
    \intertext{where}\\
    B_{ijkl}=&\,g^{pr}g^{qs}R_{piqj}R_{rksl}.
  \end{align*}
\end{lemma}
\begin{proof}
  Exercise (long computation).
\end{proof}

Based on the above considerations, we obtain the evolution equation
for $R_{ijkl}$, see \cite[Theorem 7.1]{HamiltonThree},
\begin{theorem}
  The Riemannian curvature tensor satisfies the evolution equation
  \begin{align*}
    \delt R_{ijkl}=&\,\Delta
    R_{ijkl}+2(B_{ijkl}-B_{ijlk}-B_{iljk}+B_{ikjl})\\
    &\,-g^{pq}(R_{pjkl}R_{qi}+R_{ipkl}R_{qj}
    +R_{ijpl}R_{qk}+R_{ijkp}R_{ql}).
  \end{align*}
\end{theorem}

\begin{corollary}
  The Ricci tensor and the scalar curvature fulfill the following
  evolution equations
  \begin{align}
    \label{eq:ricci-tensor}
    \delt R_{ik}=&\,\Delta R_{ik} +2g^{pr}g^{qs}R_{piqk}R_{rs}
    -2g^{pq}R_{pi}R_{qk},\umbruch\\ 
    \label{eq:scalar-curvature}
    \delt R=&\,\Delta R+2g^{ij}g^{kl}R_{ik}R_{jl}.
  \end{align}
\end{corollary}

\section{Spherical Space Forms}
\label{sec:space-forms}
Comparable to the results in Section \ref{sec:convex-hypersurfaces},
there are results for Ricci flow that show that manifolds converge,
after appropriate rescaling, smoothly to quotients of the round
sphere.

The classical result in this direction is by R. Hamilton
\cite{HamiltonThree} 
\begin{theorem}\label{thm:hamilton-three}
  Let $(M,g)$ be a compact 3-manifold of positive Ricci curvature.
  Then there exists a smooth solution $(M,g(t))$ to
  \[\begin{cases}
    \delt g(t)=-2\Ric,\\
    g(0)=g
  \end{cases}\] on some finite time interval $[0,T)$. As $t\nearrow T$,
  after appropriate rescaling, $(M,g(t))$ converges to a manifold of
  constant positive curvature, i.\,e.\ of constant positive sectional
  curvatures.
\end{theorem}
\begin{remark}\neueZeile
  \begin{enumerate}[(i)]
  \item In three dimensions,
    \[R_{ijkl}=g_{ik}R_{jl}-g_{il}R_{jk}-g_{jk}R_{il}+g_{jl}R_{ik}
    -\tfrac12R(g_{ik}g_{jl}-g_{il}g_{jk}),\] i.\,e.\ the Riemannian
    curvature tensor is determined by the Ricci tensor. In appropriate
    coordinates $R_{ij}=\diag(\lambda,\mu,\nu)$.
  \item We list a few important estimates that are all obtained by
    applying the maximum principle to appropriate test functions.
  \item In view of \eqref{eq:scalar-curvature}, $R>0$ is preserved
    during the flow.
  \item $\epsilon Rg_{ij}\le R_{ij}\le\beta Rg_{ij}$ is preserved
    during the flow for $0<\epsilon\le\frac13<\beta<1$.
  \item Positive sectional curvatures are preserved.
  \item The pinching estimate is a crucial estimate. It states that
    for some $\delta>0$
    \[(\lambda-\mu)^2+(\lambda-\nu)^2+(\mu-\nu)^2\le c R^{2-\delta}.\]
    It is the first estimate that is better than expected from
    scaling.
  \item For any $\eta>0$, we have the gradient estimate
    \[|\nabla R|^2\le\eta R^3+c(\eta,g(0)).\]
  \item As $t\nearrow T$, we have $\max\limits_M|\Ric|\to\infty$ and
    thus $\max\limits_MR\to\infty$.
  \item From the gradient estimate, we obtain $\max R/\min R\to 1$ as
    $t\nearrow T$.
  \item Hence, after rescaling, $(M,g(t))$ converges to a manifold
    with $\lambda=\mu=\nu$ as $t\nearrow T$.
  \end{enumerate}
\end{remark}

In contrast to mean curvature flow, as far as the results of this
section are concerned, most of the a priori estimates rely on maximum
principle.

Theorem \ref{thm:hamilton-three} has been improved by C. B\"ohm and B.
Wilking to any dimension and initial metrics of 2-positive curvature
operator \cite{BoehmWilking}. The main idea there is to carefully
study the properties of the Riemannian curvature tensor in the
ordinary differential equation related to Ricci flow and to introduce
a continuous family of pinching cones. Ricci flow has to enter these
cones transversally. Hence it has to converge to the intersection of
all that cones, which consists of the metrics of constant (sectional)
curvature.

This has been extended to manifolds $M$ such that $M\times\R$ or
$M\times\R^2$ has positive isotropic curvature (PIC) by S. Brendle and
R. Schoen, see \cite{SimonBrendlePIC,SimonBrendle14Pinching,%
  BrendleSchoen14Pinched}. PIC is defined using orthonormal
four-frames by the inequality
\[R_{1313}+R_{1414}+R_{2323}+R_{2424}-2R_{1234}>0.\]\par
It implies in particular that a compact strictly $1/4$-pinched
Riemannian manifold is diffeomorphic to a space form, i.\,e.\ it
implies the differentiable $1/4$-pinching theorem. This result
requires only pointwise strict $1/4$-pinching for the sectional
curvatures. Here, it is always understood that the sectional
curvatures are positive.\par
The crucial step is to show that PIC is preserved under Ricci flow.
It involves carefully studying the algebra in the corresponding
ordinary differential equation.

\section{Non-Compact Solutions}
\label{sec:non-comp-solut}

W.-X. Shi \cite{ShiJDG1989} has considered Ricci flow of non-compact
complete manifolds of bounded curvature. He gives a short time
existence proof in this situation. The proof relies on 
\begin{enumerate}[(i)]
\item solving appropriate Dirichlet problems on a sequence of balls of
  increasing radius,
\item interior a priori estimates, some of them interior in space and
  time.
\end{enumerate}
This is similar to what we have seen for graphical mean curvature
flow. Note however, that for Ricci flow bounded curvature does not
guarantee the existence of a solutions as neck-pinches may occur. \par
He has also proved non-compact maximum principles and has shown that
manifolds that are initially K\"ahler stay K\"ahler. For those
manifolds, the K\"ahler potential $u$ fulfills (at least locally) 
\[\dot u=\log\det u_{i\bar\jmath}.\]\par
This flow is (currently) the main tool to attack the following
conjecture of S.-T. Yau: A complete non-compact K\"ahler manifold with
positive holomorphic bisectional curvature is biholomorphic to
$\mathbb C^n$. \par
The strategy is to let the metric evolve according to K\"ahler-Ricci
flow. The biholomorphisms are then constructed with the help of the
limit manifold (possibly after rescaling) as $t\to\infty$. This
requires currently initially bounded curvature and at least one
additional condition like average quadratic curvature decay
\[\frac1{|B_r(x)|}\int\limits_{B_r(x)}R\,d\mu\le\frac c{(1+r)^2}
\quad\text{for all }x\in M,\, r>0,\] 
or maximal volume growth
\[\lim\limits_{r\to\infty}\frac{|B_r(x)|}{r^n}>0\quad\text{for some
}x\in M.\] See the survey article of A. Chau and L.-F. Tam
\cite{AlbertSurvey} for details.\par
Interior a priori estimates allow to reduce the smoothness that has to
be assumed for the initial data. M. Simon \cite{MilesC0} could start
Ricci flow for $C^0$ initial data $g_0$, i.\,e.\ he constructs a
smooth solution $g(t)$ that approximates $g_0$ in $C^0$ as
$t\searrow0$.\par
If $g_0$ is initially $C^0$-close to Euclidean space with the standard
metric and the difference becomes small at infinity, the corresponding
solutions were shown to converge to Euclidean space as $t\to\infty$
under Ricci harmonic map heat flow, see \cite{OSFSMSStabilityRicci}.
The proof is based on integral estimates involving
\[\sum\limits_{i=1}^n\frac1{\lambda_i^m}(\lambda_i^m-1)^2,\]
where $m\ge2$ and $\lambda_i$ denotes the eigenvalues of $g(t)$ with
respect to the flat background metric.

\section{Ricci Flow with Surgery}
\label{sec:ricci-flow-with}
Ricci flow with surgery was introduced by R. Hamilton
\cite{HamiltonFour} for four-manifolds of positive isotropic
curvature. Similar to mean curvature flow with surgery, it is
important to understand precisely, how singularities look like in
order to perform surgery. R. Hamilton made a lot of progress in this
direction, but he could not exclude singularities that look locally
like
\[\left(\R^2,\frac1{1+|x|^2}\delta\right)\times(\R,\delta),\]
a cigar soliton times $\R$, after blow-up. It is important to exclude
such a soliton as it does not seem to allow for surgery.\par
G. Perelman could exclude such singularities. This is part of his
recent work on Ricci flow of 3-manifolds
\cite{Perelman1,Perelman2,Perelman3}. Recent expositions of this
material include \cite{CaoZhuRicciAsian,MorganTianRicciBook,%
  BigRicciBook1,ChowLuNiRicciBook,ChowKnopfRicciBook,ToppingRicciBook}.
We will only sketch how to exclude such singularities.\par
Define for the metric $g$, a smooth function $f$ and $\tau>0$ the
functional $\mathcal W$ by
\[\mathcal W(g,f,\tau)=\int\left[\tau\left(R+|\nabla f|^2\right)
  +f-n\right]u \,d\mu(g),\] where $u=(4\pi\tau)^{-n/2}e^{-f}$ and $n$
is the dimension of the manifold. We require that $\int u=1$. This is
called compatibility of $u$ or $f$. Set
\[\mu(g,\tau):=\inf\limits_f\mathcal W(g,f,\tau),\]
where the infimum is taken over all compatible functions $f$. This
infimum is finite and attained for some smooth function $f$.\par
We have the following monotonicity: If $M$ is closed and $g$, $f$, and $\tau$
evolve according to
\[\begin{cases}
\fracp gt=-2\Ric,\\
\frac{d\tau}{dt}=-1,\\
\fracp ft=-\Delta f+|\nabla f|^2-R+\frac n{2\tau},
\end{cases}\]
then 
\[\dt\mathcal W(g,f,\tau)=2\tau\int
\left|\Ric+\Hess f-\frac{g}{2\tau}\right|^2 u\,d\mu\ge0.\]
This implies a lower bound on $\mu(g,\tau)$ during the flow. Such a
lower bound implies (longer computations) a positive lower bound on 
the volume ratio for geodesic balls,
\[\frac{\mu(B_r(p))}{r^n}>\zeta>0,\] if $|\Riem|\le r^{-2}$ on
$B_r(p)$, where the lower bound depends on $n$, $g(0)$, and upper
bounds for $r$ and $T$.\par
Now we consider a special situation in dimension $n=3$. If a solution
$g(t)$ runs into a singularity as described above, we can choose small
balls $B_r(p)$. If we fix the centers so that the balls lie on the
region, where the cigar looks cylindrical, we obtain (after rescaling)
the following situation.  (Note that $|\Riem|\le r^{-2}$ is preserved
under scaling and so is the volume ratio.) The geodesic balls then
roughly look like $\S^1\times B_r\subset\R^2\times\R^2$ and hence
violate the lower volume bound for large values of $r$ as the volume
of these geodesic balls grows like $r^2$.

\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 have two
  spatial and one time derivative, if all these derivatives are
  continuous.
  \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 $\epsilon\downarrow0$.
    \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}

\bibliographystyle{amsplain} 
%\bibliography{../os/tools/biblio.bib}
\def\weg#1{} \def\unterstrich{\underline{\rule{1ex}{0ex}}} \def\cprime{$'$}
  \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{10}

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\end{thebibliography}

\end{document}
