%
% Expandierende und kontrahierende Fl\"achen
% Oliver Schn\"urer
% FU Berlin, Dezember 2004
%
\documentclass{amsbook}
\usepackage{ifthen, citesort}
\usepackage{amssymb,amsmath}
%\usepackage[notref,notcite]{showkeys}
\usepackage{epic}
%\renewcommand{\familydefault}{phv}
%\usepackage[american,german]{babel}
%%\selectlanguage{american}

\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\divergence}{div}
\DeclareMathOperator{\divergenz}{div}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\osc}{osc}
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}
\def\S{\mathbb{S}}
\def\Z{\mathbb{Z}}
\def\dt{\frac{d}{dt}}
\def\fracp#1#2{\frac{\partial #1}{\partial #2}}
\def\Frskl
{\left(\tilde h^{rs}\tilde h^{kl}-\tilde h^{rk}\tilde h^{sl}\right)}
\def\Amod{\vert A\vert}
\def\theta{\vartheta}
\def\phi{\varphi}
\def\epsilon{\varepsilon}
\newcommand{\A}[1]{\ifthenelse{#1 = 2}{\lvert A\rvert^{#1}}{\tr A^{#1}}}
\newcommand{\Akl}[2]{\ifthenelse{#1 = 2}%
{\left(\lvert A\rvert^{#1}\right)^{#2}}%
{\left(\tr A^{#1}\right)^{#2}}}
\def\Az#1{\left(\A2\right)_{;\,#1}}
\def\B#1{\tr A^{#1}}
\def\hij{h_{ij;\,k}^2}
\def\hii{h_{ii;\,k}h_{jj;\,k}}
\def\li{\lambda_i}
\def\lj{\lambda_j}
\def\lk{\lambda_k}
\def\lx{\lambda_1}
\def\ly{\lambda_2}
\def\opr#1{\dt{#1}-F^{ij}{\left(#1\right)}_{;\,ij}}
\def\oprokl#1{\dt{#1}-F^{ij}{#1}_{;\,ij}}
\def\fracd#1#2{\displaystyle\frac{\displaystyle #1}{\displaystyle #2}}
\def\fracm#1#2{\textstyle\frac{\scriptstyle #1}{\scriptstyle #2}}
\long\def\etamatrix#1#2#3{\begin{pmatrix}\eta_{11}\\
\eta_{22}\end{pmatrix}^{\text{tr}}\begin{pmatrix}#1 &#2\\
#2& #3\end{pmatrix}\begin{pmatrix}\eta_{11}\\
\eta_{22}\end{pmatrix}}
\long\def\symmatrix#1#2#3{\begin{pmatrix}#1 &#2\\
#2& #3\end{pmatrix}}
\long\def\umbruch{{\displaybreak[1]}}
\def\glqq{``}
\def\grqq{''}

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

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

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

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

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

\hyphenation{mo-no-to-ni-ci-ty Nor-mal-en-ge-schwin-dig-keit
  ei-nem po-si-tiv}

\begin{document}
\frontmatter

\title{Expandierende und kontrahierende Fl\"achen}

%    Information for first author
\author{Oliver C. Schn\"urer}
%    Address of record for the research reported here
\address{FU Berlin, Arnimallee 2-6, 14195 Berlin, Germany}
%    Current address
\curraddr{}
\email{Oliver.Schnuerer@math.fu-berlin.de}
%    \thanks will become a 1st page footnote.
\thanks{}

%    General info
\subjclass[2000]{53C44, 35B40, 65K05, 68W20}
% 53C44 Geometric evolution equations (mean curvature flow)
% 35B40 Asymptotic behavior of solutions
% 65K05 Mathematical programming {Algorithms; for theory see 90Cxx}
% 68W20 Randomized algorithms 

\date{December 2004 - \today.}

\dedicatory{}

\keywords{}

\begin{abstract}
We show that strictly convex surfaces contracting with normal 
velocity equal to $\A2$ shrink to a point in finite time.
After appropriate rescaling, they converge to spheres.

Strictly convex surfaces that expand
by the inverse Gau{\ss} curvature flow converge
to infinity in finite time. After appropriate rescaling, 
they converge to spheres. 

We describe the algorithm to find our main test functions. 
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle

\tableofcontents

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\part{Expandierende und kontrahierende Fl\"achen}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Einleitung}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Problemstellung}\label{problemstellung}

In dieser Arbeit besch\"aftigen wir uns mit Fl\"achen im
dreidimensionalen eu\-kli\-di\-schen Raum, die sich in
Abh\"angigkeit ihrer Kr\"ummung bewegen. Wir betrachten
stets glatte, zusammenh\"angende, strikt konvexe, geschlossene, 
eingebettete Fl\"achen, die sich entweder zusammenziehen oder 
ausdehnen.

Unsere beiden wichtigsten Beispiele sind dabei die 
Flussgleichung 
\begin{equation}\label{A2 Fluss}
\dt X=-\A2\nu
\end{equation}
f\"ur kontrahierende Fl\"achen, wobei sich eine Fl\"ache
mit Normalengeschwindigkeit gleich dem Quadrat der Norm
der zweiten Fundamentalform zusammenzieht, und der
inverse Gau{\ss}kr\"ummungsfluss
\begin{equation}\label{1/K Fluss}
\dt X=\frac1K\nu
\end{equation}
f\"ur expandierende Fl\"achen. 

Wir erkl\"aren nun, wann sich eine Fl\"ache gem\"a{\ss} 
einer Evolutionsgleichung 
\begin{equation}\label{F Fluss}
\dt X=-F\nu
\end{equation}
bewegt. Sei $M$ eine abstrakte, d.\,h.\ nicht notwendigerweise
eingebettete, zweidimensionale Mannigfaltigkeit. F\"ur unsere
Zwecke k\"onnen wir hier stets die zweidimensionale Sph\"are
$\S^2\subset\R^3$ betrachten. Sei $X(\cdot,\,t):M\to\R^3$ eine
glatte Familie von regul\"aren Einbettungen in einem
Zeitintervall, $t\in[0,\,T)$, $T>0$. Dies definiert eine Familie 
von Fl\"achen $M_t:=X(M,\,t)$ in $\R^3$. Diese Fl\"achen
$M_t$ besitzen in jedem Punkt $X(x,\,t)\in M_t$ zwei Normalenvektoren
der L\"ange eins. Davon w\"ahlen wir einen aus, in unserer
Situation konvexer geschlossener Fl\"achen $M_t$ stets die \"au{\ss}ere
Normale, $\nu(x)$. Wir sagen nun, dass Fl\"achen $M_t$
die Flussgleichung
\eqref{F Fluss} l\"osen, wenn es eine zugeh\"orige Abbildung $X$
wie oben beschrieben gibt, so dass in jedem Punkt 
$(x,\,t)\in M\times[0,\,T)$ die Gleichung \eqref{F Fluss} f\"ur 
eine gegebene Funktion $F$ erf\"ullt ist. 

Die Einbettungen
$X(\cdot,\,t)$ induzieren auf $M$ in jedem Punkt eine Metrik
$$(g_{ij})_{1\le i,\,j\le 2}$$ und eine zweite Fundamentalform 
$$(h_{ij})_{1\le i,\,j\le 2}.$$ Wir identifizieren $M\times\{t\}$
mit $X(M,\,t)$ und erhalten in jedem Punkt $x\in M_t$ Metrik und
zweite Fundamentalform. In dieser Arbeit ist die Normalengeschwindigkeit
$F$ stets eine symmetrische Funktion der Eigenwerte $(\lx,\,\ly)$ von
$h_{ij}$ bez\"uglich $g_{ij}$,
$$\A2=\lx^2+\ly^2,\qquad K=\lx\cdot\ly.$$
Diese Eigenwerte hei{\ss}en Hauptkr\"ummungen und sind nur
bis auf Permutation wohldefiniert. Daher sind nur symmetrische
Funktionen der Hauptkr\"ummungen wohldefiniert. 

Wir werden Flussgleichungen \eqref{F Fluss} mit der Anfangsbedingung,
dass $M_0$ eine vorgegebene glatte, strikt konvexe, geschlossene Fl\"ache
in $\R^3$ ist, untersuchen. Strikt konvex bedeutet hier, dass die 
Hauptkr\"ummungen \"uberall auf $M_0$ jeweils strikt positiv sind. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Literatur\"uberblick}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Kontrahierende Fl\"usse}

Flussgleichungen der Form \eqref{F Fluss} und ihre Analoga
f\"ur Hyperfl\"achen werden schon seit mehreren Jahrzehnten
intensiv untersucht. 

Wir betrachten zun\"achst kontrahierende Fl\"usse, d.\,h.\
solche, f\"ur die $F$ positiv ist. William Firey 
betrachtete den Gau{\ss}kr\"ummungsfluss f\"ur Fl\"achen als Modell,
das beschreibt, wie konvexe Steine, die in einem Fluss oder am
Strand aus allen Richtungen mit gleicher Wahrscheinlichkeit kleine
St\"o{\ss}e abbekommen, allm\"ahlich schrumpfen. Er vermutete, 
dass sich die Fl\"achen in endlichen Zeit zu einem Punkt 
zusammenziehen und dass sie nach geeignetem Reskalieren zu einer
runden Sph\"are konvergieren. Unter den Annahmen, dass eine
L\"osung der Flussgleichung existiert und dass die Anfangswerte
punktsymmetrisch sind, konnte er dies auch beweisen
\cite{FireyStones}. Kaising Tso zeigte, dass L\"osungen in jeder
Dimension existieren \cite{TsoPoint} und Ben Andrews f\"uhrte
den Beweis f\"ur Fl\"achen ohne anf\"angliche Symmetrieannahme
\cite{AndrewsStones}. Die letztgenannte Arbeit ist der
vorliegenden in einigen Aspekten \"ahnlich. Daher werden wir
sp\"ater nochmals hierauf zur\"uckkommen. 

Gerhard Huisken bewies \cite{HuiskenRoundSphere}, 
dass sich konvexe Hyperfl\"achen unter
dem Fluss entlang ihrer mittleren Kr\"ummung in endlicher Zeit
zu einem Punkt zusammenziehen und, geeignet reskaliert, zu
Sph\"aren konvergieren, d.\,h.\ sie konvergieren zu 
\glqq runden Punkten\grqq. Die mittlere Kr\"ummung ist
positiv homogen vom Grade eins in den Hauptkr\"ummungen,
$$H=\lx+\ldots+\lambda_n.$$
Konvergenz zu runden Punkten f\"ur (strikt) konvexe 
Anfangswerte wurde danach noch h\"aufiger f\"ur andere
Flussgleichungen bewiesen, bei denen die Normalengeschwindigkeit
positiv homogen vom Grade eins in den Hauptkr\"ummungen ist, z.\,B.\ von
Ben Andrews \cite{Andrews2dnonconcave,AndrewsPinching,AndrewsContractCalc}
und Bennett Chow \cite{ChowNthRoot,ChowScalarRoot}.
F\"ur Hyperfl\"achen, die sich unter einer Flussgleichung 
zusammenziehen, bei denen die Normalengeschwindigkeit positiv
homogen von einem Grad strikt gr\"o{\ss}er als eins ist, haben
Ben Andrews \cite{Andrews2dnonconcave} und Felix Schulze \cite{OSFelixH2}
gezeigt, dass eine geeignete anf\"angliche \glqq Pinchingbedingung\grqq,
d.\,h.\ eine geeignete Schranke f\"ur den Quotienten 
beliebiger Haupt\-kr\"um\-mun\-gen
$\lambda_i/\lambda_j$, in jedem Punkt auf $M_0$
ausreicht, um Konvergenz gegen
einen runden Punkt nachzuweisen. 

Ist die Homogenit\"at der Normalengeschwindigkeit positiv, aber
strikt kleiner als eins, so konvergieren im allgemeinen nicht mehr
alle strikt konvexen Hyperfl\"achen gegen runde Punkte. Dies erh\"alt
Ben Andrews \cite{AndrewsPacific}, falls die Normalengeschwindigkeit
$F$ eine geeignete Potenz der Gau{\ss}kr\"ummung ist. Insbesondere
schrumpft jedes Ellipsoid als affines Bild einer Sph\"are unter dem 
affin invarianten Gau{\ss}\-kr\"um\-mungs\-fluss mit $F=K^{1/(n+2)}$
homothetisch zu einem Punkt. Analoge Resultate gibt es auch von
Koichi Anada und Masayoshi Tsutsumi f\"ur den harmonischen
mittleren Kr\"ummungsfluss \cite{AnadaStability,AnadaCalcHarm}.

Kontrahierende Hyperfl\"achen wurden auch in Mannigfaltigkeiten
betrachtet, vgl.\ z.\,B.\ \cite{AndrewsContrRiem,AndrewsBrisbane,%
HuiskenSphereMf}, ebenso f\"ur anisotrope Kr\"ummungsfl\"usse,
vgl.\ z.\,B.\ \cite{AndrewsMonotone}. F\"ur kontrahierende
Kurven gibt es weitergehende Resultate, z.\,B.\ \cite{GageHamilton,%
GraysonRoundPoints}. Hier braucht man keine Konvexit\"at f\"ur die
Anfangskurve zu fordern. F\"ur den Riccifluss wurde ebenso das
Rundwerden von Mannigfaltigkeiten untersucht \cite{HamiltonThree,%
HamiltonFour86}. Kompliziertere Verhaltensweisen in Singularit\"aten
sind f\"ur den mittleren Kr\"ummungsfluss \cite{SteveThroughSing,%
HuiskenSinestrari1,HuiskenSinestrari2,HuiskenSinestrari3} 
und f\"ur den Riccifluss
\cite{HamiltonFour,Perelman1,Perelman2,Perelman3} untersucht worden. 

Au{\ss}er dem Resultat von Ben Andrews 
(\cite{AndrewsStones} und \cite[Theorem 3/4]{AndrewsPacific}
im nicht strikt konvexen Fall) 
f\"ur den Gau{\ss}kr\"ummungsfluss
f\"ur Fl\"achen ist uns kein Resultat bekannt, das f\"ur einen
kontrahierenden Kr\"ummungsfluss mit Normalengeschwindigkeit
einer Homogenit\"at strikt gr\"o{\ss}er als eins in den 
Hauptkr\"ummungen Konvergenz konvexer Anfangswerte gegen einen
runden Punkt ohne anf\"angliche Pinchingbedingung liefert. 
In der vorliegenden Schrift zeigen wir dies beispielsweise 
f\"ur strikt konvexe Fl\"achen, die sich mit Normalengeschwindigkeit
$\A2$ zusammenziehen \eqref{A2 Fluss}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Expandierende Fl\"usse}

Bisher wurden expandierende Fl\"usse intensiv 
f\"ur Normalengeschwindigkeiten untersucht, die positiv homogen
vom Grade minus eins in den Hauptkr\"ummungen sind. Claus Gerhardt
und John Urbas haben jeweils sternf\"ormige Hyperfl\"achen
betrachtet, die f\"ur $t\to\infty$ exponentiell gegen Unendlich
und nach geeignetem Reskalieren gegen Sph\"aren konvergieren
\cite{CGFlowSpheres,UrbasExpandMZ,UrbasExpandJDG}.
John Urbas erh\"alt hier entsprechende Resultate auch f\"ur
negative Homogenit\"aten, die gr\"o{\ss}er als minus eins sind.
Wir wollen dieses Langzeitverhalten auch als Konvergenz gegen
Sph\"aren im Unendlichen bezeichnen. Knut Smoczyk fand eine
Darstellungsformel f\"ur den inversen harmonischen mittleren
Kr\"ummungsfluss \cite{KnutHarmonicMCF}. Gerhard Huisken und
Tom Ilmanen haben mittels einer schwachen Formulierung des 
inversen mittleren Kr\"ummungsflusses die Penroseungleichung
bewiesen \cite{HuiskenIlmanenPenrose}. 
Weitergehende Untersuchungen hierzu finden sich in 
\cite{SmoczykAsian2000,HuiskenIlmanenIMCF2}.

Inhomogene Fl\"usse, deren L\"osungen f\"ur $t\to\infty$ gegen 
Sph\"aren im Unendlichen konvergieren, finden sich in 
\cite{MinkowskiFlow,ChowNonhomogGauss,ChowAsian,IvochkinaNehringTomi}.

Expandierende Kr\"ummungsfl\"usse einer Homogenit\"at strikt
kleiner als minus eins scheinen bisher noch \"uberhaupt nicht
auf Konvergenz gegen Sph\"aren nach Reskalieren
untersucht worden zu sein. Unter solchen Fl\"ussen expandieren
Sph\"aren in endlicher Zeit nach Unendlich. In dieser Arbeit 
zeigen wir, dass beispielsweise L\"osungen des
inversen Gau{\ss}kr\"ummungsflusses \eqref{1/K Fluss}
f\"ur strikt konvexe An\-fangs\-fl\"a\-chen
in endlicher Zeit gegen runde Sph\"aren im Unendlichen 
konvergieren. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Existenz einer L\"osung}

Die Normalengeschwindigkeiten $F$ in \eqref{A2 Fluss} und 
\eqref{1/K Fluss} erf\"ullen die Bedingung
$$\fracp F\li>0$$
genau in den Punkten, in denen $M_t$ strikt konvex ist. Daher
erhalten wir f\"ur glatte strikt konvexe Anfangswerte einen
parabolischen Kr\"ummungsfluss, der auf einem maximalen
Zeitintervall $[0,\,T)$, $0<T\le\infty$, eine glatte Familie
strikt konvexer, kompakter Fl\"achen $M_t$ besitzt, die
\eqref{A2 Fluss} bzw.\ \eqref{1/K Fluss} erf\"ullen
\cite{HuiskenPolden}. Dies bezeichnet man als Kurzzeitexistenzresultat.

Ist f\"ur alle Zeiten $0\le t<T$ und alle $x\in M$ die Norm des
Einbettungsvektors $|X(x,\,t)|$ gleichm\"a{\ss}ig beschr\"ankt und
erf\"ullen auf $M\times[0,\,T)$ die Hauptkr\"ummungen $\li$ f\"ur
eine Konstante $c>0$ die Ungleichung $0<\frac1c\le\li\le c$, 
$i=1,\,2$, so erhalten wir aufgrund der H\"olderschranken f\"ur zweite
Ableitungen von Nicolai Krylov, Mikhail Safonov, Lawrence Evans, 
Ben Andrews \cite{Andrews2dKrylov} und der Schauderabsch\"atzungen
durch Betrachtung der Gleichung f\"ur die im Folgenden noch
definierte St\"utzfunktion, dass sich die Familie $M_t$ noch in
glatter Weise bis zur Zeit $t=T$ fortsetzen l\"asst. Auf die Fl\"ache
$M_T$ k\"onnen wir nun das Kurzzeitexistenzresultat anwenden und 
erhalten eine glatte Familie strikt konvexer Fl\"achen $M_t$
auf einem Zeitintervall $[0,\,T+\epsilon)$, $\epsilon>0$. 
Dies widerspricht der Annahme, dass $T$ maximal war. Daher
existiert eine L\"osung auf einem Zeitintervall $[0,\,T)$ mit
$T=\infty$, was aus anderen Gr\"unden f\"ur \eqref{A2 Fluss}
und \eqref{1/K Fluss} leicht auszuschlie{\ss}en ist, oder
$0<T<\infty$ und es gilt
\begin{align*}
\limsup\limits_{t\uparrow T}\sup\limits_{M_t}
|X(\cdot,\,t)|=&\infty,\umbruch\\
\liminf\limits_{t\uparrow T}\inf\limits_{M_t}
\min\limits_{i=1,\,2}\li=&0,\umbruch
\intertext{oder}
\limsup\limits_{t\uparrow T}\sup\limits_{M_t}
\max\limits_{i=1,\,2}\li=&\infty.
\end{align*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Resultate und Beweis\"uberblick}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Kontrahierende Fl\"achen}

F\"ur kontrahierende Fl\"achen lautet unser wich\-tig\-stes Resultat
\begin{theorem}\label{dt A2 thm}
Sei $M\subset\R^3$ eine glatte, strikt konvexe, geschlossene
Fl\"ache. Dann gibt es $T$, $0<T<\infty$, und eine glatte
Familie $M_t$, $t\in[0,\,T)$, von strikt konvexen geschlossenen Fl\"achen
in $\R^3$, die die Flussgleichung
$$\dt X=-\A2\nu$$
mit $M_0=M$ l\"osen. F\"ur $t\uparrow T$ konvergieren die Fl\"achen
$M_t$ gegen einen Punkt $Q\in\R^3$. Die reskalierten Fl\"achen
$(M_t-Q)\cdot(6(T-t))^{-1/3}$ konvergieren in glatter Weise 
f\"ur $t\uparrow T$ gegen die Einheitssph\"are $\S^2\subset\R^3$.
\end{theorem}

Das Theorem besagt also, dass zentrierte Sph\"aren, die sich
homothetisch zusammenziehen und deren Radius durch $(6(T-t))^{-1/3}$
gegeben ist, das typische Verhalten f\"ur diesen Fluss zeigen.

Im ersten Teil des Beweises zeigen wir ganz \"ahnlich wie 
Kaising Tso in \cite{TsoPoint}, dass sich die Fl\"achen $M_t$ in
endlicher Zeit zu einem Punkt zusammenziehen: Schon der Vergleich
mit einer anf\"anglich $M$ umschlie{\ss}enden Sph\"are zeigt,
dass L\"osungen aufgrund des Maximumprinzips nur f\"ur endliche
Zeit existieren k\"onnen. Solange die Fl\"achen eine Sph\"are
mit positivem Radius umschlie{\ss}en, erhalten wir obere
Schranken an $\A2$. Da positive untere Schranken an die 
Hauptkr\"ummungen erhalten bleiben, folgt die Konvergenz 
zu einem Punkt. 

In Beweisen, dass Hyperfl\"achen nach Reskalieren gegen Sph\"aren
konvergieren, spielt h\"aufig eine Gr\"o{\ss}e, deren
Skalierungsverhalten \glqq besser\grqq{} als skalierungsinvariant
ist, eine Schl\"usselrolle. Hier haben wir das folgende Resultat
\begin{theorem}\label{dt A2 mon}
Unter den Voraussetzungen von Theorem \ref{dt A2 thm} ist
$$\max\limits_{M_t}\frac{(\lx+\ly)(\lx-\ly)^2}{\lx\ly}$$
w\"ahrend des Flusses nichtwachsend. 
\end{theorem}

Aufgrund des Skalierungsverhaltens dieser Testgr\"o{\ss}e 
erhalten wir sofort, dass die einzigen glatten, strikt 
konvexen, geschlossenen L\"osungen, die sich homothetisch
zusammenziehen, sich homothetisch kontrahierende Sph\"aren sind. 
Auch solch eine Klassifikation selbst\"ahnlicher L\"osungen
scheint f\"ur die hier betrachteten Fluss\-glei\-chun\-gen neu
zu sein. 

Heuristisch erwarten wir, dass Theorem \ref{dt A2 mon} sofort
Konvergenz gegen Sph\"aren nach Reskalieren liefert, denn
wir erwarten, dass die beiden Hauptkr\"ummungen \"uberall
gegen Unendlich konvergieren. Damit wird die Fl\"ache \"uberall
fast umbilisch. Da die einzigen umbilischen, strikt konvexen, 
vollst\"andigen Fl\"achen in $\R^3$ Sph\"aren sind, sollte also
aus obigen Argumenten Konvergenz gegen Sph\"aren folgen. 

Im Gegensatz zu Flussgleichungen der Homogenit\"at eins
(d.\,h.\ die Normalengeschwindigkeit ist positiv homogen
vom Grade eins in den Hauptkr\"ummungen) 
ist die hier gefundene monotone Gr\"o{\ss}e
keine Modifikation einer skalierungsinvarianten Gr\"o{\ss}e
wie in \cite{HuiskenRoundSphere}. Aufgrund der Gegenbeispiele
in \cite{Andrews2dnonconcave} erwarten wir hier \"uberhaupt
nicht, dass es entsprechende, f\"ur den Konvergenzbeweis
gegen Sph\"aren geeignete skalierungsinvariante Gr\"o{\ss}en
gibt.

Theorem \ref{dt A2 mon} folgt unmittelbar aus dem
Maximumprinzip. In\-te\-gral\-ab\-sch\"atz\-un\-gen 
sind hier nicht n\"otig. 

Wir finden auch noch andere Testgr\"o{\ss}en, die eine
zu Theorem \ref{dt A2 mon} analoge Aussage liefern. 
Durch die Betrachtung komplizierterer Gr\"o{\ss}en 
k\"onnen wir bessere Konvergenzraten erhalten. 

Eine weitere Folgerung aus Theorem \ref{dt A2 mon} ist,
dass der Pinchingquotient $\lx/\ly$ (und damit auch
$\ly/\lx$, da $\lx$ und $\ly$ nur bis auf Permutation 
definiert sind)
w\"ahrend der Evolution beschr\"ankt bleibt. 

Der Rest des Beweises \"ahnelt nun dem von Ben Andrews f\"ur den
Gau{\ss}\-kr\"um\-mungs\-fluss \cite{AndrewsStones}. Basierend
auf einer besser als skalierungsinvarianten Absch\"atzung
f\"ur $|\lx-\ly|$ aus Theorem \ref{dt A2 mon} zeigen wir
zun\"achst, dass die Fl\"achen zwischen je zwei zentrierten
Sph\"aren liegen, deren Radien auch nach geeignetem Reskalieren f\"ur
$t\uparrow T$ noch gegeneinander konvergieren. Mittels einer
Harnackungleichung erhalten wir dann, dass sich auch die
Normalengeschwindigkeit nur wenig von der einer selbst\"ahnlich
kontrahierenden Sph\"are unterscheidet. Dies gilt dann auch f\"ur
die Haupt\-kr\"um\-mun\-gen und damit haben wir so gute 
Pinchingabsch\"atzungen, dass Konvergenz auch nach Reskalieren
aus bekannten Resultaten \cite{Andrews2dnonconcave} folgt. 

Zu Theorem \ref{dt A2 thm} und Theorem \ref{dt A2 mon}
analoge Resultate haben wir auch f\"ur weitere Flussgleichungen
mit Homogenit\"aten strikt gr\"o{\ss}er als eins bewiesen.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Algorithmen}

Die Testgr\"o{\ss}e in Theorem \ref{dt A2 mon} haben wir mit
Hilfe eines Computerprogrammes gefunden, das testet, ob eine
symmetrische, als Quotient zweier Polynome gegebene Funktion, 
geeignet ist, um Konvergenz zu Sph\"aren zu liefern, 
d.\,h.\ m\"ogliche Testgr\"o{\ss}en sollen insbesondere
in umbilischen Punkten $(\lx=\ly)$ ein globales Minimum haben
und ein geeignetes Skalierungsverhalten aufweisen. Wir 
rechnen dann in Normalkoordinaten f\"ur solche Kandidaten
$w$ die Evolutionsgleichung, d.\,h.\ $\oprokl w$, in der
Notation von Kapitel \ref{diff geom evol sec}, 
in einem kritischen Punkt von $w$ aus und werten diese an
Zufallszahlen f\"ur die Hauptkr\"ummungen und die Ableitungen
der zweiten Fundamentalform aus, die so gew\"ahlt sind,
dass $w_{;\,i}=0$ erf\"ullt ist, d.\,h.\ wir ben\"utzen
auch, dass wir in einem kritischen Punkt von $w$ sind. Dies
wiederholen wir einige Male. Erhalten wir stets ein Vorzeichen,
das die Anwendbarkeit des Maximumprinzips nicht ausschlie{\ss}t, 
untersuchen wir diesen Kandidaten von Hand genauer. Bisher
waren alle so gewonnenen Kandidaten geeignet, um Konvergenz
gegen runde Punkte zu zeigen. Wir k\"onnen auf diese Weise
viele m\"ogliche Testgr\"o{\ss}en untersuchen.

Die weiteren Betrachtungen h\"angen jedoch nicht von der
Richtigkeit des Programmes ab. Wir benutzen es lediglich, um
geeignete Testgr\"o{\ss}en zu \glqq raten\grqq. 

Im Gegensatz zu den analogen Betrachtungen f\"ur 
expandierende Fl\"usse scheint es f\"ur kontrahierende Fl\"achen zu 
den meisten Flussgleichungen monotone Gr\"o{\ss}en zu geben.

Der angegebene Algorithmus l\"a{\ss}t sich auch auf 
Hyperfl\"achen anwenden. F\"ur dreidimensionale Hyperfl\"achen
lieferte er uns jedoch keine monotonen Gr\"o{\ss}en. 

Wir wollen noch bemerken, dass es in den Evolutionsgleichungen
f\"ur Fl\"achen auch mit Hilfe des Sturmschen Algorithmusses
anstatt mit Zufallszahlen m\"oglich w\"are, das Vorzeichen
der in der Evolutionsgleichung auftretenden Terme 
algebraisch zu untersuchen. 

Die hier betrachteten Testgr\"o{\ss}en lassen sich leicht
als Funktionen der Elemente einer algebraischen Basis, 
$H$ und $\A2$ f\"ur Fl\"achen, darstellen. Die zugeh\"origen
Evolutionsgleichungen erh\"alt man daraus mit Hilfe einfacher
algebraischer Formeln. Auf diese Weise k\"onnen wir sie leicht
mit Hilfe eines Computeralgebraprogrammes ausrechnen lassen. 
F\"ur unsere beiden Modellgleichungen, \eqref{A2 Fluss} und
\eqref{1/K Fluss}, haben wir jedoch ausf\"uhrliche und ohne
Computer nachvollziehbare Berechnungen der jeweiligen
Evolutionsgleichungen in dieser Arbeit. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Expandierende Fl\"achen}

In Teil \ref{1/K part} der Arbeit erweitern wir unsere
Betrachtungen auf expandierende Fl\"achen. Dabei interessieren
wir uns insbesondere f\"ur solche Flussgleichungen, die 
eine Homogenit\"at kleiner oder gleich minus zwei haben. F\"ur
Homogenit\"aten strikt kleiner als minus eins konvergieren 
Sph\"aren in endlicher Zeit homothetisch nach Unendlich.

Unser Modelltheorem f\"ur expandierende Fl\"achen lautet
\begin{theorem}\label{dt K thm}
Sei $M\subset\R^3$ eine glatte, strikt konvexe, geschlossene
Fl\"ache. Dann gibt es $T$, $0<T<\infty$, und eine glatte
Familie $M_t$, $t\in[0,\,T)$, 
von strikt konvexen geschlossenen Fl\"achen
in $\R^3$, die die Flussgleichung
$$\dt X=\frac1K\nu$$
mit $M_0=M$ l\"osen. F\"ur $t\uparrow T$ konvergieren die Fl\"achen
$M_t$ gegen Unendlich. Die reskalierten Fl\"achen
$(M_t-Q)\cdot(T-t)$ konvergieren in glatter Weise 
f\"ur $t\uparrow T$ gegen die Einheitssph\"are $\S^2\subset\R^3$.
\end{theorem}

Auch hier weisen wir also nach, dass das Verhalten von
Sph\"aren unter dieser Evolutionsgleichung das typische ist.

Hier ben\"otigen wir schon ganz zu Anfang des Beweises das
folgende 
\begin{theorem}\label{dt K mon}
Unter den Voraussetzungen von Theorem \ref{dt K thm} ist
$$\max\limits_{M_t}\frac{(\lx-\ly)^2}{\lx^2\ly^2}$$
w\"ahrend des Flusses nichtwachsend. 
\end{theorem}

Der Beweis davon benutzt wiederum nur das Maximumprinzip. 

Auch diese monotone Gr\"o{\ss}e liefert eine Klassifikation
selbst\"ahnlicher L\"osungen wie im Falle kontrahierender
Fl\"achen, da die Gr\"o{\ss}e ein entsprechendes 
Ska\-lier\-ungs\-ver\-hal\-ten besitzt. 

Im expandierenden Fall ben\"otigen wir diese Gr\"o{\ss}e
schon um zu zeigen, dass die Fl\"achen w\"ahrend des
Flusses gepincht bleiben, d.\,h.\ dass $\frac\lx\ly+\frac\ly\lx$
beschr\"ankt bleibt. Wir erhalten positive
untere Schranken an die Geschwindigkeit, solange die
betrachteten Fl\"achen in einem Ball enthalten sind. 
Dies sichert, dass die Evolutionsgleichung so lange eine
L\"osung besitzt, bis ein Teil der Fl\"ache nach Unendlich
konvergiert. Durch Vergleich mit Sph\"aren erhalten wir,
dass dies schon in endlicher Zeit passiert. 
Nach einem Spiegelungsargument von Aleksandrov
muss dann aber auch die gesamte Fl\"ache gegen Unendlich
konvergieren und die zugeh\"origen Oszillationsabsch\"atzungen
f\"ur die St\"utzfunktion liefern bereits Konvergenz
gegen runde Sph\"aren im Unendlichen. 

Durch Betrachtung komplizierterer Testgr\"o{\ss}en k\"onnen
wir auch hier die Konvergenzraten verbessern. Daf\"ur 
argumentieren wir dann wieder \"ahnlich wie in Teil \ref{A2 part}.

Auch hier erhalten wir \"ahnliche monotone Gr\"o{\ss}en f\"ur
andere Flussgleichungen, es handelt sich dabei um solche einer 
Homogenit\"at kleiner oder gleich minus zwei. 

Der zum Auf\/finden der monotonen Gr\"o{\ss}en verwendete
Algorithmus ist nahezu identisch zu dem im Falle 
kontrahierender Fl\"achen oben bereits beschriebenen.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Perspektiven}

In der geometrischen Analysis ist es wichtig, geeignete 
Testfunktionen zu finden. H\"aufig wei{\ss} man dabei,
von welchen Gr\"o{\ss}en die Testfunktionen abh\"angen
d\"urfen und welche Eigenschaften sie idealerweise zu
erf\"ullen haben. In solchen Situationen kann man dann
wie hier den Computer zum Einsatz bringen um
herauszufinden, ob eine Vermutung richtig sein kann,
\"ahnlich wie hier beim Auf\/finden der monotonen Gr\"o{\ss}en,
oder einfach nur, um kompliziertere Rechnungen 
durch\-zu\-f\"uh\-ren.

Durch das Testen mit Zufallszahlen haben wir uns schon
fr\"uher plausibel ge\-macht, dass Lemma 4.1 in \cite{OSMM}
gelten sollte, aber auch, dass es nicht sinnvoll ist,
die I-Methode, vgl.\ 
\cite{GigliolaRefinedNLS,GigliolaWellposed,GigliolaStabil}, 
weiter zu iterieren. 

Im Abschnitt \ref{conv rate} untersuchen wir eine Testgr\"o{\ss}e,
deren Evolutionsgleichung keine \glqq konstanten Terme\grqq\ mehr
enth\"alt. F\"ur alle anderen Evolutionsgleichungen treten solche
Terme aber noch auf. Wir erwarten, dass man bei aufw\"andigerer
Suche auch in diesen F\"allen monotone Gr\"o{\ss}en finden 
k\"onnte, die eine bessere Konvergenzrate liefern und in deren 
Evolutionsgleichung keine konstanten Terme mehr auftreten.
M\"oglicherweise reicht es hierf\"ur nicht mehr aus, 
lediglich Quotienten
von Polynomen in den Hauptkr\"ummungen zu betrachten.

F\"ur den mittleren Kr\"ummungsfluss sind nach punktf\"ormigen
Singularit\"aten \cite{HuiskenRoundSphere} auch zylinderf\"ormige
untersucht worden \cite{HuiskenSinestrari3}. Vielleicht lassen
sich \"ahnliche Singularit\"aten auch f\"ur Flussgleichungen
h\"oherer Homogenit\"at untersuchen.

Die Arbeiten \cite{HuiskenYauMass} und
\cite{JanMetzgerFoliations} zeigen
die Verbindungen zwischen Bl\"atterungen und physikalischen
Gr\"o{\ss}en auf. So etwas k\"onnte auch f\"ur die durch
den inversen mittleren Kr\"ummungsfluss gegebene Bl\"atterung
m\"oglich sein. Zumindest im eu\-kli\-di\-schen Fall ist es m\"oglich,
mit Hilfe der hier untersuchten Gr\"o{\ss}en das Verhalten 
solche Bl\"atterungen genauer zu studieren.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Gliederung}

Der Rest der Arbeit gliedert sich wie folgt.
In Kapitel \ref{diff geom evol sec} erkl\"aren wir unsere
differentialgeometrischen Notationen und
leiten einige Evolutionsgleichungen her.

Im zweiten Teil betrachten wir Fl\"achen, die sich mit
Normalengeschwindigkeit $\A2$ oder \"ahnlichen
Geschwindigkeiten zusammenziehen. In Kapitel \ref{A2 sec}
beweisen wir das Theorem \ref{dt A2 thm} unter 
Benutzung der monotonen Gr\"o{\ss}e aus Theorem \ref{dt A2 mon}.
Dann beschreiben wir in Kapitel \ref{algor sec} unseren
Algorithmus und in Kapitel \ref{comp alg sec} 
erl\"autern wir, wie man Evolutionsgleichungen mit
Computeralgebra ausrechnen kann. Am Ende dieses Teiles
betrachten wir noch einige weitere kontrahierende Fl\"usse
und beweisen auch hier Konvergenz gegen runde Punkte.

Im dritten Teil betrachten wir den inversen 
Gau{\ss}kr\"ummungsfluss. In Kapitel \ref{IGCF} 
beweisen wir Theorem \ref{dt K thm} mit Hilfe
der monotonen Gr\"o{\ss}e aus Theorem \ref{dt K mon}.
Unser Algorithmus zum Auf\/finden monotoner Gr\"o{\ss}en
bei expandierenden Fl\"achen
unterscheidet sich etwas von dem f\"ur den kontrahierenden
Fall. Wir beschreiben ihn in Kapitel \ref{K algor sec}.
In Kapitel \ref{K o nor vel} betrachten wir Fl\"usse,
deren L\"osungen wie beim inversen Gau{\ss}kr\"ummungsfluss 
in endlicher Zeit gegen runde Sph\"aren im Unendlichen 
konvergieren. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Danksagung}

Diese Arbeit entstand als der Autor an der Freien Universit\"at
in Berlin in der Arbeitsgruppe von Klaus Ecker war. Wir m\"ochten
uns recht herzlich f\"ur die Unterst\"utzung w\"ahrend dieser
Zeit bedanken.

Wir besch\"aftigten uns schon fr\"uher mit \"ahnlichen
Problemen: W\"ahrend unseres Aufenthaltes als
Feodor-Lynen Stipendiat der Alexander von Humboldt-Stiftung
bei Shing-Tung Yau in Harvard, Cambridge, Massachussetts, U.S.A., 
und w\"ahrend unseres Stipendiums am Max-Planck-Institut f\"ur
Mathematik in den Naturwissenschaften bei J\"urgen Jost
in Leipzig. Wir m\"ochten auch hier f\"ur die Unterst\"utzung 
danken. 

Speziell m\"ochten wir uns noch bedanken bei
\begin{itemize}
\item Klaus Ecker, der uns auf die parabolische
  Version der Aleksandrovreflexion aufmerksam machte,
\item J\"org H\"arterich, Gerhard Huisken und Felix Otto
  f\"ur Diskussionen \"uber Verbesserungen der Konvergenzrate, 
\item Kashif Rasul, dessen C-Compiler Optionen unser
  Programm schneller laufen lie{\ss}en,
\item Olaf Schn\"urer, der uns an gute algebraische
  Basen erinnerte, 
\item Felix Schulze f\"ur Diskussionen \"uber den
  Konvergenzbeweis in \cite{AndrewsStones},
\item John Stalker, der uns auf den Sturmschen Algorithmus
  aufmerksam mach\-te. 
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Differential Geometry and Evolution Equations}
\label{diff geom evol sec}

In this chapter, we recall standard equations from
differential geometry and evolution equations, see e.\,g.\
\cite{HuiskenPolden,HuiskenRoundSphere,OSMM,OSKnutCrelle}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Differential Geometry and Notation}

We use $X=X(x,\,t)=\left(X^\alpha\right)_{1\le\alpha\le3}$ 
to denote the embedding vector of a manifold 
$M$ into $\R^3$ and $\dt X=\dot X$ for its total time derivative. 
Set $M_t:=X(M,\,t)\subset\R^3$. We will assume that
$M_t$ is strictly convex, closed, smooth, and connected.
We choose $\nu=\nu(x)=\left(\nu^\alpha\right)_{1\le\alpha\le3}$ 
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 
$(g_{ij})_{1\le i,\,j\le 2}$ and
a second fundamental form $(h_{ij})_{1\le i,\,j\le 2}$
on $M_t$. These tensors
are symmetric. The principal
curvatures $\lx,\,\ly$ are the eigenvalues of the second fundamental
form with respect to that metric. A surface is called strictly convex,
if all principal curvatures are strictly positive.
We will assume this throughout the paper.
The inverse of the second fundamental
form is denoted by $\left(\tilde h^{ij}\right)_{1\le i,\,j\le 2}$. 

Latin indices range from $1$ to $2$ and refer to geometric
quantities on the surface, Greek indices range from $1$ to 
$3$ and refer to components in the ambient space $\R^3$. 
In $\R^3$, we will always choose Euclidean coordinates.
We use the Einstein summation
convention. 
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 of $\R^3$. 

Denoting by $\langle\cdot,\,\cdot\rangle$ the Euclidean scalar
product in $\R^3$, 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.\ $h_{ij;\,k}$, 
to indicate covariant differentiation with respect to the induced
metric, e.\,g.
$$X^\alpha_{;\,ij}=X^\alpha_{,\,ij}-\Gamma^k_{ij}X^\alpha_{,\,k},$$
where 
$$\Gamma^k_{ij}=\tfrac12g^{kl}(g_{il,\,j}+g_{jl,\,i}-g_{ij,\,l})$$
are the Christoffel symbols of the metric $(g_{ij})$. 

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

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}

Symmetric functions of the principal
curvatures are well-defined, we will use the mean curvature
$H=\lx+\ly$, the square of the norm of the second fundamental form
$\A2=\lx^2+\ly^2$, $\B k=\lx^k+\ly^k$, and the Gau{\ss} curvature
$K=\lx\ly$. 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, $(h_{ij})=\diag(\lx,\,\ly)$, e.\,g.
$$\sum\lk\hij=\sum\limits_{i,\,j,\,k=1}^2\lk\hij
=h^{kl}h^i_{j;\,k}h^j_{i;\,l}
=h_{rs}h_{ij;\,k}h_{ab;\,l}g^{ia}g^{jb}g^{rk}g^{sl}.$$
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 $(\lx,\,\ly)$
or $(h_{ij},\,g_{ij})$. 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 $F=\A2$, we have $F^{ij}=2h^{ij}=2\li g^{ij}$, and 
for $F=-K^{-1}$, we have $F^{ij}=K^{-1}\tilde h^{ij}=
K^{-1}\li^{-1}g^{ij}$. 

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}

\subsection{The Support Function}
Let $M$ be a smooth, strictly convex, compact surface in $\R^3$
and $X$ its embedding vector.
Then $\nu$ is a diffeomorphism from $M$ to $\S^2$. 
Assume that $M$ encloses the origin. We define
the support function $u:\S^2\to\R_+$ of $M$ as
$$u(z):=\langle X(\nu^{-1}(z)),\,t\rangle.$$
The support functions $u(\cdot,\,t)$ of a family of surfaces 
flowing according to 
$$\dt X=-F\nu$$
fulfill the evolution equation (see \cite{AndrewsHarnack})
$$\dot u=-F.$$
In order to write $F$ in terms of $u$, we note that
the eigenvalues of $u_{;\,ij}+u\sigma_{ij}$ with respect
to the standard metric $(\sigma_{ij})$ are given by
$\frac1\lx\equiv\kappa_1$ and $\frac1\ly\equiv\kappa_2$. 
Thus $\kappa_i$ are the radii of curvature.
Here we used $u_{;\,ij}$ to
denote covariant derivatives of $u$ with respect to 
the standard metric on the sphere.
So we obtain for a family of surfaces flowing according
to $\dt X=\frac1K\nu$ the following evolution equation
for the support function
$$\dot u=\frac{\det(u_{;\,ij}+u\sigma_{ij})}{\det(\sigma_{ij})}.$$
and the evolution equation 
$$\dot u=-\frac{\kappa_1^2+\kappa_2^2}{\kappa_1^2\kappa_2^2}
=-\left(\sigma^{ij}(u_{;jk}+u\sigma_{jk})
\sigma^{kl}(u_{;li}+u\sigma_{li})\right)
\frac{\det(\sigma_{ij})^2}{\det(u_{;ij}+u\sigma_{ij})^2}$$
for the support function corresponds to $\dt X=-\A2\nu$,
where $(\sigma^{ij})$ denotes the inverse of $(\sigma_{ij})$.

We wish to remark that the calculations in \cite{AndrewsHarnack} 
use support functions. This motivates the definition of the
dual function in \eqref{dual function}.

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. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Evolution Equations}

For a family $M_t$ solving the evolution equation
$$\dt X=-F\nu$$
as described in Section \ref{problemstellung}, we
have the following evolution equations
\begin{align}
\label{g evol}
\dt g_{ij}=&-2Fh_{ij},\displaybreak[1]\\
\label{h evol}
\dt h_{ij}=&F_{;\,ij}-Fh_i^kh_{kj},\displaybreak[1]\\
\label{nu evol}
\dt\nu^\alpha=&g^{ij}F_{;\,i}X^\alpha_{;\,j}.
\end{align}

As a direct consequence of these results, we obtain
the following evolution equations for a family $M_t$ 
of surfaces (or hypersurfaces) flowing according to
$$\dt X=-F\nu.$$
\begin{lemma}\label{evol equ lem}
We have
\begin{align*}
\oprokl F=&FF^{ij}h^k_ih_{kj},\umbruch\\
\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},\umbruch\\
\dt X^\alpha-F^{ij}X^\alpha_{;\,ij}
=&\left(F^{ij}h_{ij}-F\right)\nu^\alpha,\umbruch\\
\opr{\abs{X}^2}=&2\left(F^{ij}h_{ij}-F\right)\langle X,\,\nu\rangle
-2F^{ij}g_{ij},\umbruch\\
\opr{\nu^\alpha}=&F^{ij}h^k_ih_{kj}\cdot\nu^\alpha,\umbruch
\intertext{and}
\oprokl{\langle X,\,\nu\rangle}=&-F^{ij}h_{ij}-F+F^{ij}h^k_ih_{kj}
\langle X,\,\nu\rangle.
\end{align*}
\end{lemma}
\begin{proof}
We have, see \cite[Lemma 5.4]{OSMathZ}, 
$$\fracp{F}{g_{kl}}=-F^{il}h^k_i$$
and compute the evolution equation of the normal velocity $F$
\begin{align*}
\oprokl F=&-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} and the
fact that $g_{ij;\,k}\equiv0$.

The computations for the evolution equation for $h_{ij}$ are
more involved as we have to interchange covariant derivatives.
\begin{align*}
\dt h_{ij}-F^{ij}h_{ij;\,kl}=&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*}
Combining \eqref{F Fluss} and \eqref{Gauss formula} 
implies the evolution equation for $X^\alpha$.

Then, we get
\begin{align*}
\opr{\abs{X}^2}=&2X^\alpha\delta_{\alpha\beta}
\left(\dt X^\beta-F^{ij}X^\beta_{;\,ij}\right)
-2F^{ij}X^\alpha_{;\,i}\delta_{\alpha\beta}X^\beta_{;\,j}\umbruch\\
=&2X^\alpha\delta_{\alpha\beta}\nu^\beta\left(F^{ij}h_{ij}-F\right)
-2F^{ij}g_{ij}.
\end{align*}

We now come to the evolution equation 
for the unit normal $\nu^\alpha$.
\begin{align*}
\opr{\nu^\alpha}=&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*}

Finally, we have
\begin{align*}
\oprokl{\langle X,\,\nu\rangle}=&
X^\alpha\delta_{\alpha\beta}\left(\opr{\nu^\beta}\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}&&
\text{by \eqref{Weingarten equation}}\\
=&F^{ij}h^k_ih_{kj}\langle X,\,\nu\rangle-F^{ij}h_{ij}-F.
\end{align*}
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mainmatter
\part{Surfaces Contracting with Normal Velocity $\abs{A}^2$}
\label{A2 part}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Introduction}

We consider compact strictly convex surfaces $M_t$ in $\R^3$ 
that contract with normal velocity equal to the square
of the norm of the second fundamental form
\begin{equation}\label{flow eqn}
\dt X=-\A2\nu.
\end{equation}
This is a parabolic flow equation. We obtain a solution
on a maximal time interval $[0,\,T)$, $0<T<\infty$. 
For $t\uparrow T$, the surfaces converge to a point. After
appropriate rescaling, they converge to a round sphere.
We say that the surfaces $M_t$ converge to a ``round point''.
The key step in the proof, Theorem \ref{A2 mon thm}, is to show that
\begin{equation}\label{mon groe}
\max\limits_{M_t}\left(\frac{(\lx+\ly)(\lx-\ly)^2}{\lx\ly}\right)
\end{equation}
is non-increasing in time. 

Our main theorem in this part is
\begin{theorem}\label{main thm}
For any smooth strictly convex closed surface $M$ in $\R^3$, there
exists a smooth family of surfaces $M_t$, $t\in[0,T)$,
solving \eqref{flow eqn} with $M_0=M$. For $t\uparrow T$,
$M_t$ converges to a point $Q$. The rescaled surfaces
$(M_t-Q)\cdot(6(T-t))^{-1/3}$ converge smoothly to the
unit sphere $\S^2$.
\end{theorem}

Note that we do not have to impose any pinching assumption
on the initial surface and that the normal velocity is
homogeneous of degree two, i.\,e.\ especially of some
degree larger than one. 

We will also consider other normal velocities for which
similar results hold. Therefore, we have to find 
monotone quantities like in \eqref{mon groe}.

In Table \ref{table}, we have collected some normal
velocities $F$ ($1^{\text{st}}$ column) and quantities
$w$ ($2^{\text{nd}}$ column) such that $\max_{M_t}w$ is
non-increasing in time for surfaces contracting
with normal velocity $F$. In each case, we obtain
convergence to round points for smooth compact strictly 
convex initial surfaces $M_0$. 

\begin{table}
\def\platz{\raisebox{0em}[2.2em][1.5em]{\rule{0em}{2em}}}
$$
\begin{array}{|c||c|}\hline
\A2 & \platz\fracd{(\lx+\ly)(\lx-\ly)^2}{\lx\ly}\\\hline
K {\text{ \cite{AndrewsStones}}} & \platz(\lx-\ly)^2\\\hline
H^2 & \platz\fracd{(\lx+\ly)^3(\lx-\ly)^2}
{\left(\lx^2+\ly^2\right)\lx\ly}\\\hline
H^3 & \platz\fracd{\left(\lx^2+\lx\ly+\ly^2\right)(\lx+\ly)^2(\lx-\ly)^2}
{\left(\lx^2-\lx\ly+\ly^2\right)\lx\ly}\\\hline
H^4 & \platz\fracd{\left(\lx^2+\lx\ly+\ly^2\right)(\lx+\ly)^6(\lx-\ly)^2}
{\lx^2\ly^2}\\\hline
\begin{minipage}{1.7cm}
\centerline{$\A2+\beta H^2,$}
\centerline{$0\le\beta\le5$} 
\end{minipage}
& \platz\fracd{(\lx+\ly)(\lx-\ly)^2}{\lx\ly}\\\hline
\A3 & \platz\fracd{\left(3\lx^2+2\lx\ly+3\ly^2\right)(\lx-\ly)^2}
{\lx\ly}\\\hline
\begin{minipage}{2.1cm}
\centerline{$\B{\alpha},$}
\centerline{$\alpha=2,\,4,\,5,\,6$}
\end{minipage}
& \platz\fracd{\left(\lx^{\alpha-2}+\ly^{\alpha-2}\right)
(\lx+\ly)(\lx-\ly)^2}
{\lx\ly}\\\hline
H\A2 & \platz\fracd{(\lx+\ly)^2(\lx-\ly)^2}{\lx\ly}\\\hline
\abs A^4 & \platz\fracd{\left(\lx^4+2\lx^3\ly+4\lx^2\ly^2
+2\lx\ly^3+\ly^4\right)(\lx-\ly)^2}{(\lx+\ly)\lx\ly}\\\hline
\end{array}$$
\caption{Monotone quantities}\label{table}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Surfaces Flowing by $|A|^2$}\label{A2 sec}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Convergence to a Point}
It is known, that \eqref{flow eqn} is a parabolic evolution equation
for strictly convex initial data and that is has a solution on a maximal
time interval. 

We show that $M_t$ stays uniformly strictly convex.
The following lemma is similar to results in \cite{AndrewsContractCalc}.
\begin{lemma}\label{A2 conv pre}
For a smooth compact strictly convex surface $M$ in $\R^3$, 
flowing according to $\dot X=-\A2\nu$, the minimum of the
principal curvatures is non-decreasing.
\end{lemma}
\begin{proof}
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 $t>t_0$. 
Lemma \ref{evol equ lem} yields
$$\dt h_{ij}-F^{kl}h_{ij;\,kl}=2\A3 h_{ij}-3\A2h^k_ih_{kj}
+2g^{kr}g^{ls}h_{kl;\,i}h_{rs;\,j}.$$
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 
\cite{ChowLuMP,HamiltonThree,HamiltonFour86}
gives the result.
\end{proof}

The next result shows that $\A2$ stays uniformly bounded as
long as $M_t$ encloses a ball of fixed positive radius. A similar
estimate is used in \cite{TsoPoint}.
\begin{lemma}\label{vel bound}
For a strictly convex solution of \eqref{flow eqn}, $\A2$ is
uniformly bounded in terms of the radius $R$ of an enclosed 
sphere $B_{R}(x_0)$,
$\max_{M_0}\,\frac{\A2}{\langle X-x_0,\,\nu\rangle-\frac12R}$,
and $\max_{M_0}|X-x_0|$. More precisely, we have
\begin{equation}\label{A2 bound}
\sup\limits_{t}\max\limits_{M_t}\,\A2\le\max\left\{
\max_{M_0}|X-x_0|\cdot\max_{M_0}\frac{\A2}{\langle X-x_0,\,\nu\rangle
-\frac12R},\,\frac{18}{R^2}\right\}.
\end{equation}
\end{lemma}
\begin{proof}
We may assume that $x_0=0$. Let $\alpha=\frac12R$. Then
$\alpha$ is a positive lower bound for $\langle X,\,\nu\rangle-\alpha$.
Lemma \ref{evol equ lem} yields the evolution equations
\begin{align*}
\dt X^\beta-F^{ij}X^\beta_{;\,ij}=&\A2\nu^\beta,\displaybreak[1]\\
\dt\nu^\beta-F^{ij}\nu^\beta_{;\,ij}=&2\A3\nu^\beta,\displaybreak[1]\\
\dt\langle X,\,\nu\rangle-F^{ij}\langle X,\,\nu\rangle_{;\,ij}=&
-3\A2+2\A3\langle X,\,\nu\rangle,\displaybreak[1]\\
\opr{\A2}=&2\A2\A3.
\end{align*}
In a critical point of $\frac{\A2}{\lvert X,\,\nu\rvert-\alpha}$,
we obtain
$$\dt\log\frac{\A2}{\langle X,\,\nu\rangle-\alpha}
-F^{ij}\left(\log\frac{\A2}{\langle X,\,\nu\rangle-\alpha}\right)_{;\,ij}=
\frac1{\langle X,\,\nu\rangle-\alpha}\left(3\A2-2\A3\alpha\right).$$
Note that $\langle X,\,\nu\rangle-\alpha\le\max_{M_0}
\abs X$ as a sphere of radius $\max_{M_0}\abs X$, 
centered at the origin, will enclose
any $M_t$.
We only have to prove that we preserve the bound in 
Equation \eqref{A2 bound}, when 
$\max_{M_t}\frac{\A2}{\langle X,\,\nu\rangle-\alpha}$ increases.
Then we have $0\le3\A2-2\A3\alpha$ at a point, where
$\max_{M_t}\frac{\A2}{\langle X,\,\nu\rangle-\alpha}$ 
is attained. This inequality and elementary calculations 
for convex surfaces give 
$$\A2\le2^{1/3}\cdot\Akl3{2/3}\le
2\frac{\Akl32}{\Akl22}\le\frac9{2\alpha^2}$$
at such a maximum point and the Lemma follows. 
\end{proof}

We obtain that the second fundamental form of the surface stays bounded
as long as $M_t$ encloses some ball. The estimates of Krylov, Safonov,
Evans (see also \cite{Andrews2dKrylov}), 
and Schauder imply that the solution stays smooth. Then,
similarly as in \cite{TsoPoint}, 
the positive lower bound on the minimum principal curvature
implies that the surfaces converge to a point in finite time.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A Monotone Quantity}\label{mon sec}
\begin{theorem}\label{A2 mon thm}
For a family of smooth compact strictly convex surfaces $M_t$ 
in $\R^3$ flowing according to $\dot X=-\A2\nu$, 
\begin{equation}\label{A2 quantity}
\max\limits_{M(t)}\frac{(\lambda_1+\lambda_2)(\lambda_1-\lambda_2)^2}
{2\lambda_1\lambda2}
=\max\limits_{M(t)}\frac{H\cdot\left(2\A2-H^2\right)}{H^2-\A2}
\equiv\max\limits_{M(t)}w
\end{equation}
is non-increasing in time.
\end{theorem}

An immediate consequence of this theorem is
\begin{corollary}
The only homothetically shrinking smooth compact strictly convex
surfaces $M_t$, solving the flow equation $\dot X=-\A2\nu$ in 
$\R^3$, are spheres.
\end{corollary}
\begin{proof}
The quantity $\frac{(\lx+\ly)(\lx-\ly)^2}{\lx\ly}$ 
is positive homogeneous of degree one
in the principal curvatures and non-negative. If $M$ is 
homothetically shrinking, Theorem \ref{A2 mon thm} implies that
$(\lx+\ly)(\lx-\ly)^2=0$ everywhere. Thus $M(t)$ is umbilic and 
\cite[Lemma 7.1]{Spivak4} implies that $M(t)$ is a sphere.
\end{proof}

\begin{proof}[Proof of Theorem \ref{A2 mon thm}]
We apply Lemma \ref{evol equ lem} and get the 
following general evolution equation
\begin{align}\label{gen ev eq}
\begin{split}
\dt\B\alpha-F^{kl}\left({\B\alpha}\right)_{;\,kl}=&
\alpha\sum\limits_i F^{ii}\lambda_i^2\B\alpha
+\alpha\left(F-\sum\limits_i F^{ii}\lambda_i\right)
\B{\alpha+1}\\
&-\alpha\sum\limits_{r=0}^{\alpha-2}\sum\limits_{i,\,j,\,k} 
F^{kk}\lambda_i^{\alpha-2-r}
\lambda_j^r\hij\\
&+\alpha\sum\limits_{k,\,l,\,r,\,s,\,i} 
F^{kl,\,rs}h_{kl;\,i}h_{rs;\,i}\lambda_i^{\alpha-1}
\end{split}
\end{align}
for solutions to the flow equation
$$\dt X=-F\nu.$$
Using \eqref{gen ev eq} for $F=\A2$ yields
\begin{align}
\begin{split}\label{H evol}
\oprokl H=&-\left(\A2\right)^2+2H\A3+2\sum\hij
\end{split}
\intertext{and}
\begin{split}\label{A2 evol}
\opr{\A2}=&2\A2\A3.
\end{split}
\end{align}
For the rest of the proof, we consider a critical point of 
$\left.w\right|_{M_t}$ for some $t>0$, where $w>0$.
It suffices to show that $\tilde w:=\log w$ is non-increasing
in such a point. Then our theorem follows. 

We rewrite $\tilde w$ 
\begin{align*}
\tilde w=&\log H+\log\left(2\A2-H^2\right)-\log(H^2-\A2)\\
\equiv&\log A+\log B-\log C.
\end{align*}
In a critical point of $\tilde w$, we obtain
\begin{align*}
\begin{split}
\oprokl{\tilde w}=&\frac1A\left(\oprokl A\right)
+\frac1B\left(\oprokl B\right)\\
&-\frac1C\left(\oprokl C\right)
-\frac1{AB}F^{ij}(A_{;\,i}B_{;\,j}+A_{;\,j}B_{;\,i})\\
\end{split}\\
\intertext{and}
\begin{split}
0=&\frac1HH_{;\,k}+\frac1{2\A2-H^2}\left(2\A2-H^2\right)_{;\,k}
-\frac1{H^2-\A2}\left(H^2-\A2\right)_{;\,k}\\
=&\frac{2\lx^2+\lx\ly+\ly^2}{\lx\left(\lx^2-\ly^2\right)}h_{11;\,k}
+\frac{2\ly^2+\lx\ly+\lx^2}{\ly\left(\ly^2-\lx^2\right)}h_{22;\,k}.
\end{split}
\end{align*}
So we deduce that
$$h_{22;\,1}=\frac{\ly}{\lx}\frac{2\lx^2+\lx\ly+\ly^2}{2\ly^2+\lx\ly+\lx^2}
h_{11;\,1}\equiv a_1h_{11;\,1}$$
and a similar formula holds for $h_{11;\,2}$, 
$h_{11;\,2}=a_2\cdot h_{22;\,2}$. 
We now combine all these results and obtain in a straightforward
calculation
\begin{align*}
\oprokl{\tilde w}=&
\left(\frac1H-\frac{2H}{2\A2-H^2}-\frac{2H}{H^2-\A2}\right)
\cdot\left(\oprokl H\right)\displaybreak[1]\\
&+\left(\frac{2}{2\A2-H^2}+\frac1{H^2-\A2}\right)
\cdot\left(\opr{\A2}\right)\displaybreak[1]\\
&+\left(\frac{6}{2\A2-H^2}+\frac{2}{H^2-\A2}\right)F^{ij}H_{;\,i}H_{;\,j}
\displaybreak[1]\\
&-\frac2{H\cdot\left(2\A2-H^2\right)}
F^{ij}\left(\left(\A2\right)_{;\,i}H_{;\,j}
+\left(\A2\right)_{;\,j}H_{;\,i}\right)
\displaybreak[1]\\
=&-\frac{\lambda_1^4+\lambda_1^3\lambda_2+4\lambda_1^2\lambda_2^2
+\lambda_1\lambda_2^3+\lambda_2^4}{(\lambda_1+\lambda_2)(\lambda_1-
\lambda_2)^2\lambda_1\lambda_2}\cdot\\
&\quad\cdot\left(-\left(\A2\right)^2+2H\A3\right)\displaybreak[1]\\
&+\frac{(\lambda_1+\lambda_2)^2}{2(\lambda_1-\lambda_2)^2
\lambda_1\lambda_2}\cdot
2\A2\A3\displaybreak[1]\\
&-2\frac{\lambda_1^4+\lambda_1^3\lambda_2+4\lambda_1^2\lambda_2^2
+\lambda_1\lambda_2^3+\lambda_2^4}{(\lambda_1+\lambda_2)
(\lambda_1-\lambda_2)^2\lambda_1\lambda_2}
\sum\hij\displaybreak[1]\\
&+2\frac{\lambda_1^2+4\lambda_1\lambda_2+\lambda_2^2}
{(\lambda_1-\lambda_2)^2\lambda_1\lambda_2}
\sum\lk\hii\displaybreak[1]\\
&-\frac{8}{(\lambda_1-\lambda_2)^2(\lambda_1+\lambda_2)}
\sum\lk(\li+\lj)\hii\displaybreak[1]\\
=&-\frac{\lambda_1^8+3\lambda_1^7\lambda_2+4\lambda_1^6\lambda_2^2
+9\lambda_1^5\lambda_2^3-2\lambda_1^4\lambda_2^4}
{(\lambda_1+\lambda_2)(\lambda_1-\lambda_2)^2\lambda_1\lambda_2}
\displaybreak[1]\\
&-\frac{9\lambda_1^3\lambda_2^5
+4\lambda_1^2\lambda_2^6+3\lambda_1\lambda_2^7+\lambda_2^8}
{(\lambda_1+\lambda_2)(\lambda_1-\lambda_2)^2\lambda_1\lambda_2}
\displaybreak[1]\\
&+\frac{\lambda_1^7+2\lambda_1^6\lambda_2+2\lambda_1^5\lambda_2^2
+3\lambda_1^4\lambda_2^3+3\lambda_1^3\lambda_2^4+2\lambda_1^2\lambda_2^5
+2\lambda_1\lambda_2^6+\lambda_2^7}
{(\lambda_1-\lambda_2)^2\lambda_1\lambda_2}\displaybreak[1]\\
&-2\frac{\lx^4+\lx^3\ly+4\lx^2\ly^2+\lx\ly^3+\ly^4}
{(\lx+\ly)(\lx-\ly)^2\lx\ly}\cdot\\
&\qquad\cdot\left(\left(1+3a_1^2\right)\cdot h_{11;\,1}^2
+\left(1+3a_2^2\right)\cdot h_{22;\,2}^2\right)\displaybreak[1]\\
&+2\frac{\lambda_1^2+4\lambda_1\lambda_2+\lambda_2^2}
{(\lambda_1-\lambda_2)^2\lambda_1\lambda_2}\cdot
\left(\lambda_1(1+a_1)^2\cdot h_{11;\,1}^2
+\lambda_2(1+a_2)^2\cdot h_{22;\,2}^2\right)\displaybreak[1]\\
&-\frac{16}{(\lambda_1-\lambda_2)^2(\lambda_1+\lambda_2)}\cdot
\left(\lx(\lx+\ly a_1)(1+a_1)\cdot h_{11;\,1}^2\right.\\
&\quad\left.+\ly(\ly+\lx a_2)(1+a_2)\cdot 
h_{22;\,2}^2\right)\displaybreak[1]\\
=&-4\frac{K^2}H\displaybreak[1]\\
&-2\frac{\left(5\lambda_1^8-4\lambda_1^7\lambda_2
+46\lambda_1^6\lambda_2^2+48\lambda_1^5\lambda_2^3
+72\lambda_1^4\lambda_2^4\right)\lambda_2}
{\left(\lambda_1+\lambda_2\right)\left(\lambda_1-\lambda_2\right)^2
\left(\lambda_1^2+\lambda_1\lambda_2+2\lambda_2^2\right)^2\lambda_1^3}
h_{11;\,1}^2\displaybreak[1]\\
&-2\frac{\left(44\lambda_1^3\lambda_2^5+34\lambda_1^2\lambda_2^6
+8\lambda_1\lambda_2^7+3\lambda_2^8\right)\lambda_2}
{\left(\lambda_1+\lambda_2\right)\left(\lambda_1-\lambda_2\right)^2
\left(\lambda_1^2+\lambda_1\lambda_2+2\lambda_2^2\right)^2\lambda_1^3}
h_{11;\,1}^2\displaybreak[1]\\
&-2\frac{\left(5\lambda_2^8-4\lambda_2^7\lambda_1
+46\lambda_2^6\lambda_1^2+48\lambda_2^5\lambda_1^3
+72\lambda_2^4\lambda_1^4\right)\lambda_1}
{\left(\lambda_2+\lambda_1\right)\left(\lambda_2-\lambda_1\right)^2
\left(\lambda_2^2+\lambda_2\lambda_1+2\lambda_1^2\right)^2\lambda_2^3}
h_{22;\,2}^2\displaybreak[1]\\
&-2\frac{\left(44\lambda_2^3\lambda_1^5+34\lambda_2^2\lambda_1^6
+8\lambda_2\lambda_1^7+3\lambda_1^8\right)\lambda_1}
{\left(\lambda_2+\lambda_1\right)\left(\lambda_2-\lambda_1\right)^2
\left(\lambda_2^2+\lambda_2\lambda_1+2\lambda_1^2\right)^2\lambda_2^3}
h_{22;\,2}^2\displaybreak[1]\\
\le&0.
\end{align*}
We finally apply the maximum principle and our theorem follows.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Direct Consequences}

We obtain a pinching estimate
\begin{corollary}\label{pinching cor}
For a smooth compact strictly convex surface $M_t$ in $\R^3$, 
flowing according to $\dot X=-\A2\nu$, there exists 
$c=c(M_0)$ such that $0<\frac1c\le\frac{\lambda_1}{\lambda_2}\le c$.
\end{corollary}
\begin{proof}
Choose $\epsilon>0$ such that $\lambda_1,\,\lambda_2>\epsilon$
at $t=0$. Theorem \ref{A2 mon thm} and Lemma \ref{A2 conv pre}
imply that 
$$2\epsilon\frac{\left(\frac\lx\ly-1\right)^2}{\frac\lx\ly}
=2\epsilon\frac{(\lx-\ly)^2}{\lx\ly}
\le\frac{(\lx+\ly)(\lx-\ly)^2}{\lx\ly}\le c.$$
We obtain the bound on $\frac{\lambda_1}{\lambda_2}$ claimed above. 
\end{proof}

Let $\rho_+$ be the minimal radius of enclosing spheres and
$\rho_-$ the maximal radius of enclosed spheres. The quotient
of these radii can be estimated as follows
\begin{corollary}\label{radii cor}
Under the assumptions of Corollary \ref{pinching cor},
$\rho_+/\rho_-$ is bounded above by a constant depending only
on the constant $c(M_0)$ in Corollary \ref{pinching cor}. 
\end{corollary}
\begin{proof}
Combine Corollary \ref{pinching cor}, 
\cite[Theorem 5.1]{AndrewsContractCalc},
and \cite[Lemma 5.4]{AndrewsContractCalc}.
\end{proof}

We also obtain a bound for $\abs{\lx-\ly}$
\begin{corollary}\label{diff bound}
For a smooth compact strictly convex surface $M_t$ in $\R^3$,
flowing according to $\dot X=-\A2\nu$, there exists a
constant $c=c(M_0)$ such that 
$$\abs{\lx-\ly}\le c\cdot\Akl2{1/4}
\le c\cdot\sqrt{H}.$$
\end{corollary}
\begin{proof}
This is a direct consequence of Theorem \ref{A2 mon thm}
and Corollary \ref{pinching cor}.
\end{proof}

As in \cite{AndrewsStones}, this estimate on $\abs{\lx-\ly}$ is
``better'' than scaling invariant. It is crucial for the rest
of the proof of Theorem \ref{main thm}.

Let us recall a form of the maximum principle for evolving hypersurfaces.
\begin{lemma}\label{encl lem}
Let $M_t$ and $\tilde M_t$ be two smooth compact strictly convex solutions
to \eqref{flow eqn} on some time interval $\left[0,\,T^*\right)$.
If $M_0$ encloses $\tilde M_0$, then $M_t$ encloses $\tilde M_t$
at any time $t\in\left[0,\,T^*\right)$, for which both solutions exist.
\end{lemma}
\begin{proof}
This is a standard consequence of the maximum principle. 
\end{proof}

The next result describes the evolution of spheres. 
\begin{lemma}\label{sphere evol}
Spheres $\partial B_{r(t)}(x_0)$ solve \eqref{flow eqn} for
$t\in[0,\,T)$ with 
$$r(t)=(6(T-t))^{1/3}\quad\text{and}\quad T=\frac16r^3(0).$$
\end{lemma}
\begin{proof}
The evolution equation for the radius of a sphere is
$$\dot r(t)=-\frac2{r^2(t)}.$$
\end{proof}

As a consequence, we can estimate the life span of a solution 
in terms of inner and outer radii.
\begin{lemma}\label{life span lem}
Let $\rho_+(t)$ and $\rho_-(t)$ be the inner and outer radii of $M_t$,
respectively. Assume that $M_t$ is a smooth compact strictly convex
solution of \eqref{flow eqn} on a maximal time interval $[0,\,T)$.
Then we have for $t\in[0,\,T)$
$$\frac16\rho_-^3(t)\le T-t\le\frac16\rho_+^3(t).$$
\end{lemma}
\begin{proof}
As $M_t$ contracts to a point, we deduce from Lemma \ref{encl lem}
that $T-t$ is bounded below by the life span of $\partial B_{\rho_-(t)}$
evolving according to \eqref{flow eqn}. So the lower bound follows
from Lemma \ref{sphere evol}. The upper bound is obtained similarly.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Convergence to a Round Point}
We closely follow the corresponding part of \cite{AndrewsStones}.

\begin{proposition}\label{prop int H}
Define $q(t):=\frac1{4\pi}\int\limits_{M_t}KX$.
Then 
$$\left\lvert\langle X-q,\nu\rangle-\frac1{8\pi}
\int\limits_{M_t}H\right\rvert\le\frac1{4\pi}
\cdot\sup_{M_t}\abs{\lx-\ly}\cdot\mathcal H^{2}(M_t),$$
where $\mathcal H^2(M_t)$ denotes the area of $M_t$.
\end{proposition}
\begin{proof}
This is \cite[Proposition 4]{AndrewsStones}.
\end{proof}

We define $r_+(t)$ to be the minimal radius of a sphere, 
centered at $q(t)$, that encloses $M_t$. Similarly, we
define $r_-(t)$ to be the maximal radius of a sphere,
centered at $q(t)$, that is enclosed by $M_t$.

\begin{lemma}\label{radii bound}
Under the assumptions of Theorem \ref{main thm}, 
for $T-t$ sufficiently small, $r_+$ and
$r_-$ are estimated as follows
\begin{align*}
r_+(t)\le&(6(T-t))^{1/3}\cdot\left(1+
c\cdot(T-t)^{1/6}\right),\displaybreak[1]\\
r_-(t)\ge&(6(T-t))^{1/3}\cdot\left(1-
c\cdot(T-t)^{1/6}\right),
\intertext{and}
1\le\frac{r_+}{r_-}\le&1+c\cdot(T-t)^{1/6}.
\end{align*}
\end{lemma}
\begin{proof}
Denote the bounded component of $\R^3\setminus M_t$
by $E_t$. The transformation formula for integrals
implies that
$$\frac1{4\pi}\int\limits_{M_t}KX=\frac1{4\pi}
\int\limits_{\S^2}X\left(\nu^{-1}(\cdot)\right).$$
So we see that $q(t)\in E_t$.
We have 
\begin{align*}
r_+=&\max_{M_t}\,\langle X-q(t),\,\nu\rangle,&
r_-=&\min_{M_t}\,\langle X-q(t),\,\nu\rangle,\\ 
\rho_+=&\min_{p\in\R^3}\max_{M_t}\,\langle X-p,\,\nu\rangle,
\qquad\text{and}&
\rho_-=&\max_{p\in E_t}\min_{M_t}\,\langle X-p,\,\nu\rangle.
\end{align*}

Recall the first variation formula for a vector 
field $Y$ along $M_t$ \cite{TomMCF}
$$\int\limits_{M_t}H
\langle Y,\,\nu\rangle=\int\limits_{M_t}\divergenz_{M_t} Y$$
and get for $p\in E_t$ such that $\rho_+=\max_{M_t}
\langle X-p,\,\nu\rangle$
$$\int\limits_{M_t}H\ge\frac1{\rho_+}\int\limits_{M_t}
H\cdot\langle X-p,\,\nu\rangle=\frac1{\rho_+}\int\limits_{M_t}
\divergence_{M_t}X=\frac1{\rho_+}\int\limits_{M_t}2=\frac2{\rho_+}
\mathcal H^2(M_t).$$
We employ Proposition \ref{prop int H} and deduce that 
\begin{align*}
r_-\ge&\frac1{8\pi}\int\limits_{M_t}H\cdot\left\{1-
2\left(\int_{M_t}H\right)^{-1}\cdot
\sup\limits_{M_t}\abs{\lx-\ly}\cdot\mathcal H^2(M_t)\right\}\\
\ge&\frac1{8\pi}\int\limits_{M_t}H\cdot
\left\{1-\rho_+\cdot\sup\limits_{M_t}\abs{\lx-\ly}\right\}.
\end{align*}
We estimate as follows
\begin{align*}
\rho_+\cdot\sup\limits_{M_t}\abs{\lx-\ly}\le&
c\cdot\rho_+\cdot\Akl2{1/4}&&\text{by Corollary \ref{diff bound}}\\
\le&c\cdot\rho_+\cdot\left(c+\frac c{\rho_-^2}\right)^{1/4}&&
\text{by Lemma \ref{vel bound}}\\
\le&c\cdot(T-t)^{1/6}&&
\end{align*}
by Corollary \ref{radii cor} and Lemma \ref{life span lem}
for $(T-t)$ small. So we obtain
\begin{equation}\label{r- bound}
r_-(t)\ge\frac1{8\pi}\int\limits_{M_t}H\cdot\left(1-
c\cdot(T-t)^{1/6}\right).
\end{equation}
Similar calculations yield
\begin{equation}\label{r+ bound}
r_+(t)\le\frac1{8\pi}\int\limits_{M_t}H\cdot\left(1+
c\cdot(T-t)^{1/6}\right).
\end{equation}
We employ Lemma \ref{life span lem}
$$r_-\le\rho_-\le(6(T-t))^{1/3}\le\rho_+\le r_+$$
and obtain for $(T-t)$ small
$$(6(T-t))^{1/3}\cdot\left(1-c\cdot(T-t)^{1/6}\right)
\le\frac1{8\pi}\int\limits_{M_t}H\le
(6(T-t))^{1/3}\cdot\left(1+c\cdot(T-t)^{1/6}\right).$$
Using \eqref{r- bound} and \eqref{r+ bound} gives the
claimed estimates on $r_-$ and $r_+$, and $r_+/r_-$ is
bounded as stated above.
\end{proof}

\begin{corollary}
Under the assumptions of Theorem \ref{main thm}, we have the
estimate
$$|q(t)-Q|\le c\cdot(T-t)^{1/3+1/18}.$$
Therefore, we obtain estimates of the same form as in 
Lemma \ref{radii bound}, if we define $r_+$ and $r_-$
using $Q$ instead of $q(t)$. 
\end{corollary}
\begin{proof}
Fix $t_0\in[0,\,T)$. A sphere of radius
$(6(T-t_0))^{1/3}\left(1+c(T-t_0)^{1/6}\right)$, centered
at $q(t_0)$ as defined in Proposition \ref{prop int H},
will enclose $M_t$ for all $t_0\le t<T$. Thus its radius 
at time $t=T$ is an upper bound for $|q(t_0)-Q|$. 
According to Lemma \ref{sphere evol}, the radius of that
sphere evolves as follows
$$r(t)=\left(6\left(\frac16r^3(0)-(t-t_0)\right)\right)^{1/3}.$$
Therefore, we get
\begin{align*}
|q(t_0)-Q|\le&r(T)\umbruch\\
=&\left(6\left(\frac166(T-t_0)\left(1+c(T-t_0)^{1/6}\right)^{3}
-(T-t_0)\right)\right)^{1/3}\umbruch\\
=&(6(T-t_0))^{1/3}\cdot\left(\left(1+c(T-t_0)^{1/6}\right)^3
-1\right)^{1/3}\umbruch\\
\le&(6(T-t_0))^{1/3}\cdot c\cdot(T-t_0)^{1/18}.
\end{align*}
\end{proof}

Next, we want to check, that we can apply a Harnack inequality
\cite[Theorem 5.17]{AndrewsHarnack}. For $F=F(\lambda_i)$,
$\lambda_i>0$, we define
\begin{equation}\label{dual function}
\Phi(\kappa_i):=-F\left(\kappa_i^{-1}\right).
\end{equation}
We say that $\Phi$ is $\alpha$-concave, if $\Phi=\sgn\alpha\cdot
B^\alpha$ for some $B$, where $B$ is positive and concave.
The function $\Phi$ is called the dual function to $F$.

\begin{lemma}\label{alpha concave lem}
The dual function to $F=\A2=\lx^2+\ly^2$ is
$\alpha$-concave for $\alpha\le-2$. 
\end{lemma}
\begin{proof}
We define for $\lambda_i>0$
$$\Phi=\Phi(\lambda_i)=\Phi(h_{ij},\,g_{ij})=
-\left(\frac1{\lx^2}+\frac1{\ly^2}\right)
=\frac{2K-H^2}{K^2}
=\frac{2\det h^i_j-\left(\tr h^i_j\right)^2}{\left(\det h^i_j\right)^2}$$
and obtain
\begin{align*}
\Phi^{ij}\equiv\fracp\Phi{h_{ij}}
=&\frac1{K^2}\left\{\left[2H^2-2K\right]\tilde h^{ij}
-2Hg^{ij}\right\},\displaybreak[1]\\
\Phi^{ij,\,kl}=&\frac1{K^2}\left\{\left[2K-4H^2\right]
\tilde h^{ij}\tilde h^{kl}
+\left[2K-2H^2\right]\tilde h^{ik}\tilde h^{jl}\right.\\
&\quad\left.4H\left[g^{ij}\tilde h^{kl}+g^{kl}\tilde h^{ij}\right]
-2g^{ij}g^{kl}\right\}.
\end{align*}
According to \cite[(5.4)]{AndrewsHarnack}, it suffices to show that
\begin{equation}\label{alpha concave}
\Phi^{ij,\,kl}\eta_{ij}\eta_{kl}\le\frac{\alpha-1}{\alpha\Phi}
\Phi^{ij}\eta_{ij}\Phi^{kl}\eta_{kl}
\end{equation}
for symmetric matrices $(\eta_{ij})$. Terms involving $\eta_{12}$
clearly have the right sign. The remaining terms are a quadratic
form in $(\eta_{11},\,\eta_{22})$. Thus \eqref{alpha concave} is
fulfilled, if
\begin{align*}
\A2\cdot&\left\{\left[2K-4H^2\right]\begin{pmatrix}\frac1{\lx^2}
&\frac1{\lx\ly}\\\frac1{\lx\ly}&\frac1{\ly^2}\end{pmatrix}
+\left[2K-2H^2\right]\begin{pmatrix}\frac1{\lx^2}&0\\
0&\frac1{\ly^2}\end{pmatrix}\right.\\
&\left.+4H\begin{pmatrix}\frac2{\lx}&\frac1{\lx}+\frac1{\ly}\\
\frac1{\lx}+\frac1{\ly}&\frac2{\ly^2}\end{pmatrix}
-2\begin{pmatrix}1&1\\1&1\end{pmatrix}\right\}\le\\
\le&-\frac{\alpha-1}{\alpha}\left\{\left[2H^2-2K\right]\begin{pmatrix}
\frac1{\lx}\\\frac1{\ly}\end{pmatrix}
-2H\begin{pmatrix}1\\1\end{pmatrix}\right\}\otimes
\left\{\left[2H^2-2K\right]\begin{pmatrix}
\frac1{\lx}\\\frac1{\ly}\end{pmatrix}
-2H\begin{pmatrix}1\\1\end{pmatrix}\right\}\\
\end{align*}
or equivalently
\begin{align*}
-6\left(\lx^2+\ly^2\right)\begin{pmatrix}\frac{\ly^2}{\lx^2}&0\\
0&\frac{\lx^2}{\ly^2}\end{pmatrix}\le
-4\frac{\alpha-1}{\alpha}\begin{pmatrix}\frac{\ly^4}{\lx^2}&
\lx\ly\\\lx\ly&\frac{\lx^4}{\ly^2}\end{pmatrix}.
\end{align*}
As $\frac{\alpha-1}{\alpha}\le\frac32$ for $\alpha\le-2$,
we obtain that $\Phi$ is $\alpha$-concave. 
\end{proof}

We are now able to improve our velocity bounds.
\begin{lemma}\label{A2 est}
Under the assumptions of Theorem \ref{main thm}, we obtain 
$$2(6(T-t))^{-2/3}\cdot\left(1-c\cdot(T-t)^{1/12}\right)
\le\A2\le2(6(T-t))^{-2/3}\cdot\left(1+c\cdot(T-t)^{1/12}\right)$$
everywhere on $M_t$ for $(T-t)$ sufficiently small.
\end{lemma}
\begin{proof}
We may assume that $T-t>0$ is so small that we can use the 
results obtained before. Parameterize $M_t$ by $\S^2$ such that
the normal image of $M_t$ at $X(z,\,t)$ equals $z\in\S^2$.
Let us define the support function $s$ of $M_t$ as
$$s(z,\,t):=\langle X(z,\,t),\,z\rangle.$$
Its evolution equation, see e.\,g.\ \cite{AndrewsHarnack}, is
\begin{equation}\label{supp fct evol}
\dt s(z,\,t)=-\A2(z,\,t).
\end{equation}
The $\alpha$-concavity proved in Lemma \ref{alpha concave lem}
allows us to use \cite[Theorem 5.17]{AndrewsHarnack}.
We obtain for $0<t_1<t_2<T$ and $z\in\S^2$, for
two points $(z,\,t_1)$ and $(z,\,t_2)$ with the same normal,
\begin{equation}\label{A2 Harnack}
\frac{\A2(z,\,t_2)}{\A2(z,\,t_1)}\ge
\left(\frac{t_1}{t_2}\right)^{2/3}.
\end{equation}
Let us assume that $q(t)$ is the origin for some fixed
time $t$. As $M_t$ lies between $\partial B_{r_+(t)}(0)$
and $\partial B_{r_-(t)}(0)$, $M_{t+\tau}$ lies outside
$B_{\left(r_-^3(t)-6\tau\right)^{1/3}}(0)$ for any
$0<\tau<T-t$, so
\begin{equation}\label{supp fct bounds}
r_-(t)\le s(\cdot,\,t)\le r_+(t)\quad\text{and}\quad
\left(r_-^3-6\tau\right)^{1/3}\le s(\cdot,\,t+\tau).
\end{equation}
Set $\tau=r_-^{5/2}(t)\cdot(r_+(t)-r_-(t))^{1/2}$ and observe that
$t+\tau<T$, if $(r_+-r_-)^{1/2}\le\frac16r_-^{1/2}$
(by Lemma \ref{life span lem}), or, if
$T-t$ is sufficiently small
(by Lemma \ref{radii bound}). We estimate
\begin{align*}
\A2(z,\,t)\le&\inf\limits_{0\le\tilde\tau\le\tau}
\left\{\left(\frac{t+\tilde\tau}{t}\right)^{2/3}\cdot
\A2(z,\,t+\tilde\tau)\right\}&&\text{by \eqref{A2 Harnack}}
\displaybreak[1]\\
\le&\left(\frac{t+\tau}t\right)^{2/3}\cdot\frac1{\tau}\cdot
\int\limits_t^{t+\tau}\A2(z,\,\tilde\tau)d\tilde\tau&&
\displaybreak[1]\\
\le&\left(1+\frac{\tau}t\right)^{2/3}\cdot\frac1\tau\cdot
(s(z,\,t)-s(z,\,t+\tau))&&\text{by \eqref{supp fct evol}}
\displaybreak[1]\\
\le&\left(1+\frac{\tau}t\right)\cdot\frac1{\tau}\cdot
\left(r_+(t)-\left(r_-^3(t)-6\tau\right)^{1/3}\right)
&&\text{by \eqref{supp fct bounds}}\displaybreak[1]\\
=&\frac{1+\frac\tau t}
{r_-^2\cdot\left(\frac{r_+}{r_-}-1\right)^{1/2}}
\cdot\left(\frac{r_+}{r_-}-\left(1
-6\cdot\left(\frac{r_+}{r_-}-1\right)^{1/2}\right)^{1/3}\right).&&
\end{align*}
The maximal existence time $T$ is bounded in terms of the
radius of a sphere enclosed by $M_0$. So we may also assume
that $t$ is bounded below by a positive constant. A very crude
estimate gives 
$$\tau\le r_-^{5/2}\cdot r_+^{1/2}
\le r_+^3\le c\cdot(T-t),$$ 
so we obtain
$$\frac{1+\frac\tau t}{r_-^2}\le
(6(T-t))^{-2/3}\cdot\left(1+c\cdot(T-t)^{1/6}\right).$$
Observe that for $\abs x\le\frac12$, we have
$$-(1-x)^{1/3}\le-1+\tfrac13x+\tfrac13x^2\quad
\text{and }\quad
(1+x)^{1/3}\ge1+\tfrac13x-\tfrac13x^2.$$
We conclude for small $(T-t)$
\begin{align*}
\A2(z,\,t)\le&(6(T-t))^{-2/3}\cdot\left(1+c\cdot(T-t)^{1/6}\right)
\cdot\left(2+13\cdot\left(\frac{r_+}{r_-}-1\right)^{1/2}\right)\\
\le&2\cdot(6(T-t))^{-2/3}\cdot\left(1+c\cdot(T-t)^{1/12}\right).
\end{align*}
For the lower bound on $\A2$, we proceed similarly and use
$\tau=r_-^{5/2}(t)\cdot(r_+(t)-r_-(t))^{1/2}$, 
$$\left(r_-^3+6\tau\right)^{1/3}\le s(z,\,t-\tau)\quad\text{and}
\quad s(z,\,t)\le r_+(t).$$
\end{proof}

We have the following estimate for the principal curvatures
\begin{lemma}\label{lambda bounds}
Under the assumptions of Theorem \ref{main thm}, we obtain
$$(6(T-t))^{-1/3}\cdot\left(1-c\cdot(T-t)^{1/12}\right)\le
\lx,\,\ly\le(6(T-t))^{-1/3}\cdot\left(1+c\cdot(T-t)^{1/12}\right)$$
on $M_t$ for small $(T-t)$. 
\end{lemma}
\begin{proof}
As $H^2=2\A2-(\lx-\ly)^2$, we obtain
\begin{align}\label{lx expr}
\begin{split}
\lx=&\tfrac12(\lx+\ly)+\tfrac12(\lx-\ly)\\
=&\tfrac12\sqrt{2\A2-(\lx-\ly)^2}+\tfrac12(\lx-\ly).
\end{split}
\end{align}
Combining Lemmata \ref{diff bound} and \ref{A2 est}, 
we get $\abs{\lx-\ly}\le c\cdot(T-t)^{-1/6}$. We
use Lemma \ref {A2 est} and \eqref{lx expr}. The 
claimed inequality follows.
\end{proof}

\begin{proof}[Proof of Theorem \ref{main thm}:]
Lemma \ref{lambda bounds} implies, that, everywhere on $M_t$, 
the quotient $\lx/\ly$ tends to $1$ as $t\uparrow T$. Then 
we can apply known results, see e.\,g.\ 
\cite[Theorem 2]{Andrews2dnonconcave}, to conclude that 
the rescaled surfaces converge smoothly to the unit sphere
$\S^2\subset\R^3$.
\end{proof}

A standard way of rescaling \cite{AndrewsContractCalc} 
is to consider the embeddings $\tilde X(\cdot,\,t)$,
$$\tilde X(z,\,t):=(6(T-t))^{-1/3}(X(z,\,t)-Q)$$ 
with $Q$ as in Theorem \ref{main thm}. Define 
the time function $\tau(t):=\frac16\log T-\frac16\log(T-t)$.
Then we have, using suggestive notation, 
the following evolution equation
$$\frac d{d\tau}\tilde X=-\abs{\tilde A}^2\tilde\nu+2\tilde X$$
and our a priori estimates imply, that, for $\tau\to\infty$, 
$\tilde M_\tau$ converges exponentially to $\S^2$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Convergence Rate}\label{conv rate}

In order to find out what the optimal convergence rate might
be, we consider the evolution equation 
$$\dt X=-\A2\nu+2X,$$
that appropriately rescaled solutions of $\dt X=-\A2\nu$ fulfill.
As in \cite[Appendix]{OSKnutCrelle}, we represent the surfaces $M_t$ 
as graphs over the sphere with embeddings 
$$\S^2\ni x\mapsto x\cdot u(x),$$
where $u:\S^2\to\R_+$. Let $(\sigma_{ij})$ be the standard metric
on the sphere and $\left(\sigma^{ij}\right)$ its inverse. 
Then we get as in \cite{OSKnutCrelle}, 
using indices to denote covariant
derivatives on $\S^2$,
\begin{align*}
g_{ij}=&u^2(\sigma_{ij}+\phi_i\phi_j),\quad\text{where }\phi=\log u,
\umbruch\\
h_{ij}=&\frac1{uw}g_{ij}-\frac uw\phi_{ij},\quad\text{where }
w=\sqrt{1+\phi_i\sigma^{ij}\phi_j},\umbruch\\
\fracp ut=&-\A2 w+2u.
\end{align*}
We linearize our equation around the stationary solution 
$u=1$ and take $u=1+\epsilon v$. Then
\begin{align*}
\frac d{d\epsilon} w|_{\epsilon=0}=&0,\umbruch\\
\frac d{d\epsilon} g_{ij}|_{\epsilon=0}=&2v\sigma_{ij},\umbruch\\
\frac d{d\epsilon} h_{ij}|_{\epsilon=0}=&v\sigma_{ij}-v_{ij}.
\end{align*}
So the linearized equation becomes 
$$\fracp vt=2\Delta v+6v.$$
There are spherical harmonics $u$ solving $\Delta u=-l(l+1)u$
for $l\in\N$ \cite{JonesSphericalHarmonics}. 
We do not need to consider $l=0$ as a corresponding
eigenfunction is positive (or negative) everywhere. Thus the 
resulting surface does not contract to a point for $t\uparrow T$
in the unrescaled setting. Similarly, we can exclude $l=1$ as
the respective eigenfunctions correspond to translations and
translated surfaces converge to infinity in the rescaled setting.

Using the ansatz eigenfunction 
multiplied with $e^{-\lambda t}$, we get 
solutions for the linearized equation for $\lambda=2l(l+1)-6$.
This is positive for $l=2$, $\lambda=6$. So we should not
expect a convergence rate better than $||X|-1|\le c\cdot e^{-6t}$ after 
rescaling, corresponding to estimates like
$$r_+\le(6(T-t))^{1/3}\cdot(1+c\cdot(T-t)).$$
Here, we have assumed that $q(t)=0$.

We can still improve our convergence rate by considering
$$\max_{M_t}\frac{\left(\lx^2+\ly^2\right)^2(\lx-\ly)^2}{4\lx^2\ly^2}
=\max_{M_t}\frac{\left(2\A2-H^2\right)\Akl22}{\left(H^2-\A2\right)^2}
\equiv\max_{M_t}w.$$
Under the flow equation $\dt X=-\A2\nu$, we obtain, using 
calculations as above, that $w$ fulfills in a 
critical point of $w$
\begin{align*}
\oprokl w=&
-\frac{\left(\lx^2+\ly^2\right)^2}
{\left(\lx^3-\lx\ly^2+2\ly^3\right)^2\lx^5\ly}
\cdot\left(5\lx^9-3\lx^8\ly-6\lx^7\ly^2+26\lx^6\ly^3\right.\\
&\qquad\left.-20\lx^5\ly^4
+8\lx^4\ly^5+6\lx^3\ly^6-2\lx^2\ly^7-\lx\ly^8+3\ly^9\right)
\cdot h_{11;\,1}^2\\
&+(\ldots)\cdot h_{22;\,2}^2\umbruch\\
\le&0. 
\end{align*}
This implies convergence rates like
$$r_+\le(6(T-t))^{1/3}\cdot\left(1+c\cdot(T-t)^{1/3}\right)$$
similar to \cite{AndrewsStones}, where the author also has a
scaling invariant upper bound for $|\lx-\ly|$.
As there are no negative constant terms left for this choice
of $w$, it might be, that there is no monotone quantity as
studied in this paper, that allows to further improve this convergence 
rate.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Ende \A2-Fluss

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Finding Monotone Quantities}\label{algor sec}

\section{The Algorithm}
We use a sieve algorithm and start with symmetric rational functions 
of the principal curvatures
as candidates for test functions, e.\,g.\ 
$$w=\frac{p_1(\lx,\,\ly)}{p_2(\lx,\,\ly)}
=\frac{(\lx+\ly)(\lx-\ly)^2}{\lx\ly}.$$
Here, $p_1\neq0$ and $p_2\neq0$ are homogeneous polynomials. 

In the end, we want to find functions $w$ such that 
$W:=\sup_{M_t}w$ is monotone and ensures convergence to round
spheres. 

We check, whether these test functions $w$ fulfill the
following conditions. 
\begin{enumerate}
\item\label{step one} 
 \begin{enumerate}
 \item $p_1(\lx,\,\ly),\,p_2(\lx,\,\ly)\ge0$ for $0<\lx,\,\ly$,
 \item $p_1(\lx,\,\ly)=0$ for $\lx=\ly>0$.
 \end{enumerate}
\item\label{step two} $\deg p_1>\deg p_2$.
\item\label{step three} $\fracp{w(1,\ly)}{\ly}<0$ for $0<\ly<1$ and 
  $\fracp{w(1,\ly)}{\ly}>0$ for $\ly>1$.
\item\label{step four} $\dt w-F^{ij}w_{;\,ij}\le0$ 
 \begin{enumerate}
 \item\label{step four a} for terms without derivatives of $(h_{ij})$,
 \item\label{step four b} for terms involving derivatives of $(h_{ij})$,
   if $w_{;\,i}=0$ for $i=1,\,2$.
 \end{enumerate}
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Motivation and Randomized Tests}
We restrict 
our attention to non-negative polynomials $p_i$. For all flow equations 
considered, spheres contract to points and stay spherical. So
we can only find monotone quantities, if $\deg p_1\le\deg p_2$
or $p_1(\lambda,\,\lambda)=0$. 

If $\deg p_1<\deg p_2$,
we obtain that $W$ is decreasing on any self-similarly
shrinking surface. So this does not imply convergence to a
sphere. The counterexamples in \cite{Andrews2dnonconcave} show 
for normal velocities of homogeneity larger than $1$, that 
the pinching ratio $\sup_{M_t}\ly/\lx$ (for $\ly>\lx$) 
will increase on some time interval
during the flow for appropriate initial surfaces. Therefore,
we require in step \eqref{step two}, that $\deg p_1>\deg p_2$. 

Condition \eqref{step three} ensures that the quantity decreases, if 
the eigenvalues approach each other. This excludes 
especially local zeroes of $w(1,\,\ly)$ for $\ly\neq 1$. 

In steps \eqref{step one}, \eqref{step three}, and \eqref{step four}, 
inequalities are tested by evaluating both sides at
random numbers. If an inequality is not violated for 
sufficiently many tuples of random numbers, we move to the
next step and keep the candidate, otherwise we start with
another candidate. 

The essential part is to test, whether $\dt w-F^{ij}w_{;\,ij}\le0$
holds at a point, where $w_{;\,i}=0$. 

We assume in step \eqref{step four a}, 
that $h_{ij;\,k}=0$. 
Recall the algebraic fact, that all products of $H$ and $\A2$
of given homogeneity form a basis for the symmetric homogeneous
polynomials of that degree. We represent the polynomials
$p_i$ in this basis. At random values for
$\lx,\,\ly$, we compute $\oprokl H$ and $\opr{\A2}$.
This can be combined according to the rules of 
differentiation and yields $\oprokl w$,
evaluated at these numbers.

If not all components of $h_{ij;\,k}$ vanish (step 
\eqref{step four b}), we also have to
choose random numbers for $h_{ij;\,k}$ that fulfill the
extremal condition $w_{;\,i}=0$. As above, we evaluate 
$\oprokl w$ at the random numbers chosen.
Here we can ignore all terms that do not contain derivatives 
of the second fundamental form. The evaluation of the
remaining terms is more involved than in the last step,
but can be done similarly according to the various rules
of differentiation. 

Now we iterate steps \eqref{step four a} and 
\eqref{step four b}. If all tests yield 
$\oprokl w\le0$ at critical points of $w$, it seems likely 
that we have found an appropriate test quantity. Indeed,
if this inequality is fulfilled, the maximum principle 
implies that $W=\max_{M_t}w$ is non-increasing in time.

We implemented this algorithm in a C-program and used it
to find all the new test functions of that form in this paper.
 
Obviously, the computing time depends on the
number of tests performed. The following computing times
are measured for the quantity 
\eqref{A2 quantity} on an Intel Pentium 4, 2.4 GHz, running
Linux 2.4.24 and GNU C-compiler 2.95.4. 
The number of tests per second
for step \eqref{step four a} is $1.6\cdot 10^5$ 
and $5.8\cdot 10^3$ for step \eqref{step four b}.
The other steps are comparable to step \eqref{step four a}.
In steps \eqref{step one} to \eqref{step three}, 
the calculations do not depend on the normal velocity.

It seems worth noting, that, after testing with enough random
numbers in an appropriate range, every candidate for a monotone
quantity that we checked, turned out to be a useful test quantity.
In that sense, algorithm and program seem to be correct.

We are convinced that it is possible to implement this
algorithm for surfaces without using random numbers. Evolution
equations can be computed algebraically and Sturm's algorithm
can be used to test for non-negativity.

We expect that similar algorithms will be used to find monotone
test functions for other (geometric) problems. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Computing Evolution Equations}\label{comp alg sec}

It is straightforward to use a computer algebra system 
to obtain evolution equations of test quantities $w$,
evaluated at a critical point of $w$. More precisely, let
$M_t$ be a family of surfaces in $\R^3$, moving with 
normal velocity $F=F(\lx,\,\ly)$, where $F>0$ for 
contracting surfaces. Assume that the test quantity $w$
is a function of $H$ and $\A2$. Then $w$ fulfills the
evolution equation
$$\opr w=C_w(\lx,\,\ly)+G_w(\lx,\,\ly)h_{11;\,1}^2
+G_w(\ly,\,\lx)h_{22;\,2}^2$$
in a critical point of $w$. It remains to 
compute $C_w$ (``constant terms'') and $G_w$ (``gradient terms''). 
We use Lemma \ref{evol equ lem} and the following equation
of \cite{AndrewsContractCalc,CGJDG1996}
$$F^{ij,\,kl}\eta_{ij}\eta_{kl}=\sum\fracp{^2F}{\li\partial\lj}
\eta_{ii}\eta_{jj}+\sum\limits_{i\neq j}
\frac{\fracp F\li-\fracp F\lj}{\li-\lj}(\eta_{ij})^2,$$
for symmetric matrices $(\eta_{ij})$ and $\lx\neq\ly$
or $\lx=\ly$, when we interpret the last term as a limit.
For $w=H$, we obtain
\begin{align*}
C_H=&\left(\fracp F\lx \lx^2+\fracp F\ly \ly^2\right)H
+\left(F-\fracp F\lx\lx-\fracp F\ly\ly\right)\A2,\umbruch\\
G_H=&\fracp{^2F}{\lx\partial\lx}+2\fracp{^2F}{\lx\partial\ly}a_1
+\fracp{^2F}{\ly\partial\ly}a_1^2
+2\frac{\fracp F\lx-\fracp F\ly}{\lx-\ly}a_1^2,
\end{align*}
where
$$a_1=-\fracp{w\left(\lx+\ly,\,\lx^2+\ly^2\right)}{\lx}
\left(\fracp{w\left(\lx+\ly,\,\lx^2+\ly^2\right)}{\ly}\right)^{-1}$$
is such that $h_{22;\,1}=a_1h_{11;\,1}$ in a critical point
of $w$.
Similarly, we get for $w=\A2$
\begin{align*}
C_{\A2}=&2\left(\fracp F\lx\lx^2+\fracp F\ly\ly^2\right)\A2
+2\left(F-\fracp F\lx\lx-\fracp F\ly\ly\right)\A3,\umbruch\\
G_{\A2}=&-2\left(\fracp F\lx\left(1+a_1^2\right)+2\fracp F\ly a_1^2\right)
\umbruch\\
&+2\left(\fracp{^2F}{\lx\partial\lx}+2\fracp{^2F}{\lx\partial\ly}a_1
+\fracp{^2F}{\ly\partial\ly}a_1^2\right)\lx
+4\frac{\fracp F\lx-\fracp F\ly}{\lx-\ly}a_1^2\ly.
\end{align*}
We also need some mixed terms
\begin{align*}
-F^{ij}H_{;i}H_{;j}=&-\fracp F\lx(1+a_1)^2,\umbruch\\
-F^{ij}\Az i\Az j=&-4\fracp F\lx(\lx+a_1\ly)^2,\umbruch\\
-2F^{ij}H_{;i}\Az j=&-F^{ij}
\left(H_{;i}\Az j+H_{;j}\Az i\right)\umbruch\\
=&-4\fracp F\lx(1+a_1)(\lx+a_1\ly).
\end{align*}
Combining these expressions yields
\begin{align*}
C_w=&\fracp wHC_H+\fracp w{\A2}C_{\A2},\umbruch\\
G_w=&\fracp wHG_H+\fracp w{\A2}G_{\A2}
-\fracp{^2w}{H\partial H}\fracp F\lx(1+a_1)^2\\
&-4\fracp{^2w}{\A2\partial\A2}\fracp F\lx(\lx+a_1\ly)^2
-4\fracp{^2w}{H\partial\A2}\fracp F\lx(1+a_1)(\lx+a_1\ly).
\end{align*}
This formulae allow to easily compute evolution equations
in critical points.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Other Normal Velocities}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$H^3$-Flow}

In this and the following sections, we will consider strictly convex
surfaces contracting according to $\dt X=-F\nu$ for several normal
velocities $F$. We will not repeat parts of the argument that are
very similar to the respective parts in the proof for $F=\A2$.

As the theorems for these flow equations agree essentially with
Theorem \ref{main thm}, we will state them in concise form as
follows.

\begin{theorem}\label{H3 thm}
A smooth compact strictly convex surface in $\R^3$, contracting
with normal velocity $H^3$, converges to a round point in finite time.
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{A Monotone Quantity}

\begin{theorem}
For a family of smooth compact strictly convex surfaces $M_t$ 
in $\R^3$, flowing according to $\dt X=-H^3\nu$, 
$$\max\limits_{M_t}
\frac{\left(\lx^2+\lx\ly+\ly^2\right)(\lx+\ly)^2(\lx-\ly)^2}
{2\left(\lx^2-\lx\ly+\ly^2\right)\lx\ly}$$
is non-increasing in time.
\end{theorem}
\begin{proof}
We compute
\begin{align*}
F=&H^3,\displaybreak[1]\\
F^{ij}=&3H^2g^{ij},\displaybreak[1]\\
\oprokl H=&H^3\A2+6H\sum\hii,\displaybreak[1]\\
\opr{\A2}=&6H^2\Akl22-4H^3\A3\\
&-6H^2\sum\hij+12H\sum\lk\hii,\displaybreak[1]\\
\tilde w=&\log\left(\frac{\left(\lx^2+\lx\ly+\ly^2\right)
(\lx+\ly)^2(\lx-\ly)^2}
{2\left(\lx^2-\lx\ly+\ly^2\right)\lx\ly}\right)\\
=&\log\left(\frac{-H^6+\A2H^4+2\Akl22H^2}{-H^4+4\A2H^2-3\Akl22}\right).
\end{align*}
In a critical point of $\tilde w$, we obtain
\begin{align*}
h_{22;\,1}=&\frac{\ly}{\lx}\cdot
\frac{3\lx^6-2\lx^5\ly+4\lx^4\ly^2+4\lx^3\ly^3+4\lx^2\ly^4-2\lx\ly^5+\ly^6}
{3\ly^6-2\ly^5\lx+4\ly^4\lx^2+4\ly^3\lx^3+4\ly^2\lx^4-2\ly\lx^5+\lx^6}
\cdot h_{11;\,1},
\end{align*}
\begin{align*}
\oprokl{\tilde w}=&\left(\tfrac{-6H^5+4\A2H^3+4\Akl22H}
{-H^6+\A2H^4+2\Akl22H^2}-\tfrac{-4H^3+8\A2H}{-H^4+4\A2H^2-3\Akl22}
\right)\cdot\\
&\quad\cdot\left(\A2H^3+6H\sum\hii\right)\displaybreak[1]\\
&+\left(\tfrac{H^4+4\A2H^2}{-H^6+\A2H^4+2\Akl22H^2}
-\tfrac{4H^2-6\A2}{-H^4+4\A2H^2-3\Akl22}\right)\cdot\\
&\quad\cdot\left(6\Akl22H^2-4H^3\A3\right.\\
&\qquad\left.-6H^2\sum\hij+12H\sum\lk\hii\right)\displaybreak[1]\\
&+\left(\tfrac{-30H^4+12\A2H^2+4\Akl22}{-H^6+\A2H^4+2\Akl22H^2}
-\tfrac{-12H^2+8\A2}{-H^4+4\A2H^2-3\Akl22}\right)\cdot\\
&\quad\cdot\left(-3H^2\sum\hii\right)\displaybreak[1]\\
&+\left(\tfrac{4H^2}{-H^6+\A2H^4+2\Akl22H^2}
-\tfrac{-6}{-H^4+4\A2H^2-3\Akl22}\right)\cdot\\
&\quad\cdot\left(-12H^2\sum\li\lj\hii\right)\displaybreak[1]\\
&+\left(\tfrac{4H^3+8H\A2}{-H^6+\A2H^4+2\Akl22H^2}
-\tfrac{8H}{-H^4+4\A2H^2-3\Akl22}\right)\cdot\\
&\quad\cdot\left(-6H^2\sum(\li+\lj)\hii\right)\displaybreak[1]\\
=&-12\frac{(\lx+\ly)^2\lx^3\ly^3}
{\left(\lx^2-\lx\ly+\ly^2\right)
\left(\lx^2+\lx\ly+\ly^2\right)}\displaybreak[1]\\
&-\frac{12(\lx+\ly)^4\ly}{\left(\lx^6-2\lx^5\ly+4\lx^4\ly^2
+4\lx^3\ly^3+4\lx^2\ly^4-2\lx\ly^5+3\ly^6\right)^2}\cdot\displaybreak[1]\\
&\quad\cdot\frac1{\left(\lx^2-\lx\ly+\ly^2\right)
\left(\lx^2+\lx\ly+\ly^2\right)(\lx-\ly)^2\lx^3}\cdot\displaybreak[1]\\
&\quad\cdot\left(2\lx^{16}-10\lx^{15}\ly+10\lx^{14}\ly^2
+42\lx^{13}\ly^3-135\lx^{12}\ly^4\right.\\
&\qquad\left.+274\lx^{11}\ly^5-272\lx^{10}\ly^6+258\lx^9\ly^7
-144\lx^8\ly^8+262\lx^7\ly^9\right.\\
&\qquad\left.-166\lx^6\ly^{10}+98\lx^5\ly^{11}+6\lx^4\ly^{12}
-20\lx^3\ly^{13}+14\lx^2\ly^{14}\right.\\
&\qquad\left.-4\lx\ly^{15}+\ly^{16}\right)
\cdot h_{11;\,1}^2\displaybreak[1]\\
&-(\ldots)\cdot h_{22;\,2}\displaybreak[1]\\
\le&0.
\end{align*}
We applied Sturm's algorithm \cite{SturmThm} 
to obtain the last inequality.

Here and in the rest of the paper, we have sometimes used
a computer algebra program for the calculations involving
longer polynomials. Moreover, we use $(\ldots)$ in 
$(\ldots)\cdot h_{22;\,2}^2$ to abbreviate terms that 
are, up to interchanging $\lx$ and $\ly$,
equal to the respective factors in front of $h_{11;\,1}^2$. 

We have applied the following two identities in order to rewrite
terms involving derivatives of the second fundamental form
\begin{align*}
\sum\lambda_i^\alpha\lambda_j^\beta\lambda_k^\gamma\hij=&
\left(\lx^\alpha\lx^\beta\lx^\gamma
+\lx^\alpha\ly^\beta\ly^\gamma a_1^2
+\ly^\alpha\lx^\beta\ly^\gamma a_1^2
+\ly^\alpha\ly^\beta\lx^\gamma a_1^2\right)\cdot h_{11;\,1}^2\\
&+(\ldots)\cdot h_{22;\,2}^2,\displaybreak[1]\\
\sum\lambda_i^\alpha\lambda_j^\beta\lambda_k^\gamma\hii=&
\left(\lx^\alpha+\ly^\alpha a_1\right)
\left(\lx^\beta+\ly^\beta a_1\right)\lx^\gamma h_{11;\,1}^2
+(\ldots)\cdot h_{22;\,2}^2.
\end{align*}
They hold for all $\alpha,\,\beta,\,\gamma\in\R$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Velocity Bounds}

The following lemma is known, see \cite{FelixMZ}. Nevertheless, 
we include it, as we will use some of the 
calculations of its proof later on.
\begin{lemma}\label{H lower bound}
For a family of compact smooth strictly convex surfaces 
$M_t\subset\R^3$, $0\le t<T$, flowing according to
$\dt X=-F\nu$ with $F=H^\alpha$, $\alpha>1$, a positive
lower bound on the principal curvatures, $\lx,\,\ly\ge\epsilon
>0$, is preserved during the evolution.
\end{lemma}
\begin{proof}
We wish to apply the maximum principle for tensors
\cite{ChowLuMP,HamiltonFour86,HamiltonThree}. So we define
$$M_{ij}=h_{ij}-\epsilon g_{ij}.$$
We use \eqref{g evol} and Lemma \ref{evol equ lem} to obtain
\begin{align*}
\dt M_{ij}-F^{kl}M_{ij;\,kl}=&F^{kl}h^a_kh_{al}\cdot h_{ij}
-F^{kl}h_{kl}h^a_ih_{aj}-Fh^k_ih_{kj}\\
&+F^{kl,\,rs}h_{kl;\,i}h_{rs;\,j}+2\epsilon Fh_{ij}.
\end{align*}
We now specialize to the normal velocity $F=H^\alpha$. 
It is easy to see that 
$$F^{kl,\,rs} h_{kl;\,i} h_{rs;\,j}\ge0.$$
We have to test the zero eigenvalue condition. 
Assume that $\xi$ is a zero eigenvalue of $(M_{ij})$,
$h_{ij}\xi^j=\epsilon g_{ij}\xi^j$, with $g_{ij}\xi^i\xi^j=1$.
We may assume, that in our coordinate system, we have $\xi=(1,\,0)$
and $(h_{ij})=\left(\begin{smallmatrix}\epsilon&0\\0&\lambda
\end{smallmatrix}\right)$ with $\lambda\ge\epsilon>0$.
Direct calculations yield 
\begin{align*}
\left(\dt M_{ij}-F^{kl}M_{ij;\,kl}\right)\xi^i\xi^j\ge&
\alpha H^{\alpha-1}\A2\epsilon-\alpha H^\alpha\epsilon^2
-H^\alpha\epsilon^2+2\epsilon H^\alpha\epsilon
\displaybreak[1]\\
=&H^{\alpha-1}\left(\alpha\epsilon\lambda(\lambda-\epsilon)
+\epsilon^3+\epsilon^2\lambda\right)>0.
\end{align*}
The lemma follows from the maximum principle.
\end{proof}

The next lemma appears also in \cite{FelixMZ}. Once again,
we will use the following calculations later on.
\begin{lemma}\label{H upper bound}
For a family of compact smooth strictly convex surfaces 
$M_t\subset\R^3$, $0\le t<T$, flowing according to
$\dt X=-F\nu$ with $F=H^\beta$, $\beta>0$, the velocity
$F$ is bounded as in Lemma \ref{vel bound},
in terms of $\beta$, the initial data, and the
radius $R$ of an enclosed sphere. 
\end{lemma}
\begin{proof}
We proceed as above and get
\begin{align*}
\dt\log\left(\frac F{\langle X,\,\nu\rangle-\alpha}\right)\qquad&\\
-F^{ij}\left(\log\left(\frac F{\langle X,\,\nu\rangle-\alpha}\right)
\right)_{;\,ij}=&\frac1{\langle X,\,\nu\rangle-\alpha}
\left(F+F^{ij}h_{ij}-\alpha F^{ij}h^k_ih_{kj}\right).
\end{align*}
So far, we did not use the fact, that $F=H^\beta$.
In an increasing maximum of 
$\frac{F}{\langle X,\,\nu\rangle-\alpha}$, we get the inequality
$F+F^{ij}h_{ij}-\alpha F^{ij}h^k_ih_{kj}\ge0$ and deduce there 
$$\frac1\alpha\frac{1+\beta}\beta\ge\frac{\A2}H\ge\tfrac12H.$$
Our Lemma follows.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Concavity}

\begin{lemma}
The dual function to 
$F=H^\alpha$, $\alpha>0$, defined before
Lemma \ref{alpha concave lem}, is $-\alpha$-concave.
\end{lemma}
\begin{proof}
We use the same notation as before and obtain
\begin{align*}
\Phi=&-H^\alpha K^{-\alpha},\displaybreak[1]\\
\Phi^{ij}=&-\alpha H^{\alpha-1}K^{-\alpha}g^{ij}
+\alpha H^\alpha K^{-\alpha}\tilde h^{ij},\displaybreak[1]\\
\Phi^{ij,\,kl}=&-\alpha(\alpha-1)H^{\alpha-2}K^{-\alpha}g^{ij}g^{kl}\\
&+\alpha^2H^{\alpha-1}K^{-\alpha}g^{ij}\tilde h^{kl}
+\alpha^2H^{\alpha-1}K^{-\alpha}\tilde h^{ij}g^{kl}\\
&-\alpha^2H^\alpha K^{-\alpha}\tilde h^{ij}\tilde h^{kl}
-\alpha H^\alpha K^{-\alpha}\tilde h^{ik}\tilde h^{jl}.
\end{align*}
We have to prove that
$$\Phi^{ij,\,kl}\eta_{ij}\eta_{kl}\le
\frac{\alpha+1}{\alpha\Phi}\Phi^{ij}\eta_{ij}\Phi^{kl}\eta_{kl}$$
or 
\begin{align*}
0\le&-(\alpha+1)\begin{pmatrix}\frac{\ly^2}{\lx^2}&1\\
1&\frac{\lx^2}{\ly^2}\end{pmatrix}
+(\alpha-1)\begin{pmatrix}1&1\\1&1\end{pmatrix}\\
&-\alpha H\begin{pmatrix}\frac2\lx&\frac1\lx+\frac1\ly\\
\frac1\lx+\frac1\ly&\frac2\ly\end{pmatrix}
+\alpha H^2\begin{pmatrix}\frac1{\lx^2}&\frac1{\lx\ly}\\
\frac1{\lx\ly}&\frac1{\ly^2}\end{pmatrix}\\
&+H^2\begin{pmatrix}\frac1{\lx^2}&0\\0&\frac1{\ly^2}\end{pmatrix}
\displaybreak[1]\\
=&\begin{pmatrix}2\frac\ly\lx&-2\\-2&2\frac\lx\ly\end{pmatrix}.
\end{align*}
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Some Constants}
We wish to obtain precise bounds on the normal
velocity $F$ near $t=T$. This proof is almost identical for all our
test functions in Table \ref{table} with a factor $\lx\ly$
in the denominator. So it seems appropriate to state this proof
only once with appropriate constants depending on the normal
velocity and the test function. These constants are
\begin{itemize}
\item $c_h$: the homogeneity of $F$ in terms of the principal
  curvatures.
\item $c_1$: the value of $F$ at $\lx=\ly=1$.
\item $c_\alpha$: a positive constant, such that the dual function
  to $F$, defined before Lemma \ref{alpha concave lem},
  is a $-c_\alpha$-concave function. It turns out, that, for
  all flows considered here, we can choose $c_\alpha=c_h$.
\item $c_d$: a constant depending on the difference of the 
  degrees of the numerator $d_n$ and the denominator $d_d$
  of the test function $w$,
  $c_d:=\frac12(2-d_n+d_d)$. This constant is defined such that
  \begin{equation}\label{diff est}
  \abs{\lx-\ly}\le c\cdot H^{c_d}
  \end{equation} for a pinched surface for which $\max_{M_t}w$
  is non-increasing. 
\end{itemize}

For the flow equations considered here, these constants are as
in Table \ref{c tab}. We assume there that $\alpha\ge2$ and
$\beta\ge0$.

It is important for us that $c_d<1$ as it implies that 
Inequality \eqref{diff est} is not scaling invariant.
\begin{table}
\def\platzz{\raisebox{0em}[1.2em][0.7em]{\rule{0em}{2em}}}
$$\begin{array}{|c||c|c|c|c|c|c|c|}\hline
F & \platzz\B{\alpha} & H^2 & H^3 & H^4 & H\A2 &
\abs{A}^4 & \A2+\beta H^2\\\hline\hline
c_h & \platzz\alpha & 2 & 3 & 4 & 3 & 4 & 2\\\hline
c_1 & \platzz 2 & 4 & 8 & 16 & 4 & 4 & 2+4\beta\\\hline
c_\alpha & \platzz\alpha & 2 & 3 & 4 & 3 & 4 & 2\\\hline
c_d & \platzz\tfrac12(3-\alpha) & \tfrac12 & 0 & -2 & 0 
& -\tfrac12 & \tfrac12\\\hline
\frac{1-c_d}{1+c_h} & \platzz \frac{\alpha-1}{2(\alpha+1)}
&\frac16 &\frac14 &\frac35 &\frac14 & \tfrac3{10} &\frac16\\\hline
\end{array}$$
\caption{Some constants}\label{c tab}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pinching}
We show that our surfaces are pinched during the evolution,
i.\,e.\ that there exists a constant $c>0$, depending on our test
quantity, especially on the upper bound for it, and on the positive
lower bound for the principal curvatures, $\epsilon$, such that
$$0<\frac1c\le\frac\li\lj\le c\quad\text{for }i,\,j\in\{1,\,2\}.$$

The following proof does not apply directly to the case $F=K$ 
considered in \cite{AndrewsStones}, but the result is also true
in that case.

By direct inspection, we see that all our test quantities $w$ are
such that $w\cdot\frac{\lx\ly}{(\lx-\ly)^2}$ is bounded below by
a positive constant, depending especially on $\epsilon$. So we see
that $$\frac{\left(\frac\lx\ly-1\right)^2}{\frac\lx\ly}=
\frac{(\lx-\ly)^2}{\lx\ly}\le c.$$
Thus $\frac\lx\ly$ is bounded above and the surface is pinched.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Evolution of Spheres}
The radius of contracting spheres fulfills the ordinary 
differential equation
$$\dot r(t)=-\frac{c_1}{r(t)^{c_h}}.$$
A solution is given by
$$r(t)=(c_1\cdot(1+c_h)\cdot(T-t))^{\fracm1{1+c_h}},$$
so that inner and outer radii are related to the life span $T-t$
of $M_t$ as follows
$$\rho_-\le(c_1\cdot(1+c_h)\cdot(T-t))^{\fracm1{1+c_h}}\le
\rho_+(t).$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Bounds for Radii}
In order to prove bounds for the radii $_+$ and $r_-$, we
proceed as in the proof of Lemma \ref{radii bound} and use
$\abs{\lx-\ly}\le c\cdot H^{c_d}$
$$r_-\ge\frac1{8\pi}\int\limits_{M_t}H\cdot\left\{
1-c\cdot\rho_+\cdot H^{c_d}\right\}.$$
For each $F$ in Table \ref{c tab}, there exists $c_F>0$
such that
$$0<\frac1{c_F}H\le F^{1/c_h}\le c_FH,$$
so we can apply the variants of Lemma \ref{vel bound} for other 
curvature functions. For $t$ close to $T$, we get
\begin{align*}
r_-\ge&\frac1{8\pi}\int\limits_{M_t}H\cdot
\left\{1-c\cdot(T-t)^{\fracm{1-c_d}{1+c_h}}\right\}\displaybreak[1]
\intertext{and similarly, we obtain}
r_+\le&\frac1{8\pi}\int\limits_{M_t}H\cdot
\left\{1+c\cdot(T-t)^{\fracm{1-c_d}{1+c_h}}\right\}.
\end{align*}
Then we get for small $T-t$
\begin{align*}
&(c_1\cdot(1+c_n)\cdot(T-t))^{\fracm1{1+c_h}}\cdot
\left\{1-c\cdot(T-t)^{\fracm{1-c_d}{1+c_h}}\right\}\le\\
\le\frac1{8\pi}\int\limits_{M_t}H\le&
(c_1\cdot(1+c_h)\cdot(T-t))^{\fracm1{1+c_h}}\cdot
\left\{1+c\cdot(T-t)^{\fracm{1-c_d}{1+c_h}}\right\},\displaybreak[1]\\
r_-\ge&(c_1\cdot(1+c_h)\cdot(T-t))^{\fracm1{1+c_h}}\cdot
\left\{1-c\cdot(T-t)^{\fracm{1-c_d}{1+c_h}}\right\},\displaybreak[1]\\
r_+\le&(c_1\cdot(1+c_h)\cdot(T-t))^{\fracm1{1+c_h}}\cdot
\left\{1+c\cdot(T-t)^{\fracm{1-c_d}{1+c_h}}\right\},\displaybreak[1]
\intertext{and}
1\le&\frac{r_+}{r_-}\le1+c\cdot(T-t)^{\fracm{1-c_d}{1+c_h}}.
\end{align*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Precise Velocity Bounds}
We use the notation of Lemma \ref{A2 est}.
The Harnack inequality \cite[Theorem~5.17]{AndrewsHarnack}
implies for $t_2>t_1>0$
$$\frac{F(z,\,t_2)}{F(z,\,t_1)}\ge
\left(\frac{t_1}{t_2}\right)^{\fracm{c_\alpha}{1+c_\alpha}}.$$
Note that spheres evolve such that we get
$$\left(r_-(t)^{1+c_h}-c_1(1+c_h)\tau\right)^{\fracm1{1+c_h}}
\le s(\cdot,\,t+\tau).$$
We set 
$$\tau:=r_-(t)^{1+c_h}\cdot\left(\frac{r_+(t)}{r_-(t)}-1\right)^{1/2}$$
and get
\begin{align*}
F(z,\,t)\le&\frac{1+c\cdot(T-t)}{r_-(t)^{c_h}
\left(\frac{r_+(t)}{r_-(t)}-1\right)^{1/2}}\cdot\\
&\cdot\left(\frac{r_+(t)}{r_-(t)}-\left(1-c_1\cdot(1+c_h)\cdot
\left(\frac{r_+(t)}{r_-(t)}-1\right)^{1/2}\right)^{\fracm1{1+c_h}}\right).
\end{align*}
Use for $0\le x\le c(c_h)$
$$-(1-x)^{\fracm1{1+c_h}}\le-1+\frac1{1+c_h}x+\frac1{1+c_h}x^2.$$
We get
\begin{align}\label{F bound}
F(z,\,t)\le&\frac{1+c\cdot(T-t)}{r_-(t)^{c_h}}
\cdot\left(c_1+c\left(\frac{r_+}{r_-}-1\right)^{1/2}\right)
\displaybreak[1]\\
\le&c_1\cdot\left(c_1\cdot(1+c_h)\cdot(T-t)\right)^{-\frac{c_h}{1+c_h}}
\cdot\left(1+c\cdot(T-t)^{\fracm12\cdot\fracm{1-c_d}{1+c_h}}\right)
\nonumber
\end{align}
and a similar lower bound follows. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Convergence of Principal Curvatures}
We consider $F=H^\alpha$ and obtain
\begin{align*}
\lx=&\tfrac12(\lx+\ly)+\tfrac12(\lx-\ly)\displaybreak[1]\\
=&\tfrac12 F^{1/\alpha}+\tfrac12(\lx-\ly)\displaybreak[1]\\
\le&\left(2^\alpha\cdot(1+\alpha)
\cdot(T-t)\right)^{-\fracm1{1+\alpha}}\cdot
\left(1+c\cdot(T-t)^{\fracm12\cdot\fracm{1-c_d}{1+\alpha}}\right).
\end{align*}
A similar lower bound is proved analogously.
Theorem \ref{H3 thm} follows.

% Ende H^3-Fluss

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$H^2$-Flow}

\begin{theorem}\label{H2 thm}
A smooth compact strictly convex surface in $\R^3$, contracting
with normal velocity $H^2$, converges to a round point in finite time.
\end{theorem}

\begin{theorem}
For a family of smooth compact strictly convex surfaces $M_t$ 
in $\R^3$, flowing according to $\dt X=-H^2\nu$, 
$$\max\limits_{M_t}\frac{(\lx+\ly)^3(\lx-\ly)^2}
{2\left(\lx^2+\ly^2\right)\lx\ly}$$
is non-increasing in time.
\end{theorem}
\begin{proof}
We set 
\begin{align*}
w=&\frac{(\lx+\ly)^3(\lx-\ly)^2}
{2\left(\lx^2+\ly^2\right)\lx\ly}
=\frac{H^3\left(2\A2-H^2\right)}
{\A2\left(H^2-\A2\right)}.
\end{align*}
In a critical point of $\tilde w$, we get, based on
computer algebra calculations, 
\begin{align*}
\oprokl w=&-2\frac{(\lx+\ly)^4(\lx-\ly)^2\lx\ly}
{\left(\lx^2+\ly^2\right)^2}\umbruch\\
&-\frac{(\lx+\ly)^4}{\left(\lx^2+\ly^2\right)^2
\left(\lx^4-\lx^3\ly+7\lx^2\ly^2-\lx\ly^3+2\ly^4\right)^2
\lx^4}\cdot\umbruch\\
&\quad\cdot\left(5\lx^{12}-24\lx^{11}\ly+112\lx^{10}\ly^2
-164\lx^9\ly^3+529\lx^8\ly^4\right.\\
&\qquad\left.-448\lx^7\ly^5+952\lx^6\ly^6-312\lx^5\ly^7
+391\lx^4\ly^8-72\lx^3\ly^9\right.\\
&\qquad\left.+56\lx^2\ly^{10}-4\lx\ly^{11}
+3\ly^{12}\right)\cdot h_{11;\,1}^2
+(\ldots)\cdot h_{22;\,2}^2\umbruch\\
\le&0.
\end{align*}
We apply the maximum principle. Our claim follows.
\end{proof}
Theorem \ref{H2 thm} follows. 

% Ende H^2-Fluss

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$H^4$-Flow}

\begin{theorem}\label{H4 thm}
A smooth compact strictly convex surface in $\R^3$, contracting
with normal velocity $H^4$, converges to a round point in finite time.
\end{theorem}

\begin{theorem}
For a family of smooth compact strictly convex surfaces $M_t$ 
in $\R^3$, flowing according to $\dt X=-H^4\nu$, 
$$\max\limits_{M_t}\frac{\left(\lx^2+\lx\ly+\ly^2\right)
(\lx+\ly)^6(\lx-\ly)^2}
{2\lx^2\ly^2}$$
is non-increasing in time.
\end{theorem}
\begin{proof}
We proceed as above.
\begin{align*}
F=&H^4,\displaybreak[1]\\
F^{ij}=&4H^3g^{ij},\displaybreak[1]\\
F^{ij,\,kl}=&12H^2g^{ij}g^{kl},\displaybreak[1]\\
\tilde w=&\log\left(\frac{-H^{10}+H^8\A2+2H^6\Akl22}
{H^4-2H^2\A2+\Akl22}\right)\displaybreak[1]\\
=&\log\left(\frac{\left(\lx^2+\lx\ly+\ly^2\right)(\lx+\ly)^6(\lx-\ly)^2}
{2\lx^2\ly^2}\right).
\end{align*}
In a critical point of $\tilde w$, we get
\begin{align*}
h_{22;\,1}=&a_1h_{11;\,1}\\
=&\frac{\ly}{\lx}\frac{8\lx^4+3\lx^3\ly+2\lx^2\ly^2-3\lx\ly^3+2\ly^4}
{8\ly^4+3\ly^2\lx+2\ly^2\lx^2-3\ly\lx^3+2\lx^4}h_{11;\,1},\displaybreak[1]\\
\oprokl H=&H^4\A2+12H^2(1+a_1)^2\cdot h_{11;\,1}^2
+(\ldots)\cdot h_{22;\,2}^2,
\displaybreak[1]\\
\opr{\A2}=&8H^3\Akl22-6H^4\A3\\
&-8H^3\left(1+3a_1^2\right)\cdot h_{11;\,1}^2
+24H^2\lx(1+a_1)^2\cdot h_{11;\,1}^2\\
&+(\ldots)\cdot h_{22;\,2}^2.
\end{align*}
\begin{align*}
-F^{ij}H_{;\,i}H_{;\,j}=
&-4H^3(1+a_1)^2\cdot h_{11;\,1}
+(\ldots)\cdot h_{22;\,2}^2,\displaybreak[1]\\
-F^{ij}\left(\left(\A2\right)_{;\,i}H_{;\,j}
+\left(\A2\right)_{;\,j}H_{;\,i}\right)=
&-16H^3(\lx+\ly a_1)(1+a_1)\cdot h_{11;\,1}^2\\
&+(\ldots)\cdot h_{22;\,2}^2,\displaybreak[1]\\
-F^{ij}\left(\A2\right)_{;\,i}\left(\A2\right)_{;\,j}=
&-16H^3(\lx+\ly a_1)^2\cdot h_{11;\,1}^2
+(\ldots)\cdot h_{22;\,2}^2,\\
\end{align*}

\begin{align*}
\oprokl{\tilde w}=&\left(\frac{-10H^9+8H^7\A2+12H^5\Akl22}
{-H^{10}+H^8\A2+2H^6\Akl22}\right.\\
&\qquad\left.-\frac{4H^3-4H\A2}{H^4-2H^2\A2+\Akl22}\right)\cdot\\
&\quad\cdot\left(\oprokl H\right)\displaybreak[1]\\
&+\left(\frac{H^8+4H^6\A2}{-H^{10}+H^8\A2+2H^6\Akl22}
-\frac{-2H^2+2\A2}{H^4-2H^2\A2+\Akl22}\right)\cdot\\
&\quad\cdot\left(\opr{\A2}\right)\displaybreak[1]\\
&+\left(\frac{-90H^8+56H^6\A2+60H^4\Akl22}
{-H^{10}+H^8\A2+2H^6\Akl22}\right.\\
&\qquad\left.-\frac{12H^2-4\A2}{H^4-2H^2\A2+\Akl22}\right)\cdot\\
&\quad\cdot\left(-F^{ij}H_{;\,i}H_{;\,j}\right)\displaybreak[1]\\
&+\left(\frac{8H^7+24H^5\A2}{-H^{10}+H^8\A2+2H^6\Akl22}
-\frac{-4H}{H^4-2H^2\A2+\Akl22}\right)\cdot\\
&\quad\cdot\left(-F^{ij}\left(H_{;\,i}\A2_{;\,j}+H_{;\,j}\A2_{;\,i}
\right)\right)\displaybreak[1]\\
&+\left(\frac{4H^6}{-H^{10}+H^8\A2+2H^6\Akl22}
-\frac{2}{H^4-2H^2\A2+\Akl22}\right)\cdot\\
&\quad\cdot\left(-F^{ij}\A2_{;\,i}\A2_{;\,j}\right)\displaybreak[1]\\
=&\frac{-3(\lx-\ly)^2(\lx+\ly)^3\lx\ly}{\lx^2+\lx\ly+\ly^2}\umbruch\\
&-\frac{4\ly(\lx+\ly)^5}{\lx^3(\lx^2+\lx\ly+\ly^2)(\lx-\ly)^2}\cdot
\displaybreak[1]\\
&\quad\cdot\frac1{(2\lx^4-3\lx^3\ly+2\lx^2\ly^2+3\lx\ly^2+8\ly^4)^2}
\cdot\displaybreak[1]\\
&\quad\cdot\left(4\lx^{10}+202\lx^9\ly-447\lx^8\ly^2+809\lx^7\ly^3
-16\lx^6\ly^4+696\lx^5\ly^5\right.\\
&\qquad\left.-511\lx^4\ly^6+161\lx^3\ly^7-78\lx^2\ly^8+4\lx\ly^9+40\ly^{10}
\right)\cdot h_{11;\,1}^2\displaybreak[1]\\
&+(\ldots)\cdot h_{22;\,2}^2\displaybreak[1]\\
\le&0.
\end{align*}
Here we used once more a computer algebra system and Sturm's theorem
to obtain the last inequality.
\end{proof}
Theorem \ref{H4 thm} follows. 

There is also a monotone quantity (involving polynomials of higher
homogeneity) that allows to consider the flows $H^k$, $1<k\le5$,
simultaneously. For details, we refer to \cite{OSFelixH2}.

% Ende H^4-Fluss

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$\vert A\vert^2+\beta H^2$-Flow}

\begin{theorem}\label{A2H2 thm}
A smooth compact strictly convex surface in $\R^3$, contracting
with normal velocity $\A2+\beta H^2$, $0\le\beta\le5$, 
converges to a round point in finite time.
\end{theorem}

\begin{theorem}
For a family of smooth compact strictly convex surfaces $M_t$ 
in $\R^3$, flowing according to $\dot X=-\left(\A2+\beta H^2\right)\nu$, 
$0\le\beta\le5$,
$$\max\limits_{M_t}
\frac{(\lambda_1+\lambda_2)(\lambda_1-\lambda_2)^2}{2\lambda_1\lambda_2}$$
is non-increasing in time.
\end{theorem}
\begin{proof}
Similarly as above, we obtain for $F=\A2+\beta H^2$ 
\begin{align*}
\oprokl H=&-\left(\A2\right)^2+2H\A3
+\beta\A2 H^2\umbruch\\
&+2\sum\hij+2\beta\sum\hii,\displaybreak[1]\\
\opr{\A2}=&2\A2\A3\umbruch\\
&+\beta\left(4\left(\A2\right)^2H-2H^2\A3\right)\umbruch\\
&+\beta\left(-4H\sum\hij+4\sum\hii\lambda_k\right).
\end{align*}

As in the proof of Theorem \ref{A2 mon thm}, we set
$$
\tilde w=\log H+\log\left(2\A2-H^2\right)
-\log\left(H^2-\A2\right)$$
and obtain in a critical point of $\tilde w$, where
$h_{22;\,1}=a_1h_{11;\,1}$
\begin{align*}
-F^{ij}H_{;\,i}H_{;\,j}=&-2(\lx+\beta H)(1+a_1)^2\cdot h_{11;\,1}^2
+(\ldots)\cdot h_{22;\,2}^2,\umbruch\\
-F^{ij}\left(\left(\A2\right)_{;\,i}H_{;\,j}
+\left(\A2\right)_{;\,j}H_{;\,i}\right)=&
-8(\lx+\beta H)(\lx+\ly a_1)(1+a_1)\cdot h_{11;\,1}^2\\
&+(\ldots)\cdot h_{22;\,2}^2,\umbruch\\
h_{22;\,1}=&\frac\ly\lx\frac{2\lx^2+\lx\ly+\ly^2}
{2\ly^2+\ly\lx+\lx^2}h_{11;\,1}.
\end{align*}

In a critical point of $\tilde w$, we obtain 
the evolution equation
\begin{align*}
\oprokl{\tilde w}=
&\left(\frac1H-\frac{2H}{2\A2-H^2}-\frac{2H}{H^2-\A2}\right)
\cdot\left(\oprokl H\right)\displaybreak[1]\\
&+\left(\frac{2}{2\A2-H^2}+\frac1{H^2-\A2}\right)
\cdot\left(\opr{\A2}\right)\displaybreak[1]\\
&+\left(\frac{6}{2\A2-H^2}+\frac{2}{H^2-\A2}\right)F^{ij}H_iH_j
\displaybreak[1]\\
&-\frac2{H\cdot\left(2\A2-H^2\right)}
F^{ij}\left(\left(\A2\right)_iH_j+\left(\A2\right)_jH_i\right)
\displaybreak[1]\\
=&-4\frac{K^2}H-2\beta HK\displaybreak[1]\\
&-2\frac{\left(5\lambda_1^8-4\lambda_1^7\lambda_2
+46\lambda_1^6\lambda_2^2+48\lambda_1^5\lambda_2^3
+72\lambda_1^4\lambda_2^4\right)\lambda_2}
{\left(\lambda_1+\lambda_2\right)\left(\lambda_1-\lambda_2\right)^2
\left(\lambda_1^2+\lambda_1\lambda_2+2\lambda_2^2\right)^2\lambda_1^3}
h_{11;\,1}^2\displaybreak[1]\\
&-2\frac{\left(44\lambda_1^3\lambda_2^5+34\lambda_1^2\lambda_2^6
+8\lambda_1\lambda_2^7+3\lambda_2^8\right)\lambda_2}
{\left(\lambda_1+\lambda_2\right)\left(\lambda_1-\lambda_2\right)^2
\left(\lambda_1^2+\lambda_1\lambda_2+2\lambda_2^2\right)^2\lambda_1^3}
h_{11;\,1}^2\displaybreak[1]\\
&+2\beta\frac{\left(\lambda_1^4-16\lambda_1^3\lambda_2
-6\lambda_1^2\lambda_2^2-8\lambda_1\lambda_2^3-3\lambda_2^4\right)
\left(\lambda_1+\lambda_2\right)^3\lambda_2}
{\left(\lambda_1-\lambda_2\right)^2
\left(\lambda_1^2+\lambda_1\lambda_2+2\lambda_2^2\right)^2\lambda_1^3}
h_{11;\,1}^2\displaybreak[1]\\
&+(\ldots)\cdot h_{22;\,2}^2.\\
\end{align*}

For $\beta=5$, the factor in front of $h_{11;\,1}^2$ equals
\begin{align*}
&-\frac{4\lambda_2^2}{(\lambda_1+\lambda_2)(\lambda_1-\lambda_2)^2
\left(\lambda_1^2+\lambda_1\lambda_2+2\lambda_2^2\right)^2\lambda_1^3}\cdot\\
&\quad\cdot\left(28\lambda_1^7+183\lambda_1^6\lambda_2
+334\lambda_1^5\lambda_2^2+371\lambda_1^4\lambda_2^3
+272\lambda_1^3\lambda_2^4+157\lambda_1^2\lambda_2^5
+54\lambda_1\lambda_2^6+9\lambda_2^7\right).
\end{align*}
Our claim follows. 

In order to see that the range for $\beta$
is sharp for applying the maximum principle,
we observe that the terms without derivatives of the
second fundamental form require that $\beta\ge0$. 
For $\lambda_2=1$ and $\lambda_1\to\infty$,
the factor in front of $h_{11;\,1}^2$ behaves like
$-112\lambda_1^{-3}$ for $\beta=5$ and like
$-10\lambda_1^{-2}$ for $\beta=0$, so we need the
upper bound $\beta\le 5$.
\end{proof}

\begin{lemma}
For a family of compact smooth strictly convex surfaces 
$M_t\subset\R^3$, $0\le t<T$, flowing according to
$\dt X=-F\nu$ with $F=\A2+\beta H^2$, $\beta\ge0$, a positive
lower bound on the principal curvatures, $\lx,\,\ly\ge\epsilon
>0$, is preserved during the evolution.
\end{lemma}
\begin{proof}
We proceed similarly as in Lemma \ref{H lower bound}.
Dropping the term involving second derivatives of $F$ yields
\begin{align*}
\left(\dt M_{ij}-F^{kl}M_{ij;\,kl}\right)\xi^i\xi^j\ge&
2\left(\A3+\beta H\A2\right)\epsilon
-\left(\A2+\beta H^2\right)\epsilon^2\displaybreak[1]\\
=&\epsilon^4+\epsilon\lambda^3+\epsilon\lambda^2(\lambda-\epsilon)
+\beta\left(\epsilon^4+\epsilon^2\lambda^2+2\epsilon\lambda^3\right)>0.
\end{align*}
\end{proof}

Similar calculations as before give an upper bound on the
velocity for $F=\A2+\beta H^2$, $\beta\ge0$. 

\begin{lemma}
The dual function to 
$F=\A2+\beta H^2$, $\beta\ge0$, is $\alpha$-concave
for $\alpha\le-2$.
\end{lemma}
\begin{proof}
We have 
\begin{align*}
\Phi=&\frac{2K-(1+\beta)H^2}{K^2},\displaybreak[1]\\
\Phi^{ij}=&\frac1{K^2}\left(\left(-2K+2(1+\beta)H^2\right)\tilde h^{ij}
-2(1+\beta)Hg^{ij}\right),\displaybreak[1]\\
\Phi^{ij,\,kl}=&\frac1{K^2}\left(\left(2K-4(1+\beta)H^2\right)
\tilde h^{ij}\tilde h^{kl}-\left(-2K+2(1+\beta)H^2\right)
\tilde h^{ik}\tilde h^{jl}\right.\\
&\left.+4(1+\beta)H\left(g^{ij}\tilde h^{kl}+g^{kl}\tilde h^{ij}\right)
-2(1+\beta)g^{ij}g^{kl}\right).
\end{align*}
We wish to show for $\alpha\le-2$ and for symmetric matrices 
$(\eta_{ij})$, that 
$$\Phi^{ij,\,kl}\eta_{ij}\eta_{kl}\le\frac{\alpha-1}{\alpha\Phi}
\Phi^{ij}\eta_{ij}\Phi^{kl}\eta_{kl}.$$
Terms involving $\eta_{12}^2$ have the right sign.

Consider $\alpha=-2$. Then it suffices to prove the inequality
\begin{align*}
&\left((1+\beta)H^2-2K\right)\begin{pmatrix}
6\frac{\ly^2}{\lx^2}+6\beta\frac{\ly^2}{\lx^2}+4\beta\frac{\ly}{\lx}&
2\beta\\2\beta&
6\frac{\lx^2}{\ly^2}+6\beta\frac{\lx^2}{\ly^2}+4\beta\frac{\lx}{\ly}
\end{pmatrix}\ge\\
\ge&\tfrac32\begin{pmatrix}
2(1+\beta)\frac{\ly^2}{\lx}+2\beta\ly\\
2(1+\beta)\frac{\lx^2}{\ly}+2\beta\lx
\end{pmatrix}\otimes
\begin{pmatrix}
2(1+\beta)\frac{\ly^2}{\lx}+2\beta\ly\\
2(1+\beta)\frac{\lx^2}{\ly}+2\beta\lx
\end{pmatrix}\\
\ge&\tfrac32\begin{pmatrix}
\scriptstyle\left(2(1+\beta)\frac{\ly^2}{\lx}+2\beta\ly\right)^2&
\scriptstyle\left(2(1+\beta)\frac{\lx^2}{\ly}+2\beta\lx\right)
\left(2(1+\beta)\frac{\ly^2}{\lx}+2\beta\ly\right)\\
\scriptstyle\left(2(1+\beta)\frac{\lx^2}{\ly}+2\beta\lx\right)
\left(2(1+\beta)\frac{\ly^2}{\lx}+2\beta\ly\right)&
\scriptstyle\left(2(1+\beta)\frac{\lx^2}{\ly}+2\beta\lx\right)^2
\end{pmatrix}
\end{align*}
in order to obtain $\alpha$-concavity for all
$\alpha\le-2$. This inequality is fulfilled, if
$$\left\{6\lx\ly+\beta\left(4\lx^2+12\lx\ly+4\ly^2\right)
+\beta^2\left(4\lx^2+8\lx\ly+4\ly^2\right)\right\}
\begin{pmatrix}\frac{\ly}{\lx}&-1\\ -1&\frac{\lx}{\ly}\end{pmatrix}$$
is positive semi-definite.
\end{proof}

We want to derive precise bounds on the principal curvatures. To 
this end, we use \eqref{diff est} and \eqref{F bound}
\begin{align*}
&(2+4\beta)\left((2+4\beta)3(T-t)\right)^{-2/3}\cdot
\left(1-c\cdot(T-t)^{1/12}\right)\umbruch\\
\le&F=\A2+\beta H^2\umbruch\\
\le&\lx^2+(\lx+\abs{\lx-\ly})^2+\beta(\lx+(\lx+\abs{\lx-\ly}))^2
\umbruch\\
\le&(2+4\beta)\lx^2+c\cdot F^{1/2}\cdot F^{1/4}+c\cdot F^{1/2}
\umbruch\\
\le&(2+4\beta)\lx^2+c\cdot(T-t)^{-2/3}\cdot
\left((T-t)^{1/6}+(T-t)^{1/3}\right).
\end{align*}
We get
$$\lx\ge((2+4\beta)3(T-t))^{-1/3}\cdot
\left(1+c\cdot(T-t)^{1/12}\right)$$
and a similar upper bound follows analogously. 

We obtain Theorem \ref{A2H2 thm}.

% Ende \A2+\beta H^2-Fluss

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$\tr A^3$-Flow}

\begin{theorem}\label{A3 thm}
A smooth compact strictly convex surface in $\R^3$, contracting
with normal velocity $\A3$, converges to a round point in finite time.
\end{theorem}

\begin{theorem}
For a family of smooth compact strictly convex surfaces $M_t$ 
in $\R^3$, flowing according to $\dot X=-\A3\nu$, 
$$\max\limits_{M_t}\frac{\left(3\lx^2+2\lx\ly+3\ly^2\right)(\lx-\ly)^2}
{\lx\ly}$$
is non-increasing in time.
\end{theorem}
\begin{proof}
Calculations as above yield
\def\adreizaehler{-H^4+4\Akl22}
\def\adreinenner{H^2-\A2}
\begin{align*}
\tilde w=&\log\left(\frac{\left(3\lx^2+2\lx\ly+3\ly^2\right)(\lx-\ly)^2}
{2\lx\ly}\right)\displaybreak[1]\\
=&\log\left(\frac{-H^4+4\Akl22}{H^2-\A2}\right),\displaybreak[1]\\
\oprokl H=&3\A4H-2\A3\A2+3\sum(\li+\lj)\hij\displaybreak[1]\\
=&3\A4H-2\A3\A2+3\left(2\lx+2\lx a_1^2+4\ly a_1^2\right)\cdot 
h_{11;\,1}^2\displaybreak[1]\\
&+(\ldots)\cdot h_{22;\,2}^2,
\displaybreak[1]\\
\opr{\A2}=&6\A4\A2-4\Akl32\umbruch\\
&-6\sum\lk^2\hij+6\sum\lk(\li+\lj)\hij
\displaybreak[1]\\
=&6\A4\A2-4\Akl32
-6\left(\lx^2+\lx^2a_1^2+2\ly^2a_1^2\right)\cdot h_{11;\,1}^2
\displaybreak[1]\\
&+6\left(2\lx^2+4\lx\ly a_1^2+2\ly^2a_1^2\right)\cdot h_{11;\,1}^2
+(\ldots)\cdot h_{22;\,2}^2,
\displaybreak[1]\\
-F^{ij}H_{;\,i}H_{;\,j}=&-3\lx^2(1+a_1)^2\cdot h_{11;\,1}^2
+(\ldots)\cdot h_{22;\,2}^2,\displaybreak[1]\\
-F^{ij}\left(\A2\right)_{;i}\left(\A2\right)_{;j}=&
-12\lx^2(\lx+\ly a_1)^2\cdot h_{11;\,1}^2+(\ldots)\cdot h_{22;\,2}^2,
\displaybreak[1]\\
h_{22;\,1}=&\frac\ly\lx\frac{9\lx^3+\lx^2\ly+3\lx\ly^2+3\ly^3}
{9\ly^3+\ly^2\lx+3\ly\lx^2+3\lx^3}\cdot h_{11;\,1},\displaybreak[1]\\
\oprokl{\tilde w}=&\left(\frac{-4H^3}\adreizaehler
-\frac{2H}\adreinenner\right)\cdot\left(\oprokl H\right)\umbruch\\
&+\left(\frac{8\A2}\adreizaehler
+\frac1\adreinenner\right)\cdot\\
&\quad\cdot\left(\opr{\A2}\right)\umbruch\\
&+\left(\frac{-12H^2}\adreizaehler-\frac2\adreinenner\right)\cdot
\left(-F^{ij}H_{;\,i}H_{;\,j}\right)\umbruch\\
&+\frac8\adreizaehler\left(-F^{ij}\left(\A2\right)_{;\,i}
\left(\A2\right)_{;\,j}\right)\umbruch\\
=&-\frac{2\left(\lx^4-2\lx^3\ly+18\lx^2\ly^2-2\lx\ly^3+\ly^4\right)\lx\ly}
{3\lx^2+2\lx\ly+3\ly^2}
\umbruch\\
&-\frac{6\ly}{\left(3\lx^2+2\lx\ly+3\ly^2\right)
\left(\lx-\ly\right)^2\lx^3}\cdot\\
&\quad\cdot\frac1{\left(3\lx^3+3\lx^2\ly+\lx\ly^2+9\ly^3\right)^2}\cdot\\
&\quad\cdot\left(63\lx^{12}+381\lx^{11}\ly-1389\lx^{10}\ly^2
+2883\lx^9\ly^3+36\lx^8\ly^4\right.\\
&\qquad\left.+1218\lx^7\ly^5+2294\lx^6\ly^6+582\lx^5\ly^7+855\lx^4\ly^8
\right.\\
&\qquad\left.+945\lx^3\ly^9+135\lx^2\ly^{10}+135\lx\ly^{11}
+54\ly^{12}\right)\cdot h_{11;\,1}^2\umbruch\\
&+(\ldots)\cdot h_{22;\,2}^2\umbruch\\
\le&0.
\end{align*}
\end{proof}

\begin{lemma}
For a family of compact smooth strictly convex surfaces 
$M_t\subset\R^3$, $0\le t<T$, flowing according to
$\dt X=-F\nu$ with $F=\B\alpha$, $\alpha\ge2$, a positive
lower bound on the principal curvatures, $\lx,\,\ly\ge\epsilon
>0$, is preserved during the evolution.
\end{lemma}
\begin{proof}
We proceed similarly as in Lemma \ref{H lower bound}.
Once again, the term involving second derivatives of $F$ 
is non-negative. As before, we obtain
\begin{align*}
\left(\dt M_{ij}-F^{kl}M_{ij;\,kl}\right)\xi^i\xi^j\ge&
\alpha\B{\alpha+1}\epsilon-\alpha\B\alpha\epsilon^2
-\B\alpha\epsilon^2+2\epsilon\B\alpha\epsilon\displaybreak[1]\\
=&\epsilon\left(\alpha\left(\lambda^{\alpha+1}
-\epsilon\lambda^{\alpha}\right)
+\epsilon^{\alpha+1}+\epsilon\lambda^\alpha\right)>0.
\end{align*}
\end{proof}

Similar calculations as in Lemma \ref{H upper bound}, using
$F+F^{ij}h_{ij}-\alpha F^{ij}h^k_ih_{kj}\ge0$ for 
$F=\B{\beta}$, $\beta\ge2$, yield for some $c_\beta>0$
$$\frac1{c_\beta}\left(\B\beta\right)^{1/\beta}\le
\frac1{c_\beta}H\le\frac{\B{\beta+1}}{\B\beta}\le
\frac{\beta+1}{\beta}\frac1\alpha$$
and an estimate as in Lemma \ref{H upper bound} follows.

\begin{lemma}
For $\alpha>0$, the dual function to $F=\B{\alpha}=
\lx^\alpha+\ly^\alpha$ is $-\alpha$-concave.
\end{lemma}
\begin{proof}
We set $\Phi=-\B{\alpha}\cdot K^{-\alpha}$ and have to show
that
$$\Phi^{ij,\,kl}\eta_{ij}\eta_{kl}\le
\frac{-\alpha-1}{-\alpha\Phi}\Phi^{ij}\eta_{ij}
\Phi^{kl}\eta_{kl}$$
for symmetric matrices $(\eta_{ij})$. Direct computations yield that
this inequality is equivalent to
\begin{align*}
&-\alpha(\alpha-1)K^{-\alpha}\etamatrix{\lx^{\alpha-2}}0{\ly^{\alpha-2}}
-2\alpha K^{-\alpha}
\sum\limits_{r=0}^{\alpha-2}\lx^r\ly^{\alpha-2-r}\eta_{12}^2
\displaybreak[1]\\
&+\alpha^2K^{-\alpha}\etamatrix{2\lx^{\alpha-2}}{\frac1{\lx\ly}\B{\alpha}}
{2\ly^{\alpha-2}}\displaybreak[1]\\
&-\alpha^2\B{\alpha}K^{-\alpha}\etamatrix{\frac1{\lx^2}}{\frac1{\lx\ly}}
{\frac1{\ly^2}}\displaybreak[1]\\
&-\alpha\B{\alpha}K^{-\alpha}\etamatrix{\frac1{\lx^2}}0{\frac1{\ly^2}}
-2\alpha\B{\alpha}K^{-\alpha}\frac1{\lx\ly}\eta_{12}^2\displaybreak[1]\\
\le&-\frac{\alpha(\alpha+1)K^{-\alpha}}{\B{\alpha}}
\etamatrix{\frac{\ly^{2\alpha}}{\lx^2}}{\lx^{\alpha-1}\ly^{\alpha-1}}
{\frac{\lx^{2\alpha}}{\ly^2}}.
\end{align*}
Further computations show that this is fulfilled, if
$$0\le\begin{pmatrix}\lx^{\alpha-2}\ly^\alpha
&-\lx^{\alpha-1}\ly^{\alpha-1}\\
-\lx^{\alpha-1}\ly^{\alpha-1}&
\lx^\alpha\ly^{\alpha-2}\end{pmatrix}.$$
\end{proof}

Then we proceed as before.
Similar calculations as for $F=\A2+\beta H^2$ 
give for $\alpha=3$ 
$$\lx,\,\ly\le(2(1+\alpha)\cdot(T-t))^{-\fracm1{1+\alpha}}
\cdot\left(1+c\cdot(T-t)^{\fracm14\fracm{\alpha-1}{\alpha+1}}\right)$$
and a corresponding lower bound holds.
This estimate holds also for $F=\B{\alpha}$, 
$\alpha=2,\,3,\,4,\,5,\,6$.

This finishes the proof of Theorem \ref{A3 thm}.

% Ende \A3-Fluss

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$\tr A^{\alpha}$-Flow}

\begin{theorem}\label{Aalpha thm}
A smooth compact strictly convex surface in $\R^3$, contracting
with normal velocity $\A4$, $\A5$, or $\A6$, 
converges to a round point in finite time.
\end{theorem}

\begin{theorem}
For a family of smooth compact strictly convex surfaces $M_t$ 
in $\R^3$, flowing according to $\dot X=-\B{\alpha+2}\nu$, 
$\alpha=2,\,3,\,4$,
$$\max\limits_{M_t}
\frac{\left(\lx^\alpha+\ly^\alpha\right)(\lx+\ly)
(\lx-\ly)^2}{\lx\ly}$$
is non-increasing in time.
\end{theorem}
\begin{proof}
One might conjecture, that this quantity is also monotone
for other values of $\alpha$. For $\alpha=0$, corresponding to
$F=\A2$, we have already checked that in Theorem \ref{A2 mon thm}. 
Further computations for $\alpha=1,\,5,\,6,\,7$ suggest, however, 
that this quantity is not monotone for these values of $\alpha$.

We obtain
\begin{align*}
\oprokl H=&(\alpha+2)\B{\alpha+3}H-(\alpha+1)\B{\alpha+2}\A2
\displaybreak[1]\\
&+(\alpha+2)\sum\limits_{r=0}^\alpha\sum\limits_{i,\,j,\,k=1}^2
\lambda_i^r\lambda_j^{\alpha-r}
\hij,\displaybreak[1]\\
\opr{\A2}=&2(\alpha+2)\B{\alpha+3}\A2-2(\alpha+1)\B{\alpha+2}\A3
\displaybreak[1]\\
&-2(\alpha+2)\sum\limits_{i,\,j,\,k=1}^2\lk^{\alpha+1}\hij
\displaybreak[1]\\
&+2(\alpha+2)\sum\limits_{r=0}^\alpha
\sum\limits_{i,\,j,\,k=1}^2\lambda_i^r\lambda_j^{\alpha-r}
\hij\lk,\umbruch\\
\opr{\B\alpha}=&\alpha(\alpha+2)\B{\alpha+3}\B\alpha
-\alpha(\alpha+1)\B{\alpha+2}\B{\alpha+1}\umbruch\\
&-\alpha(\alpha+2)\sum\limits_{r=0}^{\alpha-2}\lk^{\alpha+1}
\li^{\alpha-2-r}\lj^r\hij\umbruch\\
&+\alpha(\alpha+2)\sum\limits_{r=0}^\alpha\li^r\lj^{\alpha-r}
\lk^{\alpha-1}\hij,
\end{align*}
\begin{align*}
\tilde w=&\log\left(\frac{\left(\lx^\alpha+\ly^\alpha\right)(\lx+\ly)
(\lx-\ly)^2}{\lx\ly}\right)\displaybreak[1]\\
\equiv&\log A+\log B+\log C-\log D,\displaybreak[1]\\
\oprokl {\tilde w}=&\frac1A\left(\oprokl A\right)
+\frac1B\left(\oprokl B\right)\displaybreak[1]\\
&+\frac1C\left(\oprokl C\right)
-\frac1D\left(\oprokl D\right)\displaybreak[1]\\
&+\frac2{B^2}F^{ij}B_{;\,i}B_{;\,j}+\frac2{C^2}
F^{ij}C_{;\,i}C_{;\,j}
+\frac1{BC}F^{ij}(B_{;\,i}C_{;\,j}+B_{;\,j}C_{;\,i})
\displaybreak[1]\\
&-\frac1{BD}F^{ij}(B_{;\,i}D_{;\,j}+B_{;\,j}D_{;\,i})
-\frac1{CD}F^{ij}(C_{;\,i}D_{;\,j}+C_{;\,j}D_{;\,i})\displaybreak[1]\\
=&\frac1{\tr A^\alpha}\cdot\left(\opr{\tr A^\alpha}\right)\\
&+\left(\frac1H-\frac{2H}{2\A2-H^2}-\frac{2H}{H^2-\A2}\right)\cdot
\left(\oprokl H\right)\displaybreak[1]\\
&+\left(\frac2{2\A2-H^2}+\frac1{H^2-\A2}\right)\cdot
\left(\oprokl{\A2}\right)\displaybreak[1]\\
&+\left(\frac2{2\A2-H^2}+\frac2{H^2-\A2}+\frac2{H^2}
+\frac{8H^2}{\left(2\A2-H^2\right)^2}\right.\\
&\quad\left.
-\frac{4H}{H\left(2\A2-H^2\right)}
-\frac{4H}{H\left(H^2-\A2\right)}\right.\\
&\quad\left.
+\frac{8H^2}{\left(2\A2-H^2\right)\left(H^2-\A2\right)}\right)
\cdot F^{ij}H_{;\,i}H_{;\,j}\displaybreak[1]\\
&+\left(\frac8{\left(2\A2-H^2\right)^2}+\frac4{\left(2\A2-H^2\right)
\left(H^2-\A2\right)}\right)\cdot\\
&\quad\cdot
F^{ij}\left(\A2\right)_{;\,i}\left(\A2\right)_{;\,j}\displaybreak[1]\\
&+\left(\frac1{H\left(H^2-\A2\right)}
-\frac{8H}{\left(2\A2-H^2\right)^2}
+\frac2{H\left(2\A2-H^2\right)}\right.\\
&\quad\left.
-\frac{6H}{\left(2\A2-H^2\right)\left(H^2-\A2\right)}\right)\cdot\\
&\quad\cdot F^{ij}\left(H_{;\,i}\left(\A2\right)_{;\,j}
+H_{;\,j}\left(\A2\right)_{;\,i}\right).
\end{align*}
Plugging this into a computer algebra program yields\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for $\alpha=2$, corresponding to $F=\A4$,
\begin{align*}
h_{22;\,1}=&\frac{\ly}{\lx}
\frac{4\lx^4+\lx^3\ly+\lx^2\ly^2+\lx\ly^3+\ly^4}
{4\ly^4+\ly^3\lx+\ly^2\lx^2+\ly\lx^3+\lx^4}h_{11;\,1},\displaybreak[1]\\
\oprokl{\tilde w}=&\frac{-24\lx^4\ly^4}
{\left(\lx+\ly\right)\left(\lx^2+\ly^2\right)}\displaybreak[1]\\
&+\frac{-4\ly}{\left(\lx-\ly\right)^2\left(\lx+\ly\right)
\lx^3\left(\lx^2+\ly^2\right)}\cdot\umbruch\\
&\quad\cdot\frac1
{\left(\lx^4+\lx^3\ly+\lx^2\ly^2+\lx\ly^3+4\ly^4\right)^2}\cdot\umbruch\\
&\quad\cdot\left(11\lx^{16}+24\lx^{15}\ly+39\lx^{14}\ly^2
-328\lx^{13}\ly^3+482\lx^{12}\ly^4\right.\\
&\qquad\left.+192\lx^{11}\ly^5+215\lx^{10}\ly^6+236\lx^9\ly^7
+432\lx^8\ly^8+200\lx^7\ly^9\right.\\
&\qquad\left.+173\lx^6\ly^{10}+144\lx^5\ly^{11}+158\lx^4\ly^{12}
+32\lx^3\ly^{13}+21\lx^2\ly^{14}\right.\\
&\qquad\left.+12\lx\ly^{15}+5\ly^{16}\right)\cdot h_{11;\,1}^2
\displaybreak[1]\\
&+(\ldots)\cdot h_{22;\,2}^2,
\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for $\alpha=3$ $\left(F=\A5\right)$
\begin{align*}
h_{22;\,1}=&\frac{\ly}{\lx}
\frac{5\lx^4-4\lx^3\ly+2\lx^2\ly^2+\ly^4}
{5\ly^4-4\ly^3\lx+2\ly^2\lx^2+\lx^4}h_{11;\,1},\displaybreak[1]\\
\oprokl{\tilde w}=&\frac{-4\left(2\lx^4-7\lx^3\ly+12\lx^2\ly^2
-7\lx\ly^3+2\ly^4\right)\lx^2\ly^2}{\left(\lx^2-\lx\ly+\ly^2\right)}
\displaybreak[1]\\
&+\frac{-10\ly}{\lx^3\left(\lx-\ly\right)^2
\left(\lx^4+2\lx^2\ly^2-4\lx\ly^3+5\ly^4\right)^2
\left(\lx^2-\lx\ly+\ly^2\right)}\cdot\\
&\quad\cdot\left(7\lx^{16}-65\lx^{15}\ly+397\lx^{14}\ly^2
-1295\lx^{13}\ly^3+2464\lx^{12}\ly^4\right.\\
&\qquad\left.-2981\lx^{11}\ly^5+2645\lx^{10}\ly^6-2007\lx^9\ly^7
+1510\lx^8\ly^8\right.\\
&\qquad\left.-1011\lx^7\ly^9+583\lx^6\ly^{10}-309\lx^5\ly^{11}
+176\lx^4\ly^{12}-71\lx^3\ly^{13}\right.\\
&\qquad\left.+23\lx^2\ly^{14}-5\lx\ly^{15}+3\ly^{16}\right)
\cdot h_{11;\,1}^2\displaybreak[1]\\
&+(\ldots)\cdot h_{22;\,2}^2,
\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and for $\alpha=4$ $\left(F=\A6\right)$
\begin{align*}
h_{22;\,1}=&\frac{\ly}{\lx}
\frac{6\lx^6+\lx^5\ly-3\lx^4\ly^2+2\lx^2\ly^4+\lx\ly^5+\ly^6}
{6\ly^6+\ly^5\lx-3\ly^4\lx^2+2\ly^2\lx^4+\ly\lx^5+\lx^6}h_{11;\,1},
\displaybreak[1]\\
\oprokl{\tilde w}=&\frac{-10\left(\lx^8-2\lx^6\ly^2+6\lx^4\ly^4-2\lx^2\ly^6
+\ly^8\right)\lx^2\ly^2}{\left(\lx+\ly\right)\left(\lx^4+\ly^4\right)}
\displaybreak[1]\\
&+\frac{-6\ly}{\lx^3\left(\lx-\ly\right)^2\left(\lx+\ly\right)
\left(\lx^4+\ly^4\right)}\cdot\umbruch\\
&\quad\cdot\frac1{\left(\lx^6+\lx^5\ly+2\lx^4\ly^2-3\lx^2\ly^4
+\lx\ly^5+6\ly^6\right)^2}\cdot\umbruch\\
&\quad\cdot\left(17\lx^{24}-124\lx^{23}\ly+218\lx^{22}\ly^2
+646\lx^{21}\ly^3-642\lx^{20}\ly^4\right.\\
&\qquad\left.-2586\lx^{19}\ly^5+2536\lx^{18}\ly^6+3576\lx^{17}\ly^7
-2411\lx^{16}\ly^8\right.\\
&\qquad\left.-2928\lx^{15}\ly^9+1524\lx^{14}\ly^{10}
+1724\lx^{13}\ly^{11}+548\lx^{12}\ly^{12}\right.\\
&\qquad\left.-276\lx^{11}\ly^{13}-696\lx^{10}\ly^{14}
-8\lx^9\ly^{15}+499\lx^8\ly^{16}+236\lx^7\ly^{17}\right.\\
&\qquad\left.+146\lx^6\ly^{18}-66\lx^5\ly^{19}-2\lx^4\ly^{20}
+46\lx^3\ly^{21}+48\lx^2\ly^{22}\right.\\
&\qquad\left.+16\lx\ly^{23}+7\ly^{24}\right)\cdot h_{11;\,1}^2
\displaybreak[1]\\
&+(\ldots)\cdot h_{22;\,2}^2.
\end{align*}
In each case, Sturm's algorithm yields, that the 
right-hand side is non-positive.
\end{proof}
Theorem \ref{Aalpha thm} follows. 

% Ende \A{\alpha}-Fluss

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$H|A|^2$-Flow}

\begin{theorem}\label{A2H thm}
A smooth compact strictly convex surface in $\R^3$, contracting
with normal velocity $H\A2$, converges to a round point in finite time.
\end{theorem}

\begin{theorem}
For a family of smooth compact strictly convex surfaces $M_t$ 
in $\R^3$, flowing according to $\dt X=-H\A2\nu$, 
$$\max\limits_{M_t}\frac{(\lx+\ly)^2(\lx-\ly)^2}{2\lx\ly}$$
is non-increasing in time.
\end{theorem}
\begin{proof}
We proceed as above. 
\begin{align*}
w=&\log\left(\frac{(\lx+\ly)^2(\lx-\ly)^2}{2\lx\ly}\right)
\displaybreak[1]\\
=&\log\left(\frac{-H^4+2\A2H^2}{H^2-\A2}\right),\displaybreak[1]\\
h_{22;\,1}=&\frac{3\lx^2+\ly^2}{3\ly^2+\lx^2}\frac\ly\lx
h_{11;\,1},\displaybreak[1]\\
\oprokl H=&2H^2\A3-H\Akl22\\&+2\sum(\li+\lj)\hii
+2H\sum\hij\displaybreak[1]\\
=&2H^2\A3-H\Akl22\\
&+4(\lx+\ly a_1)(1+a_1)\cdot h_{11;\,1}^2
+2H(1+3a_1^2)\cdot h_{11;\,1}^2\\
&+(\ldots)\cdot h_{22;\,2}^2,\displaybreak[1]\\
\opr{\A2}=&2\Akl23-2\A2\sum\hij+4\sum(\li+\lj)\lk\hii\displaybreak[1]\\
=&2\Akl23-2\A2(1+3a_1^2)\cdot h_{11;\,1}^2\\&
+8\lx(\lx+\ly a_1)(1+a_1)\cdot h_{11;\,1}^2\\
&+(\ldots)\cdot h_{22;\,2}^2,\displaybreak[1]\\
-F^{ij}H_{;\,i}H_{;\,j}=&-\sum\left(\A2+2H\lk\right)\hii\displaybreak[1]\\
=&-\A2(1+a_1)^2\cdot h_{11;\,1}^2-2H\lx(1+a_1)^2\cdot h_{11;\,1}^2\\
&+(\ldots)\cdot h_{22;\,2}^2,
\end{align*}
\begin{align*}
-F^{ij}\left(\left(\A2\right)_{;\,i}H_{;\,j}
+\left(\A2\right)_{;\,j}H_{;\,i}\right)
=&-2\sum\left(\A2+2H\lk\right)(\li+\lj)\hii\\
=&-4\A2(\lx+\ly a_1)(1+a_1)\cdot h_{11;\,1}^2\\
&-8H(\lx+\ly a_1)(1+a_1)\lx
h_{11;\,1}^2\\
&+(\ldots)\cdot h_{22;\,2}^2,
\end{align*}
\begin{align*}
\oprokl w=&\left(\frac{-4H^3+4\A2H}{-H^4+2\A2H^2}-\frac{2H}{H^2-\A2}\right)
\left(\oprokl H\right)\\
&+\left(\frac{2H^2}{-H^4+2\A2H^2}+\frac1{H^2-\A2}\right)
\left(\opr{\A2}\right)\\
&+\left(\frac{-12H^2+4\A2}{-H^4+2\A2H^2}-\frac2{H^2-\A2}\right)
\left(-F^{ij}H_{;\,i}H_{;\,j}\right)\\
&+\frac{4H}{-H^4+2\A2H^2}
\left(-F^{ij}\left(\left(\A2\right)_{;\,i}H_{;\,j}
+\left(\A2\right)_{;\,j}H_{;\,i}\right)\right)\displaybreak[1]\\
=&-8\lx^2\ly^2\\
&-\frac{4\ly}{\left(\lx^2+3\ly^2\right)^2(\lx-\ly)^2\lx^3}\cdot\\
&\qquad\cdot\left(6\lx^8-39\lx^7\ly+91\lx^6\ly^2+\lx^5\ly^3
+91\lx^4\ly^4+3\lx^3\ly^5\right.\\
&\qquad\quad\left.+33\lx^2\ly^6+3\lx\ly^7+3\ly^8\right)
\cdot h_{11;\,1}^2\displaybreak[1]\\
&+(\ldots)\cdot h_{22;\,2}^2\umbruch\\
\le&0.
\end{align*}
\end{proof}

\begin{lemma}
For a family of compact smooth strictly convex surfaces 
$M_t\subset\R^3$, $0\le t<T$, flowing according to
$\dt X=-F\nu$ with $F=H\A2$, a positive
lower bound on the principal curvatures, $\lx,\,\ly\ge\epsilon
>0$, is preserved during the evolution.
\end{lemma}
\begin{proof}
We proceed similarly as in Lemma \ref{H lower bound}
and compute
\begin{align*}
F^{kl,\,rs}h_{kl;\,1}h_{rs;\,1}=&
(6\lx+2\ly)\cdot h_{11;\,1}^2+4(\lx+\ly)\cdot h_{11;\,1}h_{22;\,1}\\
&+4(\lx+\ly)\cdot h_{11;\,2}^2+(2\lx+6\ly)\cdot h_{22;\,1}^2\ge0,\umbruch\\
\left(\dt M_{ij}-F^{kl}M_{ij;\,kl}\right)\xi^i\xi^j\ge&
\left(\Akl22+2H\A3\right)\cdot h_{ij}\xi^i\xi^j\\
&-4H\A2h^k_ih_{kj}\xi^i\xi^j+2\epsilon H\A2h_{ij}\xi^i\xi^j
\displaybreak[1]\\
=&\epsilon^5+3\epsilon\lambda^4>0.
\end{align*}
\end{proof}

Similar calculations as in Lemma \ref{H upper bound} using
$F+F^{ij}h_{ij}-\alpha F^{ij}h^k_ih_{kj}\ge0$ for 
$F=H\A2$ give
\begin{align*}
\alpha\left(\Akl22+2H\A3\right)\le&4H\A2,\displaybreak[1]\\
\tfrac1c\left(H\A2\right)^{1/3}\le\tfrac12H
\le\frac{\A2}H\le&\frac4\alpha
\end{align*}
and an estimate as in Lemma \ref{H upper bound} follows.

\begin{lemma}
The dual function to $F=H\A2$ is $-3$-concave.
\end{lemma}
\begin{proof}
We set $\Phi=-H\A2 K^{-3}$ and want to prove that
$$\Phi^{ij,\,kl}\eta_{ij}\eta_{kl}\le\frac4{3\Phi}\Phi^{ij}\eta_{ij}
\Phi^{kl}\eta_{kl}.$$
We compute
\begin{align*}
\Phi^{ij}=&-\A2 K^{-3}g^{ij}-2H K^{-3}h^{ij}+3H\A2K^{-3}\tilde h^{ij},
\displaybreak[1]\\
\Phi^{ij,\,kl}=&-2K^{-3}\left(g^{ij}h^{kl}+h^{ij}g^{kl}\right)
+3\A2K^{-3}\left(g^{ij}\tilde h^{kl}+\tilde h^{ij}g^{kl}\right)\\
&+6HK^{-3}\left(h^{ij}\tilde h^{kl}+\tilde h^{ij}h^{kl}\right)\\
&-2HK^{-3}g^{ik}g^{jl}-9H\A2 K^{-3}\tilde h^{ij}\tilde h^{kl}
-3H\A2 K^{-3}\tilde h^{ik}\tilde h^{jl}.
\end{align*}
We have to check that
\begin{align*}
&-2H\A2\symmatrix{2\lx}{\lx+\ly}{2\ly}
+3H\Akl22\symmatrix{\frac2\lx}{\frac1\lx+\frac1\ly}{\frac2\ly}\\
&+6H^2\A2\symmatrix{2}{\frac\lx\ly+\frac\ly\lx}{2}
-2H^2\A2\symmatrix101\\
&-9H^2\Akl22\symmatrix{\frac1{\lx^2}}{\frac1{\lx\ly}}{\frac1{\ly^2}}
-3H^2\Akl22\symmatrix{\frac1{\lx^2}}0{\frac1{\ly^2}}\\
\le&-\frac43\symmatrix
{\scriptstyle\left(2\ly^2+\lx\ly+\frac{3\ly^3}\lx\right)^2}
{\scriptstyle\left(2\ly^2+\lx\ly+\frac{3\ly^3}\lx\right)\cdot
\left(2\lx^2+\lx\ly+\frac{3\lx^3}\ly\right)}
{\scriptstyle\left(2\lx^2+\lx\ly+\frac{3\lx^3}\ly\right)^2}.
\end{align*}
This is equivalent to
$$0\le\left(2\lx^4+\tfrac{20}3\lx^3\ly+\tfrac{44}3\lx^2\ly^2
+\tfrac{20}3\lx\ly^3+2\ly^4\right)
\symmatrix{\frac\ly\lx}{-1}{\frac\lx\ly}.$$
\end{proof}

Calculations as before show that
$$(16(T-t))^{-1/4}\cdot\left(1-c\cdot(T-t)^{1/8}\right)
\le\lx,\,\ly\le
(16(T-t))^{-1/4}\cdot\left(1+c\cdot(T-t)^{1/8}\right).$$
This finishes the proof of Theorem \ref{A2H thm}.

% Ende H\A2-Fluss

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$\abs A^4$-Flow}

\begin{theorem}\label{A4 thm}
A smooth compact strictly convex surface in $\R^3$, contracting
with normal velocity $\abs A^4$, 
converges to a round point in finite time.
\end{theorem}

\begin{theorem}
For a family of smooth compact strictly convex surfaces $M_t$ 
in $\R^3$, flowing according to $\dt X=-\abs A^4\nu=-\Akl22\nu$, 
$$\max\limits_{M_t}\frac{\left(\lx^4+2\lx^3\ly+4\lx^2\ly^2
+2\lx\ly^3+\ly^4\right)(\lx-\ly)^2}{(\lx+\ly)\lx\ly}$$
is non-increasing in time.
\end{theorem}
\begin{proof}
We calculate
\def\zaehlera22{{-H^6+2\A2 H^4-\Akl22 H^2+2\Akl23}}
\def\nennera22{{H^3-\A2 H}}
\begin{align*}
\tilde w=&\log\left(\frac{\left(\lx^4+2\lx^3\ly+4\lx^2\ly^2
+2\lx\ly^3+\ly^4\right)(\lx-\ly)^2}{(\lx+\ly)\lx\ly}\right)
\displaybreak[1]\\
=&\log\left(\frac{\zaehlera22}{\nennera22}\right),\displaybreak[1]\\
\oprokl H=&4H\A2\A3-3\Akl23\displaybreak[1]\\
&+8\sum\li\lj\hii+4\A2\sum\hij,\displaybreak[1]\\
\opr{\A2}=&2\Akl22\A3+16\sum\li\lj\lk\hii,
\end{align*}
\begin{align*}
\oprokl {\tilde w}=&\left(\frac{-6H^5+8\A2 H^3-2\Akl22 H}{\zaehlera22}
-\frac{3H^2-\A2}{\nennera22}\right)\cdot\\
&\quad\cdot\left(\oprokl H\right)\displaybreak[1]\\
&+\left(\frac{2H^4-2\A2 H^2+6\Akl22}{\zaehlera22}
-\frac{-H}{\nennera22}\right)\cdot\\
&\quad\cdot\left(\opr{\A2}\right)\displaybreak[1]\\
&+\left(\frac{-30H^4+24\A2 H^2-2\Akl22}{\zaehlera22}
-\frac{6H}{\nennera22}\right)\cdot\\
&\quad\left(-F^{ij}H_{;\,i}H_{;\,j}\right)\displaybreak[1]\\
&+\frac{-2H^2+12\A2}{\zaehlera22}\cdot\\
&\quad\cdot
\left(-F^{ij}\left(\A2\right)_{;\,i}\left(\A2\right)_{;\,j}\right)
\displaybreak[1]\\
&+\left(\frac{8H^3-4\A2 H}{\zaehlera22}
-\frac{-1}{\nennera22}\right)\cdot\\
&\quad\cdot\left(-F^{ij}\left(H_{;\,i}\left(\A2\right)_{;\,j}
+H_{;\,j}\left(\A2\right)_{;\,i}\right)\right).
\end{align*}
We use a computer algebra program and obtain
\begin{align*}
h_{22;\,1}=&\frac{\ly}{\lx}\frac
{4\lx^6+9\lx^5\ly+11\lx^4\ly^2+10\lx^3\ly^3+2\lx^2\ly^4+3\lx\ly^5+\ly^6}
{4\ly^6+9\lx\ly^5+11\lx^2\ly^4+10\lx^3\ly^3+2\lx^4\ly^2+3\lx^5\ly+\lx^6}
\cdot h_{11;\,1},\displaybreak[1]\\
\oprokl {\tilde w}=&\frac{-12
\left(3\lx^2+4\lx\ly+3\ly^2\right)\left(\lx^2+\ly^2\right)
\lx^3\ly^3}{\left(\lx+\ly\right)
\left(\lx^4+2\lx^3\ly+4\lx^2\ly^2+2\lx\ly^3+\ly^4\right)}
\displaybreak[1]\\
&+\frac{-4\ly}{\left(\lx^4+2\lx^3\ly+4\lx^2\ly^2+2\lx\ly^3+\ly^4\right)
\left(\lx-\ly\right)^2(\lx+\ly)\lx^3}\cdot\displaybreak[1]\\
&\quad\cdot\frac1{\left(\lx^6+3\lx^5\ly+2\lx^4\ly^2+10\lx^3\ly^3
+11\lx^2\ly^4+9\lx\ly^5+4\ly^6\right)^2}\cdot\displaybreak[1]\\
&\quad\cdot\left(11\lx^{22}+90\lx^{21}\ly-113\lx^{20}\ly^2
-840\lx^{19}\ly^3-1507\lx^{18}\ly^4\right.\\
&\qquad\left.+66\lx^{17}\ly^5+7465\lx^{16}\ly^6
+23136\lx^{15}\ly^7+45494\lx^{14}\ly^8\right.\\
&\qquad\left.+70100\lx^{13}\ly^9+84982\lx^{12}\ly^{10}
+85120\lx^{11}\ly^{11}+70882\lx^{10}\ly^{12}\right.\\
&\qquad\left.+52148\lx^9\ly^{13}+33938\lx^8\ly^{14}
+20928\lx^7\ly^{15}+11263\lx^6\ly^{16}\right.\\
&\qquad\left.+5490\lx^5\ly^{17}+2363\lx^4\ly^{18}+744\lx^3\ly^{19}
+193\lx^2\ly^{20}\right.\\
&\qquad\left.+42\lx\ly^{21}+5\ly^{22}\right)
\cdot h_{11;\,1}^2\displaybreak[1]\\
&+(\ldots)\cdot h_{22;\,2}.
\end{align*}
We apply Sturm's algorithm and obtain that the right-hand side
is non-positive.
\end{proof}

\begin{lemma}
For a family of compact smooth strictly convex surfaces 
$M_t\subset\R^3$, $0\le t<T$, flowing according to
$\dt X=-F\nu$ with $F=\abs A^4$, a positive
lower bound on the principal curvatures, $\lx,\,\ly\ge\epsilon
>0$, is preserved during the evolution.
\end{lemma}
\begin{proof}
The term involving second derivatives of $F$ is non-negative,
so we have
\begin{align*}
\left(\dt M_{ij}-F^{kl}M_{ij;\,kl}\right)\xi^i\xi^j\ge&
\A2\epsilon\left(4\A3-3\epsilon\A2\right)\displaybreak[1]\\
=&\A2\epsilon\left(\lambda^3+3\lambda^2(\lambda-\epsilon)
+\epsilon^3\right)>0.
\end{align*}
\end{proof}

Similar calculations as in Lemma \ref{H upper bound} using
$F+F^{ij}h_{ij}-\alpha F^{ij}h^k_ih_{kj}\ge0$ for 
$F=\abs A^4$ give
\begin{align*}
5\A2-4\alpha\A3\ge&0,\displaybreak[1]\\
\tfrac1c\left(\abs A^4\right)^{1/4}\le\frac{\A3}{\A2}\le\frac5{4\alpha}
\end{align*}
and an estimate as in Lemma \ref{H upper bound} follows.

The following lemma implies that the dual function to 
$F=\abs A^4$ is $-4$-concave.
\begin{lemma}
If the dual function to $F$ is $\alpha$-concave for
some $\alpha<0$, then the dual function to $F^\beta$
is $\alpha\cdot\beta$-concave for $\beta>0$.
\end{lemma}
\begin{proof}
We use similar notation as before. Assume that 
$$-G^{ij,\,kl}\eta_{ij}\eta_{kl}\le\frac{\alpha-1}{\alpha(-G)}
G^{ij}\eta_{ij}G^{kl}\eta_{kl}.$$
We have to show for $\Phi=-G^\beta$
$$\Phi^{ij,\,kl}\eta_{ij}\eta_{kl}\le\frac{\alpha\beta-1}
{\alpha\beta\Phi}\Phi^{ij}\eta_{ij}\Phi^{kl}\eta_{kl}.$$
Direct calculations yield
\begin{align*}
\Phi^{ij}=&-\beta G^{\beta-1}G^{ij},\displaybreak[1]\\
\Phi^{ij,\,kl}=&-\beta G^{\beta-1}G^{ij,\,kl}
-\beta(\beta-1)G^{\beta-2}G^{ij}G^{kl},\displaybreak[1]\\
\Phi^{ij,\,kl}\eta_{ij}\eta_{kl}=&
-\beta G^{\beta-1}G^{ij,\,kl}\eta_{ij}\eta_{kl}
-\beta(\beta-1)G^{\beta-2}\left(G^{ij}\eta_{ij}\right)^2
\displaybreak[1]\\
\le&-\beta\frac{\alpha-1}{\alpha}G^{\beta-2}
\left(G^{ij}\eta_{ij}\right)^2
-\beta(\beta-1)G^{\beta-2}\left(G^{ij}\eta_{ij}\right)^2
\displaybreak[1]\\
=&-\frac{\alpha\beta-1}{\alpha\beta G^\beta}\beta^2
G^{2\beta-2}\left(G^{ij}\eta_{ij}\right)^2\displaybreak[1]\\
=&\frac{\alpha\beta-1}{\alpha\beta\Phi}
\left(\Phi^{ij}\eta_{ij}\right)^2.
\end{align*}
\end{proof}

Calculations as before show that
$$\lx,\,\ly\le
(20(T-t))^{-4/5}\cdot\left(1+c\cdot(T-t)^{3/{20}}\right).$$
A corresponding lower estimate is also true.
Theorem \ref{A4 thm} follows.

% Ende  A^4=\Akl22-Fluss

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\part{Surfaces Expanding by the Inverse Gau{\ss} Curvature Flow}
\label{1/K part}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Introduction}

We consider a family of compact strictly convex surfaces $M_t$ in $\R^3$ 
that expand by the inverse Gau{\ss} curvature flow
\begin{equation}\label{K flow eqn}
\dt X=\frac1K\nu.
\end{equation}
This is a parabolic flow equation. We obtain a solution
on a maximal time interval $[0,\,T)$, $0<T<\infty$. 
For $t\uparrow T$, the surfaces converge to infinity. After
appropriate rescaling, they converge to a round sphere.
We say that the surfaces $M_t$ converge to ``round spheres at infinity''.
The key step in the proof, Theorem \ref{K invers mon thm}, is to show that
\begin{equation}\label{K mon groe}
\max\limits_{M_t}\left(\frac{(\lx-\ly)^2}{\lx^2\ly^2}\right)
\end{equation}
is non-increasing in time. \par

Our main theorem in this part is
\begin{theorem}\label{K main thm}
For any smooth strictly convex closed surface $M$ in $\R^3$, 
there exists a smooth family of surfaces $M_t$, $t\in[0,T)$,
solving \eqref{K flow eqn} with $M_0=M$. For $t\uparrow T$,
$M_t$ converges to infinity. The rescaled surfaces
$M_t\cdot(T-t)$ converge smoothly to the
unit sphere $\S^2$.
\end{theorem}

We will also consider other normal velocities for which
similar results hold. Therefore, we have to find 
quantities like \eqref{K mon groe} that are monotone
during the flow.
In the following Table, we have collected some normal
velocities $F$ ($1^{\text{st}}$ column) and quantities
$w$ ($2^{\text{nd}}$ column) such that $\max_{M_t}w$ is
non-increasing in time for surfaces contracting
with normal velocity $F$. It is common to use positive
functions $F$ for contracting flows. Thus the negative
sign corresponds to the fact that these surfaces expand.
In each case, we obtain
convergence to round spheres at infinity for smooth 
compact strictly convex initial surfaces $M_0$. 

\begin{table}
\def\platz{\raisebox{0em}[2.2em][1.5em]{\rule{0em}{2em}}}
$$
\begin{array}{|c||c|}\hline
-\fracd1K & \platz\fracd{(\lx-\ly)^2}{\lx^2\ly^2}\\\hline
-\fracd{H^2}{K^2} & \platz\fracd{(\lx-\ly)^2}{(\lx+\ly)\lx\ly}\\\hline
-\fracd{\A2}{K^2} & \platz\fracd{(\lx-\ly)^2}{(\lx+\ly)\lx\ly}\\\hline
-\fracd{H^3}{K^3} & \platz\fracd{(\lx+\ly)(\lx-\ly)^2}
{\left(\lx^2+\ly^2\right)\left(\lx^2+\lx\ly+\ly^2\right)\lx^3\ly^3}\\
\hline
\end{array}$$
\caption{Monotone quantities}\label{K table}
\end{table}

This part concerns mainly the expansion of surfaces by the inverse 
Gau{\ss} curvature flow. This flow is homogeneous of degree 
minus two
in the principal curvatures. We want to stress that we don't
have to assume any pinching condition on the initial surface.
To our knowledge, the expansion of hypersurfaces by normal
velocities homogeneous in the principal curvatures of degree
smaller than minus one has not been studied before.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Inverse Gau{\ss} Curvature Flow}\label{IGCF}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A Monotone Quantity}\label{K mon sec}

\begin{theorem}\label{K invers mon thm}
For a family of smooth compact strictly convex surfaces $M_t$ 
in $\R^3$ flowing according to $\dot X=\frac1K\nu$, 
\begin{equation}\label{K invers quantity}
\max\limits_{M(t)}\frac{(\lambda_1-\lambda_2)^2}
{4\lambda_1^2\lambda_2^2}
=\max\limits_{M(t)}\frac{2\A2-H^2}{\left(H^2-\A2\right)^2}
\equiv\max\limits_{M(t)}w
\end{equation}
is non-increasing in time.
\end{theorem}

An immediate consequence of this theorem is
\begin{corollary}
The only homothetically expanding smooth compact strictly convex
surfaces $M_t$, solving the flow equation $\dot X=\frac1K\nu$ in 
$\R^3$, are spheres.
\end{corollary}
\begin{proof}
The quantity $\frac{(\lx-\ly)^2}{\lx^2\ly^2}$ 
is negative homogeneous of degree two
in the principal curvatures and non-negative. If $M$ is 
homothetically expanding, Theorem \ref{K invers mon thm} implies that
$(\lx-\ly)^2=0$ everywhere. Thus $M(t)$ is umbilic and 
\cite[Lemma 7.1]{Spivak4} implies that $M(t)$ is a sphere.
\end{proof}

\begin{proof}[Proof of Theorem \ref{K invers mon thm}]
First and second derivatives of $F=-1/K$ with respect 
to the second fundamental form are given by
\begin{align}\label{K F abl}
\begin{split}
F^{ij}=&\frac1K\tilde h^{ij},\\
F^{ij,\,kl}=&-\frac1K\left(\tilde h^{ij}\tilde h^{kl}+
\tilde h^{ik}\tilde h^{jl}\right).
\end{split}
\end{align}
Lemma \ref{evol equ lem} yields
\begin{align}\label{K H evol}
\oprokl H=&\frac1K\left(H^2-3\A2\right)
-\frac1K\sum\frac1{\li\lj}\left(\hii+\hij\right)
\intertext{and}
\begin{split}\label{K A2 evol}
\opr{\A2}=&\frac1K\left(2H\A2-6\A3\right)-\frac1K\sum\frac2\lk\hij\\
&-\frac1K\sum\frac{2\lk}{\li\lj}
\left(\hii+\hij\right).
\end{split}
\end{align}
For the rest of the proof, we consider a critical point of 
$\left.w\right|_{M_t}$ for some $t>0$, where $w>0$
and $\lx\neq\ly$.
It suffices to show that $w$ is non-increasing
in such a point. Then our theorem follows. 

We differentiate $w$
\begin{align*}
w=&\frac{-H^2+2\A2}{\left(H^2-\A2\right)^2},\umbruch\\
w_{;\,i}=&\frac{2H\left(H^2-3\A2\right)H_{;\,i}+2\A2\Az i}
{\left(H^2-\A2\right)^3},\umbruch\\
w_{;\,ij}=&\frac1{\left(H^2-\A2\right)^4}\left\{
2H\left(H^2-\A2\right)\left(H^2-3\A2\right)H_{;\,ij}\right.\\
&\quad\left.+2\A2\left(H^2-\A2\right)\Az{ij}\right.\\
&\quad\left.\left(6\left(H^2-\A2\right)^2-12H^2\left(H^2-3\A2\right)
\right)H_{;\,i}H_{;\,j}\right.\\
&\quad\left.+2\left(H^2+2\A2\right)\Az i\Az j\right.\\
&\quad\left.-12H\A2\left(H_{;\,i}\Az j+H_{;\,j}\Az i\right)\right\}.
\end{align*}
In a critical point of $w$ with $\lx\neq\ly$, we get
\begin{align}
0=&4\ly^2(\lx-\ly)h_{11;\,k}+4\lx^2(\ly-\lx)h_{22;\,k}
\quad\text{for }k=1,\,2,\umbruch\nonumber\\
h_{22;\,1}=&\frac{\ly^2}{\lx^2}h_{11;\,1}
\equiv a_1h_{11;\,1}^2\quad\text{and}\quad
h_{11;\,2}=\frac{\lx^2}{\ly^2}h_{22;\,2}
\equiv a_2h_{22;\,2}^2.\label{K extremal cond}
\end{align}
The evolution equation of $w$ is
\begin{align*}
\left(H^2-\A2\right)^4\cdot&\left(\oprokl w\right)=\\
=&2H\left(H^2-\A2\right)\left(H^2-3\A2\right)\left(\oprokl H\right)
\umbruch\\
&+2\A2\left(H^2-\A2\right)\left(\opr{\A2}\right)\umbruch\\
&+\left(6\left(H^2-\A2\right)^2-12H^2\left(H^2-3\A2\right)\right)
\left(-F^{ij}H_{;\,i}H_{;\,j}\right)\umbruch\\
&+2\left(H^2+2\A2\right)\left(-F^{ij}\Az i\Az j\right)\umbruch\\
&-12H\A2\left(-F^{ij}\left(H_{;\,i}\Az j+H_{;\,j}\Az i\right)\right).
\end{align*}
We consider the terms on the right-hand side individually and use
\eqref{K H evol}, \eqref{K A2 evol}, and \eqref{K extremal cond}
\begin{align*}
\oprokl H=&-\frac{2\left(\lx^2-\lx\ly+\ly^2\right)}{\lx\ly}\umbruch\\
&-\frac{2\left(\lx^2a_1^2+\lx\ly a_1^2+\lx\ly a_1+\ly^2\right)}
{\lx^3\ly^3}h_{11;\,1}^2\umbruch\\
&-\frac{2\left(\ly^2a_2^2+\ly\lx a_2^2+\ly\lx a_2+\lx^2\right)}
{\ly^3\lx^3}h_{22;\,2}^2\umbruch\\
=&-\frac{2\left(\lx^2-\lx\ly+\ly^2\right)}{\lx\ly}\umbruch\\
&-\frac{2(\lx+\ly)\left(\lx^2+\ly^2\right)}{\lx\ly}
\left(\frac1{\lx^5}h_{11;\,1}^2+\frac1{\ly^5}h_{22;\,2}^2\right),
\umbruch\\
2H\left(H^2-\A2\right)&\left(H^2-3\A2\right)\cdot
\left(\oprokl H\right)=\umbruch\\
=&-8(\lx+\ly)\left(\lx^2-\lx\ly+\ly^2\right)\lx\ly
\left(\oprokl H\right)\umbruch\\
=&16(\lx+\ly)\left(\lx^2-\lx\ly+\ly^2\right)^2\umbruch\\
&+16(\lx+\ly)^2\left(\lx^2-\lx\ly+\ly^2\right)\left(\lx^2+\ly^2\right)
\cdot\\
&\quad\cdot
\left(\frac1{\lx^5}h_{11;\,1}^2+\frac1{\ly^5}h_{22;\,2}^2\right)\umbruch\\
=&8(\lx+\ly)
\left(2\lx^4-4\lx^3\ly+6\lx^2\ly^2-4\lx\ly^3+2\ly^4\right)\umbruch\\
&+\left(4\lx^7\ly+4\lx^6\ly^2+4\lx^5\ly^3+8\lx^4\ly^4\right.\\
&\qquad\left.+4\lx^3\ly^5+4\lx^2\ly^6+4\lx\ly^7\right)\cdot\\
&\quad\cdot\frac4{\lx\ly}\left(\frac1{\lx^5}h_{11;\,1}^2
+\frac1{\ly^5}h_{22;\,2}^2\right),\umbruch\\
\opr{\A2}=&-\frac{2(\lx+\ly)\left(2\lx^2-3\lx\ly+2\ly^2\right)}{\lx\ly}
\umbruch\\
&-\frac{2\left(2\lx^2 a_1^2+2\lx\ly a_1^2+3\ly^2a_1^2
+2\lx\ly a_1+3\ly^2\right)}{\lx^2\ly^3}h_{11;\,1}^2\umbruch\\
&-\frac{2\left(2\ly^2a_2^2+2\ly\lx a_2^2+3\lx^2a_2^2
+2\ly\lx a_2+3\lx^2\right)}{\ly^2\lx^3}h_{22;\,2}^2\umbruch\\
=&-\frac{2(\lx+\ly)\left(2\lx^2-3\lx\ly+2\ly^2\right)}{\lx\ly}\umbruch\\
&-\frac{2\left(3\lx^4+2\lx^3\ly+2\lx^2\ly^2+2\lx\ly^3+3\ly^4\right)}
{\lx\ly}\cdot\\
&\quad\cdot\left(\frac1{\lx^5}h_{11;\,1}^2+\frac1{\ly^5}h_{22;\,2}^2\right),
\umbruch\\
2\A2\left(H^2-\A2\right)\cdot&\left(\opr{\A2}\right)=\umbruch\\
=&4\left(\lx^2+\ly^2\right)\lx\ly\left(\opr{\A2}\right)\umbruch\\
=&-8(\lx+\ly)\left(\lx^2+\ly^2\right)\left(2\lx^2-3\lx\ly+2\ly^2\right)
\umbruch\\
&-8\left(\lx^2+\ly^2\right)
\left(3\lx^4+2\lx^3\ly+2\lx^2\ly^2+2\lx\ly^3+3\ly^4\right)\cdot\umbruch\\
&\quad\cdot\left(\frac1{\lx^5}h_{11;\,1}^2+\frac1{\ly^5}h_{22;\,2}^2\right)
\umbruch\\
=&-8(\lx+\ly)\left(2\lx^4-3\lx^3\ly+4\lx^2\ly^2-3\lx\ly^3+2\ly^4\right)
\umbruch\\
&-\left(6\lx^7\ly+4\lx^6\ly^2+10\lx^5\ly^3+8\lx^4\ly^4+10\lx^3\ly^5
\right.\\
&\qquad\left.+4\lx^2\ly^6+6\lx\ly^7\right)\frac4{\lx\ly}
\left(\frac1{\lx^5}h_{11;\,1}^2+\frac1{\ly^5}h_{22;\,2}^2\right).
\end{align*}
For the remaining terms, we employ \eqref{K F abl}
\begin{align*}
-F^{ij}H_{;\,i}H_{;\,j}=&-\frac{(a_1+1)^2}{\lx^2\ly}h_{11;\,1}^2
-\frac{(a_2+1)^2}{\ly^2\lx}h_{22;\,2}^2\umbruch\\
=&-\frac{\left(\lx^2+\ly^2\right)^2}{\lx\ly}
\left(\frac1{\lx^5}h_{11;\,1}^2+\frac1{\ly^5}h_{22;\,2}^2\right),
\umbruch\\
&\hspace*{-20ex}
\left(6\left(H^2-\A2\right)^2-12H^2\left(H^2-3\A2\right)\right)
\left(-F^{ij}H_{;\,i}H_{;\,j}\right)
=\umbruch\\
=&-\left(\lx^4+\lx^3\ly+\lx^2\ly^2+\lx\ly^3+\ly^4\right)\cdot\\
&\quad\cdot6\left(\lx^2+\ly^2\right)^2\frac{4}{\lx\ly}
\left(\frac1{\lx^5}h_{11;\,1}^2+\frac1{\ly^5}h_{22;\,2}^2\right)\umbruch\\
=&-\left(6\lx^8+6\lx^7\ly+18\lx^6\ly^2+18\lx^5\ly^3+24\lx^4\ly^4
+18\lx^3\ly^5\right.\\
&\qquad\left.+18\lx^2\ly^6+6\lx\ly^7+6\ly^8\right)\cdot\umbruch\\
&\quad\cdot\frac{4}{\lx\ly}
\left(\frac1{\lx^5}h_{11;\,1}^2+\frac1{\ly^5}h_{22;\,2}^2\right),\umbruch\\
-F^{ij}\Az i\Az j=&-\frac{4(\ly a_1+\lx)^2}{\lx^2\ly}h_{11;\,1}^2
-\frac{4(\lx a_2+\ly)^2}{\ly^2\lx}h_{22;\,2}^2\umbruch\\
=&-\left(\lx^3+\ly^3\right)^2\cdot\frac4{\lx\ly}
\left(\frac1{\lx^5}h_{11;\,1}^2+\frac1{\ly^5}h_{22;\,2}^2\right),\umbruch\\
2\left(H^2+2\A2\right)\cdot&\left(-F^{ij}\Az i\Az j\right)=\umbruch\\
=&-2\left(3\lx^2+2\lx\ly+3\ly^2\right)\left(\lx^3+\ly^3\right)^2\cdot\\
&\quad\cdot\frac4{\lx\ly}\left(\frac1{\lx^5}h_{11;\,1}^2
+\frac1{\ly^5}h_{22;\,2}^2\right)\umbruch\\
=&-\left(6\lx^8+4\lx^7\ly+6\lx^6\ly^2+12\lx^5\ly^3
+8\lx^4\ly^4+12\lx^3\ly^5\right.\\
&\qquad\left.+6\lx^2\ly^6+4\lx\ly^7+6\ly^8\right)
\frac4{\lx\ly}\left(\frac1{\lx^5}h_{11;\,1}^2
+\frac1{\ly^5}h_{22;\,2}^2\right),\umbruch\\
&\hspace*{-20ex}-F^{ij}\left(H_{;i}\Az j+H_{;j}\Az i\right)=\umbruch\\
=&-\frac{4(\ly a_1+\lx)(a_1+1)}{\lx^2\ly}h_{11;\,1}^2
-\frac{4(\lx a_2+\ly)(a_2+1)}{\ly^2\lx}h_{22;\,2}^2\umbruch\\
=&-\left(\lx^3+\ly^3\right)\left(\lx^2+\ly^2\right)
\cdot\frac4{\lx\ly}\left(\frac1{\lx^5}h_{11;\,1}^2
+\frac1{\ly^5}h_{22;\,2}^2\right),\umbruch\\
&\hspace*{-20ex}-12H\A2\left(-F^{ij}
\left(H_{;\,i}\Az j+H_{;\,j}\Az i\right)\right)=\umbruch\\
=&12(\lx+\ly)\left(\lx^2+\ly^2\right)^2
\left(\lx^3+\ly^3\right)\cdot\umbruch\\
&\quad\cdot\frac4{\lx\ly}
\left(\frac1{\lx^5}h_{11;\,1}^2
+\frac1{\ly^5}h_{22;\,2}^2\right)\umbruch\\
=&\left(12\lx^8+12\lx^7\ly+24\lx^6\ly^2
+36\lx^5\ly^3+24\lx^4\ly^4+36\lx^3\ly^5\right.\\
&\qquad\left.+24\lx^2\ly^6+12\lx\ly^7+12\ly^8\right)\cdot\\
&\quad\cdot\frac4{\lx\ly}\left(\frac1{\lx^5}h_{11;\,1}^2
+\frac1{\ly^5}h_{22;\,2}^2\right).
\end{align*}
Combining these expressions, we see that many terms cancel. 
We obtain
\begin{align*}
&\hspace*{-20ex}\left(H^2-\A2\right)^4\left(\oprokl w\right)=\umbruch\\
=&-8(\lx+\ly)(\lx-\ly)^2\lx\ly
-\frac{32\ly^3}{\lx^2}h_{11;\,1}^2
-\frac{32\lx^3}{\ly^2}h_{22;\,2}^2,\umbruch\\
\oprokl w=&-\frac{(\lx+\ly)(\lx-\ly)^2}{2\lx^3\ly^3}
-\frac2{\lx^6\ly}h_{11;\,1}^2
-\frac2{\lx\ly^6}h_{22;\,2}^2\umbruch\\
\le&0.
\end{align*}
We finally apply the maximum principle and our theorem follows.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Convergence to Infinity}\label{K inf sec}

It is known, that \eqref{K flow eqn} is a parabolic evolution equation
for strictly convex initial data and that it has a solution on a maximal
time interval $[0,\,T)$. Here, we want to show that 
some points on $M_t$ converge
to infinity for $t\uparrow T$, i.\,e.
$$\lim\limits_{t\uparrow T}\sup\limits_{M_t}|X|=\infty.$$

We show that the principal curvatures of $M_t$ stay uniformly 
bounded above.
\begin{lemma}\label{K upper bound}
For a smooth compact strictly convex surface $M$ in $\R^3$, 
flowing according to $\dot X=\frac1K\nu$, the maximum of the
principal curvatures is non-increasing.
\end{lemma}
\begin{proof}
Consider $M_{ij}=h_{ij}-\mu g_{ij}$ with $\mu>0$ so large
that $M_{ij}$ is negative semi-definite for some time $t_0$. We wish
to show that $M_{ij}$ is negative semi-definite for $t>t_0$. 
Lemma \ref{evol equ lem} gives
$$\dt h_{ij}-F^{kl}h_{ij;\,kl}=\frac1KH h_{ij}-\frac1Kh^k_ih_{kj}
-\frac1K\left(\tilde h^{kl}\tilde h^{rs}
+\tilde h^{kr}\tilde h^{ls}\right)h_{kl;\,i}h_{rs;\,j}.$$
In the evolution equation for $M_{ij}$, we drop the negative definite 
terms involving derivatives of the second fundamental form
$$\dt M_{ij}-F^{kl}M_{ij;\,kl}\le 
\frac1K\left(Hh_{ij}-h^k_ih_{kj}-2\mu h_{ij}\right).$$
Let $\xi$ be a zero eigenvalue of $M_{ij}$ with $\abs{\xi}=1$,
$M_{ij}\xi^j=h_{ij}\xi^j-\mu g_{ij}\xi^j=0$.
So we obtain in a point with $M_{ij}\le0$
$$\left(Hh_{ij}-h^k_ih_{kj}-2\mu h_{ij}\right)\xi^i\xi^j=
H\mu-3\mu^2\le2\mu\mu-3\mu^2\le0$$
and the maximum principle for tensors 
\cite{ChowLuMP,HamiltonThree} implies the claimed result.
\end{proof}

We obtain a pinching estimate
\begin{lemma}\label{K pinching lem}
For a smooth compact strictly convex surface $M_t$ in $\R^3$, 
flowing according to $\dot X=\frac1K\nu$, there exists 
$c=c(M_0)$ such that $0<\frac1c\le\frac{\lambda_1}{\lambda_2}\le c$.
\end{lemma}
\begin{proof}
Choose $\mu>0$ such that $\lambda_1,\,\lambda_2\le\mu$
at $t=0$. Theorem \ref{K invers mon thm} and Lemma \ref{K upper bound}
imply that 
$$\frac1{\mu^2}\frac{\left(\frac\lx\ly-1\right)^2}{\frac\lx\ly}
=\frac1{\mu^2}\frac{(\lx-\ly)^2}{\lx\ly}\le
\frac{(\lx-\ly)^2}{\lx^2\ly^2}\le c.$$
We obtain the bound on $\frac{\lambda_1}{\lambda_2}$ claimed above. 
\end{proof}

It is only here that we use the monotone quantity of 
Theorem \ref{K invers mon thm}. For our purposes, this quantity
is better than scaling invariant. As it becomes apparent in 
the proof of Lemma \ref{K 1st C0 est}, however, the Aleksandrov
reflection principle can be used instead for the rest of the proof. 
This simplifies the
proof compared to \cite{AndrewsStones} or
Part \ref{A2 part}, where similar monotone
quantities are used. Later on, see Theorem \ref{K impr K invers mon thm}
and Theorem \ref{K impr conv thm}, 
we will use monotone quantities
to improve the convergence rate that follows from Aleksandrov
reflection. 

The next result shows that $K$ stays uniformly bounded below 
by a positive constant as
long as $M_t$ is enclosed by a ball of fixed positive radius. A similar
estimate is used in \cite{TsoPoint}.
\begin{lemma}\label{K vel bound}
For a strictly convex solution of \eqref{K flow eqn}, $K$ is
uniformly bounded below by a positive constant 
in terms of the radius $R$ of an enclosing 
sphere $B_{R}(x_0)$ and
$\max_{M_0}\,\frac{K^{-1}}{2R-\langle X-x_0,\,\nu\rangle}$.
More precisely, we have everywhere
\begin{equation}\label{K bound}
K\ge\frac1{3R}\left(\max\limits_{M_0}\frac{K^{-1}}{2R-\langle X-x_0,\,
\nu\rangle}\right)^{-1}\quad\text{or}\quad H\ge\frac1{2R}.
\end{equation}
In both cases, we obtain a positive lower bound on the principal
curvatures. 
\end{lemma}
\begin{proof}
We may assume that $x_0=0$. Observe that
$R$ is a positive lower bound for $2R-\langle X,\,\nu\rangle$.
Thus $\frac{K^{-1}}{2R-\langle X,\,\nu\rangle}$ 
is finite for strictly convex surfaces. 
If $\frac{K^{-1}}{2R-\langle X,\,\nu\rangle}$ is bounded above by
its maximum over $M_0$, we get
$$\inf\limits_t\inf\limits_{M_t}K\ge
\frac1{3R}\left(\max\limits_{M_0}\frac{K^{-1}}
{2R-\langle X,\,\nu\rangle}\right)^{-1}$$
and \eqref{K bound} follows.
According to Lemma \ref{evol equ lem}, we get
\begin{align*}
\dt X^\beta-F^{ij}X^\beta_{;\,ij}=&\frac3K\nu^\beta,\displaybreak[1]\\
\dt\nu^\beta-F^{ij}\nu^\beta_{;\,ij}=&\frac HK\nu^\beta,\displaybreak[1]\\
\dt\langle X,\,\nu\rangle-F^{ij}\langle X,\,\nu\rangle_{;\,ij}=&
-\frac1K+\frac HK\langle X,\,\nu\rangle,\displaybreak[1]\\
\opr{\frac1K}=&\frac H{K^2}.
\end{align*}
In a critical point of $\frac{K^{-1}}{2R-\lvert X,\,\nu\rvert}$,
we obtain
$$\dt\log\frac{K^{-1}}{2R-\langle X,\,\nu\rangle}
-F^{ij}\left(\log\frac{K^{-1}}{2R-\langle X,\,\nu\rangle}\right)_{;\,ij}=
\frac1{2R-\langle X,\,\nu\rangle}\frac1K(2RH-1).$$
We only have to prove that we preserve the bound in 
Equation \eqref{K bound} for $t>0$, when 
$$\max_{M_t}\frac{K^{-1}}{2R-\langle X,\,\nu\rangle}$$ 
increases.
Then we have $0\le2RH-1$ at a point, where
$\max_{M_t}\frac{K^{-1}}{2R-\langle X,\,\nu\rangle}$ 
is attained. Finally, the positive lower bound on $K$ or
$H$ and our pinching estimate, Lemma \ref{K pinching lem},
imply a positive lower bound on the principal curvatures. 
\end{proof}

Let us recall a form of the maximum principle for evolving hypersurfaces.
\begin{lemma}\label{K encl lem}
Let $M_t$ and $\tilde M_t$ be two smooth compact strictly convex solutions
to \eqref{K flow eqn} on some time interval $\left[0,\,T^*\right)$.
If $M_0$ encloses $\tilde M_0$, then $M_t$ encloses $\tilde M_t$
at any time $t\in\left[0,\,T^*\right)$,
for which both solutions exist.
\end{lemma}
\begin{proof}
This is a standard consequence of the maximum principle. 
\end{proof}

The next result describes the evolution of spheres. 
\begin{lemma}\label{K sphere evol}
Spheres $\partial B_{r(t)}(x_0)$ solve \eqref{K flow eqn} for
$t\in[0,\,T)$ with $r(t)=(T-t)^{-1}$ and $T=r^{-1}(0)$.
\end{lemma}
\begin{proof}
The evolution equation for the radius of a sphere is
$$\dot r(t)=r^2(t).$$
\end{proof}

\begin{lemma}
Let $M_t$ be a family of smooth compact strictly convex 
solutions to \eqref{K flow eqn} on a maximal time interval
$[0,\,T)$. Then $T<\infty$.
\end{lemma}
\begin{proof}
Spheres solving \eqref{K flow eqn} tend to infinity in 
finite time, see Lemma \ref{K sphere evol}. So Lemma
\ref{K encl lem} implies that $T<\infty$. 
\end{proof}

\begin{lemma}
Let $M_t$ be a family of smooth compact strictly convex
surfaces solving \eqref{K flow eqn} on a maximal time interval
$[0,\,T)$. Then
$$\lim\limits_{t\uparrow T}\sup\limits_{M_t}\abs X=\infty.$$
\end{lemma}
\begin{proof}
Assume that $M_t\subset B_R(0)$, $0\le t<T$, for some $R>0$.
Then Lemmata \ref{K upper bound} and \ref{K vel bound} imply
that the principal curvatures of $M_t$ stay uniformly bounded
above and below by positive constants. Thus \eqref{K flow eqn}
can be rewritten as a uniformly parabolic equation and the 
estimates of Krylov, Safonov, Evans (see also
\cite{Andrews2dKrylov}), and Schauder imply uniform a priori
estimates up to $t=T$. This allows to continue the solution $M_t$
smoothly past $t=T$, a contradiction. Note finally, that
\eqref{K flow eqn} is an expanding flow, so $\sup_{M_t}\abs X$
is monotone in $t$ and the limit exists. 
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Convergence to a Sphere}\label{K conv sphere sec}

\begin{lemma}\label{K 1st C0 est}
Under the assumptions of Theorem \ref{K main thm}, 
we get
$$\lim\limits_{t\uparrow T}\inf\limits_{M_t}|X|=\infty,$$
more precisely, there exists $c=c(M_0)$ such that
$$\sup\limits_{M_t}|X|-c\le(T-t)^{-1}\le\inf\limits_{M_t}|X|+c$$
and
\begin{equation}\label{K resc Hsd conv}
M_t\cdot (T-t)\subset B_{1+c(T-t)}(0)\setminus B_{1-c(T-t)}(0),
\end{equation}
so the rescaled surfaces $M_t\cdot(T-t)$ converge to the
unit sphere $\S^2$ in Hausdorff distance.
\end{lemma}
\begin{proof}
We may shift the origin such that $0$ lies inside $M_0$.
This does not affect the convergence rate claimed above.
\par
Define the support function $u:\S^2\times[0,\,T)\to\R_+$ for a 
convex surface by
$$u(z,\,t)=\langle X\left(\nu^{-1}(z),\,t\right),\,z\rangle.$$
It fulfills the evolution equation, see e.\,g.\ \cite{AndrewsHarnack},
$$\dot u=\frac{\det(u_{;\,ij}+u\sigma_{ij})}{\det(\sigma_{ij})},$$
where $u_{;\,ij}$ denotes covariant derivatives on $\S^2$
and $\sigma_{ij}$ is the standard metric on $\S^2$.
\par
We apply the Aleksandrov reflection principle of
Bennett Chow, Robert Gulliver \cite{ChowGulliverSphere},
and James McCoy \cite[Theorem 3.1]{JamesMcCoyArea} and
obtain a uniform bound on the oscillation (and the gradient)
of $u(\cdot,\,t)$ for all $t\in[0,\,T)$, that depends only
on the initial data. 
\par
As $\sup_{M_t}|X|\to\infty$ for $t\uparrow T$, we obtain that
$\inf_{M_t}|X|\to\infty$ for $t\uparrow T$, more precisely, we
have
$$\sup\limits_{M_t}|X|\le\inf\limits_{M_t}|X|+c.$$\par
It remains to show that 
$$\inf\limits_{M_t}|X|\le(T-t)^{-1}\le\sup\limits_{M_t}|X|.$$
Consider the surfaces $\partial M_{(T-t)^{-1}}(0)$, solving
\eqref{K flow eqn}. For $t=T$, $M_t$ and 
$\partial B_{(T-t)^{-1}}(0)$ converge to infinity. We claim
that for all $t\in[0,\,T)$, $\partial B_{(T-t)^{-1}}(0)
\cap M_t\neq\emptyset$. Otherwise, for some $t_0$, $M_{t_0}$
encloses $\partial B_{(T-t)^{-1}}(0)$ or is contained in
$B_{(T-t)^{-1}}(0)$. Both cases are similar. We only
consider the first case. Choose $\epsilon>0$ such that
$M_{t_0}$ encloses also $\partial B_{(T-t_0-\epsilon)^{-1}}(0)$,
a slightly larger sphere. For $t\in[t_0,\,T-\epsilon)$,
$\partial B_{(T-t-\epsilon)^{-1}}(0)$ solves \eqref{K flow eqn}.
As $\partial B_{(T-t-\epsilon)^{-1}}(0)$ converges to
infinity for $t\uparrow T-\epsilon$, Lemma \ref{K encl lem}
implies that $M_t$ has to converge to infinity for
$t\uparrow T-\epsilon$, a contradiction.
\end{proof}

In terms of the support function $u$, this Lemma implies that
\begin{equation}\label{K supp fct est}
u(x,\,t)-c\le(T-t)^{-1}\le u(x,\,t)+c.
\end{equation}
Note that this estimate is sharp for spheres
$\partial B_{(T-t)^{-1}}(Q)$ solving \eqref{K flow eqn}, 
if $Q$ is different from the origin. The method of 
\cite{AndrewsStones}, see also Part \ref{A2 part}, where the
origin is replaced by some $q(t)$, can be adapted to the
present situation. In order to improve the estimate
\eqref{K supp fct est}, however, we need a monotone
quantity similar to \eqref{K mon groe} with a better
scaling behavior. We address this issue in Section 
\ref{K impr conv sec}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Smooth Convergence to a Sphere}
\label{K smooth conv sec}

\begin{lemma}\label{K upper bound two}
Under the assumptions of Theorem \ref{K main thm}, there
exists $c=c(M_0)$, such that
$$K\le c\cdot(T-t)^2.$$
\end{lemma}
\begin{proof}
For $\mu\gg1$ to be fixed below, we consider
$$w:=|X|^2-\mu\frac1K.$$
We may assume that $\mu$ is so large that 
$w<0$ on $M_0$. Our aim is
to show that $w$ stays negative during the flow.
We use the evolution equations of the proof of Lemma \ref{K vel bound}.
The evolution equation of $w$ is given by
$$\oprokl w=6\frac1K\langle X,\,\nu\rangle-2\frac H{K^2}
-\mu\frac H{K^2}.$$
Let $t_0\in[0,T)$ be minimal such that $\max_{M_t}w=0$.
Choose $x_0\in M_t$ such that $w(x_0,\,t_0)=0$. At this
point, we apply the parabolic maximum principle and 
obtain
\begin{align*}
0\le&6K\langle X,\,\nu\rangle-(2+\mu)H\umbruch\\
\le&6K|X|-(2+\mu)H\umbruch\\
=&6\sqrt K\sqrt\mu-(2+\mu)H&\text{as~}&w(x_0,\,t_0)=0,\umbruch\\
\le&(6\sqrt\mu-2-\mu)H&\text{as~}&K\le H^2.
\end{align*}
We fix $\mu$ sufficiently large and obtain that $w\le0$ during
the flow. In view of Lemma \ref{K 1st C0 est},
this implies the upper bound on the Gau{\ss} 
curvature claimed above.
\end{proof}

Combining this result with the Lemmata \ref{K pinching lem},
\ref{K vel bound}, and \ref{K 1st C0 est}, we obtain
\begin{equation}\label{K bounded}
\frac1c\cdot(T-t)^2\le K\le c\cdot(T-t)^2.
\end{equation}

We rescale our surfaces similarly as in \cite{AndrewsContractCalc}.
Consider the embeddings $\tilde X(\cdot,\,t)$,
$$\tilde X(z,\,t):=(T-t)\cdot X(z,\,t).$$
Define a new time function
$$\tau(t):=-\log\left(\frac{T-t}T\right).$$
We use a tilde to denote geometric quantities of the
rescaled surfaces. For $\tilde X$, we obtain the 
evolution equation
$$\frac d{d\tau}\tilde X=\frac1{\tilde K}\tilde\nu-\tilde X.$$
Our a priori estimates and the estimates of Krylov, Safonov,
Evans, and Schauder imply uniform bounds on all derivatives
of the support function $\tilde u$ of $\tilde X$. 
Applying interpolation inequalities as in
\cite[Lemma C.2]{OSKnutAIHP} to
\begin{align*}
\abs{\tilde u-1}\le&c\cdot(T-t),\umbruch\\
\abs{D^k\tilde u}\le&c_k,\umbruch
\intertext{we get}
\abs{D^k\tilde u}\le&c(k,\,\epsilon)\cdot(T-t)^{1-\epsilon}
\end{align*}
for any $\epsilon>0$. 

This finishes the proof of Theorem \ref{K main thm}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Improved Convergence Rate}\label{K impr conv sec}

\begin{theorem}\label{K impr K invers mon thm}
For a family of smooth compact strictly convex surfaces $M_t$ 
in $\R^3$ flowing according to $\dot X=\frac1K\nu$, 
\begin{equation}
\max\limits_{M(t)}\frac{\left(\lx^2+\ly^2\right)(\lambda_1-\lambda_2)^2}
{8(\lx+\ly)\lx^3\ly^3}
\equiv\max\limits_{M(t)}w\equiv\max\limits_{M_t}
\frac{\A2\left(2\A2-H^2\right)}{H\left(H^2-\A2\right)^3}
\end{equation}
is non-increasing in time.
\end{theorem}
\begin{proof}
We use Section \ref{comp alg sec} and obtain in a critical point of $w$
\begin{align*}
\oprokl w=&\frac{-3\left(\lx^4+2\lx^3\ly-2\lx^2\ly^2+2\lx\ly^3+\ly^4\right)
(\lx-\ly)^2}{8(\lx+\ly)^2\lx^4\ly^4}\umbruch\\ 
&-\frac1{\left(3\lx^3+3\lx^2\ly-\lx\ly^2+3\ly^3\right)^2
(\lx+\ly)^2\lx^7\ly^2}\cdot\\
&\quad\cdot\left(18\lx^9-9\lx^8\ly+12\lx^7\ly^2
+72\lx^6\ly^3-12\lx^5\ly^4+70\lx^4\ly^5\right.\\
&\qquad\left.+60\lx^3\ly^6+18\lx\ly^8+27\ly^9\right)h_{11;\,1}^2\umbruch\\
&-\frac1{\left(3\ly^3+3\ly^2\lx-\ly\lx^2+3\lx^3\right)^2
(\ly+\lx)^2\ly^7\lx^2}\cdot\\
&\quad\cdot\left(18\ly^9-9\ly^8\lx+12\ly^7\lx^2
+72\ly^6\lx^3-12\ly^5\lx^4+70\ly^4\lx^5\right.\\
&\qquad\left.+60\ly^3\lx^6+18\ly\lx^8+27\lx^9\right)h_{22;\,2}^2\umbruch\\
\le&0.
\end{align*}
We finally apply the maximum principle. 
\end{proof}

This allows to improve our bound on $\abs{\lx-\ly}$.
\begin{lemma}\label{K diff bound}
For a smooth compact strictly convex surface $M_t$ in $\R^3$,
flowing according to $\dot X=\frac1K\nu$, there exists a
constant $c=c(M_0)$ such that $\abs{\lx-\ly}\le c\cdot K^{5/4}
\le c\cdot (T-t)^{5/2}$.
\end{lemma}
\begin{proof}
This is a direct consequence of Theorem \ref{K impr K invers mon thm},
Lemma \ref{K pinching lem} and Lemma \ref{K upper bound two}.
\end{proof}

We now closely follow the corresponding sections of \cite{AndrewsStones}
and Part \ref{A2 part}.

\begin{proposition}\label{K prop int H}
Define $q(t):=\frac1{4\pi}\int\limits_{M_t}KX$.
Then 
$$\left\lvert\langle X-q,\nu\rangle-\frac1{8\pi}
\int\limits_{M_t}H\right\rvert\le\frac1{4\pi}
\cdot\sup_{M_t}\abs{\lx-\ly}\cdot\mathcal H^{2}(M_t),$$
where $\mathcal H^2(M_t)$ denotes the area of $M_t$.
\end{proposition}
\begin{proof}
This is \cite[Proposition 4]{AndrewsStones}.
\end{proof}

We will call $q(t)$ the pseudocenter of $M_t$.

We define $r_+(t)$ to be the minimal radius of a sphere, 
centered at $q(t)$, that encloses $M_t$. Similarly, we
define $r_-(t)$ to be the maximal radius of a sphere,
centered at $q(t)$, that is enclosed by $M_t$.
Let $\rho_-(t)$ be the maximal radius of a sphere
(with arbitrary center) enclosed by $M_t$ and $\rho_+(t)$
be the minimal radius of spheres enclosing $M_t$.

\begin{lemma}\label{K radii bound}
Under the assumptions of Theorem \ref{K main thm}, 
for $T-t$ sufficiently small, $r_+$ and
$r_-$ are estimated as follows
$$(T-t)^{-1}\cdot\left(1-c\cdot(T-t)^{3/2}\right)\le r_-(t)
\le r_+(t)\le(T-t)^{-1}\cdot\left(1+c\cdot(T-t)^{3/2}\right).$$
\end{lemma}
\begin{proof}
Denote the bounded component of $\R^3\setminus M_t$
by $E_t$. The transformation formula for integrals
implies that
$$\frac1{4\pi}\int\limits_{M_t}KX=\frac1{4\pi}
\int\limits_{\S^2}X\left(\nu^{-1}(\cdot)\right).$$
So we see that $q(t)\in E_t$.
We have 
\begin{align*}
r_+=&\max_{M_t}\,\langle X-q(t),\,\nu\rangle,&
r_-=&\min_{M_t}\,\langle X-q(t),\,\nu\rangle,\\ 
\rho_+=&\min_{p\in\R^3}\max_{M_t}\,\langle X-p,\,\nu\rangle,
\qquad\text{and}&
\rho_-=&\max_{p\in E_t}\min_{M_t}\,\langle X-p,\,\nu\rangle.
\end{align*}

Recall the first variation formula for a vector 
field $Y$ along $M_t$
$$\int\limits_{M_t}H
\langle Y,\,\nu\rangle=\int\limits_{M_t}\divergenz_{M_t} Y$$
and get for $p\in E_t$ such that $\rho_+=\max_{M_t}
\langle X-p,\,\nu\rangle$
$$\int\limits_{M_t}H\ge\frac1{\rho_+}\int\limits_{M_t}
H\cdot\langle X-p,\,\nu\rangle=\frac1{\rho_+}\int\limits_{M_t}
\divergence_{M_t}X=\frac1{\rho_+}\int\limits_{M_t}2=\frac2{\rho_+}
\mathcal H^2(M_t).$$
We employ Proposition \ref{K prop int H} and deduce that 
\begin{align*}
r_-\ge&\frac1{8\pi}\int\limits_{M_t}H\cdot\left\{1-
2\left(\int_{M_t}H\right)^{-1}\cdot
\sup\limits_{M_t}\abs{\lx-\ly}\cdot\mathcal H^2(M_t)\right\}\\
\ge&\frac1{8\pi}\int\limits_{M_t}H\cdot
\left\{1-\rho_+\cdot\sup\limits_{M_t}\abs{\lx-\ly}\right\}\\
\ge&\frac1{8\pi}\int\limits_{M_t}H\cdot
\left\{1-c\cdot(T-t)^{3/2}\right\},
\end{align*}
where we have used the Lemmata \ref{K diff bound} and 
\ref{K 1st C0 est}.
So we obtain
\begin{equation}\label{K r- bound}
r_-(t)\ge\frac1{8\pi}\int\limits_{M_t}H\cdot\left(1-
c\cdot(T-t)^{3/2}\right)
\end{equation}
and similarly
\begin{equation}\label{K r+ bound}
r_+(t)\le\frac1{8\pi}\int\limits_{M_t}H\cdot\left(1+
c\cdot(T-t)^{3/2}\right).
\end{equation}
As in the proof of Lemma \ref{K 1st C0 est}, we get
$$r_-\le\rho_-\le(T-t)^{-1}\le\rho_+\le r_+$$
and 
$$(T-t)^{-1}\cdot\left(1-c\cdot(T-t)^{3/2}\right)
\le\frac1{8\pi}\int\limits_{M_t}H\le
(T-t)^{-1}\cdot\left(1+c\cdot(T-t)^{3/2}\right).$$
Using \eqref{K r- bound} and \eqref{K r+ bound} gives the
claimed estimates on $r_-$ and $r_+$.
\end{proof}

\begin{lemma}\label{K Q est}
Under the assumptions of Theorem \ref{K main thm}, 
$q(t)$ as defined in Proposition \ref{K prop int H}
is a smooth function of $t$ in $[0,\,T)$ and converges
to some point $Q\in\R^3$ for $t\uparrow T$,
$$\abs{q(t)-Q}\le c\cdot (T-t)^{1/2}.$$
\end{lemma}
\begin{proof}
The definition of $q(t)$ involves only quantities that depend
smoothly on $t$, so it remains to prove convergence for
$t\uparrow T$.\par
For $0<t_1<t_2<T$, we want to estimate $\abs{q(t_1)-q(t_2)}$
from above. We may assume that $q(t_1)\neq q(t_2)$. Consider
the line passing through $q(t_1)$ and $q(t_2)$. It intersects
the surface $M_{t_2}$ in two points, denoted by $p_l(t_2)$ and
$p_r(t_2)$.
 
\setlength{\unitlength}{0.00075in}
%
\begingroup\makeatletter\ifx\SetFigFont\undefined%
\gdef\SetFigFont#1#2#3#4#5{%
  \reset@font\fontsize{#1}{#2pt}%
  \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
  \selectfont}%
\fi\endgroup%
{\renewcommand{\dashlinestretch}{30}
\begin{picture}(6324,1135)(0,-10)
\drawline(12,748)(6312,748)
\drawline(3072,838)(3252,658)
\drawline(3252,838)(3072,658)
\drawline(2622,838)(2802,658)
\drawline(2802,838)(2622,658)
\drawline(552,1108)(551,1106)(549,1102)
        (546,1096)(541,1086)(536,1073)
        (529,1057)(521,1038)(513,1018)
        (504,996)(496,972)(489,947)
        (482,920)(476,891)(470,860)
        (466,825)(463,788)(462,748)
        (463,708)(466,671)(470,636)
        (476,605)(482,576)(489,549)
        (496,524)(504,500)(513,478)
        (521,458)(529,439)(536,423)
        (541,410)(546,400)(549,394)
        (551,390)(552,388)
\drawline(5772,1108)(5773,1106)(5775,1102)
        (5778,1096)(5783,1086)(5788,1073)
        (5795,1057)(5803,1038)(5811,1018)
        (5820,996)(5828,972)(5835,947)
        (5842,920)(5848,891)(5854,860)
        (5858,825)(5861,788)(5862,748)
        (5861,708)(5858,671)(5854,636)
        (5848,605)(5842,576)(5835,549)
        (5828,524)(5820,500)(5811,478)
        (5803,458)(5795,439)(5788,423)
        (5783,410)(5778,400)(5775,394)
        (5773,390)(5772,388)
\put(462,73){\makebox(0,0)[cb]{\smash{{\SetFigFont{12}{14.4}
{\rmdefault}{\mddefault}{\updefault}$p_l(t_2)$}}}}
\put(2712,73){\makebox(0,0)[cb]{\smash{{\SetFigFont{12}{14.4}
{\rmdefault}{\mddefault}{\updefault}$q(t_1)$}}}}
\put(3162,73){\makebox(0,0)[cb]{\smash{{\SetFigFont{12}{14.4}
{\rmdefault}{\mddefault}{\updefault}$q(t_2)$}}}}
\put(5862,73){\makebox(0,0)[cb]{\smash{{\SetFigFont{12}{14.4}
{\rmdefault}{\mddefault}{\updefault}$p_r(t_2)$}}}}
\end{picture}
}
\begin{center}Figure: Convergence of pseudocenters
\end{center}

We may assume that 
$$\langle p_r(t_2)-q(t_2),\,q(t_2)-q(t_1)\rangle>0.$$
This corresponds to $p_r(t_2)$ and $q(t_2)$ lying on the same
side of $q(t_1)$ as shown in the figure. We estimate
\begin{align*}
2c(T-t_1)^{1/2}\ge&r_+(t_1)-r_-(t_1) &&\text{by Lemma \ref{K radii bound}}
\umbruch\\
=&\osc u_{q(t_1)}(\cdot,\,t_1)&&\umbruch\\
\ge&\osc u_{q(t_1)}(\cdot,\,t_2)&&
\text{by \cite[Theorem 3.1]{JamesMcCoyArea}}\umbruch\\
\ge&\abs{p_r(t_2)-q(t_1)}-\abs{p_l(t_2)-q(t_1)}&&\umbruch\\
=&\abs{p_r(t_2)-q(t_2)}+\abs{q(t_2)-q(t_1)}&&\\
&-\left(\abs{p_l(t_2)-q(t_2)}-\abs{q(t_2)-q(t_1)}\right)&&\umbruch\\
\ge&2\abs{q(t_2)-q(t_1)}+r_-(t_2)-r_+(t_2)&&\umbruch\\
\ge&2\abs{q(t_2)-q(t_1)}-2c(T-t_2)^{1/2},&&
\end{align*}
where $u_{q(t_1)}(\cdot,\,t_2)$ is the support function of
$M_{t_2}-q(t_1)$. Thus we can apply Cauchy's convergence criterion.
Finally, we let $t_2\uparrow T$ and the claimed bound follows.
\end{proof}

This allows to improve our convergence result.
\begin{theorem}\label{K impr conv thm}
Under the assumptions of Theorem \ref{K main thm}, there exists
$Q\in\R^3$ such that the Hausdorff distance of $M_t$ to a
family of 
expanding spheres around $Q$ is bounded as follows
$$d_{\mathcal H}(M_t,\,\partial B_{(T-t)^{-1}}(Q))\le
c\cdot(T-t)^{1/2}.$$
\end{theorem}
\begin{proof}
Combine the Lemmata \ref{K Q est} and \ref{K radii bound}.
\end{proof}

\begin{remark}
Theorem \ref{K impr conv thm} implies also better estimates for
rescaled surfaces,
$$d_{\mathcal H}\left((M_t-Q)\cdot(T-t),\,\S^2\right)\le
c\cdot(T-t)^{3/2}.$$
As above, this implies that all derivatives of the support
function decay,
$$\Vert u_Q(\cdot,\,t)-1\Vert_{C^k}\le c(k,\,\epsilon)
\cdot(T-t)^{3/2-\epsilon}$$
for any $\epsilon>0$, where $u_Q(\cdot,\,t)$ is the support
function of $M_t-Q$. This implies for the principal 
curvatures $\tilde\lambda_i$, $i=1,\,2$, of $(M_t-Q)\cdot
(T-t)$
$$1-c(\epsilon)\cdot(T-t)^{3/2-\epsilon}\le
\tilde\lambda_i\le1+c(\epsilon)\cdot(T-t)^{3/2-\epsilon}.$$
\par
In the proof of Theorem \ref{K main thm}, we have used 
\eqref{K mon groe} only to prove that surfaces stay uniformly
pinched, i.\,e.\ that $\lambda_1/\lambda_2$ is uniformly bounded.
If we use it to bound $\abs{\lx-\ly}$, and then $r_+$ and $r_-$
as above, we don't get more than in \eqref{K resc Hsd conv}, where
we used the oscillation estimates of \cite{JamesMcCoyArea}. 
Our computer program, however, did not yield a scaling invariant
quantity that implies uniform pinching. This is similar to
\cite{Andrews2dnonconcave}. It might be possible to find a 
monotone quantity that allows to further improve this 
convergence rate. 
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Convergence Rate}\label{K conv rate}

In order to find out what the optimal convergence rate might
be, we proceed as in Part \ref{A2 part}, use the same notation, 
and compute the linearized equation corresponding to
$$\dt X=\frac1K\nu-X\qquad\text{or}\qquad
\fracp ut=\frac1K w-u$$
as
$$\fracp vt=\Delta v+v.$$
As in the contracting case, we only need to consider 
eigenvalues $-l(l+1)$ of the laplacian on the sphere 
for $l\in\N_+$. For $l=1$, we cannot expect convergence 
rates better than
$$r_+\le(T-t)^{-1}\cdot(1+c\cdot(T-t)),$$
if we fix $q(t)$ arbitrarily. This estimate is sharp,
if we do not adjust $q(t)$. The corresponding eigenfunctions
induce translations of the surface. Considering $l=2$, we 
expect that we cannot obtain convergence rates better than
$$r_+(t)\le(T-t)^{-1}\cdot\left(1+c\cdot(T-t)^5\right).$$

% Ende 1/K-Fluss

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Finding Monotone Quantities}\label{K algor sec}

\section{The Algorithm}
We use a sieve algorithm and start with symmetric rational functions 
of the principal curvatures
as candidates for test functions, e.\,g.\ 
$$w=\frac{p_1(\lx,\,\ly)}{p_2(\lx,\,\ly)}
=\frac{(\lx-\ly)^2}{\lx^2\ly^2}.$$
Here, $p_1\neq0$ and $p_2\neq0$ are homogeneous polynomials. 

In the end, we want to find functions $w$ such that 
$W:=\sup_{M_t}w$ is monotone and ensures convergence to round
spheres. 

We check, whether these test functions $w$ fulfill the
following conditions. 
\begin{enumerate}
\item\label{K step one} 
 \begin{enumerate}
 \item $p_1(\lx,\,\ly),\,p_2(\lx,\,\ly)\ge0$ for $0<\lx,\,\ly$,
 \item $p_1(\lx,\,\ly)=0$ for $\lx=\ly>0$.
 \end{enumerate}
\item\label{K step two} $\deg p_1<\deg p_2$.
\item\label{K step three} $\fracp{w(1,\ly)}{\ly}<0$ for $0<\ly<1$ and 
  $\fracp{w(1,\ly)}{\ly}>0$ for $\ly>1$.
\item\label{K step four} $\dt w-F^{ij}w_{;\,ij}\le0$ 
 \begin{enumerate}
 \item\label{K step four a} for terms without derivatives of $(h_{ij})$,
 \item\label{K step four b} for terms involving derivatives of $(h_{ij})$,
   if $w_{;\,i}=0$ for $i=1,\,2$.
 \end{enumerate}
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Motivation and Randomized Tests}
For all flow equations 
considered, spheres contract to points and stay spherical. So
we can only find monotone quantities, if $\deg p_1\ge\deg p_2$
or $p_1(\lambda,\,\lambda)=0$. 

If $\deg p_1\ge\deg p_2$,
we obtain that $W$ is non-increasing on any self-similarly
expanding surface. So this does not imply convergence to a
sphere. 

Condition \eqref{K step three} ensures that the quantity decreases, if 
the eigenvalues approach each other. 

In step \eqref{K step four a} and \eqref{K step four b}, we check that 
we can apply the maximum principle. Here, we have to use various
differentiation rules.

In steps \eqref{step one}, \eqref{step two}, and \eqref{step four}, 
inequalities are tested by evaluating both 
sides at random numbers. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Other Normal Velocities}\label{K o nor vel}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Homogeneity less than minus one}

In this section, we prove 
\begin{theorem}\label{K sammel thm}
For any smooth compact strictly convex surface $M$ in $\R^3$, 
there exists a smooth family of surfaces $M_t$, $t\in[0,T)$,
solving one of the following flow equations
\begin{itemize}
\item ${\displaystyle\dt X=\frac{H^2}{K^2}\nu,}$\\[0.5ex]\rule{1mm}{0mm}
\item ${\displaystyle\dt X=\frac{\A2}{K^2}\nu,}$\\[0.5ex]\rule{1mm}{0mm}
\item ${\displaystyle\dt X=\frac{H^3}{K^3}\nu,}$\\[0.5ex]\rule{1mm}{0mm}
\end{itemize}
with $M_0=M$.
For $t\uparrow T$, $M_t$ converges to infinity. 
The rescaled surfaces $M_t\cdot r^{-1}(t)$ converge smoothly to the
unit sphere $\S^2$, where $r(t)$ is the radius of an expanding
sphere that converges to infinity for $t\uparrow T$, more
precisely, $r(t)$ is as follows
\begin{itemize}
\item ${\displaystyle\dt X=\frac{H^2}{K^2}\nu:
\quad r(t)=(4(T-t))^{-1},}$\\[0.5ex]\rule{1mm}{0mm}
\item ${\displaystyle\dt X=\frac{\A2}{K^2}\nu:
\quad r(t)=(2(T-t))^{-1},}$\\[0.5ex]\rule{1mm}{0mm}
\item ${\displaystyle\dt X=\frac{H^3}{K^3}\nu:
\quad r(t)=(16(T-t))^{-1/2}.}$
\end{itemize}
\end{theorem}

In Part \ref{A2 part}, we have studied convex surfaces 
contracting according to 
$$\dt X=-F\nu,$$
where $F$ is positive homogeneous of some degree larger than 
or equal to two. 
There, we got the impression, that appropriate monotone quantities
that assure convergence to a sphere after rescaling are available
for almost every normal velocity considered.  
In contrast to this, such monotone quantities seem to be rare 
objects for expanding surfaces with normal velocity of homogeneity
less than or equal to minus two. 
\par
In both cases, we restricted our attention to normal velocities
and possible candidates for monotone quantities in
$\Z(\lx,\,\ly)$ with small coefficients which are 
symmetric in $\lx$ and $\ly$. 
\par
In the case of expanding surfaces, however, we also find
monotone quantities for surfaces expanding with a normal 
velocity that is positive homogeneous of degree minus one in 
the principal curvatures. 

The proof of Theorem \ref{K sammel thm} is similar to the proof
of Theorem \ref{K main thm}. Therefore, we will present in the
following only those results that are not almost identical 
to their counterparts in the proof of Theorem \ref{K main thm}.

\begin{theorem}
For a family $M_t$ of smooth compact strictly convex surfaces in 
$\R^3$, flowing according to $\dt X=\frac{H^2}{K^2}\nu$, 
$$\max_{M_t}\frac{(\lx-\ly)^2}{2(\lx+\ly)\lx\ly}
\equiv\max_{M_t}w$$
is non-increasing in time.
\end{theorem}
\begin{proof}
According to Section \ref{comp alg sec}, we obtain in a
critical point of $w$
\begin{align*}
\oprokl w=&-\frac{5(\lx-\ly)^2}{\lx^2\ly^2}\\
&-\frac{128}{(\lx+3\ly)^2\lx^4}\cdot h_{11;\,1}^2
+(\ldots)\cdot h_{22;\,2}^2.
\end{align*}
Here and in the following, we write $(\ldots)$ to denote a 
term that equals the factor in front of $h_{11;\,1}^2$
with $\lx$ and $\ly$ interchanged. 

We finally apply the maximum principle. 
\end{proof}

The factor $2$ in the denominator is useful to rewrite $w$
in terms of the algebraic basis consisting of $H$ and $\A2$
$$\frac{(\lx-\ly)^2}{2(\lx+\ly)\lx\ly}=
\frac{2\A2-H^2}{H\cdot\left(H^2-\A2\right)}.$$

\begin{theorem}
For a family $M_t$ of smooth compact strictly convex surfaces in 
$\R^3$, flowing according to $\dt X=\frac{\A2}{K^2}\nu$, 
$$\max_{M_t}\frac{(\lx-\ly)^2}{2(\lx+\ly)\lx\ly}
\equiv\max_{M_t}w$$
is non-increasing in time.
\end{theorem}
\begin{proof}
According to Section \ref{comp alg sec}, we obtain in a
critical point of $w$
\begin{align*}
\oprokl w=&-\frac{2\left(2\lx^2+\lx\ly+2\ly^2\right)(\lx-\ly)^2}
{(\lx+\ly)^2\lx^2\ly^2}\\ 
&-\frac{4\left(21\lx^4+24\lx^3\ly+18\lx^2\ly^2+\ly^4\right)}
{(\lx+3\ly)^2(\lx+\ly)^2\lx^6}
\cdot h_{11;\,1}^2
+(\ldots)\cdot h_{22;\,2}^2
\end{align*}
and apply the maximum principle. 
\end{proof}

\begin{theorem}
For a family $M_t$ of smooth compact strictly convex surfaces in 
$\R^3$, flowing according to $\dt X=\frac{H^3}{K^3}\nu$, 
$$\max_{M_t}\frac{(\lx+\ly)^6(\lx-\ly)^2}{16\left(\lx^2+\ly^2\right)
\left(\lx^2+\lx\ly+\ly^2\right)\lx^3\ly^3}
\equiv\max_{M_t}w$$
is non-increasing in time.
\end{theorem}
\begin{proof}
According to Section \ref{comp alg sec}, we obtain in a
critical point of $w$
\begin{align*}
\oprokl w=&-\frac{(\lx+\ly)^8(\lx-\ly)^2}{4\left(\lx^2+\ly^2\right)^2
\left(\lx^2+\lx\ly+\ly^2\right)^2\lx^6\ly^6}\cdot\\
&\quad\cdot(\lx^6-\lx^5\ly+8\lx^4\ly^2+2\lx^3\ly^3+8\lx^2\ly^4
-\lx\ly^5+\ly^6)\\ 
&-\frac{3(\lx+\ly)^8}{8\left(3\lx^6+11\lx^4\ly^2+2\lx^3\ly^3
+9\lx^2\ly^4-2\lx\ly^5+\ly^6\right)^2}
\cdot\\
&\quad\cdot\frac1{\left(\lx^2+\lx\ly+\ly^2\right)^2
\left(\lx^2+\ly^2\right)^2\lx^8\ly^6}\cdot\\
&\quad\cdot\left(2\lx^{18}-4\lx^{17}\ly+57\lx^{16}\ly^2
-108\lx^{15}\ly^3+508\lx^{14}\ly^4\right.\\
&\qquad\left.-428\lx^{13}\ly^5+2152\lx^{12}\ly^6-156\lx^{11}\ly^7
+4784\lx^{10}\ly^8\right.\\
&\qquad\left.+172\lx^9\ly^9+4942\lx^8\ly^{10}-612\lx^7\ly^{11}
+2676\lx^6\ly^{12}\right.\\
&\qquad\left.-772\lx^5\ly^{13}+872\lx^4\ly^{14}-340\lx^3\ly^{15}
+126\lx^2\ly^{16}\right.\\
&\qquad\left.-56\lx\ly^{17}+9\ly^{18}\right)\cdot h_{11;\,1}^2\\
&+(\ldots)\cdot h_{22;\,2}^2
\end{align*}
and apply the maximum principle. 
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{K upper princ curv bnd}
Under the assumptions of Theorem \ref{K sammel thm},
$\dt X=-F\nu$, the maximum principal curvature of $M_t$
is non-increasing in time.
\end{lemma}
\begin{proof}
Consider $M_{ij}:=h_{ij}-\mu g_{ij}$, where $\mu$ is chosen
such that $M_{ij}\le0$ initially. We compute
according to \eqref{g evol} and Lemma \ref{evol equ lem}
\begin{align*}
\dt M_{ij}-F^{kl}M_{ij;\,kl}=&
F^{kl}h^a_kh_{al}\cdot h_{ij}-F^{kl}h_{kl}\cdot h^a_ih_{aj}
-Fh^k_ih_{kj}+2\mu Fh_{ij}\\
&+F^{kl,\,rs}h_{kl;\,i}\xi^ih_{rs;\,j}\xi^j.
\end{align*}
Assume that $\xi$ is a zero eigenvalue of $(M_{ij})$,
$h_{ij}\xi^j=\mu g_{ij}\xi^j$, with $g_{ij}\xi^i\xi^j=1$. 
We may assume, that in our coordinate system, we have
$\xi=(1,\,0)$ and $(h_{ij})=\left(\begin{smallmatrix}
\mu&0\\ 0&\lambda\end{smallmatrix}\right)$ with $0<\lambda\le\mu$.
Normal velocities $F$ as in Theorem \ref{K sammel thm} are
concave, so
$$F^{kl,\,rs}h_{kl;\,i}\xi^ih_{rs;\,j}\xi^j\le0.$$
We estimate
\begin{itemize}
\item $\left(\dt M_{ij}-F^{kl}M_{ij;\,kl}\right)\xi^i\xi^j\le
-\frac{(\lambda+\mu)(3\mu-\lambda)}
{\lambda^2}\le0$ for $F=-\frac{H^2}{K^2}$,
\item $\left(\dt M_{ij}-F^{kl}M_{ij;\,kl}\right)\xi^i\xi^j\le
-\frac{-2\lambda\mu+\lambda^2+3\mu^2}{\lambda^2}\le0$
for $F=-\frac{\A2}{K^2}$,
\item $\left(\dt M_{ij}-F^{kl}M_{ij;\,kl}\right)\xi^i\xi^j\le
-\frac{2(\lambda+\mu)^2(2\mu-\lambda)}{\mu\lambda^3}\le0$
for $F=-\frac{H^3}{K^3}$.
\end{itemize}
Now the claim follows directly from the maximum principle
for tensors.
\end{proof}

\begin{lemma}
Under the assumptions of Theorem \ref{K sammel thm}, we obtain
a lower bound on the principal curvatures similarly to
Lemma \ref{K vel bound}.
\end{lemma}
\begin{proof}
We proceed as in the proof of Lemma \ref{K vel bound}.
In a critical point of $\frac{-F}{2R-\langle X,\,\nu\rangle}$, 
we compute the following evolution equation for a family
of surfaces flowing according to $\dt X=-F\nu$
\begin{align*}
\dt\log\frac{-F}{2R-\langle X,\,\nu\rangle}
-F^{ij}\left(\log\frac{-F}
{2R-\langle X,\,\nu\rangle}\right)_{;\,ij}=&
\frac1{2R-\langle X,\,\nu\rangle}\cdot\\
&\cdot\left(2R\cdot F^{ij}h^k_ih_{kj}-F^{ij}h_{ij}-F\right).
\end{align*}
We compute this explicitly for $F$ as in Theorem \ref{K sammel thm}
and obtain in an increasing maximum of 
$\frac{-F}{2R-\langle X,\,\nu\rangle}$
\begin{itemize}
\item $8R-\frac HK\ge0$ for $F=-\frac{H^2}{K^2}$,
\item $4HR-\frac{\A2}K\ge0$ for $F=-\frac{\A2}{K}$,
\item $6R-\frac HK\ge0$ for $F=-\frac{H^3}{K^3}$.
\end{itemize}
Our monotone quantities and Lemma \ref{K upper princ curv bnd}
imply that our surfaces are pinched, i.\,e.\ that 
$\frac\lx\ly+\frac\ly\lx$ is uniformly bounded above. So
we obtain there that $\lambda_i\ge\frac1{cR}$.
\end{proof}

\begin{lemma}
Consider one of the flow equations of Theorem \ref{K sammel thm}, 
$\dt X=-F\nu$, and a solution as in this theorem.
Let $\gamma$ be such that $F$ is positive
homogeneous of degree $-\gamma$. Then there exists a
constant $\alpha\gg1$ such that $\abs F\ge\frac1\alpha
|X|^\gamma$ during the flow. More precisely, there
exists $\alpha\gg1$ such that $|X|^\gamma+\alpha F$ remains
non-positive during the flow, if this quantity is negative 
initially.  
\end{lemma}
\begin{proof}
For $F$ positive homogeneous of degree minus two, 
we get $F^{ij}h_{ij}=-2F$ and obtain at a point,
where $|X|^2+\alpha F=0$,
\begin{align*}
\dt\left(|X|^2+\alpha F\right)
-F^{ij}\left(|X|^2+\alpha F\right)_{;\,ij}=&
-6F\langle X,\,\nu\rangle-2F^{ij}g_{ij}
+\alpha FF^{ij}h^k_ih_{kj}\umbruch\\
\le&-6F\sqrt\alpha\sqrt{-F}
+\alpha FF^{ij}h^k_ih_{kj}.
\end{align*}
It is straightforward to check that for $\alpha\gg1$ sufficiently
large and $F$ as assumed above, 
the right-hand side is non-positive. 
\par
If $F=-\frac{H^3}{K^3}$, we obtain similarly as above
\begin{align*}
\dt\left(|X|^3+\alpha F\right)
-F^{ij}\left(|X|^3+\alpha F\right)_{;\,ij}\le&
12|X|^2\frac{H^3}{K^3}-6\alpha\frac{H^5}{K^5}\le 0.
\end{align*}
\par
In both cases, the lemma follows from the maximum principle.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Homogeneity minus one}

\begin{table}
\def\platz{\raisebox{0em}[2.2em][1.5em]{\rule{0em}{2em}}}
$$\begin{array}{|c||c|}\hline
-\fracd1H & \platz\fracd{(\lx-\ly)^2}{(\lx+\ly)\lx\ly}\\\hline
-\fracd1H & \platz\fracd{\left(\lx^2+\ly^2\right)(\lx-\ly)^2}
{(\lx+\ly)\lx^3\ly^3}\\\hline
-\fracd HK & \platz\fracd{(\lx-\ly)^2}{\lx^2\ly^2}\\\hline
\end{array}$$
\caption{More monotone quantities}\label{K tab2}
\end{table}

In Table \ref{K tab2}, we have collected some normal velocities $F$,
positive homogeneous of degree minus one, and test functions $w$,
such that $\max_{M_t}w$ is non-increasing during the flow of
a closed strictly convex surface flowing according to
$$\dt X=-F\nu.$$

\begin{theorem}
For a family $M_t$ of smooth compact strictly convex surfaces in 
$\R^3$, flowing according to $\dt X=\frac1H\nu$, 
$$\max_{M_t}\frac{(\lx-\ly)^2}{2(\lx+\ly)\lx\ly}
\equiv\max_{M_t}w$$
and 
$$\max_{M_t}
\frac{\left(\lx^2+\ly^2\right)(\lx-\ly)^2}{8(\lx+\ly)\lx^3\ly^3}
\equiv\max_{M_t}\hat w$$
are non-increasing in time.
\end{theorem}
\begin{proof}
According to Section \ref{comp alg sec}, we obtain in a
critical point of $w$
\begin{align*}
\oprokl w=&-\frac{\left(\lx^2+4\lx\ly+\ly^2\right)(\lx-\ly)^2}
{2(\lx+\ly)^3\lx\ly}\\
&-\frac{2\left(5\lx^2+2\lx\ly+\ly^2\right)\ly}
{(\lx+3\ly)^2(\lx+\ly)\lx^5}\cdot h_{11;\,1}^2\\
&+(\ldots)\cdot h_{22;\,2}^2
\end{align*}
and apply the maximum principle. 
\par
Similarly, we obtain in a critical point of $\hat w$
\begin{align*}
\oprokl{\hat w}=&-\frac{\left(3\lx^4-2\lx^2\ly^2+3\ly^4\right)(\lx-\ly)^2}
{8(\lx+\ly)^3\lx^3\ly^3}\umbruch\\
&-\frac1{2\left(3\lx^3+3\lx^2\ly-\lx\ly^2+3\ly^3\right)^2
(\lx+\ly)^3\lx^7\ly}\cdot\\
&\quad\cdot\left(9\lx^{10}-9\lx^8\ly^2+96\lx^7\ly^3
-38\lx^6\ly^4+96\lx^5\ly^5+30\lx^4\ly^6\right.\\
&\qquad\left.+45\lx^2\ly^8+27\ly^{10}\right)\cdot h_{11;\,1}^2\\
&+(\ldots)\cdot h_{22;\,2}^2
\end{align*}
and apply the maximum principle once again. 
\end{proof}

\begin{theorem}
For a family $M_t$ of smooth compact strictly convex surfaces in 
$\R^3$, flowing according to $\dt X=\frac HK\nu$, 
$$\max_{M_t}\frac{(\lx-\ly)^2}{4\lx^2\ly^2}
\equiv\max_{M_t}w$$
is non-increasing in time.
\end{theorem}
\begin{proof}
According to Section \ref{comp alg sec}, we obtain in a
critical point of $w$
\begin{align*}
\oprokl w=&-\frac{(\lx-\ly)^2}{\lx^2\ly^2}
-\frac2{\lx^6}\cdot h_{11;\,1}^2+(\ldots)\cdot h_{22;\,2}.
\end{align*}
and apply the maximum principle. 
\end{proof}

For these flow equations, convergence to infinity and
convergence to a sphere after rescaling have been proved
before for hypersurfaces 
\cite{CGFlowSpheres,UrbasExpandMZ,UrbasExpandJDG}. 
Monotone quantities as mentioned above might at most be useful
to improve the convergence rate. Inverse mean curvature
was used to prove the Penrose inequality in general 
relativity \cite{HuiskenIlmanenPenrose}. Our techniques
might also apply to surfaces expanding in the asymptotically 
flat manifolds considered there.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\backmatter
\bibliographystyle{/home/napier/schnuere/os/tools/amsplain}
%\bibliography{/home/napier/schnuere/os/tools/biblio}
\def\unterstrich{\underline{\rule{1ex}{0ex}}} \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}

\bibitem{SteveThroughSing}
Steven~J. Altschuler and Matthew~A. Grayson, \emph{Shortening space curves and
  flow through singularities}, J. Differential Geom. \textbf{35} (1992), no.~2,
  283--298.

\bibitem{AnadaCalcHarm}
Koichi Anada, \emph{Contraction of surfaces by harmonic mean curvature flows
  and nonuniqueness of their self similar solutions}, Calc. Var. Partial
  Differential Equations \textbf{12} (2001), no.~2, 109--116.

\bibitem{AnadaStability}
Koichi Anada and Masayoshi Tsutsumi, \emph{Stability of solutions of nonlinear
  parabolic equations for harmonic mean curvature flows}, Nonlinear Anal.
  \textbf{51} (2002), no.~2, Ser. A: Theory Methods, 305--319.

\bibitem{Andrews2dKrylov}
Ben{}{{}}{} Andrews, \emph{{Fully nonlinear parabolic equations in two space
  variables}}, \relax\\{\tt arXiv:math.AP/0402235}.

\bibitem{Andrews2dnonconcave}
Ben{{}}{{}}{} Andrews, \emph{{Moving surfaces by non-concave curvature
  functions}}, {\tt arXiv:math.DG/0402273}.

\bibitem{AndrewsPinching}
Ben{}{}{{}} Andrews, \emph{{Pinching estimates and motion of hypersurfaces by
  curvature functions}}, \relax\\{\tt arXiv:math.DG/0402311}.

\bibitem{AndrewsContractCalc}
Ben{{}}{} Andrews, \emph{Contraction of convex hypersurfaces in {E}uclidean
  space}, Calc. Var. Partial Differential Equations \textbf{2} (1994), no.~2,
  151--171.

\bibitem{AndrewsContrRiem}
Ben{{}} Andrews, \emph{Contraction of convex hypersurfaces in {R}iemannian
  spaces}, J. Differential Geom. \textbf{39} (1994), no.~2, 407--431.

\bibitem{AndrewsBrisbane}
Ben{} Andrews, \emph{{Evolving convex hypersurfaces}}, {Martin, Gaven (ed.) et
  al., Proceedings of the miniconference on analysis and applications, held at
  the University of Queensland, Brisbane, Australia, September 20-23, 1993},
  Canberra: Australian National University, Centre for Mathematics and its
  Applications. Proc. Cent. Math. Appl. Aust. Natl. Univ. 33, 1-24, 1994.

\bibitem{AndrewsHarnack}
Ben{}{} Andrews, \emph{Harnack inequalities for evolving hypersurfaces}, Math.
  Z. \textbf{217} (1994), no.~2, 179--197.

\bibitem{AndrewsMonotone}
Ben{{}}{{}} Andrews, \emph{{Monotone quantities and unique limits for evolving
  convex hypersurfaces}}, Int. Math. Res. Not. \textbf{1997} (1997), no.~20,
  1002--1031.

\bibitem{AndrewsStones}
Ben Andrews, \emph{Gauss curvature flow: the fate of the rolling stones},
  Invent. Math. \textbf{138} (1999), no.~1, 151--161.

\bibitem{AndrewsPacific}
Ben{}{}{} Andrews, \emph{Motion of hypersurfaces by {G}auss curvature}, Pacific
  J. Math. \textbf{195} (2000), no.~1, 1--34.

\bibitem{MinkowskiFlow}
Kai-Seng Chou and Xu-Jia Wang, \emph{A logarithmic {G}auss curvature flow and
  the {M}inkowski problem}, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire
  \textbf{17} (2000), no.~6, 733--751.

\bibitem{ChowNthRoot}
Bennett Chow, \emph{Deforming convex hypersurfaces by the {$n$}th root of the
  {G}aussian curvature}, J. Differential Geom. \textbf{22} (1985), no.~1,
  117--138.

\bibitem{ChowScalarRoot}
Bennett{} Chow, \emph{Deforming convex hypersurfaces by the square root of the
  scalar curvature}, Invent. Math. \textbf{87} (1987), no.~1, 63--82.

\bibitem{ChowGulliverSphere}
Bennett Chow and Robert Gulliver, \emph{Aleksandrov reflection and nonlinear
  evolution equations. {I}. {T}he {$n$}-sphere and {$n$}-ball}, Calc. Var.
  Partial Differential Equations \textbf{4} (1996), no.~3, 249--264.

\bibitem{ChowLuMP}
Bennett Chow and Peng Lu, \emph{The maximum principle for systems of parabolic
  equations subject to an avoidance set}, Pacific J. Math. \textbf{214} (2004),
  no.~2, 201--222.

\bibitem{ChowAsian}
Bennett Chow and Dong-Ho Tsai, \emph{Expansion of convex hypersurfaces by
  nonhomogeneous functions of curvature}, Asian J. Math. \textbf{1} (1997),
  no.~4, 769--784.

\bibitem{ChowNonhomogGauss}
Bennett Chow and Dong-Ho{} Tsai, \emph{Nonhomogeneous {G}auss curvature flows},
  Indiana Univ. Math. J. \textbf{47} (1998), no.~3, 965--994.

\bibitem{GigliolaRefinedNLS}
Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka, and Terry Tao,
  \emph{A refined global well-posedness result for {S}chr\"odinger equations
  with derivative}, SIAM J. Math. Anal. \textbf{34} (2002), no.~1, 64--86
  (electronic).

\bibitem{GigliolaStabil}
Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka, and Terry{{}}
  Tao, \emph{Polynomial upper bounds for the orbital instability of the 1{D}
  cubic {NLS} below the energy norm}, Discrete Contin. Dyn. Syst. \textbf{9}
  (2003), no.~1, 31--54.

\bibitem{GigliolaWellposed}
Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka, and Terry{} Tao,
  \emph{Sharp global well-posedness for {K}d{V} and modified {K}d{V} on
  {$\mathbb R$} and {$\mathbb T$}}, J. Amer. Math. Soc. \textbf{16} (2003),
  no.~3, 705--749 (electronic).

\bibitem{FireyStones}
William~J. Firey, \emph{Shapes of worn stones}, Mathematika \textbf{21} (1974),
  1--11.

\bibitem{GageHamilton}
Michael Gage and Richard~S. Hamilton, \emph{The heat equation shrinking convex
  plane curves}, J. Differential Geom. \textbf{23} (1986), no.~1, 69--96.

\bibitem{CGFlowSpheres}
Claus Gerhardt, \emph{Flow of nonconvex hypersurfaces into spheres}, J.
  Differential Geom. \textbf{32} (1990), no.~1, 299--314.

\bibitem{CGJDG1996}
Claus{} Gerhardt, \emph{Closed {W}eingarten hypersurfaces in {R}iemannian
  manifolds}, J. Differential Geom. \textbf{43} (1996), no.~3, 612--641.

\bibitem{GraysonRoundPoints}
Matthew~A. Grayson, \emph{{The heat equation shrinks embedded plane curves to
  round points}}, J. Differ. Geom. \textbf{26} (1987), 285--314.

\bibitem{HamiltonThree}
Richard~S. Hamilton, \emph{Three-manifolds with positive {R}icci curvature}, J.
  Differential Geom. \textbf{17} (1982), no.~2, 255--306.

\bibitem{HamiltonFour86}
Richard~S.{} Hamilton, \emph{Four-manifolds with positive curvature operator},
  J. Differential Geom. \textbf{24} (1986), no.~2, 153--179.

\bibitem{HamiltonFour}
Richard~S. Hamilton, \emph{Four-manifolds with positive isotropic curvature},
  Comm. Anal. Geom. \textbf{5} (1997), no.~1, 1--92.

\bibitem{HuiskenRoundSphere}
Gerhard Huisken, \emph{Flow by mean curvature of convex surfaces into spheres},
  J. Differential Geom. \textbf{20} (1984), no.~1, 237--266.

\bibitem{HuiskenSphereMf}
Gerhard{} Huisken, \emph{Contracting convex hypersurfaces in {R}iemannian
  manifolds by their mean curvature}, Invent. Math. \textbf{84} (1986), no.~3,
  463--480.

\bibitem{HuiskenIlmanenPenrose}
Gerhard Huisken and Tom Ilmanen, \emph{The inverse mean curvature flow and the
  {R}iemannian {P}enrose inequality}, J. Differential Geom. \textbf{59} (2001),
  no.~3, 353--437.

\bibitem{HuiskenIlmanenIMCF2}
Gerhard Huisken and Tom{} Ilmanen, \emph{Higher regularity of the inverse mean
  curvature flow}, 2002,
  \texttt{http://www.math.ethz.ch/$\sim$ilmanen/papers/pub.html}.

\bibitem{HuiskenPolden}
Gerhard Huisken and Alexander Polden, \emph{Geometric evolution equations for
  hypersurfaces}, Calculus of variations and geometric evolution problems
  (Cetraro, 1996), Lecture Notes in Math., vol. 1713, Springer, Berlin, 1999,
  pp.~45--84.

\bibitem{HuiskenSinestrari2}
Gerhard Huisken and Carlo{{}} Sinestrari, \emph{Convexity estimates for mean
  curvature flow and singularities of mean convex surfaces}, Acta Math.
  \textbf{183} (1999), no.~1, 45--70.

\bibitem{HuiskenSinestrari1}
Gerhard Huisken and Carlo Sinestrari, \emph{Mean curvature flow singularities
  for mean convex surfaces}, Calc. Var. Partial Differential Equations
  \textbf{8} (1999), no.~1, 1--14.

\bibitem{HuiskenSinestrari3}
Gerhard Huisken and Carlo{} Sinestrari, 2004, in preparation.

\bibitem{HuiskenYauMass}
Gerhard Huisken and Shing-Tung Yau, \emph{Definition of center of mass for
  isolated physical systems and unique foliations by stable spheres with
  constant mean curvature}, Invent. Math. \textbf{124} (1996), no.~1-3,
  281--311.

\bibitem{TomMCF}
Tom Ilmanen, \emph{Lectures on mean curvature flow and related equations},
  1998, \relax\\ {\tt http://www.math.ethz.ch/$\sim$ilmanen/papers/pub.html},
  pp.~ii+1--61.

\bibitem{IvochkinaNehringTomi}
Nina~M. Ivochkina, Thomas Nehring, and Friedrich Tomi, \emph{Evolution of
  starshaped hypersurfaces by nonhomogeneous curvature functions}, Algebra i
  Analiz \textbf{12} (2000), no.~1, 185--203.

\bibitem{JonesSphericalHarmonics}
Michael~N. Jones, \emph{Spherical harmonics and tensors for classical field
  theory}, Electronic \& Electrical Engineering Research Studies: Applied and
  Engineering Mathematics Series, vol.~2, Research Studies Press Ltd.,
  Chichester, 1985.

\bibitem{JamesMcCoyArea}
James~A. McCoy, \emph{The surface area preserving mean curvature flow}, Asian
  J. Math. \textbf{7} (2003), no.~1, 7--30.

\bibitem{JanMetzgerFoliations}
Jan Metzger, \emph{{Foliations of asymptotically flat 3-manifolds by 2-surfaces
  of prescribed mean curvature}}, AEI-2004-090, \texttt{arXiv:math.DG/0410413}.

\bibitem{Perelman3}
Grisha Perelman{{}}, \emph{{Finite extinction time for the solutions to the
  Ricci flow on certain three-manifolds}}, {\tt arXiv:math.DG/0307245}.

\bibitem{Perelman2}
Grisha Perelman{}, \emph{{Ricci flow with surgery on three-manifolds}}, {\tt
  arXiv:math.DG/0303109}.

\bibitem{Perelman1}
Grisha Perelman, \emph{{The entropy formula for the Ricci flow and its
  geometric applications}}, \rule{0mm}{1mm}\\{\tt arXiv:math.DG/0211159}.

\bibitem{OSMathZ}
Oliver~C. Schn\"urer{{{}}}, \emph{{{T}he {D}irichlet problem for {W}eingarten
  hypersurfaces in {L}orentz manifolds}}, Math. Z. \textbf{242} (2002), no.~1,
  159--181.

\bibitem{OSMM}
Oliver~C. Schn\"urer{}, \emph{{T}ranslating solutions to the second boundary
  value problem for curvature flows}, Manuscripta Math. \textbf{108} (2002),
  no.~3, 319--347.

\bibitem{OSKnutCrelle}
Oliver~C. Schn\"urer and Knut Smoczyk{}, \emph{{E}volution of hypersurfaces in
  central force fields}, J. Reine Angew. Math. \textbf{550} (2002), 77--95.

\bibitem{OSKnutAIHP}
Oliver~C. Schn\"urer and Knut Smoczyk, \emph{{N}eumann and second boundary
  value problems for {H}essian and {G}au{\ss} curvature flows}, Ann. Inst. H.
  Poincar\'e. Anal. Non Lin\'eaire \textbf{20} (2003), no.~6, 1043--1073.

\bibitem{FelixMZ}
Felix Schulze, \emph{Evolution of convex hypersurfaces by powers of the mean
  curvature}, 2003, to appear in Mathematische Zeitschrift.

\bibitem{OSFelixH2}
Felix Schulze{, appendix by Oliver C. Schn\"urer}, \emph{Convexity estimates
  for flows of hypersurfaces by powers of the mean curvature}, 2004, in
  preparation.

\bibitem{SmoczykAsian2000}
Knut Smoczyk, \emph{Remarks on the inverse mean curvature flow}, Asian J. Math.
  \textbf{4} (2000), no.~2, 331--335.

\bibitem{KnutHarmonicMCF}
Knut{} Smoczyk, \emph{A representation formula for the inverse harmonic mean
  curvature flow}, 2003, MPI-MIS Preprint 85/2003, {\tt
  http://www.mis.mpg.de/}.

\bibitem{Spivak4}
Michael Spivak, \emph{{A comprehensive introduction to differential geometry.
  Vol. IV. 2nd ed.}}, {Berkeley: Publish Perish, Inc. VII, 561 p.}, 1979.

\bibitem{SturmThm}
Charles Sturm, \emph{M\'emoire sur la r\'esolution des \'equations numeriques},
  Bull. Sci. Math. Ferussac \textbf{11} (1829), 419--422.

\bibitem{TsoPoint}
Kaising Tso, \emph{Deforming a hypersurface by its {G}auss-{K}ronecker
  curvature}, Comm. Pure Appl. Math. \textbf{38} (1985), no.~6, 867--882.

\bibitem{UrbasExpandMZ}
John I.~E. Urbas, \emph{On the expansion of starshaped hypersurfaces by
  symmetric functions of their principal curvatures}, Math. Z. \textbf{205}
  (1990), no.~3, 355--372.

\bibitem{UrbasExpandJDG}
John I.~E.{} Urbas, \emph{An expansion of convex hypersurfaces}, J.
  Differential Geom. \textbf{33} (1991), no.~1, 91--125.

\end{thebibliography}

\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
