% Title:   Evolution of hypersurfaces in central force fields
% Date:    November 2000
% Authors: Oliver C. Schn\"urer and Knut Smoczyk
% Address: Max Planck Institute for Mathematics in the Sciences (MPI MIS)
%          Inselstr. 22-26
%          D-04103 Leipzig, Germany
% email:   schnuere@mis.mpg.de
%          smoczyk@mis.mpg.de

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


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\begin{document}    
  
\title{\huge 
Evolution of hypersurfaces\\ 
in central force 
 fields}  
\author{\setcounter{footnote}{0}{\sc   
Oliver C.\ Schn\"urer\footnote  
{Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany} 
\ and   
\setcounter{footnote}{0}  
Knut Smoczyk\footnotemark}\setcounter{footnote}{-1}
\footnote{Mathematics Subject Classification 2000: Primary 53C44, Secondary
35K15  }}  
\date{November 2000, revised June 2002}

\maketitle  
\begin{abstract}  
We consider flows of hypersurfaces in $\real{n+1}$ decreasing the
energy induced by radially symmetric potentials.
These flows are similar to the mean curvature flow but different
phenomena occur. 
We show for a natural class of potentials that hypersurfaces  
converge smoothly to a uniquely determined sphere if they 
satisfy a strengthened starshapedness condition at the  
beginning.  
\end{abstract}  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
%\tableofcontents    
  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\section{Introduction}  
Consider a smooth family  $F_t:M\to\real{n+1}\setminus\{0\}$   
of oriented hypersurfaces $M_t:=F_t(M)$ that evolve according to 
\begin{equation}\label{main} 
\dt F=-(H-\phi(s)\langle F,\nu\rangle)\nu\equiv -f\nu, 
\end{equation} 
where we have omitted the index $t$. Here, $H$ denotes the mean curvature 
of $M_t$ w.\ r.\ t.\  the  
outer unit normal $\nu$ (i.\ e.\  $H>0$ for the standard sphere)
and $\phi$ is a smooth, radially 
symmetric function (reflecting the presence of a central force) depending 
on $s:=\frac{\vert F\vert^2}{2}$.  
More precisely  
$\phi$ is related to the potential 
$v$ on $\real{n+1}\setminus\{0\}$ by 
\begin{equation}\nonumber  
v(s):=\exp{\left(-\frac{n}{2}\int\limits_1^s\frac{\eta(\sigma)} 
{\sigma}d\sigma\right)}, 
~\phi(s)=-\left.\frac{\frac{\partial}{\partial\sigma}v(\sigma)} 
  {v(\sigma)}\right|_{\sigma=s}=\frac{n\eta(s)}{2s} 
\end{equation}  
where $\eta:\real{+}\to\real{}$ is a  
smooth function. 
The total potential energy $V(F)$ 
of the hypersurface $F(M)$ will then be defined  
as  
\begin{equation}\nonumber  
V(F):=\int_Mv(s(F))d\mu.  
\end{equation}  
A physical interpretation of our flow is as follows. Suppose we   
have an electrically charged membrane with a constant charge per  
area. If this surface is in a radially symmetric potential,  
then the energy of the system  
is given by the energy in the exterior potential  
plus other terms, for example are there forces between different  
parts of the surface, which we neglect.  
The negative gradient flow for this energy is given by 
\begin{equation}\nonumber 
\dt F=-v(H-\phi(s)\langle F,\nu\rangle)\nu, 
\end{equation} 
so we see that the stationary solutions of (\ref{main}) coincide 
with the stationary solutions of the negative gradient flow. 
We will prove the following main theorem 
\bt\label{theo: main} 
Let $\t F:S^n\to\real{n+1}$ be a smooth embedding of a strictly starshaped 
hypersurface such that  
\begin{equation}\nonumber 
0<s_-\le s=\frac{\vert \t F\vert^2}{2}\le s_+ <\infty. 
\end{equation} 
 Further assume that $\eta$ satisfies 
\begin{eqnarray} 
\exists\, s_0&\in&[s_-,s_+]\text{~with~}\eta(s_0)=1,\label{main con 3}\\ 
\eta'(s)&<&-\frac{2}{n}c_\eta<0\quad\forall\, s\in[s_-,s_+]\label{main con 2} 
\end{eqnarray} 
 for a constant $c_\eta$ 
and that the strengthened starshapedness condition
\begin{equation}\label{main con 1} 
\frac{f}{\langle\t F,\nu\rangle}<c_\eta 
\end{equation} 
holds for $\t F$. 
Then (\ref{main}) admits an embedded 
solution $F_t$ for all $t\ge 0$ with $F_0=\t F$ 
and $S_t:=F_t(S^n)$ converges exponentially in the $C^\infty$-topology to a  
stable sphere centered at the origin with radius $r_0=\sqrt{2s_0}$. 
\et 
 
\brem 
{\rm
Condition (\ref{main con 1}) can be easily satisfied. 

Let $\alpha>0$, $\beta>0$ and assume that $\phi$ has the 
 form $\phi(s)=\frac{\alpha}{s^{1+\beta}}$. 
Let $\tilde F$ be as in the main theorem,  
$r_0=\max\,\bigl|\tilde F\bigr|$.  
At the point where this maximum is attained 
we have with $s_0=\frac{1}{2}r_0^2$ 
\begin{equation}\nonumber 
c_\eta>\frac{f}{\left\langle\tilde F,\nu\right\rangle} 
  \ge\frac{1}{s_0}\left(\frac{n}{2} 
  -\frac{\alpha}{s_0^\beta}\right). 
\end{equation} 
On the other hand $\eta'(s_0)<-\frac{2}{n}c_\eta$ implies 
\begin{equation}\nonumber 
\frac{\alpha\beta}{s_0^{1+\beta}}>c_\eta 
\end{equation} 
so we obtain 
\begin{equation}\nonumber 
s_0<\left(\frac{2}{n}\alpha(1+\beta)\right)^\frac{1}{\beta} 
\end{equation} 
and see that $r_0\le R(n,\alpha,\beta)$. 
}\erem 
 
 
We have chosen the special ansatz for the potential $v$ as the  
stability of a stationary sphere of radius $r_0=\sqrt{2s_0}$  
is equivalent to the simple condition $\eta'(s_0)<0$.  
  
We wish to explain the condition (\ref{main con 1})  
assumed for the embedding $\tilde F$. First we show, that the  
 flow does not preserve convexity nor starshapedness during  
the evolution (so a somehow different initial condition is  
needed for the smooth long-time existence),   
then we give a geometric interpretation of   
(\ref{main con 1}).  
  
We assume a potential $v$ that tends to infinity for $s\to 0$  
and decays for $s\to\infty$. Then a convex hypersurface does   
not need to stay convex  
during its evolution. We take a sphere of   
large radius but assume that the origin is very close to the   
surface. At the beginning of the evolution the surface is   
repelled apart from the origin where it is very close to   
the origin but moves relatively slowly at a large distance   
 from the origin. As a sphere of large radius is nearly flat   
near a fixed point this destroys convexity.   
  
A more elaborated example will show that starshapedness with   
respect to the origin is not preserved, too. We describe our   
surface in polar coordinates as follows. The height above the   
unit sphere is given by a positive constant times the    
characteristic function of a small geodesic ball around a   
 fixed point. The example is obtained if we smooth out this   
situation slightly. Figure 1 shows a cross-section of    
this hypersurface.   
  
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\begin{center}  
 Figure 1: Starshaped example  
\end{center}  
  
Rotating it around the horizontal   
axis gives the whole hypersurface. If the neck is thin, $n-1$   
principal curvatures become very large there. The remaining   
negative principal curvature there is small compared to the   
others when the neck becomes small. Of course the non-spherical   
part has to be made longer simultaneously so that the curvature   
near the tip remains bounded. This construction yields for   
a small potential compared to the mean curvature that the   
motion of the surface is especially large at the neck where   
we assume that it shrinks so fast that not only the    
starshapedness is lost but also the evolution will become   
singular in finite time.   
   
If we represent the hypersurface as ${\rm graph\,}   
\left.u\right|_{S^n}$ (this is possible for starshaped   
hypersurfaces with respect to the origin), then the time  
derivative of $\log u$ equals $-\frac{f}{\left\langle  
F,\nu\right\rangle}$. The change of coordinates  
$u\mapsto\log u$, however, is natural as the induced  
metric of the hypersurface in this new coordinates is very  
similar to the induced metric for graphs in Euclidean space,  
in fact, this shows the conformal equivalence of $S^n\times\real{}$ 
and $\real{n+1}\setminus\{0\}$. More details can be found in 
the appendix.  
  
Another interesting feature of our flow can be obtained for  
$\phi=1$, a case which is not considered in our main theorem.  
Then the stationary solutions are characterized by  
$H=\langle F,\nu\rangle$ and these hypersurfaces shrink   
homothetically under the evolution of the mean curvature   
 flow $\frac{d}{dt}F=-H\nu$.  
  
We consider a potential with appropriate asymptotics as for   
example the potential induced by $\eta(s)=\frac{2}{n}\frac{1}{s}$.  
Then there are two possibilities to reduce the energy; the surface   
may move apart from the source of the potential or it may    
contract and thus reduce its area and its energy as the charge   
is assumed to be proportional to the area. When the surface    
encloses the origin, i.\ e.\ the source of the potential,   
a charged point, these two effects are opposite to each other and   
so it is reasonable to conjecture that the surface tends to a   
sphere for which both forces compensate each other when the   
topology allows this.   
  
We wish to mention some further papers on related problems.  
In \cite{Gerhardt}, \cite{Ivochkina et al.}, \cite{Smoczyk 2}  
and \cite{Urbas} the evolution of starshaped hypersurfaces for  
various curvature driven flow equations has been considered. 
In contrast to outward directed flows, where starshapedness  
usually is preserved, this fails for inward directed flows like 
the mean curvature flow. For these flows convexity is naturally 
preserved \cite{Huisken}. But even for inward directed flows 
we have nice properties for starshaped hypersurfaces \cite{Smoczyk 2}. 
 
The paper is organized as follows:  
In section \ref{Euler-Lagrange equations}   
we introduce notations from differential   
geometry and compute the Euler-Lagrange equations, in  
section \ref{Inclusion principle}  
we show how spheres can be used as barriers for our flow. In  
section \ref{Convergence to a stable sphere}  
we derive evolution equations for geometric  
quantities, deduce a priori estimates and prove the smooth   
convergence to a stable sphere. Finally, we state   
additional properties of our flow in the appendix.  
 
We wish to thank J\"urgen Jost and the Max Planck Institute for 
Mathematics in the Sciences (MPI MIS), Leipzig, for their hospitality.   
\newpage
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
\section{Euler-Lagrange equations}\label{Euler-Lagrange equations} 
We are interested in the first and second variation of $V$. To this end  
assume that $F_t:(-\epsilon,\epsilon)\times M\to\real{n+1}$ is a smooth  
 family of immersions of orientable hypersurfaces such that  
\begin{equation}\nonumber  
\dt F_t=-f\nu,  
\end{equation}  
with a smooth function $f$ depending on $t\in(-\epsilon,\epsilon)$ and  
$\nu$ denoting the outward unit normal.  
We also recall the Gau\ss\ formula, the 
equations of Gau\ss, Weingarten, Codazzi and Simons 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bp\label{Gauss Weingarten Codazzi Simons} 
\begin{eqnarray}  
\dddd Rijkl&=&\dd hik\dd hjl-\dd hil\dd hjk,\label{gauss}\\  
\cdd Fij&=&-\dd hij\nu,\label{weingarten}\\  
\cd{\nu}i&=&\du hil\cd Fl,\label{weingarten 2}\\  
\cd{\dd hij}k&=&\cd{\dd hik}j,\label{codazzi}\\  
\cdd Hij&=&\lap\dd hij-H\du hil\dd hlj+\vert A\vert^2\dd hij,\label{simons}\\  
2\uu hij\cdd Hij&=&\lap\vert A\vert^2-2\vert\nabla A\vert^2-2Z. 
\label{simons 2}  
\end{eqnarray}  
\ep 
Here $\dd hij$ is the second fundamental form  
and  
\begin{eqnarray}  
H&=&\uu gij\dd hij,\nonumber\\  
\vert A\vert^2&=&\uu gij\uu gkl\dd hik\dd hjl,\nonumber\\  
C&=&\uu gij\uu gkl\uu gst\dd hik\dd hjs\dd hlt,\nonumber\\ 
Z&=&HC-\vert A\vert^4.\nonumber 
\end{eqnarray}   
$\uu gij$ is the inverse of the induced metric   
\begin{equation}\nonumber  
\dd gij:=\left\langle\part F{}xi\, ,\, \part F{}xj\right\rangle.  
\end{equation}  
Doubled indices are always summed from $1$ to $n$, 
indices are raised and lowered with respect to the induced 
metric and $\nabla$ denotes  
the Levi-Civita connection on $M$ w.\ r.\ t.\ $\dd gij$.  
In the sequel we won't distinguish between vectors $V\in T_xM$ and 
$DF(V)\in T_{F(x)}\real{n+1}$ and we also use $\langle\cdot, \cdot\rangle$ 
both for the scalar product on $\real{n+1}$ and on $M$. 
The calculations in \cite{Smoczyk} give  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bp\:  
\begin{eqnarray}  
\dt\dd gij&=&-2f\dd hij~,\label{evol metric}\\  
\dt d\mu&=&-fHd\mu~,\label{evol vol}\\  
\dt\nu&=&DF(\nabla f)=\uu gij\cd fi\cd Fj~,\label{evol normal}\\  
\dt\dd hij&=&\cdd fij-f\du hik\dd hkj~,\label{evol second}\\  
\dt H&=&\lap f+f\vert A\vert^2~,\label{evol mean}\\  
\dt\vert A\vert^2&=&2\uu hij\cdd fij+2fC.\label{evol A2}  
\end{eqnarray}  
\ep  
In addition we get  
\begin{equation}\label{evol v}  
\dt v=-\frac{n\eta}{2s}v\dt s=f\phi\langle F,\nu\rangle v. 
\end{equation} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bl\: \label{lemma variation}  
The first and second variation for $V$ and compactly supported  
$f\in C_0^\infty(M)$ are given by  
\begin{equation}\nonumber  
\dt V=-\int_M fv(H-\phi\langle F,\nu\rangle)d\mu,  
\end{equation}  
\begin{eqnarray}  
\frac{d^2}{dt^2}V&=&-\int_M(H-\phi\langle F,\nu\rangle)\dt(fvd\mu)  
\nonumber\\  
&+&\int_Mv\bigl(\vert\nabla f\vert^2-f^2(\vert A\vert^2+\phi 
+\phi'\langle F,\nu  
\rangle^2)\bigr)d\mu.\nonumber  
\end{eqnarray}  
\el  
\begin{proof}  
The first variation formula is a direct consequence of (\ref{evol vol}) and  
(\ref{evol v}). For the second variation we use (\ref{evol normal}),  
(\ref{evol mean}) and compute  
\begin{eqnarray}  
-vf\dt(H-\phi\langle F,\nu\rangle)  
&=&-vf\bigl(  
\lap f+f\vert A\vert^2+f\phi'\langle F,\nu\rangle^2+f\phi-\phi 
\langle \nabla s,  
\nabla f\rangle\bigr)\nonumber\\  
&=&-\text{div}(vf\nabla f)+v\bigl(  
\vert\nabla f\vert^2-f^2(\vert A\vert^2+\phi+\phi'\langle F,\nu\rangle^2)  
\bigr)\nonumber  
\end{eqnarray}  
and the second variation formula follows from partial integration.  
\end{proof}   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bc\label{cor stability}\:  
The Euler-Lagrange equation for stationary hypersurfaces w.\ r.\ t.\  
$V$ is given by  
\begin{equation}\nonumber  
H-\phi\langle F,\nu\rangle=0.  
\end{equation}  
A stationary hypersurface is stable (resp. strictly stable) if and only if  
\begin{equation}\nonumber  
\int_M\bigl(\vert\nabla f\vert^2-f^2(\vert A\vert^2+\phi+\phi'\langle F,\nu  
\rangle^2)\bigr)vd\mu\ge 0 ~~(\text{resp.~}>0)  
\end{equation}  
 for all smooth, compactly supported $f\not\equiv 0$.  
\ec  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bc\:   
Assume that $s_0$ satisfies   
$\eta(s_0)=1$   
and that   
$\eta'(s_0)<0$.   
Then the sphere with center at the origin and radius   
$r_0=\sqrt{2s_0}$  
is strictly stable.  
\ec  
\begin{proof}   
Left to the reader.  
\end{proof}   
The negative gradient flow for $V$ is given by  
$\dt F=-v(H-\phi\langle F,\nu\rangle)\nu$. Since this is stationary  
if and only if $H-\phi\langle F,\nu\rangle=0$ we can as well consider  
the modified gradient flow  
\begin{equation}\label{evol F}  
\dt F=-f\nu  
\end{equation}  
with  
\begin{equation}\nonumber  
 f:=H-\phi\langle F,\nu\rangle.  
\end{equation}  
We will set $M_t:=F_t(M)$. Then the results in \cite{Huisken Polden} imply  
\bp\: \label{shorttime existence}  
(\ref{evol F}) is a system of quasilinear parabolic equations and there  
exists a maximal time $0<T\le\infty$ such that (\ref{evol F}) admits  
a smooth solution on $[0,T)$.   
\ep 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
\section{Inclusion principle}\label{Inclusion principle} 
\def\graph{{\rm graph\,}}   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bl\label{F remains embedded}   
Assume that the initial value $F_0$ of a smooth solution   
$F_t$, $0\le t<T$, of the flow equation (\ref{evol F})   
is embedded. Then $F_t$, $0<t<T$, is also embedded.   
\el    
\begin{proof}   
Let $t_0$ be the first time when the immersion fails   
to be an embedding, $F_{t_0}(x_0)=F_{t_0}(y_0)$,    
$x_0\not=y_0$. We choose a local coordinate system    
such that $F_{t_0}(x_0)=F_{t_0}(y_0)=0$ and the unit   
normals at the origin are $\pm e_{n+1}$. So we may   
write $F_{t_0}$ around $x_0$ and $y_0$ as    
$\graph (u^1)$ and $\graph (u^2)$, respectively.   
The representation as a graph is possible for a small   
time interval $[t_0-\varepsilon,t_0+\varepsilon]$,   
$\varepsilon>0$, too. We may assume that $u^1>u^2$ for   
$t<t_0$. If we replace $\nu$ by $-\nu$ in the flow    
equation, $H$, $\langle F,\nu\rangle$ and $\nu$ change   
sign, so we may assume that the normals at $(x_0,t_0)$   
and $(y_0,t_0)$ both equal $e_{n+1}$.   
We rewrite the geometric evolution equation as a   
parabolic evolution equation for $u^1$ and $u^2$     
and obtain   
\begin{equation}   
\frac{\partial}{\partial t}(u^1-u^2)=   
a^{ij}(u^1_{ij}-u^2_{ij})+b^i(u^1_i-u^2_i)+c(u^1-u^2).   
\end{equation}   
We remark that this equation is parabolic as $H$ is   
elliptic for every smooth solution, i.\ e.\ especially  
 for any function $\tau u^1+(1-\tau)u^2$, $0\le\tau\le1$.   
 From the strong maximum principle we deduce, that $u^1$ and   
$u^2$ coincide locally and furthermore, that these two    
 functions have locally to be equal before $t=t_0$. This   
contradicts our assumption about $t_0$.   
\end{proof}   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bc\label{Inclusion remains}   
Let $F$ be a smooth immersed solution of (\ref{evol F})   
and $\tilde F$ be an immersed solution of this    
evolution equation. If $\tilde F$ is contained in    
a connected component of $\real{n+1}\setminus F$ or in the    
closure of such a component at the beginning of the    
evolution, then this remains true during the   
evolution.   
\ec   
\begin{proof}   
As the geometric situation at the first point of contact is   
similar to the situation in the proof of Lemma   
\ref{F remains embedded}, we can argue as we have done there.  
If $F$ and $\tilde F$ touch for $t=0$ then the strong maximum  
principle yields that they are disjoint at least for small  
positive times unless there are connected components of $F$  
and $\tilde F$ which coincide for $t=0$.   
\end{proof}   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
The following barrier argument will become important in the sequel. 
\bl\label{barrier} 
Assume $s_-,s_0,s_+$ are positive numbers such that $s_-<s_0<s_+$ and that 
$\eta$ satisfies 
\begin{eqnarray} 
\eta(s_0)&=&1,\nonumber\\ 
\eta'(s)&<&0\quad\forall\, s\in[s_-,s_+].\nonumber 
\end{eqnarray} 
Then under our flow 
all centered spheres of radius $r=\sqrt{2s}$ with $s\in[s_-,s_+]$ 
converge to the stable sphere with radius $r_0=\sqrt{2s_0}$. 
Moreover, if $M_t, t\ge 0$, is a smooth 
evolution of hypersurfaces such that $M_0$ 
is contained in  
\begin{equation}\nonumber 
\{y\in\real{n+1}\vert r_-:=\sqrt{2s_-} 
<\vert y\vert<r_+:=\sqrt{2s_+}\}, 
\end{equation} 
then 
$M_t$ will be contained in 
\begin{equation}\nonumber 
\{y\in\real{n+1}\vert r_-(t)<\vert y\vert<r_+(t)\}, 
\end{equation} 
where $r_-(t)$ and  
$r_+(t)$ are the radii for the evolutions of the inner and outer sphere.   
If $M_t$ exist for all $t\ge 0$, then $M_t$ converges pointwise to the 
stable sphere of radius $r_0$. 
\el 
\begin{proof} 
The first part is an easy consequence of 
\begin{equation}\label{20} 
\dt s=n(\eta-1) 
\end{equation} 
which holds for centered spheres, see the independently proven Lemma 
\ref{lemma evol s}. The remainder then follows from the  
inclusion principle \ref{Inclusion remains}. 
\end{proof} 
\brem
{\rm
 From (\ref{20}), $\eta(s_0)=1$ and $\eta'(s_0)<0$ we deduce that the
convergence in Lemma \ref{barrier} is even exponentially.
}\erem


   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
\section{Convergence to a stable sphere}\label{Convergence to a stable sphere}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bl\label{lemma evol f}\:   
The evolution equation for $f$ is  
\begin{equation}\nonumber  
\dt f=\lap f-\phi\langle \nabla s,\nabla f\rangle+f(\vert A\vert^2+\phi+\phi'  
\langle F,\nu\rangle^2).  
\end{equation}  
\el  
\begin{proof}  
This has been calculated in the proof of Lemma \ref{lemma variation}.  
\end{proof}  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bl\label{lemma evol s}\:  
$s$ satisfies  
\begin{equation}\nonumber  
\dt s=\lap s-\phi\vert\nabla s\vert^2+2s\phi-n,  
\end{equation}  
\el  
\begin{proof}  
 First we compute  
\begin{equation}\nonumber  
\cd si=\langle F,F_i\rangle  
\end{equation}  
with $F_i:=\part F{}xi$. Then the Gau\ss\ formula implies  
\begin{equation}\nonumber  
\cdd sij=\dd gij-\langle F,\nu\rangle\dd hij.  
\end{equation}  
Therefore  
\begin{equation}\nonumber  
\lap s=n-H\langle F,\nu\rangle=n-f\langle F,\nu\rangle  
-\phi\langle F,\nu\rangle^2.  
\end{equation}  
On the other hand  
\begin{equation}\nonumber  
\dt s=\langle F,\dt F\rangle=-f\langle F,\nu\rangle  
\end{equation}  
and by writing $F=DF(\nabla s)+\fnu \nu$  
\begin{equation}\nonumber  
-\phi\vert\nabla s\vert^2  
=-\phi(2s-\langle F,\nu\rangle^2).  
\end{equation}  
The combination gives  
\begin{eqnarray}  
\dt s  
&=&  
-f\langle F,\nu\rangle  
+\lap s-n+f\langle F,\nu\rangle+\phi\langle F,\nu\rangle^2\nonumber\\  
&-&\phi\vert\nabla s\vert^2+\phi(2s-\langle F,\nu\rangle^2)\nonumber\\  
&=&\lap s-\phi\vert\nabla s\vert^2+2s\phi-n.\nonumber  
\end{eqnarray}  
\end{proof}  
Next we want to compute the evolution of $\langle F,\nu\rangle$.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bl\label{lemma evol Fnu}\:  
$\langle F,\nu\rangle$ satisfies  
\begin{eqnarray}  
\dt\langle F,\nu\rangle&=&\lap\langle F,\nu\rangle-\phi\langle \nabla s,  
\nabla\langle F,\nu\rangle\rangle+\langle F,\nu\rangle(\vert A\vert^2+\phi  
+\phi'\langle F,\nu\rangle^2)\nonumber\\  
&-&2H-2s\phi'\langle F,\nu\rangle\nonumber  
\end{eqnarray}  
\el  
\begin{proof}  
We compute  
\begin{equation}\nonumber  
\cd{\langle F,\nu\rangle}i=\du hil\cd sl  
\end{equation}  
and  
\begin{equation}\nonumber  
\cdd{\langle F,\nu\rangle}ij=\cu{\dd hij}l\cd sl+\dd hij-\langle F,\nu\rangle  
\du hil\dd hlj,  
\end{equation}  
where we used the Codazzi equation (\ref{codazzi}).  
This gives  
\begin{equation}\nonumber  
\lap\langle F,\nu\rangle=\langle \nabla s,\nabla H\rangle+H  
-\vert A\vert^2\langle F,\nu\rangle.  
\end{equation}  
On the other hand with (\ref{evol normal}) and (\ref{evol F}) we conclude 
\begin{equation}\nonumber  
\dt\langle F,\nu\rangle=-f+\langle F,DF(\nabla f)\rangle=-f+\langle \nabla s, 
\nabla f\rangle 
\end{equation}  
so that  
\begin{equation}\nonumber  
\dt\langle F,\nu\rangle=\lap\langle F,\nu\rangle-\langle \nabla s, 
\nabla (H-f)\rangle  
-H-f+\vert A\vert^2\langle F,\nu\rangle.  
\end{equation}  
Since  
\begin{equation}\label{nabla H-f}  
\nabla(H-f)=\langle F,\nu\rangle\phi'\nabla s+\phi\nabla\langle F,\nu\rangle  
\end{equation}  
and   
\begin{equation}\nonumber  
\vert\nabla s\vert^2=2s-\langle F,\nu\rangle^2  
\end{equation}  
we obtain the result.  
\end{proof}  
We want to rewrite equations (\ref{evol mean}) and (\ref{evol A2}). To  
this end we need an expression for $\cdd{(f-H)}ij$. From   
(\ref{nabla H-f}) we obtain  
\begin{eqnarray}  
\cdd{(f-H)}ij  
&=&-\cd{(\langle F,\nu\rangle\phi'\cd sj+\phi\cd{\langle F,\nu\rangle}j)}i  
\nonumber\\  
&=&-\cd{\bigl((\langle F,\nu\rangle\phi'\du\delta jl+\phi\du hjl) 
\cd sl\bigr)}i\nonumber\\  
&=&  
-(\du hik\cd sk\phi'\du\delta jl+\langle F,\nu\rangle\phi''\cd si\du\delta jl  
+\phi'\cd si\du hjl+\phi\cu{\dd hij}l)\cd sl\nonumber\\  
&-&(\langle F,\nu\rangle\phi'\du\delta jl+\phi\du hjl)  
(\dd gil-\langle F,\nu\rangle\dd hil)\nonumber\\  
&=&-\phi'(\du hil\cd sj+\du hjl\cd si)\cd sl-\phi''\langle F,\nu\rangle  
\cd si\cd sj-\phi\cu{\dd hij}l\cd sl\nonumber\\  
&-&(\phi-\phi'\langle F,\nu\rangle^2)\dd hij+\phi\langle F,\nu\rangle  
\du hil\dd hlj-\phi'\langle F,\nu\rangle\dd gij.\label{nablanabla}  
\end{eqnarray}  
Then  
\begin{eqnarray}  
\lap(f-H)  
&=&-2\phi'\uu hij\cd si\cd sj-\phi''\langle F,\nu\rangle  
(2s-\langle F,\nu\rangle^2)-\phi\langle \nabla s,\nabla H\rangle\nonumber\\  
&-&(\phi-\phi'\langle F,\nu\rangle^2)H+\phi\langle F,\nu\rangle\vert A\vert^2  
-n\phi'\langle F,\nu\rangle.\nonumber  
\end{eqnarray}  
Now (\ref{evol mean}) implies  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bl\label{lemma evol mean}\:  
\begin{eqnarray}  
\dt H  
&=&\lap H-\phi\langle \nabla s,\nabla H\rangle  
+H(\vert A\vert^2-\phi+\phi'\langle F,\nu\rangle^2)\nonumber\\  
&-&2\phi'\uu hij\cd si\cd sj+\langle F, 
  \nu\rangle(\phi''\langle F,\nu\rangle^2  
-2s\phi''-n\phi').\nonumber  
\end{eqnarray}  
\el  
(\ref{nablanabla}) also gives  
\begin{eqnarray}  
2\uu hij\cdd{(f-H)}ij  
&=&-4\phi'\uu hil\du hlj\cd si\cd sj-2\phi''\langle F,\nu\rangle\uu hij\cd si  
\cd sj\nonumber\\  
&-&\phi\langle \nabla s,\nabla\vert A\vert^2\rangle  
-2\vert A\vert^2(\phi-\phi'\langle F,\nu\rangle^2)\nonumber\\  
&+&2\phi\langle F,\nu\rangle C-2\phi'\langle F,\nu\rangle H.\nonumber  
\end{eqnarray}  
 From (\ref{evol A2}) and Simons' identity (\ref{simons 2}) we then derive  
\begin{eqnarray}  
\dt\vert A\vert^2  
&=&2\uu hij\cdd fij+2fC=2\uu hij\cdd Hij+2\uu hij\cdd{(f-H)}ij+2fC\nonumber\\  
&=&\lap\vert A\vert^2-2\vert\nabla A\vert^2-2Z\nonumber\\  
&-&4\phi'\uu hil\du hlj\cd si\cd sj-2\phi''\langle F,\nu\rangle\uu hij\cd si  
\cd sj-\phi\langle \nabla s,\nabla\vert A\vert^2\rangle\nonumber\\  
&-&2\vert A\vert^2(\phi-\phi'\langle F,\nu\rangle^2) 
+2\phi\langle F,\nu\rangle  
C-2\phi'\langle F,\nu\rangle H+2fC.\nonumber  
\end{eqnarray}  
Cancellation gives  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bl\label{lemma evol A2}\:  
\begin{eqnarray}  
\dt\vert A\vert^2  
&=&\lap\vert A\vert^2-\phi\langle \nabla s,\nabla\vert A\vert^2\rangle  
-2\vert\nabla A\vert^2\nonumber\\  
&-&4\phi'\uu hil\du hlj\cd si\cd sj-2\phi''\langle F,\nu\rangle\uu hij\cd si  
\cd sj\nonumber\\  
&+&2\vert A\vert^2(\vert A\vert^2-\phi+\phi'\langle F,\nu\rangle^2)  
-2\phi'\langle F,\nu\rangle H.\nonumber  
\end{eqnarray}  
\el  
In the next steps we want to prove that our spheres remain starshaped 
and that (\ref{main con 1}) remains true under the evolution. 
Therefore we define the quantity 
\begin{equation}\nonumber 
q:=\frac{f}{\fnu}. 
\end{equation} 
A geometric motivation for $q$ will be given in the appendix. 
Then the evolution equations for $f$ and $\fnu$ imply 
\begin{eqnarray} 
\dt q 
&=&\frac{1}{\fnu}\bigl(\lap f-\phi\langle\nabla s,\nabla f\rangle 
+f(\vert A\vert^2+\phi+\phi'\fnu^2)\bigr)\nonumber\\ 
&-&\frac{f}{\fnu^2}\bigl(\lap\fnu-\phi\langle\nabla s,\nabla\fnu\rangle 
+\fnu(\vert A\vert^2+\phi+\phi'\fnu^2)\nonumber\\ 
&-&2H-2s\phi'\fnu\bigr).\label{prelim q1} 
\end{eqnarray} 
On the other hand we compute 
\begin{eqnarray} 
\lap q 
&=&\frac{\lap f}{\fnu}+f\bigl(-\frac{1}{\fnu^2}\lap\fnu+\frac{2}{\fnu^3}\vert 
\nabla\fnu\vert^2\bigr)\nonumber\\ 
&-&\frac{2}{\fnu^2}\langle\nabla f,\nabla\fnu\rangle\nonumber\\ 
&=&\frac{\lap f}{\fnu}-\frac{f}{\fnu^2}\lap\fnu-\frac{2}{\fnu}\langle 
\nabla\fnu,\nabla q\rangle\nonumber 
\end{eqnarray} 
Inserting this in (\ref{prelim q1}) yields 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bl\label{lemma evol: q} 
\begin{eqnarray} 
\dt q 
&=&\lap q+\frac{2}{\fnu}\langle\nabla\fnu,\nabla q\rangle 
-\phi\langle\nabla s,\nabla q\rangle+2q(q+s\phi'+\phi).\label{evol: q} 
\end{eqnarray} 
\el 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bc\label{cor: bound for q} 
Under the assumptions of Theorem \ref{main} there exists a positive constant 
$\epsilon>0$ independent of $t$ such that 
\begin{eqnarray} 
q+s\phi'+\phi&\le&-\epsilon,\label{cor bound 1}\\ 
q^2&\le&(\max_{t=0}\, q^2)e^{-4\epsilon t}\label{cor bound 2} 
\end{eqnarray} 
as long as a smooth starshaped solution of (\ref{main}) exists. 
\ec 
\begin{proof} 
 From Lemma \ref{lemma evol: q} we obtain 
\begin{eqnarray} 
\dt q^2 
&=&\lap q^2-2\vert\nabla q\vert^2 
+\frac{2}{\fnu}\langle\nabla\fnu,\nabla q^2\rangle 
-\phi\langle\nabla s,\nabla q^2\rangle\nonumber\\ 
&+&4q^2(q+s\phi'+\phi). 
\end{eqnarray} 
Since $s\phi'+\phi=\frac{n}{2}\eta'$, the barrier argument, Lemma 
\ref{barrier}, and inequality (\ref{main con 2}) imply that 
\begin{equation}\nonumber 
s\phi'+\phi<-c_\eta 
\end{equation} 
as long as $S_t$ remains smooth. Consequently 
\begin{eqnarray} 
\dt q^2 
&\le&\lap q^2-2\vert\nabla q\vert^2 
+\frac{2}{\fnu}\langle\nabla\fnu,\nabla q^2\rangle 
-\phi\langle\nabla s,\nabla q^2\rangle\nonumber\\ 
&+&4q^2(q-c_\eta). 
\end{eqnarray} 
Hence the maximum principle and (\ref{main con 1}) imply that 
$q^2$ cannot develop a positive  
maximum with $\dt q^2>0$ and $q-c_\eta$ remains 
bounded above by
its initial (negative) maximum. This proves (\ref{cor bound 1}) with 
\begin{equation}\nonumber 
\epsilon:=c_\eta-\max_{t=0}\, q. 
\end{equation} 
But then the evolution equation for $q^2$ and (\ref{cor bound 1}) even 
imply that $q^2\, e^{4\epsilon t}$ remains bounded by its initial maximum. 
\end{proof} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bl\label{lemma starshapedness} 
Under the assumptions of Theorem \ref{main} the starshapedness of $S_t$ 
remains true as long as a smooth solution of (\ref{main}) exists.  
\el 
\begin{proof} 
We compute the evolution for the quantity 
\begin{equation}\nonumber 
\frac{1}{\fnu^2} 
\end{equation} 
which remains bounded above if and only if $S_t$ remains starshaped. 
 From Lemma \ref{lemma evol Fnu} we get 
\begin{eqnarray} 
\dt\frac{1}{\fnu^2} 
&=&-\frac{2}{\fnu^3}\bigl(\lap\fnu-\phi\langle\nabla s,\nabla\fnu\rangle 
\nonumber\\ 
&+&\fnu(\vert A\vert^2+\phi+\phi'\fnu^2)-2H-2s\phi'\fnu\bigr)\nonumber\\ 
&=&\lap\frac{1}{\fnu^2}-\frac{6}{\fnu^4}\vert\nabla\fnu\vert^2 
 -\phi\left\langle 
\nabla s,\nabla\frac{1}{\fnu^2}\right\rangle\nonumber\\ 
&+&\frac{2}{\fnu^2}(-\vert A\vert^2-\phi-\phi'\fnu^2+2(q+s\phi'+\phi)). 
\label{evol: fnu-2} 
\end{eqnarray} 
Now as long as $\fnu$ remains positive we can use  
(\ref{cor bound 1}) and estimate 
\begin{eqnarray} 
\dt\frac{1}{\fnu^2} 
&\le&\lap\frac{1}{\fnu^2}-\phi\left\langle 
\nabla s,\nabla\frac{1}{\fnu^2}\right\rangle 
+\frac{2}{\fnu}\left\langle\nabla\fnu, 
\nabla\frac{1}{\fnu^2}\right\rangle\nonumber\\ 
&+&\frac{2}{\fnu^2}(-\vert A\vert^2-\phi-2\epsilon)-2\phi'.\nonumber 
\end{eqnarray} 
The barrier argument, Lemma  \ref{barrier}, implies  
\begin{eqnarray} 
\phi(s(x,t)) 
&\ge&\min_{\sigma\in\left[\, \min\left\{\min\limits_{t=t_0} 
  \frac{\vert F\vert^2}{2}, s_0\right\}\, ,\, 
\max\left\{\max\limits_{t=t_0}\frac{\vert F\vert^2}{2}, s_0\right\}\,  
  \right]} 
\phi(\sigma)=:c_{\phi, t_0}, 
\label{bound phi}\\ 
\phi'(s(x,t)) 
&\ge&\min_{\sigma\in\left[\,  
\min\left\{\min\limits_{t=t_0}\frac{\vert F\vert^2}{2} 
, s_0\right\}\, ,\, 
\max\left\{\max\limits_{t=t_0}\frac{\vert F\vert^2}{2}, s_0\right\}\,  
\right]} 
\phi'(\sigma)=:c_{\phi', t_0} 
\label{bound phi'} 
\end{eqnarray} 
 for all $x\in S^n$ and all $t\ge t_0$, where $t_0\ge 0$ is a fixed time. 
 For the proof of this lemma we will actually only use $t_0=0$ but we need
the same estimates for a different $t_0$ later.
Hence for all $t\ge t_0$ the following inequality is valid for starshaped 
$S_t$ 
\begin{eqnarray} 
\dt\frac{1}{\fnu^2} 
&\le&\lap\frac{1}{\fnu^2}-\phi\left\langle 
\nabla s,\nabla\frac{1}{\fnu^2}\right\rangle 
+\frac{2}{\fnu}\left\langle\nabla\fnu, 
\nabla\frac{1}{\fnu^2}\right\rangle\nonumber\\ 
&-&\frac{2(\vert A\vert^2 
+2\epsilon+c_{\phi,t_0})}{\fnu^2}-2c_{\phi',t_0}.\label{fnu est} 
\end{eqnarray} 
The last term in the first line is nonpositive but we keep it for later
purposes.
Let $p(t)$ be the solution of 
\begin{eqnarray} 
\dt p&=&-2(2\epsilon+c_{\phi,t_0})p-2c_{\phi',t_0},\nonumber\\ 
p(t_0)&=&\max\limits_{t=t_0}\frac{1}{\fnu^2},\nonumber 
\end{eqnarray} 
i.\ e.\ for $2\epsilon+c_{\phi,t_0}\not=0$ 
\begin{equation}\nonumber 
p(t)=\left(p(t_0)+\frac{c_{\phi',t_0}} 
{2\epsilon+c_{\phi,t_0}}\right) 
e^{-2(2\epsilon+c_{\phi,t_0})(t-t_0)} 
-\frac{c_{\phi',t_0}}{2\epsilon+c_{\phi,t_0}}. 
\end{equation} 
Then for all $t\ge t_0$ and starshaped $S_t$ 
\begin{eqnarray} 
\dt\left(\frac{1}{\fnu^2}-p\right) 
&\le&\lap\left(\frac{1}{\fnu^2}-p\right)-\phi\left\langle 
\nabla s,\nabla\left(\frac{1}{\fnu^2}-p\right)\right\rangle\nonumber\\ 
&-&2(2\epsilon+c_{\phi,t_0}) 
\left(\frac{1}{\fnu^2}-p\right).\nonumber 
\end{eqnarray} 
The maximum principle then implies 
\begin{equation}\label{starineq} 
\frac{1}{\fnu^2}\le\left(\max\limits_{t=t_0}\frac{1}{\fnu^2} 
+\frac{c_{\phi',t_0}}{2\epsilon+c_{\phi,t_0}}\right) 
e^{-2(2\epsilon+c_{\phi,t_0})(t-t_0)} 
-\frac{c_{\phi',t_0}}{2\epsilon+c_{\phi,t_0}}  
\end{equation} 
 for all $t\ge t_0$. Consequently the quantity $\fnu$ cannot tend to zero 
in finite time and $S_t$ remains starshaped. 
\end{proof} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\bt\label{longtime} 
With the same assumptions as in Theorem \ref{main} a smooth solution 
exists for all $t>0$. 
\et 
\begin{proof} 
We need bounds for $\vert\nabla^kA\vert^2$ for all $k\ge 0$. 
We begin with a bound for 
\begin{equation}\nonumber 
\frac{\vert A\vert^2}{\fnu^2}. 
\end{equation} 
 From Lemma \ref{lemma evol A2} and the evolution equation 
(\ref{evol: fnu-2}) we obtain 
\begin{eqnarray} 
\dt\frac{\vert A\vert^2}{\fnu^2} 
&=&\vert A\vert^2 
\bigl(\lap\frac{1}{\fnu^2}-\frac{6}{\fnu^4}\vert\nabla\fnu\vert^2 
-\phi\left\langle 
\nabla s,\nabla\frac{1}{\fnu^2}\right\rangle\nonumber\\ 
&+&\frac{2}{\fnu^2}(-\vert A\vert^2-\phi-\phi'\fnu^2+2(q+s\phi'+\phi)) 
\bigr)\nonumber\\ 
&+&\frac{1}{\fnu^2}\bigl( 
\lap\vert A\vert^2-\phi\langle\nabla s,\nabla\vert A\vert^2\rangle  
-2\vert\nabla A\vert^2\nonumber\\  
&-&4\phi'\uu hil\du hlj\cd si\cd sj-2\phi''\langle F,\nu\rangle\uu hij\cd si  
\cd sj\nonumber\\  
&+&2\vert A\vert^2(\vert A\vert^2-\phi+\phi'\langle F,\nu\rangle^2)  
-2\phi'\langle F,\nu\rangle H 
\bigr)\nonumber\\ 
&=&\lap\frac{\vert A\vert^2}{\fnu^2} 
-2\left\langle\nabla\vert A\vert^2,\nabla 
\frac{1}{\fnu^2}\right\rangle 
-\phi\left\langle\nabla s,\nabla\frac{\vert A\vert^2} 
{\fnu^2}\right\rangle\nonumber\\ 
&-&2\frac{\vert\nabla A\vert^2}{\fnu^2}-6\frac{\vert A\vert^2\vert\nabla 
\fnu\vert^2}{\fnu^4}-4\phi'\frac{\vert\nabla\fnu\vert^2}{\fnu^2}\nonumber\\ 
&-&2\frac{\phi''}{\fnu}\uu hij\cd si\cd sj-2\phi'\frac{H}{\fnu} 
+4\frac{\vert A\vert^2}{\fnu^2}(q+s\phi').\nonumber 
\end{eqnarray} 
To proceed we define 
\begin{eqnarray} 
Q^2&:=&\vert\fnu\cd{\dd hjk}i-\cd\fnu i\dd hjk\vert^2\nonumber\\ 
&=&\fnu^2\vert\nabla A\vert^2+\vert A\vert^2\vert\nabla\fnu\vert^2 
-\fnu\langle\nabla\fnu,\nabla\vert A\vert^2\rangle.\nonumber 
\end{eqnarray} 
This implies 
\begin{eqnarray} 
&-&2\frac{\vert\nabla A\vert^2}{\fnu^2}-6\frac{\vert A\vert^2\vert\nabla 
\fnu\vert^2}{\fnu^4}-2\left\langle\nabla\vert A\vert^2,\nabla 
\frac{1}{\fnu^2}\right\rangle\nonumber\\ 
&=&-2\frac{Q^2}{\fnu^4}+\frac{2}{\fnu} 
\left\langle\nabla\fnu,\nabla\frac{\vert A 
\vert^2}{\fnu^2}\right\rangle\nonumber 
\end{eqnarray} 
and finally 
\begin{eqnarray} 
\dt\frac{\vert A\vert^2}{\fnu^2} 
&=&\lap\frac{\vert A\vert^2}{\fnu^2} 
-\phi\left\langle\nabla s,\nabla\frac{\vert A\vert^2} 
{\fnu^2}\right\rangle  
+\frac{2}{\fnu}\left\langle\nabla\fnu,\nabla\frac{\vert A 
\vert^2}{\fnu^2}\right\rangle\nonumber\\ 
&-&2\frac{Q^2}{\fnu^4}-4\phi'\frac{\vert\nabla\fnu\vert^2}{\fnu^2} 
-2\frac{\phi''}{\fnu}\uu hij\cd si\cd sj\nonumber\\ 
&-&2\phi'\frac{H}{\fnu} 
+4\frac{\vert A\vert^2}{\fnu^2}(q+s\phi').\label{eq prelim 1} 
\end{eqnarray} 
To proceed we need the estimate 
\begin{eqnarray} 
\vert\nabla\fnu\vert^2 
&=&\ud hil\uu hlj\cd si\cd sj\nonumber\\ 
&\le&\vert A\vert^2\vert\nabla s\vert^2\nonumber\\ 
&\le&2s\vert A\vert^2.\nonumber 
\end{eqnarray} 
Moreover 
\begin{eqnarray} 
\left|\frac{H}{\fnu}\right| 
&\le&\frac{H^2}{2\fnu^2}+\frac{1}{2}\nonumber\\ 
&\le&\frac{n\vert A\vert^2}{2\fnu^2}+\frac{1}{2}\nonumber. 
\end{eqnarray} 
Then we use the barrier argument, Lemma \ref{barrier}, inequality
(\ref{cor bound 1}) 
and estimate 
\begin{eqnarray} 
\dt\frac{\vert A\vert^2}{\fnu^2} 
&\le&\lap\frac{\vert A\vert^2}{\fnu^2} 
-\phi\left\langle\nabla s,\nabla\frac{\vert A\vert^2} 
{\fnu^2}\right\rangle  
+\frac{2}{\fnu}\left\langle\nabla\fnu,\nabla\frac{\vert A 
\vert^2}{\fnu^2}\right\rangle\nonumber\\ 
&+&c_1\frac{\vert A\vert^2}{\fnu^2}+c_2\label{estimate} 
\end{eqnarray} 
 for two positive constants $c_1, c_2$ independent of $t$. 
So we see that $\frac{|A|^2}{\fnu^2}$ can increase at most  
exponentially in time and since by Lemma  
\ref{lemma starshapedness} $S_t$ remains starshaped and 
$\vert\langle F,\nu\rangle\vert\le \text{const}$,
this also means that $\vert A\vert^2$ remains bounded on any 
 finite time interval. We can then proceed as in 
\cite{Huisken} to derive uniform upper 
bounds for all higher covariant derivatives of $A$ 
on any finite time interval $[0, \t T)$ for which a smooth solution of 
the flow exists. Consequently $T=\infty$. 
\end{proof}  
 
\begin{rproof} 
We are now able to prove uniform upper bounds in $t$ also. First we need 
a uniform upper bound for $\frac{1}{\fnu^2}$. Therefore we go back  
to inequality (\ref{starineq}). From Theorem \ref{longtime} we know that 
the flow exists for all $t>0$ and since by assumption (\ref{main con 3}) 
$\phi(s_0)=\frac{n\eta(s_0)}{2s_0} 
=\frac{n}{2s_0}>0$, the barrier argument, Lemma 
\ref{barrier}, implies the existence of $t_0\ge 0$  
estimated from above such that 
$c_{\phi,t_0}>0$. 
But then inequality (\ref{starineq}) 
implies that with a constant $c_3>0$ and for all $t>0$ 
\begin{equation}\label{uniform 1} 
\frac{1}{\fnu^2}\le c_3. 
\end{equation} 
Our idea is to add enough of $\frac{1}{\fnu^2}$ to 
$\frac{\vert A\vert^2}{\fnu^2}$ to 
obtain a uniform bound for $\vert A\vert^2$. Therefore, let 
\begin{equation}\nonumber 
B:=\frac{\vert A\vert^2+k}{\fnu^2} 
\end{equation} 
 for a large constant $k$ to be determined. 
Then (\ref{fnu est}) and (\ref{estimate}) give 
\begin{eqnarray} 
\dt B 
&\le&\lap B-\phi\langle\nabla s,\nabla B\rangle  
+\frac{2}{\fnu}\langle\nabla\fnu,\nabla B\rangle\nonumber\\ 
&+&(c_1-2k)\frac{\vert A\vert^2}{\fnu^2} 
-k\left(\frac{2(2\epsilon+c_{\phi,t_0})} 
{\fnu^2}+2c_{\phi',t_0}\right)+c_2 
\end{eqnarray} 
 for all $t\ge t_0$, with a fixed $t_0$, without loss of
generality $t_0=0$. We choose $k=c_1$. 
Then 
\begin{eqnarray} 
\dt B 
&\le&\lap B-\phi\langle\nabla s,\nabla B\rangle  
+\frac{2}{\fnu}\langle\nabla\fnu,\nabla B\rangle\nonumber\\ 
&-&c_1B+c_4, 
\end{eqnarray} 
where $c_4$ depends only on $c_1, c_2, c_3, \epsilon, c_{\phi,0}$ and 
$c_{\phi',0}$ but not on $t$. Consequently a maximum of $B$ 
with $\dt B>0$ 
must be smaller than $\frac{c_4}{c_1}$ and furthermore 
$B$ and $\vert A\vert^2$ 
are uniformly bounded. 
Once we've obtained a uniform bound for $\vert A\vert^2$ we use the technique 
in \cite{Huisken} to obtain uniform upper bounds for all quantities 
$\vert\nabla^kA\vert^2$. 
This shows that $F_t$ is uniformly bounded in $C^\infty$ 
 for all $t\ge0$. From the 
barrier argument, Lemma \ref{barrier},
we conclude that $\vert s-s_0\vert$ decays exponentially. Then we use
the elementary interpolation inequality
\begin{equation}\nonumber
||\nabla\psi||^2\le \text{const}\bigl(\t M,\t g\bigr)\cdot
||\psi||\cdot\left(||\nabla\psi||+||\nabla^2\psi||\right),
\end{equation}
 for $C^2$-functions $\psi$ on a compact Riemannian
manifold $(\t M, \t g)$, where $\nabla$ denotes the covariant derivative
w.\ r.\ t.\ $\t g$ and the constant depends only on $\t M$ and $\t g$.
This inequality, the $C^k$-bounds for $s-s_0$ (which follow from those for the
second fundamental form) and the uniform equivalence of the induced metrics
on $M=S^n$ for any $t$
imply an exponential decay of 
$\vert\nabla^k(s-s_0)\vert^2$ for all $k\in\natural{}$.
We thus conclude exponential convergence in $C^\infty$ to the 
stable sphere of radius $r_0=\sqrt{2s_0}$. 
\end{rproof}  
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
\section{Appendix}\label{Appendix} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\bc\:  
Assume $f>0$ for $~t=0$.  
If the absolute value of  
$~\vert A\vert^2+\phi+\phi'\langle F,\nu\rangle^2$  
remains bounded from above 
by some positive constant $c$ on the time interval  
$[0,t_0), t_0\le T$, then for $t\in[0,t_0)$  
\begin{equation}\nonumber  
\min_{x\in M_t}f(x)\ge \min_{x\in M_0}f(x)e^{-ct}>0.  
\end{equation}  
\ec  
\begin{proof}  
This is a direct consequence of the parabolic maximum principle 
applied to Lemma \ref{lemma evol f}.  
\end{proof}   
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
\subsection{A geometric motivation for $\frac{f}{\fnu}$} 
 
We wish to represent our evolution problem via graphs over the 
sphere $S^n$ as in \cite{Gerhardt}. Therefore we use an embedding of the form 
\begin{equation} 
 F=x\left(\xi^i,t\right)\cdot u\left(x\left(\xi^i,t\right),t\right), 
\nonumber 
\end{equation} 
where $x\in S^n$ and $\xi^i$ denotes local coordinates of $S^n$. 
We use covariant derivatives with respect to the metric of $S^n$ 
and compute 
\begin{eqnarray} 
 F_i&=&x_iu+xu_i,\nonumber\\ 
 F_{ij}&=&x_{ij}u+x_iu_j+x_ju_i+xu_{ij},\nonumber\\ 
g_{ij}&=&\langle F_i,F_j\rangle=u^2\sigma_{ij}+u_iu_j\nonumber\\ 
&=&u^2(\sigma_{ij}+\varphi_i\varphi_j),\quad\mbox{ where } 
  \varphi=\log u,\nonumber\\ 
g^{ij}&=&u^{-2}\left(\sigma^{ij} 
  -\frac{\varphi^i\varphi^j}{w^2}\right),\nonumber\\ 
\varphi^k&=&\sigma^{kl}\varphi_l,\nonumber\\ 
w&:=&\sqrt{1+\varphi^i\varphi_i},\nonumber 
\end{eqnarray} 
where $\sigma_{ij}$ denotes the metric of the sphere $S^n$. 
The outer unit normal is given by 
\begin{equation} 
\nu=\frac{1}{w}\left(x-\varphi^kx_k\right).\nonumber 
\end{equation} 
The Gau{\ss} formula relates the second covariant derivatives 
of the embedding vector to the second fundamental form and 
the normal, where the second derivatives are taken with respect 
to the induced metric. At the moment, however, $F_{ij}$ denotes  
the second covariant derivatives with respect to the metric 
$\sigma_{ij}$. However, these derivatives are related by 
\begin{equation} 
 F^g_{ij}=F^\sigma_{ij}+\left(\Gamma^k_{ij}(\sigma) 
  -\Gamma^k_{ij}(g)\right)F_k,\nonumber 
\end{equation} 
so we deduce from the Gau{\ss} formulae for $M$ and $S^n$ 
\begin{eqnarray} 
\langle F^\sigma_{ij},\nu\rangle&=&\langle-h_{ij}\nu,\nu\rangle 
  =-h_{ij,}\nonumber\\ 
h_{ij}&=&-\frac{1}{w}\left\langle x-\frac{1}{u}u^kx_k, 
 u_{ij}x+u_ix_j+u_jx_i-u\sigma_{ij}x\right\rangle\nonumber\\ 
&=&\frac{u}{w}\left(\sigma_{ij}+2\frac{1}{u^2}u_iu_j 
  -\frac{1}{u}u_{ij}\right)\nonumber\\ 
&=&\frac{u}{w}(\sigma_{ij}+\varphi_i\varphi_j-\varphi_{ij}) 
  \nonumber\\ 
&=&\frac{1}{uw}g_{ij}-\frac{u}{w}\varphi_{ij}.\nonumber   
\end{eqnarray} 
Next, we will derive an evolution equation for $u$ 
\begin{eqnarray}\label{F diff t} 
\frac{d}{dt}(x\cdot u)&=&\frac{\partial}{\partial t}x\cdot u 
  +x\cdot\frac{\partial}{\partial t}u 
  +x\cdot\left\langle\nabla u, 
  \frac{\partial x}{\partial t}\right\rangle,\\ 
\frac{d}{dt}F&=&-f\nu=-f\frac{1}{w} 
  \left(x-\varphi^ix_i\right).\nonumber 
\end{eqnarray} 
In view of 
\begin{equation} 
0=\frac{\partial}{\partial t}\langle x,x\rangle 
  =2\left\langle\frac{\partial}{\partial t}x,x\right\rangle, 
\nonumber 
\end{equation} 
we obtain by multiplying (\ref{F diff t}) with $x$ 
\begin{equation} 
-f\frac{1}{w}=\frac{\partial u}{\partial t} 
  +\left\langle\nabla u, \frac{\partial x}{\partial t} 
  \right\rangle.\nonumber 
\end{equation} 
 From the remaining part of (\ref{F diff t}) we get 
\begin{equation} 
 f\frac{1}{w}\varphi^ix_i=\frac{\partial}{\partial t}x\cdot u,
\nonumber 
\end{equation} 
thus 
\begin{eqnarray} 
\frac{\partial}{\partial t}u&=& 
  -f\frac{1}{w}\left(1+\left\langle\nabla\varphi, 
  \varphi^ix_i\right\rangle\right)\nonumber\\ 
&=&-fw\nonumber 
\end{eqnarray} 
and in view of $\langle F,\nu\rangle=\frac{u}{w}$ 
\begin{equation} 
\dot\varphi:= 
\frac{\partial}{\partial t}\varphi=-\frac{f}{\fnu}.\nonumber 
\end{equation} 
 
We now state the parabolic differential equations for 
$\varphi$, $\dot\varphi$ and  
$W=\frac{1}{2}(1+\varphi^k\varphi_k)$ to give the reader the 
opportunity to compare the evolution equations for $\dot\varphi$  
and $q$, Lemma \ref{lemma evol: q}, as well as those  
 for $W$ and  $\frac{1}{\fnu^2}$, equation (\ref{evol: fnu-2}), 
\begin{eqnarray} 
\dot\varphi&=&g^{ij}\varphi_{ij}-ne^{-2\varphi}+\phi,\nonumber\\ 
\ddot\varphi&=&g^{ij}\dot\varphi_{ij} 
  -e^{-2\varphi} 
  \frac{1}{w^2}(\dot\varphi^i\varphi^j+\varphi^i\dot\varphi^j) 
  \varphi_{ij}+2e^{-2\varphi}\frac{1}{w^4}\varphi^i\varphi^j \varphi_{ij}
  \varphi^k\dot\varphi_k \nonumber\\ 
  &&{}+\dot\varphi(-2\dot\varphi+2\phi 
  +\phi'e^{2\varphi}),\nonumber\\ 
\dot W&=&g^{ij}W_{ij}-e^{-2\varphi}\varphi^j_i\varphi^i_j 
  -\frac{1}{w^2}e^{-2\varphi}W^iW_i 
  +2\frac{1}{w^4}e^{-2\varphi}(W^i\varphi_i)^2\nonumber\\ 
  &&{}+(2W-1)\left(-e^{-2\varphi}(n-1)-2\dot\varphi 
  +2\phi+\phi'e^{2\varphi}\right).\nonumber 
\end{eqnarray} 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
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into spheres. J.\ Differential Geom.\ 32 
(1990), 299--314.  
 
\bibitem{Huisken} G.\ Huisken: Flow  by mean curvature of convex surfaces  
into spheres. J.\ Differential Geom.\ {\bf 20} (1984), 237--266. 
 
\bibitem{Huisken Polden} G.\ Huisken, A.\ Polden: 
Geometric evolution equations for 
hypersurfaces. Calculus of variations and geometric evolution problems 
(Cetraro, 1996), 45--84, Lecture Notes in Math., {\bf 1713}, Springer,  
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\bibitem{Ivochkina et al.}N.\ M.\ Ivochkina, Th.\ Nehring, F.\ Tomi:  
Evolution of starshaped hypersurfaces by nonhomogeneous curvature functions.  
Algebra i Analiz {\bf 12} (2000), 185--203. 
 
\bibitem{Smoczyk} K.\ Smoczyk:  
Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups  
by their mean curvature. Calc.\ Var.\ Partial Differential Equations
{\bf 4} (1996), 155--170.  
  
\bibitem{Smoczyk 2} K.\ Smoczyk: 
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\bibitem{Urbas} J.\ I.\ E.\ Urbas: On the expansion of starshaped  
hypersurfaces by symmetric functions of 
their principal curvatures. Math.\ Z.\ {\bf 205} (1990), 355--372. 
 
\end{thebibliography}  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
\begin{center}
\sc
Address:\\
Max Planck Institute\\
 for Mathematics in the Sciences (MPI MIS)\\
Inselstr. 22-26, D-04103 Leipzig, Germany\\
email: \tt Oliver.Schnuerer@mis.mpg.de,\, Knut.Smoczyk@mis.mpg.de
\end{center}

\end{document}    
    
                   













