# The Curve shortening Flow with Parallel 1-form

Hengyu Zhou  The Graduate Center,
The City University of New York,
365 fifth Ave.,
New York, NY 10016, USA<EMAIL_ADDRESS>
###### Abstract.

Let $M$ be a closed Riemannian manifold with a parallel 1-form $\Omega$. We
prove two theorems about the curve shortening flow in $M$. One is that the
curve shortening flow $\mathcal{C}_{t}$ in $M$ exists for all $t$ in
$[0,\infty)$, if it satisfies $\Omega(T)\geq 0$ on the initial curve
$\mathcal{C}_{0}$. Here $T$ is the unit tangent vector on $\mathcal{C}_{0}$.
The other one is about the convergence. It says that in a closed Riemannian
manifold $\tilde{M}$, assume the curve shortening flow $\mathcal{C}_{t}$
exists for all $t\in[0,\infty)$ and its length converges to a positive limit,
then
$\lim\limits_{t\rightarrow\infty}max_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}=0$ for
all $m=0,1,\cdots$. Here $A$ denotes the second fundamental form of
$\mathcal{C}_{t}$ in $\tilde{M}$.

###### Key words and phrases:

Curve shortening flow, mean curvature flow, parallel form.

###### 2010 Mathematics Subject Classification:

Primary 53C44, Secondary 58J35

## Introduction

We use $\nabla$ denote the Levi-Civita connection of a Riemannian manifold . A
1-form $\Omega$ is called parallel if $\nabla\Omega=0$. In this paper we
consider the long time existence and the convergence of the curve shortening
flow in the Riemannian manifold with arbitrary codimension. More precisely,
the curve shortening flow $\mathcal{C}_{t}$ is given as the solution of the
follow equation:

(0.1) $\left\\{\begin{aligned} &\frac{\partial\mathcal{C}_{t}}{\partial
t}=\vec{H}\\\ &\mathcal{C}(x,0)=\mathcal{C}_{0}(x);\end{aligned}\right.$

Here $\vec{H}=\nabla_{T}T$ is the mean curvature vector of $\mathcal{C}_{t}$
in the ambient Riemannian manifold.

The evolution of closed curves under (0.1) has received considerable study.
For example, Gage ([Gag84]), Gage-Hamilton ([GH86]) and Grayson ([Gra87]),
considered the evolution of the convex curve flowed by the mean curvature in
the plane. Furthermore, Grayson ([Gra89]) generalized the results into the
case on the embedded closed curve of Riemannian surfaces. Huisken ([Hui98])
and Andrews-Bryan ([AB11]) used the technique of distance comparison to
investigate the curve shortening flow on the plane. For the mean curvature
flow of the submanifolds with codimension $k\geq 2$, Andrews-Baker ([AB10])
proposed a machinery about the evolution of the second fundamental form $A$
with arbitrary codimension. They also obtained the convergence of the mean
curvature flow for the submanifold pinched to sphere in Euclidean space. More
recently, with an integral curvature condition ([KFLZ10]) and a pinched
condition in hyperbolic space form ([KFLZ11]) Liu, Xu, Ye and Zhao also proved
two extension theorems for the mean curvature flow of the submanifold with
high codimension. Smoczyk’s survey ([Smo12]) is a good reference.

In a different direction, Wang ([Wan02]) initialized a way to use parallel
n-forms $\Omega$, $n\geq 2$, to investigate the graphic mean curvature flow
for arbitrary codimension ($\geq 2$) in product manifolds. With the evolution
equation of $*\Omega$, ($*$ is the Hodge dual), Wang obtained its long-time
existence and an estimate of its second fundamental form along the mean
curvature flow. Moreover, Wang and his coauthors generalized the results of
graphic mean curvature flow into other settings
([MW11],[TW04],[Wan01],[SW02],). Both of ([AB10]) and ([Wan02]) did not
consider the case of the curve shortening flow, which is the topic of this
paper.

There are two special properties about the curve shortening flow whose
codimension in its ambient manifold is greater than 1. First the embeddedness
of the evolving curve is not necessarily preserved ([Alt91]). Therefore, we
only consider the case of the immersed curve. On the other hand,
$|\vec{H}|^{2}=|A|^{2}$. Based on this fact, we can overcome the barrier that
the curve shortening flow has arbitrary codimension. This makes many
computations possible, especially in Section 2.

Now let us state two main theorems:

######  Theorem 0.1.

Let $M$ be a closed Riemannian manifold, and $\Omega$ be a parallel 1-form in
$M$. If $\mathcal{C}_{0}$ is a closed curve satisfying $\Omega(T)>0$ on
$\mathcal{C}_{0}$. Here $T$ is the unit tangent vector on $\mathcal{C}_{0}$.
Then the curve shortening flow $\mathcal{C}_{t}$ in (0.1) exists for all $t$
in $[0,\infty)$.

######  Corollary 1.

Let $(N,g)$ be a closed manifold, $du^{2}$ is the canonical metric on $S^{1}$.
Assume $M=S^{1}\times N$ is the product manifold with the product metric
$\bar{g}=g+du^{2}$. Let $\Omega=du$, which is a parallel 1-form in $M$.
$\mathcal{C}_{0}$ is a closed curve in $M$. If the unit tangent vector $T$ of
$\mathcal{C}_{0}$ satisfies $\Omega(T)>0$, the curve shortening flow
$\mathcal{C}_{t}$ in (0.1) exists for all $t$ in $[0,\infty)$.

######  Remark 0.1.

Corollary 1 can be viewed the 1-dimension version of Theorem A in ([Wan02]).
The existence result here with $\Omega(T)>0$ is stronger than that of
graphical mean curvature flow with $*\Omega>\frac{1}{2}$ in ([Wan02]). It also
can be compared with the results of Tsui-Wang ([TW04]) if we choose $N=S^{n}$.

Now we consider the convergence of the curve shortening flow in any closed
Riemannian manifold. Notice that we do not require there is a parallel 1-form
in the ambient manifold. The convergence of the curve shortening flow in the
$C^{\infty}$ sense can be stated as follows.

######  Theorem 0.2.

Let $\tilde{M}$ be a closed manifold. If the curve shortening flow
$\mathcal{C}_{t}$ in $\tilde{M}$ exists for all $t\in[0,\infty)$ and the
length of $\mathcal{C}_{t}$ converges to a positive number,
$\lim\limits_{t\rightarrow\infty}max_{\mathcal{C}_{t}}|\nabla^{m}A|=0$ for all
$m=0,1,\cdots$.

######  Remark 0.2.

The definition of $\nabla^{m}A$ is very subtle. In ([Hui84],[Hui86]),
Huisken’s definitions about $\nabla^{m}A$ only work in the hypersurface’s
case. Andrews-Bakers ([AB10]) gave a rigorous definition of $\nabla^{m}A$ for
any mean curvature flow with arbitrary codimension in Riemannian manifold. The
basic idea is that $|\nabla^{m}A|$ can contain all information about the mean
curvature flow. Namely, when $|\nabla^{m}A|$ goes to 0 when $t$ goes to
infinity, the mean curvature flow $F_{t}$ will converge to a geodesic
submanifold in some natural sense. For example, one of these cases is that
$F_{t}$ has long time existence and always stay in a compact region of its
ambient Riemannian manifold. We will state the definition of $\nabla^{m}A$ in
Section 2.

######  Remark 0.3.

The condition that $M$ and $\tilde{M}$ are closed is not essential. We take
them only for the sake of the exposition. In Theorem 0.1, what we really need
is that for all finite time $t$, the curve shortening flow $\mathcal{C}_{t}$
is in a compact region in $M$, which can depend on time $t$. In Theorem 0.2,
we can only require that all covariant derivatives of the Riemann tensor in
$\tilde{M}$ are uniformly bounded.

### Outline of this paper

In section 1, we give two examples to illustrate our motivations of using
parallel form to investigate the curve shortening flow. In Section 2, we give
the definition of $\nabla^{m}A$. Then we derive the evolution equations
related to the second fundamental form $A$ (lemma 2.2 and lemma 2.3). With
those equations, we derive the evolution equation of $\Omega(T)$ for any
1-form $\Omega$ along the curve shortening flow (lemma 2.4). In particular,
when $\Omega$ is a parallel 1-form, this evolution equation states that the
lower positive bound of $\Omega(T)$ can be preserved along the curve
shortening flow. In section 3, we prove Theorem 0.1 by using the results from
section 2. In section 4, with Grayson’s idea ([Gra89]), we prove Theorem 0.2.

### Notation

We collect some geometric quantities and facts used later.

  1. (1)

$\mathcal{C}(u,t)$ denotes the solution of (0.1). $\frac{\partial}{\partial
t}$ and $\frac{\partial}{\partial u}$ are the abbreviations of
$\mathcal{C}_{*}(\frac{\partial}{\partial t})$ and
$\mathcal{C}_{*}(\frac{\partial}{\partial u})$ respectively.

  2. (2)

$\nabla$ is the Levi-Civita connection, $du^{2}$ is the canonical metric on
$S^{1}$.

  3. (3)

Let $R$ denote Riemann curvature tensor, given by
$R(X,Y)=\nabla_{Y}\nabla_{X}-\nabla_{X}\nabla_{Y}+\nabla_{[X,Y]}$.

  4. (4)

Let $s$ be the arc-length parameter of a curve $\mathcal{C}$. Its unit tangent
vector $T$ is equal to $\frac{\partial}{\partial
s}=\frac{\mathcal{C}_{*}(\frac{\partial}{\partial
u})}{<\mathcal{C}_{*}(\frac{\partial}{\partial
u}),\mathcal{C}_{*}(\frac{\partial}{\partial u})>^{\frac{1}{2}}}.$

  5. (5)

$\vec{H}=\frac{\partial}{\partial t}=A(T,T)=\nabla_{T}T$.
$|A|^{2}=|\vec{H}|^{2}$.

  6. (6)

For a curve $\mathcal{C}$, $|\vec{H}|^{2}=|A|^{2}=<\nabla_{T}T,\nabla_{T}T>$.
All curves in this paper are closed.

### Acknowledgements

The author would like to thank Professor Zeno Huang for suggesting the
problem, and for many helpful discussions and encouragement, and Professor
Yunping Jiang for his interest and suggestions. This work is completed while
the author is partially supported, as a student associate, by a CIRG award
$\\#1861$. The author is also indebted to Xiangwen Zhang, Longzhi Lin and Andy
Huang. These discussions with them greatly help to simplify the computations
in this paper.

##  1\. The Examples about Parallel form

In this section we use two examples to illustrate the motivations that we
choose parallel 1-form $\Omega$ to investigate the curve shortening flow. The
first one is the graphic mean curvature flow in the product manifolds
([Wan02]). Based on this, we can use Wang’s idea ([Wan02]) about parallel form
to propose a problem about the open curve shortening flow in $R^{1+p}$.

######  Example 1 (graphic mean curvature flow).

In ([Wan02]), Mu-Tao Wang initialized a way to use parallel form to deform a
map along the mean curvature flow in product manifolds. The basic setting is
as followed: Let $N_{1}$, $N_{2}$ be closed manifolds with constant sectional
curvatures, dimension $\geq 2$. $f$ is a smooth map from $N_{1}$ to $N_{2}$.
Let $F(p,t):N_{1}\times[0,q)\rightarrow N_{1}\times N_{2}$ satisfies the
following equation.

(1.1) $\left\\{\begin{aligned} \frac{\partial F_{t}}{\partial t}&=\vec{H}\\\
F(p,0)&=(p,f(p));\end{aligned}\right.$

Let $\Omega$ be the volume form of $N_{1}$, which is parallel in the product
manifold $N_{1}\times N_{2}$. Together with some technical conditions about
the sectional curvature of $N_{1}$ and $N_{2}$, Wang ([Wan02]) obtained that
if $*\Omega>\frac{1}{2}$ at the initial manifold, the evolution equations of
$*\Omega$ indicates the long time existence of the solution in (1.1).

######  Example 2 (open curve shortening flow).

In this example, we propose a question about open curve shortening flow in
$R^{1+p}$. $\mathcal{C}_{0}$ in $R^{1+p}$ is defined by

(1.2) $\mathcal{C}_{0}:R^{1}\rightarrow
R^{1+p}\quad\mbox{and}\qquad\mathcal{C}_{0}(x)=(x,f_{1}(x),\cdots,f_{p}(x));$

Let $|Df|^{2}=\sum_{i=1,\cdots,p}(f_{i}^{{}^{\prime}}(x))^{2}$. Then,

$T=\left(\frac{\partial}{\partial
x^{1}}+\sum_{i=1,\cdots,p}f_{i}^{{}^{\prime}}(x)\frac{\partial}{\partial
x^{i+1}}\right)\frac{1}{\sqrt{1+|Df|^{2}}};$

Let $\Omega=dx_{1}$. Again $\Omega$ is a parallel 1-form. On
$\mathcal{C}_{0}$, $\Omega(T)=\frac{1}{\sqrt{1+|Df|^{2}}}$. From lemma 2.6, if
$\mathcal{C}_{t}$ exists for $t\in[0,q)$, then $\Omega(T)$ on
$\mathcal{C}_{t}$ satisfies

$\frac{\partial}{\partial
t}\Omega(T)=\triangle^{\mathcal{C}_{t}}\Omega(T)+|A|^{2}\Omega(T)$

Here $A$ is its second fundamental form. Recall that with
$*\Omega>\frac{1}{2}$ in the initial data, Wang ([Wan02]) answered the long
time existence of graphic mean curvature flow positively. Naturally if we
assume that $\mathcal{C}_{0}$ satisfies $\Omega(T)\geq\delta>0$, does the
curve shortening flow $\mathcal{C}_{t}$ in (0.1) exist for all
$t\in[0,\infty)$? If it exists, can we give any description about the
convergence of this curve shortening flow in $R^{1+p}$?

######  Remark 1.1.

The above examples reflect our basic perspectives about the parallel form and
the mean curvature flow. (2.6) and (3.3) are some possible evidences to
support these connections between them. (2.6) can be viewed as a
generalization of Wang’s graphical mean curvature flow ([Wan02]) in the
curve’s case. (3.3) indicates that $\mu=\frac{1}{\Omega(T)}$ be thought as the
gradient function ([EH89]) in the case of the curve shortening flow. There are
no any oblivious connections between ([Wan02]) and ([EH89]). For the curve
shortening flow, however, (3.3) and (2.6) are two different forms of the same
evolution equation.

##  2\. Evolution Equations

We derive the evolution equations of the curve shortening flow. Generally, the
forms of those equations are very complicated for the mean curvature flow
whose dimension and codimension are both $>1$. ([AB10], [Wan02])

###  2.1. Definition of $\nabla^{m}A$

We briefly state the definition of $\nabla^{m}A$ which can be found in
([AB10]). Assume $M$ is a Riemannian manifold with the Levi-Civita connection
$\bar{\nabla}$, $N$ is an immersed submanifold of $M$. Suppose $I$ is a real
interval, then the tangent space $T(N\times I)$ splits into a direct product
$\mathcal{H}\otimes R\partial t$, where $\mathcal{H}=\\{u\in T(N\times
I);dt(u)=0\\}$ is the ”spatial” tangent bundle.

We consider a smooth map $F:N\times I\rightarrow M$ which is a time-dependent
immersion, i.e.,for each fixed $t\in I$, $F(\,,t):N\rightarrow M$ is an
immersion. $F^{*}TM$ is a vector bundle over $N\times I$. We can define a
metric $g_{F}$ and the connection ${}^{F}\nabla$ on $F^{*}TM$ by the pull-back
from $(\bar{\nabla},\bar{g})$ on $M$. Let $\mathcal{N}$ be the orthogonal
complement of $\mathcal{H}$ in $F^{*}TM$. We denote the $\pi,\pi^{\bot}$ be
the orthogonal projections from $F^{*}TM$ onto $\mathcal{H}$ and $\mathcal{N}$
respectively. The connections ${}^{\mathcal{H}}\nabla$ and ${}^{N}\nabla$ are
given by.

$\displaystyle{}^{\mathcal{H}}\nabla$ $\displaystyle=\pi\circ^{F}\nabla\circ
F_{*};$ $\displaystyle{}^{\mathcal{N}}\nabla$
$\displaystyle=\pi^{\bot}\circ^{F}\nabla\circ F_{*};$

For a tensor $K\in\Gamma(\mathcal{H}^{*}\otimes\mathcal{H}^{*}\otimes N)$. We
have to define $\nabla^{m}K\in\Gamma(\otimes^{m+2}\mathcal{H}^{*}\otimes N)$
by induction on $m$. Let $u_{0},u_{i},i=1,2,3,\cdots,m+1$, then $\nabla^{m}K$
and $\nabla_{\frac{\partial}{\partial t}}\nabla^{m}K$ are given by

$\displaystyle\nabla_{u_{0}}$
$\displaystyle(\nabla^{m-1}K)(u_{1},\cdots,u_{m+1})=^{\mathcal{N}}\nabla_{u_{0}}(\nabla^{m-1}K(u_{1},\cdots,u_{m+1}))-\sum\limits_{i=1}^{m+1}\nabla^{m-1}K(\cdots,^{\mathcal{H}}\nabla_{u_{0}}u_{i},\cdots);$
$\displaystyle\nabla_{\frac{\partial}{\partial t}}$
$\displaystyle(\nabla^{m}K)(u_{0},u_{1},\cdots,u_{m+1})=^{\mathcal{N}}\nabla_{\frac{\partial}{\partial
t}}(\nabla^{m}K(u_{0},\cdots,u_{m+1}))-\sum\limits_{k=0}^{m+1}\nabla^{m}K(\cdots,^{\mathcal{H}}\nabla_{\frac{\partial}{\partial
t}}u_{k},\cdots);$

###  2.2. The Evolution Equations for the Curve Shortening Flow

For the curve shortening flow $\mathcal{C}_{t}$ in (0.1), we define a new
tensor
$\tilde{R}\in\Gamma(\mathcal{H}^{*}\otimes\mathcal{H}^{*}\otimes\mathcal{N})$
by $\tilde{R}(T,T)=R(T,\vec{H})T$. We also have the definition of
$\nabla^{m}\tilde{R}$. Since
$A\in\Gamma(\mathcal{H}^{*}\otimes\mathcal{H}^{*}\otimes\mathcal{N})$, with
the definitions in the previous subsection, we can obtain a theorem about the
evolution equation of $A$ from equation (18) in ([AB10]).

######  Theorem 2.1 (Andrews-Baker).

For the curve shortening flow $\mathcal{C}_{t}$, the evolution equation of
$A(T,T)$ is given by

(2.1) $\nabla_{\frac{\partial}{\partial
t}}A(T,T)=\nabla_{T}\nabla_{T}A(T,T)+|A|^{2}\vec{H}+\tilde{R}(T,T);$

Recall that $S*T$ ([Hui84]) means any linear combination of tensors formed by
contraction on $S$ and $T$. (2.1) can be rewritten as
$\nabla_{\frac{\partial}{\partial t}}A=\nabla^{2}A+\sum A*A*A+\tilde{R}(T,T)$.
This form can be generalized into any $m$-th covariant derivative $A$ for the
curve shortening flow $\mathcal{C}_{t}$.

######  Lemma 2.2.

The evolution equation of $m$-th covariant derivative $A$ is of the form.

(2.2) $\nabla_{\frac{\partial}{\partial
t}}\nabla^{m}A=\nabla^{m+2}A+\sum\limits_{i+j+k=m}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A+\nabla^{m}\tilde{R}(T,T);$

###### Proof.

We prove the lemma through the induction on $m$. The case $m=0$ is given by
(2.1). Now suppose the results hold up to $k\leq m-1$. In ([Hui84]), the time
derivative of the Christoffel symbols $\Gamma_{jk}^{i}$ has the form $A*\nabla
A$. Therefore we obtain,

$\displaystyle\nabla_{\frac{\partial}{\partial t}}\nabla^{m}A$
$\displaystyle=\nabla(\nabla_{\frac{\partial}{\partial
t}}\nabla^{m-1}A)+\nabla^{m-1}A*A*\nabla A;$
$\displaystyle=\nabla\\{\nabla^{m+1}A)+\sum\limits_{i+j+k=m-1}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A+\nabla^{m-1}\tilde{R}(T,T)\\}$
$\displaystyle+\nabla^{m-1}A*A*\nabla A;$
$\displaystyle=\nabla^{m+2}A+\sum\limits_{i+j+k=m}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A+\nabla^{m}\tilde{R}(T,T);$

∎

Moreover, we obtain the evolution equations of
$|\nabla^{m}A|^{2},m=0,1,\cdots,$.

######  Lemma 2.3.

The evolution equations of $|\nabla^{m}A|^{2}$ are given by

(2.3) $\displaystyle\frac{\partial}{\partial
t}|A|^{2}=\triangle^{\mathcal{C}_{t}}|A|^{2}-2|\nabla
A|^{2}+2|A|^{4}+2R(T,\vec{H},T,\vec{H});$ (2.4)
$\displaystyle\begin{split}\frac{\partial}{\partial
t}|\nabla^{m}A|^{2}&=\triangle^{\mathcal{C}_{t}}|\nabla^{m}A|^{2}-2|\nabla^{m+1}A|^{2}+\sum\limits_{i+j+k=m}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A*\nabla^{m}A\\\
&+2<\nabla^{m}\tilde{R}(T,T),\nabla^{m}A>\quad\text{for}\quad m\geq
1;\end{split}$

###### Proof.

We differentiate $|\nabla^{m}A|^{2}$ with respect to $t$. For $m=0$,

$\displaystyle\frac{\partial}{\partial t}|A|^{2}$
$\displaystyle=2<\nabla_{\frac{\partial}{\partial t}}A(T,T),A(T,T)>;$
$\displaystyle=2<\nabla^{2}A(T,T)+|A|^{2}\vec{H}+R(T,\vec{H})T,A(T,T)>;$ Since
$A(T,T)=\vec{H}$ $\displaystyle=\triangle^{\mathcal{C}_{t}}|A|^{2}-2|\nabla
A|^{2}+2|A|^{4}+2R(T,\vec{H},T,\vec{H});$

For $m\geq 1$.

$\displaystyle\frac{\partial}{\partial t}|\nabla^{m}A|^{2}$
$\displaystyle=2<\nabla_{\frac{\partial}{\partial
t}}\nabla^{m}A,\nabla^{m}A>;$
$\displaystyle=2<\nabla^{m+2}A+\sum\limits_{i+j+k=m}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A+\nabla^{m}\tilde{R}(T,T),\nabla^{m}A>;$
$\displaystyle=\triangle^{\mathcal{C}_{t}}|\nabla^{m}A|^{2}-2|\nabla^{m+1}A|^{2}+\sum\limits_{i+j+k=m}\nabla^{i}A*\nabla^{j}A*\nabla^{k}A*\nabla^{m}A+$
$\displaystyle+2<\nabla^{m}\tilde{R}(T,T),\nabla^{m}A>;$

∎

Now we derive the evolution equation of $\Omega(T)$ along the curve shortening
flow $\mathcal{C}_{t}$.

######  Lemma 2.4.

Let $\Omega$ be a 1-form in a Riemannian manifold $\tilde{M}$, $s$ be the arc-
length parameter of $\mathcal{C}_{t}$. Let
$\triangle^{\mathcal{C}_{t}}=\frac{\partial^{2}}{\partial s^{2}}$ be the
Laplace operator on $\mathcal{C}_{t}$, and $T=\frac{\partial}{\partial s}$ be
the unit tangent vector. Thus we have

(2.5) $\frac{\partial}{\partial
t}\Omega(T)=\triangle^{\mathcal{C}_{t}}\Omega(T)+|A|^{2}\Omega(T)-\nabla^{2}\Omega(T,T,T)-2\nabla\Omega(T,\nabla_{T}T);$

In particular, if $\Omega$ is parallel in $\tilde{M}$, we obtain

(2.6) $\frac{\partial}{\partial
t}\Omega(T)=\triangle^{\mathcal{C}_{t}}\Omega(T)+|A|^{2}\Omega(T);$

###### Proof.

Recall $T=\frac{C_{*}(\frac{\partial}{\partial
u})}{<C_{*}(\frac{\partial}{\partial u}),C_{*}(\frac{\partial}{\partial
u})>^{\frac{1}{2}}}$ and $\frac{\partial}{\partial t}=\vec{H}$. And

(2.7) $\begin{split}[\frac{\partial}{\partial
t},T]&=\frac{[C_{*}(\frac{\partial}{\partial
t}),C_{*}(\frac{\partial}{\partial u})]}{<C_{*}(\frac{\partial}{\partial
u}),C_{*}(\frac{\partial}{\partial
u})>^{\frac{1}{2}}}+C_{*}(\frac{\partial}{\partial u})\frac{\partial}{\partial
t}(\frac{1}{<C_{*}(\frac{\partial}{\partial u}),C_{*}(\frac{\partial}{\partial
u})>^{\frac{1}{2}}});\\\ &=-C_{*}(\frac{\partial}{\partial
u})\frac{<\nabla_{C_{*}(\frac{\partial}{\partial
t})}(C_{*}(\frac{\partial}{\partial u})),C_{*}(\frac{\partial}{\partial
u})>}{<C_{*}(\frac{\partial}{\partial u}),C_{*}(\frac{\partial}{\partial
u})>^{\frac{3}{2}}};\\\ &=-C_{*}(\frac{\partial}{\partial
u})\frac{<\nabla_{C_{*}(\frac{\partial}{\partial
u})}(\vec{H}),C_{*}(\frac{\partial}{\partial
u})>}{<C_{*}(\frac{\partial}{\partial u}),C_{*}(\frac{\partial}{\partial
u})>^{\frac{3}{2}}};\\\ &=|A|^{2}T;\end{split}$

Therefore, the relation of $\nabla_{\frac{\partial}{\partial t}}T$ and
$\nabla_{T}\frac{\partial}{\partial t}$ is given by

(2.8) $\nabla_{\frac{\partial}{\partial
t}}T-\nabla_{T}\frac{\partial}{\partial t}=|A|^{2}T;$

(2.9)
$\begin{split}\triangle^{\mathcal{C}_{t}}\Omega(T)&=T\big{(}\Omega(\nabla_{T}T)+\nabla\Omega(T,T)\big{)};\\\
&=\Omega(\nabla_{T}\frac{\partial}{\partial
t})+\nabla\Omega(\frac{\partial}{\partial
t},T)+\nabla^{2}\Omega(T,T,T)+2\nabla\Omega(T,\nabla_{T}T);\end{split}$

The $t$-derivative of $\Omega(T)$

(2.10)
$\begin{split}\frac{d}{dt}\Omega(T)&=\Omega(\nabla_{\frac{\partial}{\partial
t}}T)+\nabla\Omega(\frac{\partial}{\partial t},T);\\\
&=\Omega(\nabla_{T}\frac{\partial}{\partial
t})+|A|^{2}\Omega(T)+\nabla\Omega(\frac{\partial}{\partial
t},T);\quad\end{split}$

Therefore, (2.9) and (2.10) lead to the lemma. ∎

######  Remark 2.1.

It’s easily checked $min_{\mathcal{C}_{t}}\Omega$ is a Lipschitz function of
$t$, and $min_{\mathcal{C}_{t}}\Omega(T)$ is differentiable with respect to
$t$ almost everywhere. When $\Omega$ ia a parallel 1-form in $\tilde{M}$,
$\Omega(T)>0$ for $t\in[0,q_{0})$. For
$\triangle^{\mathcal{C}_{t}}\Omega(T)\geq 0$ for the points which attain the
minimal value of $\Omega(T)$, for a.e $t\in[0,q_{0})$ we have the following:

$\frac{d}{dt}min_{\mathcal{C}_{t}}\Omega(T)\geq|A|^{2}min_{\mathcal{C}_{t}}\Omega(T)\geq
0;$

This implies that $min_{\mathcal{C}_{t}}\Omega(T)$ is nondecreasing along the
flow. Then $\Omega(T)\geq\delta_{0}$ will be preserved whenever the curve
shortening flow exists.

##  3\. Long time existence

With the evolution results in the previous section, we prove Theorem 0.1. The
short time existence of $\mathcal{C}_{t}$ is from the short time existence of
solution of the quasilinear parabolic equation for closed initial data. It’s
well known that if the mean curvature flow $F_{t}$ of the hypersurface in
Euclidean space exists for only finite time interval $[0,q)$,
$max_{F_{t}}|A|\rightarrow\infty$ when $t$ approach to $q$ ([Hui84]). For the
submanifold in Euclidean space with high codimension, such kind of lemma is
proved by Andrews-Baker ([AB10]). For the curve shortening flow’s case, we
give the proof of the following lemma only for the sake of completeness.

######  Lemma 3.1.

Let $\tilde{M}$ be a Riemannian manifold. If the curve shortening flow
$\mathcal{C}_{t}$ in $\tilde{M}$ exists for $t$ in the maximal finite interval
$[0,q)$, then $max_{\mathcal{C}_{t}}|A|^{2}\rightarrow\infty\ \text{as}\
t\rightarrow q$.

###### Proof.

If the lemma is false, there exists a constant $C_{4}<\infty$ such that
$max_{\mathcal{C}_{t}}|A|\leq C_{4}$ for $t\in[0,q)$. It follows that for all
$u\in S^{1}$ and $t_{1},t_{2}\in[0,q)$.

$\displaystyle dist(\mathcal{C}_{t_{1}}(u),\mathcal{C}_{t_{2}}(u))$
$\displaystyle\leq|\int\limits_{t_{1}}^{t_{2}}\vec{H}dt|\leq
C_{4}|t_{1}-t_{2}|;$ $\displaystyle\frac{d}{dt}\int_{\mathcal{C}_{t}}ds_{t}$
$\displaystyle\geq-C_{4}\int_{\mathcal{C}_{t}}ds_{t};$

Therefore, $\mathcal{C}_{t}$ converges uniformly to some continuous limit
$\mathcal{C}_{q}(u)$ and the length $l_{\mathcal{C}_{t}}\geq\delta>0$ for
$t\in[0,q)$. We want to show that $\mathcal{C}_{q}(u)$ actually represents a
smooth limit curve. Then we can extend the flow $\mathcal{C}_{t}$ over time
$q$. It’s a contradiction to the maximal finite interval in the assumption.

By (2.4), we are led to

(3.1) $\begin{split}\frac{\partial}{\partial
t}|\nabla^{m}A|^{2}&\leq\triangle^{\mathcal{C}_{t}}|\nabla^{m}A|^{2}-2|\nabla^{m+1}A|^{2}+C_{m}\sum_{i+j+k=m}|\nabla^{i}A||\nabla^{j}A||\nabla^{k}A||\nabla^{m}A|\\\
&+D_{m}\sum_{j\leq
m}|\nabla^{j}A||\nabla^{m}A|+\tilde{C}_{m}|\nabla^{m}A|;\end{split}$

By Cauchy inequality, we have the follows.

(3.2) $\displaystyle|\nabla^{m}A|^{2}$ $\displaystyle\leq
min_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}+\int_{\mathcal{C}_{t}}\frac{\partial}{\partial
s}|\nabla^{m}A|^{2}ds_{t};$
$\displaystyle\leq\frac{1}{l_{\mathcal{C}_{t}}}\int|\nabla^{m}A|^{2}ds_{t}+2\int_{\mathcal{C}_{t}}<\nabla^{m}A,\nabla^{m+1}A>ds_{t};$
$\displaystyle\leq(\frac{1}{\delta}+2)\int|\nabla^{m}A|^{2}ds_{t}+2\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t};$

If $\mathcal{C}_{t}$ satisfies $|\nabla^{m}A|\leq C_{m}$ for all $t<q$, then
$\mathcal{C}_{q}$ is a smooth curve. From our definition about $\nabla^{m}A$,
the proof of this fact is classical but a little tedious. Therefore we omit it
here. For the Euclidean case, please refer to the proof of Theorem 3 in
([AB10]).

By (3.2), we have to prove that
$\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\leq C_{m}$ for all $m$. We
argue $|\int_{\mathcal{C}_{t}}\nabla^{m}Ads_{t}|\leq C_{m}$ by induction on
$m$. If it holds up for $k\leq m-1$. Then by (3.2) for $k\leq m-2$,
$max_{\mathcal{C}_{t}}|\nabla^{k}A|\leq C_{k}$ for $t\in[0,q)$. (3.1) and
(3.2) imply that

$\frac{\partial}{\partial
t}\int_{ct}|\nabla^{m}A|^{2}ds_{t}\leq\tilde{C}(m)\big{(}\int_{ct}|\nabla^{m}A|^{2}ds_{t}+(\int_{ct}|\nabla^{m}A|^{2}ds_{t})^{\frac{1}{2}}\big{)};$

With the above inequality, we obtain

$(\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t})^{\frac{1}{2}}+1\leq
e^{\frac{\tilde{C}(m)}{2}t}((\int_{\mathcal{C}_{0}}|\nabla^{m}A|^{2}ds_{0})^{\frac{1}{2}}+1);$

Then we conclude that for all
$m=1,\cdots,max_{\mathcal{C}_{t}}|\nabla^{m}A|\leq C_{m}$. $C_{q}(u)$ is the
smooth limit curve of $\mathcal{C}_{t}$ for $t<q$. This will lead to the
contradiction, since we suppose $[0,q)$ is the maximal time interval for the
existence of $\mathcal{C}_{t}$. ∎

Now let $\mu=\frac{1}{\Omega(T)}$. The following lemma tells us $\mu$ can be
viewed as the gradient function $\nu$ in ([EH89]). They have the same form of
evolution equation.

######  Lemma 3.2.

Let $M$ be a closed manifold, $\Omega$ be a parallel 1-form in $M$, $s$ be the
arc-length parameter of $\mathcal{C}_{t}$. If the curve shortening flow
$\mathcal{C}_{t}$ exists for $t\in[0,q)$ and $\mathcal{C}_{0}$ satisfies
$\Omega(T)>0$, then $\mu$ satisfies

(3.3)
$(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})\mu=-|A|^{2}\mu-2\mu^{-1}|\frac{\partial\mu}{\partial
s}|^{2};$

###### Proof.

$\displaystyle(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})\mu$
$\displaystyle=-\frac{1}{\Omega(T)^{2}}(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})\Omega(T)-2\Omega(T)|\frac{\partial\Omega(T)}{\partial
s}|;$
$\displaystyle=-\frac{1}{\Omega(T)^{2}}(|A|^{2}\Omega(T))-2\Omega(T)|\frac{\partial}{\partial
s}\frac{1}{\Omega(T)}|^{2};$
$\displaystyle=-|A|^{2}\mu-2\mu^{-1}|\frac{\partial\mu}{\partial s}|^{2};$

∎

######  Lemma 3.3.

Use the assumption in lemma 3.2. Let $C_{0}=max_{x\in M}R$. If the curve
shortening flow $\mathcal{C}_{t}$ exists for $t\in[0,q)$, then

(3.4)
$(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})|A|^{2}\mu^{2}\leq-2\mu^{-1}\frac{\partial\mu}{\partial
s}\frac{\partial|A|^{2}\mu^{2}}{\partial s}+2C_{0}|A|^{2}\mu^{2};$

###### Proof.

$(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})|A|^{2}=-2|\nabla
A|^{2}+2|A|^{4}+2R(T,\vec{H},T,\vec{H});$

and together with the identity

$(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})\mu^{2}=-2|A|^{2}\mu^{2}-6|\frac{\partial\mu}{\partial
s}|^{2}$

yields

(3.5) $(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})|A|^{2}\mu^{2}\leq-2|\nabla
A|^{2}\mu^{2}-6|\frac{\partial\mu}{\partial
s}|^{2}|A|^{2}-2\frac{\partial|A|^{2}}{\partial
s}\frac{\partial\mu^{2}}{\partial s}+2R(T,\vec{H},T,\vec{H})\mu^{2};$

and

(3.6) $\displaystyle-2\frac{\partial|A|^{2}}{\partial
s}.\frac{\partial\mu^{2}}{\partial s}$
$\displaystyle\leq-\frac{\partial|A|^{2}}{\partial
s}\frac{\partial\mu^{2}}{\partial s}-4\mu|A|\frac{\partial|A|}{\partial
s}\frac{\partial\mu}{\partial s};$
$\displaystyle\leq-\mu^{-2}\frac{\partial\mu^{2}}{\partial
s}\frac{\partial|A|^{2}\mu^{2}}{\partial s}+4|\frac{\partial\mu}{\partial
s}|^{2}|A|^{2}-4\mu|A|\frac{\partial|A|}{\partial
s}\frac{\partial\mu}{\partial s}$
$\displaystyle\leq-2\mu^{-1}\frac{\partial\mu}{\partial
s}\frac{\partial|A|^{2}\mu^{2}}{\partial s}+2|\frac{\partial|A|}{\partial
s}|^{2}\mu^{2}+6|\frac{\partial\mu}{\partial s}|^{2}|A|^{2};$

Here we use the fact $\frac{\partial|A|}{\partial s}\leq|\nabla A|$. For $M$
is closed, $|\vec{H}|^{2}=|A|^{2}$, $|R(T,\vec{H},T,\vec{H})|\leq
C_{0}|A|^{2}$. By (3.5) and (3.6), we obtain

(3.7)
$(\frac{d}{dt}-\triangle^{\mathcal{C}_{t}})|A|^{2}\mu^{2}\leq-2\mu^{-1}\frac{\partial\mu}{\partial
s}\frac{\partial|A|^{2}\mu^{2}}{\partial s}+2C_{0}|A|^{2}\mu^{2};$

∎

######  Corollary 2.

Under the assumption of Theorem 0.1. Then $max_{\mathcal{C}_{t}}|A|\leq
C(q,C_{0})$ when $t<q$. Here $C(q,C_{0})$ is a constant only dependent on $q$
and $C_{0}$.

###### Proof.

Because $M$ is closed, for any unit vector field $S$ in $TM$, $|\Omega(S)|\leq
C_{M}$.

$\mu^{-1}\frac{\partial\mu}{\partial s}=\Omega(T)\frac{\partial}{\partial
s}\frac{1}{\Omega(T)}=-\mu\Omega(\vec{H});$

Since $|\mu\Omega(\vec{H})|$ is continuous for $(u,t)\in S^{1}\times[0,q)$,
the maximal principle of parabolic equation in (3.7) gives ,

(3.8) $\frac{\partial}{\partial t}max_{\mathcal{C}_{t}}|A|^{2}\mu^{2}\leq
2C_{0}max_{\mathcal{C}_{t}}|A|^{2}\mu^{2};$

From $\mu\geq\frac{1}{C_{M}}$ and (3.8), we conclude the corollary. ∎

#### The proof of Theorem 1

Now we prepare everything which we need to prove Theorem 0.1. From corollary
2, for all finite time interval $[0,q)$, $max_{\mathcal{C}_{t}}|A|$ is always
uniformly bounded (depending on $q$). If $\mathcal{C}_{t}$ exists only for a
finite interval, Lemma 3.1 will lead to a contraction. Then Theorem 1 is
followed.

##  4\. The Convergence

This section is devoted to prove Theorem 0.2. Recall Theorem 0.2 concludes
that the limit of $max_{\mathcal{C}_{t}}|\nabla^{m}A|$ is $0$ for all
$m=0,1,\cdots$ if the curve shortening flow $\mathcal{C}_{t}$ in the closed
manifold $\tilde{M}$ exists for all $t\in[0,\infty)$ and the length of
$\mathcal{C}_{t}$ converges to a positive number. The idea of its proof
originated from Section 7 in ([Gra89]). We suppose the length of
$\mathcal{C}_{t}$ satisfies $l_{\mathcal{C}_{t}}\geq l_{\infty}>0$. First,
let’s prove a technique lemma.

######  Lemma 4.1.

For all $m\geq 0$,

(4.1) $|\nabla^{m}A|^{2}\leq
a\int|\nabla^{m}A|^{2}ds_{t}+2\int|\nabla^{m+1}A|^{2}ds_{t};$

Here $a=\frac{1}{l_{\infty}}+2$. For $m\geq 1$,

(4.2)
$(\int|\nabla^{m}A|^{2}ds_{t})^{2}\leq\int|\nabla^{m-1}A|^{2}ds_{t}\int|\nabla^{m+1}A|^{2}ds_{t};$

###### Proof.

For $|\nabla^{m}A|^{2}$,

$\displaystyle|\nabla^{m}A|^{2}$ $\displaystyle\leq
min_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}+\int_{\mathcal{C}_{t}}\frac{\partial}{\partial
s}|\nabla^{m}A|^{2}ds_{t};$
$\displaystyle\leq\frac{1}{l_{\mathcal{C}_{t}}}\int|\nabla^{m}A|^{2}ds_{t}+2\int_{\mathcal{C}_{t}}<\nabla^{m}A,\nabla^{m+1}A>ds_{t};$
$\displaystyle\leq(\frac{1}{l_{\mathcal{C}_{t}}}+2)\int|\nabla^{m}A|^{2}ds_{t}+2\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t};$

Integrating $\int|\nabla^{m}A|^{2}ds_{t}$ by parts,

$\int|\nabla^{m}A|^{2}ds_{t}=\int\frac{\partial}{\partial
s}<\nabla^{m-1}A,\nabla^{m}A>ds_{t}-\int<\nabla^{m-1}A,\nabla^{m+1}A>ds_{t};$

The first term is 0. Using the Cauchy inequality we obtain (4.2). ∎

#### The proof of Theorem 0.2

Our strategy is to prove for all $m=0,1,\cdots$, $\int|\nabla^{m}A|^{2}ds_{t}$
converges to 0 as $t\rightarrow\infty$. First, we prove $m=0$ case, then argue
by induction.

From the defintion of the curve shortening flow, we get the following two
facts.

(4.3)
$\displaystyle\frac{d}{dt}\int_{\mathcal{C}_{t}}ds_{t}=-\int|\vec{H}|^{2}ds_{t}=-\int_{\mathcal{C}_{t}}|A|^{2}ds_{t};$
(4.4) $\displaystyle
l_{\mathcal{C}_{t}}-l_{\mathcal{C}_{0}}=\int\limits_{0}^{t}\frac{d}{dt}\int_{\mathcal{C}_{t}}ds_{t}dt=-\int\limits_{0}^{t}\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}dt<\infty;$

There exists a measure zero set $E\,\text{in}\,[0,\infty)$. Let
$E_{n}=[n,\infty)\cap E^{c}$. We get

(4.5) $max_{t\in E_{n}}\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}\rightarrow
0\quad\text{as}\quad n\rightarrow\infty;$

From (2.3), we differentiate $\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}$ with
respect to $t$.

(4.6) $\displaystyle\frac{d}{dt}\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}$
$\displaystyle\leq-2\int|\nabla
A|^{2}ds_{t}+2\int_{\mathcal{C}_{t}}|A|^{4}ds_{t}+D_{0}\int|A|^{2}ds_{t};$
$\displaystyle\leq-2\int|\nabla
A|^{2}ds_{t}+(a\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}+2\int_{\mathcal{C}_{t}}|\nabla
A|^{2}ds_{t})\int|A|^{2}ds_{t}$ $\displaystyle+D_{0}\int|A|^{2}ds_{t};$
$\displaystyle\leq-(2-2\int_{\mathcal{C}_{t}}|A|^{2}ds_{t})\int_{\mathcal{C}_{t}}|\nabla
A|^{2}ds_{t}+\int|A|^{2}ds_{t}(D_{0}+a\int|A|^{2}ds_{t});$

Here we use (4.1). Let $n$ be a sufficiently large number. We can assume
$(2-2\int_{\mathcal{C}_{t}}|A|^{2}ds_{t})\geq 1$ and
$D_{0}+a\int|A|^{2}ds_{t}\leq D_{0}+1$ for all $t\in E_{n}$. Furthermore,

(4.7)
$\frac{d}{dt}\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}\leq(D_{0}+1)\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}$

For both sides of (4.7) are continuous with respect to $t$, (4.7) holds up for
$t\in[n,\infty)$. $\int_{\mathcal{C}_{t}}|A|^{2}ds_{t}$ increases at most with
exponential growth when $n$ is sufficiently large. By (4.5), we obtain
$\lim_{t\rightarrow\infty}\int|A|^{2}ds_{t}=0$.

We notice that $\tilde{M}$ is closed, all $|\nabla^{m}R|\leq C_{m}\,\text{for
all}\,m$. Together with (2.4), we have the following inequality for
$m=1,\cdots$.

(4.8) $\frac{\partial}{\partial
t}|\nabla^{m}A|^{2}\leq\triangle^{\mathcal{C}_{t}}|\nabla^{m}A|^{2}-2|\nabla^{m+1}A|^{2}\\\
+\tilde{C}_{m}\sum_{i+j+k=m}|\nabla^{i}A||\nabla^{j}A||\nabla^{k}A||\nabla^{m}A|+D_{m}\sum_{j\leq
m}|\nabla^{j}A||\nabla^{m}A|+\tilde{C}_{m}|\nabla^{m}A|;$

Here all constants in this proof are only depending on $C_{m}$.

When $m\geq 1$, we argue by the induction on $m$. Suppose
$\lim\limits_{t\rightarrow\infty}\int_{\mathcal{C}_{t}}|\nabla^{k}A|^{2}ds_{t}=0$
for $k\leq m-1$. Then by (4.1), for $k\leq m-2$,
$\lim\limits_{t\rightarrow\infty}max_{\mathcal{C}_{t}}|\nabla^{k}A|^{2}=0$.

For any $\epsilon>0$, let $t_{0}$ be a positive number sufficiently large,
such that for $t\geq t_{0}$ and $k\leq m-2$,
$\int_{\mathcal{C}_{t}}|\nabla^{m-1}A|^{2}ds_{t}\leq\epsilon$ and
$max_{\mathcal{C}_{t}}|\nabla^{k}A|^{2}\leq\epsilon$. By (4.8), we obtain the
following estimates.

$\displaystyle\frac{\partial}{\partial t}|\nabla^{m}A|^{2}$
$\displaystyle\leq\triangle^{\mathcal{C}_{t}}|\nabla^{m}A|^{2}-2|\nabla^{m+1}A|^{2}+\tilde{C}_{m}(\epsilon)(|\nabla^{m}A|^{2}$
$\displaystyle+|\nabla^{m}A||\nabla^{m-1}A|)+\tilde{D}_{m}|\nabla^{m}A|;$
$\displaystyle\frac{d}{dt}\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$
$\displaystyle\leq-2\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t}+\tilde{C}_{m}(\epsilon)(\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$
$\displaystyle+\int_{\mathcal{C}_{t}}|\nabla^{m}A||\nabla^{m-1}A|ds_{t})+\tilde{D}_{m}\int_{\mathcal{C}_{t}}|\nabla^{m}A|ds_{t};$

Here $\tilde{C}_{m}(\epsilon)$ goes to 0 as $\epsilon$ converges to 0.

Let $C>3$ be a constant determined later. If
$\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\geq
C\int_{\mathcal{C}_{t}}|\nabla^{m-1}A|^{2}ds_{t}$, the following estimates are
given by (4.2).

(4.9) $\displaystyle\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$
$\displaystyle\leq\frac{1}{C}\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t};$
$\displaystyle\int_{\mathcal{C}_{t}}|\nabla^{m-1}A|^{2}ds_{t}$
$\displaystyle\leq\frac{1}{C^{2}}\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t};$

As a result,

(4.10) $\displaystyle\int_{\mathcal{C}_{t}}|\nabla^{m}A||\nabla^{m-1}A|ds_{t}$
$\displaystyle\leq\frac{1}{\sqrt{C^{3}}}\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t};$
$\displaystyle\int_{\mathcal{C}_{t}}|\nabla^{m}A|ds_{t}$
$\displaystyle\leq\frac{\sqrt{l_{\mathcal{C}_{0}}}}{\sqrt{C}}(\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t})^{\frac{1}{2}};$

The $t$-derivative of $\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$ can be
written as followed:

(4.11)
$\displaystyle\frac{d}{dt}\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$
$\displaystyle\leq-(2-\tilde{C}_{m}(\epsilon)(\frac{1}{\sqrt{C^{3}}}+\frac{1}{C}))\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t}$
$\displaystyle+\frac{\tilde{D}_{m}}{\sqrt{C}}(\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t})^{\frac{1}{2}};$

Let $C$ be sufficiently large such that
$(2-\tilde{C}_{m}(\epsilon)(\frac{1}{\sqrt{C^{3}}}+\frac{1}{C}))\geq 1$ and
$\frac{\tilde{D}_{m}}{\sqrt{C}}\leq 2\epsilon$. (4.11) becomes

(4.12)
$\frac{d}{dt}\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\leq-\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t})^{\frac{1}{2}}(\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t})^{\frac{1}{2}}-2\epsilon);$

Now we can conclude Theorem 0.2 as follows.

  1. (1)

If $\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\geq
C\int_{\mathcal{C}_{t}}|\nabla^{m-1}A|^{2}ds_{t}$,

    1. (a)

If $(\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t})^{\frac{1}{2}}\geq
3\epsilon$, we have
$(\int_{\mathcal{C}_{t}}|\nabla^{m+1}A|^{2}ds_{t})^{\frac{1}{2}}\geq
3\epsilon$. From (4.12), then
$\frac{d}{dt}\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\leq-2\epsilon^{2}$.
$\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$ will decrease until

$(\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t})\leq(3\epsilon)^{2}$

If we can find a $t_{0}$ such that

$(\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t})^{\frac{1}{2}}\leq 3\epsilon;$

then for all $t>t_{0}$,
$\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\leq(3\epsilon)^{2}$.

    2. (b)

If $(\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t})^{\frac{1}{2}}\leq
3\epsilon$, the conclusion of theorem 0.2 is obviously true.

  2. (2)

If
$\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}<C\int_{\mathcal{C}_{t}}|\nabla^{m-1}A|^{2}ds_{t}$,
we have $\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}\leq C\epsilon$.

In a word, finally $\int_{\mathcal{C}_{t}}|\nabla^{m}A|^{2}ds_{t}$ converges
to 0 when $t\rightarrow\infty$. By (4.1), $max_{\mathcal{C}_{t}}|\nabla^{m}A|$
converges to 0 for all $m$ as $t$ goes to $\infty$.

## References

  * [AB10] Ben Andrews and Charles Baker.  Mean curvature flow of pinched submanifolds to spheres.  J. Differential Geom., 85(3):357–395, 2010.
  * [AB11] Ben Andrews and Paul Bryan.  A comparison theorem for the isoperimetric profile under curve-shortening flow.  Comm. Anal. Geom., 19(3):503–539, 2011.
  * [Alt91] Steven J. Altschuler.  Singularities of the curve shrinking flow for space curves.  J. Differential Geom., 34(2):491–514, 1991.
  * [EH89] Klaus Ecker and Gerhard Huisken.  Mean curvature evolution of entire graphs.  Ann. of Math. (2), 130(3), 1989.
  * [Gag84] M. E. Gage.  Curve shortening makes convex curves circular.  Invent. Math., 76(2):357–364, 1984.
  * [GH86] M. Gage and R. S. Hamilton.  The heat equation shrinking convex plane curves.  J. Differential Geom., 23(1):69–96, 1986.
  * [Gra87] Matthew A. Grayson.  The heat equation shrinks embedded plane curves to round points.  J. Differential Geom., 26(2):285–314, 1987.
  * [Gra89] Matthew A. Grayson.  Shortening embedded curves.  Ann. of Math. (2), 129(1):71–111, 1989.
  * [Hui84] Gerhard Huisken.  Flow by mean curvature of convex surfaces into spheres.  J. Differential Geom., 20(1):237–266, 1984.
  * [Hui86] Gerhard Huisken.  Contracting convex hypersurfaces in riemannian manifolds by their mean curvature.  Invent. Math., 84(3):463–480, 1986.
  * [Hui98] Gerhard Huisken.  A distance comparison principle for evolving curves.  Asian J. Math., 2(1):127–133, 1998.
  * [KFLZ10] F. Ye K. F. Liu, H.W. Xu and E. T. Zhao.  The extension and convergence of mean curvature flow in higher codimension, 2010.  arXiv:1104.0971v1 [math.DG].
  * [KFLZ11] F. Ye K. F. Liu, H. W. Xu and E. T. Zhao.  Mean curvature flow of higher codimension in hyperbolic spaces, 2011.  arXiv:1105.5686v1 [math.DG].
  * [MW11] Ivana Medoš and Mu-Tao Wang.  Deforming symplectomorphisms of complex projective spaces by the mean curvature flow.  J. Differential Geom., 87(2):309–341, 2011.
  * [Smo12] Knut Smoczyk.  Mean Curvature Flow in Higher Codimension Introduction and Survey, volume 17 of Springer Proceedings in Mathematics, pages 231–274.  Springer Berlin Heidelberg, 2012.
  * [SW02] Knut Smoczyk and Mu-Tao Wang.  Mean curvature flows of Lagrangians submanifolds with convex potentials.  J. Differential Geom., 62(2):243–257, 2002.
  * [TW04] Mao-Pei Tsui and Mu-Tao Wang.  Mean curvature flows and isotopy of maps between spheres.  Comm. Pure Appl. Math., 57(8):1110–1126, 2004.
  * [Wan01] Mu-Tao Wang.  Mean curvature flow of surfaces in Einstein four-manifolds.  J. Differential Geom., 57(2):301–338, 2001.
  * [Wan02] Mu-Tao Wang.  Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension.  Invent. Math., 148(3):525–543, 2002.