#  Eigenvaluues monotonicity of Witten-Laplacian along the mean curvature flow

Shahroud Azami  Department of Mathematics, Faculty of Sciences, Imam Khomeini
International University, Qazvin, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we derive the evolution equation for the first eigenvalue of
the Witten-Laplace operator acting on the space of functions along the mean
curvature flow on a closed oriented manifold. We show some interesting
monotonic quantities under the mean curvature flow.

###### Key words and phrases:

Laplace operator, eigenvalue, mean curvature flow

###### 2010 Mathematics Subject Classification:

58C40, 53C44

##  1\. Introduction

In this paper we study monotonicity of the first eigenvalue of Witten-Laplace
operator along the mean curvature flow. Over the last few years mean curvature
flow and other geometric flows have been a topic of active research interest
in both mathematics and physics and many mathematicians consider the problem
under various geometric flows. There are many results on the evolution and
monotonicity of eigenvalues of the geometric operator, for instance
$p$-Lapalcian and Witten-Laplacian, on evolving manifolds with or without
curvature assumptions.
Notice that the main study of the properties of eigenvalues of the geometric
operator, especially $p$-Laplacian, on evolving closed manifolds is still very
young and began when Perelman [10] showed that the functional

$F=\int_{M}(R+|\nabla f|^{2})e^{-f}\,d\nu$

is increasing under the Ricci flow coupled to a backward heat-type equation,
where $R$ is the scalar curvature with respect to the metric $g(t)$ and $d\nu$
denotes the volume form of the metric $g=g(t)$. The nondecreasing of the
functional $F$ yields that the first eigenvalue of the geometric operator
$-4\Delta+R$ is nondecreasing along the Ricci flow. Then, Li [8] and Cao [3]
extended the geometric operator $-4\Delta+R$ to the operator $-\Delta+cR$ and
both them proved that the first eigenvalue of the geometric operator
$-\Delta+cR$ for $c\geq\frac{1}{4}$ is nondecreasing along the Ricci flow.
Zeng and et ’al [15] studied the monotonicity of eigenvalues of the operator
$-\Delta+cR$ along the Ricci-Bourguignon flow.
On the other hand in [1, 14, 16] has been investigated the evolution for the
first eigenvalue of $p$-Laplacian along the Ricci-harmonic flow, Ricci flow
and mth mean curvature flow, respectively. Mao [9], studied the monotonicity
of the first eigenvalue of the Laplace and the p-Laplace operators under a
forced mean curvature flow. Author in [2] shown that the first eigenvalue of
Witten-Laplace operator $-\Delta_{\phi}$ is monotonic along the Ricci-
Bourguignon flow with some assumption where $\phi\in C^{2}(M)$. In [4] and
[13] have been studied the evolution for the first eigenvalue of geometric
operator $-\Delta_{\phi}+\frac{R}{2}$ under the Yamabe flow and Ricci flow,
respectively, and constructed some monotonic quantities under this flow.

The mean curvature flow is a kind of heat flow equation for the immersion and
occurs in the description of the interface evolution in certain physical
models. This is related to the property that such a flow is the gradient flow
of the area functional and therefore appears naturally in problems where a
surface energy is minimized [6, 7, 12]. Motivated by the above works, in this
paper we will study the first eigenvalue of the Witten-Laplacian operator
whose metric satisfies the mean curvature flow.

##  2\. Preliminaries

In this section, we recall some basic knowledge about mean curvature flow. Let
$M$ be a closed oriented $n$-dimensional manifold with $n\geq 2$ and
$F_{0}:M\to\mathbb{R}^{n+1}$ be a smooth immersion of $M$ into the Euclidean
space. The evolution of $M_{0}=F_{0}(M)$ by unnormalized mean curvature flow
is a one-parameter family of immersions $F:M\times[0,T)\to\mathbb{R}^{n+1}$
which satisfy the partial differential equation

(2.1) $\frac{\partial F}{\partial
t}(x,t)=-H(x,t)\nu(x,t),\,\,\,\,\,\,\,F(.\,,0)=F_{0},\,\,\,\,\,x\in
M,\,\,\,t\geq 0,$

where $H(x,t)$ and $\nu(x,t)$ are the mean curvature and the normal at the
point $F(x,t)$ of the surface $M_{t}=F(.\,,t)(M)$, respectively.

From (2.1), we can conclude that $-H(x,t)\nu(x,t)=\Delta_{M_{t}}F(x,t)$, where
$\Delta_{M_{t}}$ is the Laplace-Beltrami operator on $M_{t}$. Thus, the
unnormalized mean curvature flow is a kind of heat equation for the immersion,
a parabolic problem and it has unique solution for small times. Also, we can
apply the maximum principle for the mean curvature flow. The volume-preserving
mean curvature flow defined as

(2.2) $\frac{\partial F}{\partial
t}(x,t)=(r(t)-H(x,t))\nu(x,t),\,\,\,\,\,\,\,F(.\,,0)=F_{0},\,\,\,\,\,x\in
M,\,\,\,t\geq 0,$

where $r(t)=\frac{\int_{M}Hd\mu_{t}}{\int_{M}d\mu_{t}}$ is a function of $t$.
Short time exitance and uniqueness for solution to the mean curvature flow on
$[0,T)$ have been shown in [11]. An extended of mean curvature flow defined as

(2.3) $\frac{\partial F}{\partial
t}(x,t)=-H^{k}(x,t)\nu(x,t),\,\,\,\,\,\,\,F(.\,,0)=F_{0},\,\,\,\,\,x\in
M,\,\,\,t\geq 0,$

where $k>0$ and $H^{k}$ is $k$th power of the mean curvature. This evolution
equation called unnormalized $H^{k}$\- flow. We consider a interesting
generalization of the mean curvature flow as follows

(2.4) $\frac{\partial F}{\partial
t}(x,t)=-\mathcal{S}(x,t)\nu(x,t),\,\,\,\,\,\,\,F(.\,,0)=F_{0},\,\,\,\,\,x\in
M,\,\,\,t\geq 0,$

where $\mathcal{S}(x,t)$ is a given symmetric function of the principal
curvatures of $M_{t}$ at $x,\,t$. For instance, if $S$ equal $H$, $H-r(t)$ or
$H^{k}$ then this flow is a unnormalized mean curvature flow, the volume-
preserving mean curvature flow, $H^{k}$-flow, respectively. Ritoré and
Sinestrari [11], shown that, if $k_{1},...,k_{n}$ the principal curvatures of
$M$ and $F_{0}:M\to\mathbb{R}^{n+1}$ be a smooth immersion of a closed
$n$-dimensional manifold $M$ such that at any point $x\in M$,

$\frac{\partial\mathcal{S}}{\partial
k_{i}}(k_{1}(x),...,k_{n}(x))>0,\,\,\,\,i=1,2,...n,$

then equation (2.4) has unique smooth solution on some time interval $[0,T)$.
Also, from [11] we have

######  Lemma 2.1.

If $M_{t}$ evolves by (2.4), with $\mathcal{S}$ a symmetric function of the
curvatures homogeneous of degree $\gamma>0$, then the geometric quantities
associated to $M_{t}$ satisfy the following equations:

i:

$\frac{\partial}{\partial t}g_{ij}=-2\mathcal{S}h_{ij}$,
$\frac{\partial}{\partial t}g^{ij}=2\mathcal{S}h^{ij}$,

ii:

$\frac{\partial}{\partial t}d\mu=-\mathcal{S}Hd\mu$,

iii:

$\frac{\partial}{\partial t}h_{ij}=\frac{\partial\mathcal{S}}{\partial
h_{l}^{m}}\nabla^{l}\nabla_{m}h_{ij}+\frac{\partial^{2}\mathcal{S}}{\partial
h_{r}^{s}\partial
h_{l}^{k}}\nabla_{i}h_{r}^{s}\nabla_{j}h_{l}^{k}+\frac{\partial\mathcal{S}}{\partial
h_{l}^{k}}h_{m}^{k}h_{l}^{m}h_{ij}-(\gamma+1)h_{ik}h_{j}^{k}\mathcal{S}$,

iv:

$\frac{\partial}{\partial t}H=\frac{\partial\mathcal{S}}{\partial
h_{i}^{j}}\nabla^{i}\nabla_{j}H+\frac{\partial^{2}\mathcal{S}}{\partial
h_{r}^{s}\partial
h_{l}^{k}}\nabla^{i}h_{r}^{s}\nabla_{i}h_{l}^{k}+\frac{\partial\mathcal{S}}{\partial
h_{l}^{k}}h_{m}^{k}h_{l}^{m}H-(\gamma-1)|A|^{2}\mathcal{S}$,

where $A=(h_{ij})$, $h_{ij}=<\partial_{i}F,\partial_{j}\nu>_{e}$ is the second
fundamental form associated with $F:M\to\mathbb{R}^{n+1}$ and $<.,.>_{e}$
denotes the Euclidean scalar product in $\mathbb{R}^{n+1}$.

###  2.1. Eigenvalues of Witten-Laplace operator

In this section, we will first give the definitions for the first eigenvalue
of the Witten-Laplace operator $\Delta_{\phi}$ then we will find the formula
for the evolution of the first eigenvalue of the Witten-Laplace operator under
the evolution equation (2.4) on a closed oriented manifold. Let $M$ be a
closed oriented Riemannian manifold, and $(M,g(t))$ be a smooth solution of
the evolution equation (2.4) for $t\in[0,T)$. Let $\nabla$ be the Levi-Civita
connection on $(M,g(t))$ and $f:M\to\mathbb{R}$ be a smooth function on $M$ or
$f\in W^{1,2}(M)$ where $W^{1,2}(M)$ is the Sobolev space. The Laplacian of
$f$ is defined as

(2.5) $\Delta f=div(\nabla
f)=g^{ij}(\partial_{i}\partial_{j}f-\Gamma_{ij}^{k}\partial_{k}f).$

Let $d\nu$ be the Riemannian volume measure, and $d\mu$ be the weight volume
measure on $(M,g(t))$ related to function $\phi$; i.e.

(2.6) $d\mu=e^{-\phi(x)}d\nu$

where $\phi\in C^{2}(M)$. The Witten-Laplacian is defined by

(2.7) $\Delta_{\phi}=\Delta-\nabla\phi.\nabla$

which is a symmetric operator on $L^{2}(M,\mu)$ and satisfies the following
integration by part formula:

$\int_{M}<\nabla u,\nabla
v>d\mu=-\int_{M}v\Delta_{\phi}u\,d\mu=-\int_{M}u\Delta_{\phi}v\,d\mu\qquad\qquad\forall
u,v\in C^{\infty}(M),$

The Witten-Laplacian is generalize of Laplacian operator, for instance, when
$\phi$ is a constant function, the Witten-Laplacian operator is just the
Laplace-Belterami operator.
We say that $\lambda_{1}(t)$ is an eigenvalue of the Witten-Laplace operator
$\Delta_{\phi}$ at time $t\in[0,T)$ whenever for some $f\in W^{1,2}(M)$,

(2.8) $-\Delta_{\phi}f=\lambda_{1}(t)f,$

or equivalently

(2.9) $\int_{M}<\nabla f,\nabla
h>d\mu=\lambda_{1}\int_{M}f\,h\,d\mu,\qquad\qquad\forall h\in C^{\infty}(M).$

If in above formula we set $h=f$ then

$\lambda_{1}=\frac{\int_{M}|\nabla f|^{2}d\mu}{\int_{M}f^{2}\,d\mu}.$

therefore the first eigenvalue of the Witten-Laplace operator defined as

$\lambda=\mathop{\min}\limits_{f\neq 0}\left\\{\int_{M}|\nabla
f|^{2}d\mu:\,\,f\in C^{\infty}(M),\,\,\,\int_{M}f^{2}d\mu=1\right\\}.$

Author [2] shown that the following Lemma:

######  Lemma 2.2.

If $g_{1}$ and $g_{2}$ are two metrics on Riemannian manifold $M$ which
satisfy

$\frac{1}{1+\epsilon}g_{1}\leq g_{2}\leq(1+\epsilon)g_{1},$

then

$\lambda(g_{2})-\lambda(g_{1})\leq\left((1+\epsilon)^{\frac{n}{2}+1}-(1+\epsilon)^{-\frac{n}{2}}\right)(1+\epsilon)^{\frac{n}{2}}\lambda(g_{1}).$

In particular, $\lambda$ is a continuous function respect to the
$C^{2}$-topology.

we say that $f$ is normalized eigenfunction corresponding to $\lambda$
whenever $\lambda=\int_{M}|\nabla f|^{2}d\mu$ and $\int_{M}f^{2}d\mu=1$.
Similar method of [2], at time $t_{0}\in[0,T)$, we first let $f_{0}=f(t_{0})$
be the eigenfunction for the eigenvalue $\lambda(t_{0})$ of Witten-Laplacian.
We consider the following smooth function

$l(t)=f_{0}\left[\frac{\det(g_{ij}(t_{0}))}{\det(g_{ij}(t))}\right]^{\frac{1}{2}}$

along the evolution equation (2.4). We assume that

$f(t)=\frac{l(t)}{\left(\int_{M}(l(t))^{2}d\mu\right)^{\frac{1}{2}}}$

which $f(t)$ is smooth function under the evolution equation (2.4), satisfies
$\int_{M}f^{2}d\mu=1$ and at time $t_{0}$, $f$ is the eigenfunction for
$\lambda$ of Witten-Laplacian. Now we define a smooth eigenvalue function

(2.10) $\lambda(f,t):=\int_{M}|\nabla f|^{2}d\mu$

where $\lambda(f(t_{0}),t_{0})=\lambda(t_{0}),\,\,f$ is smooth function and
satisfies

(2.11) $\int_{M}f^{2}\,d\mu=1.$

##  3\. Variation of $\lambda(t)$

In this section, we will give some useful evolution formulas for $\lambda(t)$
under the evolution equation (2.4). Now, we give a useful proposition about
the variation of eigenvalues of Witten-Laplacian under the evolution equation
(2.4).

######  Proposition 3.1.

Let $(M^{n},g(t))$ be a solution of the evolution equation (2.4) on the smooth
closed manifold $(M^{n},g_{0})$. If $\lambda(t)$ denotes the evolution of the
first eigenvalue under the evolution equation (2.4), then

(3.1)
$\frac{d}{dt}\lambda(f,t)|_{t=t_{0}}=\lambda(t_{0})\int_{M}\mathcal{S}Hf^{2}d\mu+2\int_{M}\mathcal{S}h^{ij}\nabla_{i}f\nabla_{j}fd\mu-\int_{M}|\nabla
f|^{2}\mathcal{S}Hd\mu$

###### Proof.

According to the above description, $\lambda(f,t)$ is a smooth function and by
derivation (2.10) we have

(3.2) $\frac{d}{dt}\lambda(f,t)=\int_{M}\frac{d}{dt}(|\nabla
f|^{2})d\mu+\int_{M}|\nabla f|^{2}\frac{d}{dt}(d\mu).$

On the other hand, we have

(3.3) $\frac{d}{dt}(d\mu_{t})=-\mathcal{S}Hd\mu$

and

$\displaystyle\frac{d}{dt}\bigg{(}|\nabla f|^{2}\bigg{)}$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\bigg{(}g^{ij}\nabla_{i}f\nabla_{j}f\bigg{)}=\frac{\partial}{\partial
t}(g^{ij})\nabla_{i}f\nabla_{j}f+2g^{ij}\nabla_{i}f^{\prime}\nabla_{j}f$
$\displaystyle=$ $\displaystyle-g^{il}g^{jk}\frac{\partial}{\partial
t}(g_{lk})\nabla_{i}f\nabla_{j}f+2<\nabla f^{\prime},\nabla f>.$

Plug in (3.3), (2.4) and (3) into (3.2), yields

(3.5)
$\displaystyle\frac{d}{dt}\lambda(f,t)=2\int_{M}\left\\{\mathcal{S}h^{ij}\nabla_{i}f\nabla_{j}f+<\nabla
f^{\prime},\nabla f>\right\\}d\mu-\int_{M}|\nabla f|^{2}\mathcal{S}Hd\mu.$

Now, using (2.11), from the condition

$\int_{M}f^{2}d\mu=1,$

and the time derivative, we can get

(3.6) $2\int_{M}f^{\prime}fd\mu=\int_{M}f^{2}\mathcal{S}Hd\mu,$

(2.9) and (3.6) imply that at time $t=t_{0}$, we have

(3.7) $\int_{M}<\nabla f^{\prime},\nabla
f>d\mu=\lambda(t_{0})\int_{M}f^{\prime}fd\mu=\frac{\lambda(t_{0})}{2}\int_{M}f^{2}\mathcal{S}Hd\mu.$

Replacing (3.7) in (3.5), we obtain (3.1). ∎

Now, if in the pervious proposition we suppose that $\mathcal{S}=H$,
$\mathcal{S}=H-r$ or $\mathcal{S}=H^{k}$ then we can obtain variation of
$\lambda(t)$ under the volume-preserving, unnormalized mean curvature flow and
$H^{k}$-flow as follow:

######  Corollary 3.2.

Let $(M^{n},g(t))$ be a solution of the unnormalized mean curvature flow on
the smooth closed oriented manifold $(M^{n},g_{0})$. If $\lambda(t)$ denotes
the evolution of a first eigenvalue under the unnormalized mean curvature
flow, then

(3.8)
$\frac{d\lambda}{dt}(f,t)|_{t=t_{0}}=\lambda(t_{0})\int_{M}H^{2}f^{2}d\mu+2\int_{M}Hh^{ij}\nabla_{i}f\nabla_{j}fd\mu-\int_{M}|\nabla
f|^{p}H^{2}d\mu.$

Also, for the volume-preserving mean curvature flow, we have

(3.9) $\displaystyle\qquad\frac{d\lambda}{dt}(f,t)|_{t=t_{0}}$
$\displaystyle=$
$\displaystyle\lambda(t_{0})\int_{M}(H-r)Hf^{2}d\mu+2\int_{M}(H-r)h^{ij}\nabla_{i}f\nabla_{j}fd\mu$
$\displaystyle-\int_{M}|\nabla f|^{2}(H-r)Hd\mu.$

and for unnormalized $H^{k}$-flow, we obtain

(3.10)
$\qquad\frac{d\lambda}{dt}(f,t)|_{t=t_{0}}=\lambda(t_{0})\int_{M}H^{k+1}f^{2}d\mu+2\int_{M}H^{k}h^{ij}\nabla_{i}f\nabla_{j}fd\mu-\int_{M}|\nabla
f|^{p}H^{k+1}d\mu.$

In the following theorem, we show that the evolution of the first eigenvalue
(2.10) under the unnormalized mean curvature flow with some conditions at the
initial time, is nondecreasing.

######  Theorem 3.3.

Let $(M^{n},g(t))$ be a solution of the unnormalized mean curvature flow on
the smooth closed manifold $(M^{n},g_{0})$ and $\lambda(t)$ denotes the
evolution of the first eigenvalue under the unnormalized mean curvature flow.
At the initial time $t=0$, if $H>0$, and there exists a non-negative constant
$\epsilon$ such that

(3.11) $h_{ij}\geq\epsilon Hg_{ij},\,\,\,\,\,\frac{1}{2}\leq\epsilon$

then $\lambda(t)$ is nondecreasing under the unnormalized mean curvature flow.

###### Proof.

G. Huisken in [6], by the weak maximum principle for symmetric tensors shown
that

$h_{ij}\geq\epsilon Hg_{ij}\,\,\,\,\,\text{in}\,\,M^{n}\times[0,T),$

thus $h_{ij}\geq\epsilon Hg_{ij}$ is preserved by the unnormalized mean
curvature flow which implies that

$h^{ij}\nabla_{i}f\nabla_{j}f\geq\epsilon H|\nabla
f|^{2}\,\,\,\,\,\text{in}\,\,M^{n}\times[0,T).$

Replacing this into (3.8) yields

$\displaystyle\frac{d\lambda}{dt}(f,t)|_{t=t_{0}}$ $\displaystyle\geq$
$\displaystyle\lambda(t_{0})\int_{M}H^{2}f^{2}d\mu+(2\epsilon-1)\int_{M}|\nabla
f|^{2}H^{2}d\mu$

we arrive at $\frac{d}{dt}\lambda(f(t),t)>0$ in any sufficiently small
neighborhood of $t_{0}$, then

$\lambda(f(t_{1}),t_{1})<\lambda(f(t_{0}),t_{0})\,\,\,\,\,\,\text{on}\,\,\,\,\,\,[t_{1},t_{0}].$

Since $\lambda(f(t_{0}),t_{0})=\lambda(t_{0})$ and
$\lambda(f(t_{1}),t_{1})\geq\lambda(t_{1})$ we conclude that
$\lambda(t_{1})<\lambda(t_{0})$ which show that $\lambda(t)$ is strictly
monotone increasing in any sufficiently small neighborhood of $t_{0}$. Then
$\lambda(t)$ is strictly increasing along the evolution equation (2.4) on
$[0,T)$ because of $t_{0}$ is arbitrary. ∎

Now, suppose that the evolution equation (2.4), with
$\mathcal{S}=(H^{2}-\tilde{r}(t))$ where
$\tilde{r}(t)=\frac{\int_{M}H^{2}d\mu}{\int_{M}d\mu}$, i.e.

(3.12) $\frac{\partial F}{\partial
t}(x,t)=(\tilde{r}(t)-H^{2}(x,t))\nu(x,t),\,\,\,\,\,\,\,F(.\,,0)=F_{0},\,\,\,\,\,x\in
M,\,\,\,t\geq 0.$

In the next theorem we study some monotonic quantities dependent to the first
eigenvalue (2.10) under the equation (3.12).

######  Theorem 3.4.

Let $(M^{n},g(t))$ be a solution of the evolution equation (3.12) on the
smooth closed oriented manifold $(M^{n},g_{0})$ and $\lambda(t)$ denotes the
evolution of the first eigenvalue (2.10) under the evolution equation (3.12).
If at the initial time $t=0$, $H>0$ and there exist non-negative constant
$\epsilon$ such that $h_{ij}\geq\epsilon Hg_{ij}$ then there exist
$\varphi(t)$ and $\psi(t)$ such that the quantities

$e^{-\int_{0}^{t}\left(\psi(\tau)-\varphi(\tau)+2\epsilon\psi(\tau)\right)d\tau}\lambda(t)$

is nondecreasing along the evolution equation (3.12). Similarly,

$e^{-\int_{0}^{t}\left(\varphi(\tau)-\psi(\tau)+2\varphi(\tau)\right)d\tau}\lambda(t)$

is nonincreasing along the evolution equation (3.12).

###### Proof.

Rivas and Sinestrari in [12] shown that the inequality $\epsilon Hg_{ij}\leq
h_{ij}\leq Hg_{ij}$ preserves along the evolution equation (3.12), therefore

$\epsilon H|\nabla f|^{2}\leq h^{ij}\nabla_{i}f\nabla_{j}f\leq H|\nabla
f|^{2}\,\,\,\,\,\text{in}\,\,M^{n}\times[0,T).$

Moreover, since $H>0$ in $t=0$ by the maximum principle they shown the
following inequalities also preserve.

(3.13) $0\leq C_{1}e^{-C_{2}t}\leq H\leq C_{3},$

for some constants $C_{1},C_{2}$ and $C_{3}$. Therefore,

$C_{1}^{2}e^{-2C_{2}t}-\tilde{r}(t)\leq\mathcal{S}=(H^{2}-\tilde{r}(t))\leq
C_{3}^{2}-\tilde{r}(t)$

and

$\psi(t)=(C_{1}^{2}e^{-2C_{2}t}-\tilde{r}(t))C_{1}e^{-C_{2}t}\leq\mathcal{S}H=(H^{2}-\tilde{r}(t))H\leq(C_{3}^{2}-\tilde{r}(t))C_{3}=\varphi(t).$

Now (3.1) results that

$\displaystyle\frac{d\lambda}{dt}(f,t)|_{t=t_{0}}$ $\displaystyle\geq$
$\displaystyle\lambda(t_{0})\int_{M}\mathcal{S}Hf^{2}d\mu+2\epsilon\int_{M}\mathcal{S}H|\nabla
f|^{2}d\mu-\int_{M}|\nabla f|^{2}\mathcal{S}Hd\mu$ $\displaystyle\geq$
$\displaystyle\lambda(t_{0})\psi(t_{0})+2\epsilon\psi(t_{0})\int_{M}|\nabla
f|^{2}d\mu-\varphi(t)\int_{M}|\nabla f|^{2}d\mu$ $\displaystyle\geq$
$\displaystyle\lambda(t_{0})\left(\psi(t_{0})-\varphi(t_{0})+2\epsilon\psi(t_{0})\right)$

this results that in any enough small neighborhood of $t_{0}$ as $I_{0}$, we
have

$\frac{d}{dt}\lambda(f,t)\geq\lambda(t)\left(\psi(t)-\varphi(t)+2\epsilon\psi(t)\right).$

Integrating the last inequality with respect to variable time $t$ on
$[t_{1},t_{0}]\subset I_{0}$, we get

$\ln\frac{\lambda(f(t_{0}),t_{0})}{\lambda(f(t_{1}),t_{1})}>\int_{t_{1}}^{t_{0}}\left(\psi(\tau)-\varphi(\tau)+2\epsilon\psi(\tau)\right)d\tau.$

Since $\lambda(f(t_{0}),t_{0})=\lambda(t_{0})$ and
$\lambda(f(t_{1}),t_{1})\geq\lambda(t_{1})$ we conclude that

$\ln\frac{\lambda(t_{0})}{\lambda(t_{1})}>\int_{t_{1}}^{t_{0}}\left(\psi(\tau)-\varphi(\tau)+2\epsilon\psi(\tau)\right)d\tau,$

that is the quantity
$e^{-\int_{0}^{t}\left(\psi(\tau)-\varphi(\tau)+2\epsilon\psi(\tau)\right)d\tau}\lambda(t)$
is strictly increasing in any sufficiently small neighborhood of $t_{0}$.
Since $t_{0}$ is arbitrary, then

$e^{-\int_{0}^{t}\left(\psi(\tau)-\varphi(\tau)+2\epsilon\psi(\tau)\right)d\tau}\lambda(t)$

is nondecreasing along the mean curvature flow (3.12) on $[0,T)$. Similarly,

$e^{-\int_{0}^{t}\left(\varphi(\tau)-\psi(\tau)+2\varphi(\tau)\right)d\tau}\lambda(t)$

is nonincreasing along the mean curvature flow (3.12). ∎

In the following, we need to use Hamilton’s maximum principle for tensors on
manifolds (see [5]) which is

######  Theorem 3.5 ([5]).

Suppose that on $[0,T)$ the evolution equation

$\frac{\partial}{\partial t}M_{ij}=\Delta M_{ij}+u^{k}\nabla_{k}M_{ij}+N_{ij}$

holds, where $N_{ij}=P(M_{ij},g_{ij})$, a polynomial in $M_{ij}$ formed by
contracting products of $M_{ij}$ with itself using the metric, satisfies
$N_{ij}X^{i}X^{j}\geq 0$ for any null eigenvector $X=\\{X^{i}\\}$ of $M_{ij}$.
If $M_{ij}\geq 0$ at $t=0$ then it remains so on $[0,T)$.

######  Lemma 3.6.

Let $M_{t}^{n}$ be a solution of unnormalized $H^{k}$-flow on the smooth
closed oriented manifold $M_{0}^{n}$. If there exist positive constant
$a_{1},a_{2},...,a_{n}$ such that the initial hypersurface $M_{0}^{n}$
satisfies

(3.14)
$h_{ij}=a_{i}Hg_{ij},\,\,\,\,\,\,\text{where}\,\,\sum_{i=1}^{n}a_{i}=1,\,\,\text{and}\,\,\,\,\,\,\,0\leq\frac{1}{n}-a_{i}\leq\epsilon$

for small enough $\epsilon$ only depending on $n$, then (3.14) remains along
the unnormalized $H^{k}$-flow for any $t\in[0,T)$.

###### Proof.

By (3.14), we have

$a_{i}Hg_{ij}\leq h_{ij}\leq a_{i}Hg_{ij},\,\,\,\,\,\,\text{on}\,\,\,M_{0}.$

On the other hand, by Lemma (2.6) for $\mathcal{S}=H^{k}$ we get

(3.15) $\frac{\partial H}{\partial t}=kH^{k-1}\Delta H+k(k-1)H^{k-2}|\nabla
H|^{2}+|A|^{2}H^{k}$

and

(3.16) $\frac{\partial h_{ij}}{\partial
t}=-(k+1)H^{k}h_{il}h_{j}^{l}+kH^{k-1}\Delta
h_{ij}+k(k-1)H^{k-2}\nabla_{i}H\nabla_{j}H+kH^{k-1}|A|^{2}h_{ij}.$

From the classical maximum principle, we conclude that $H^{k}\geq 0$ for all
$k\in\mathbb{N}$ and $t\in[0,T)$. If $M_{ij}=h_{ij}-\frac{1}{2}a_{i}Hg_{ij}$
then

(3.17) $\displaystyle\frac{\partial M_{ij}}{\partial t}$ $\displaystyle=$
$\displaystyle kH^{k-1}\Delta
M_{ij}+k(k-1)H^{k-2}\nabla_{i}H\nabla_{j}H-\frac{1}{2}a_{i}k(k-1)H^{k-1}|\nabla
H|^{2}g_{ij}$
$\displaystyle-(k+1)H^{k}h_{il}h_{j}^{l}-\frac{1}{2}a_{i}|A|^{2}H^{k}g_{ij}+a_{i}H^{k+1}h_{ij}+kH^{k-1}|A|^{2}h_{ij}.$

Suppose that

(3.18)
$N_{ij}=-(k+1)H^{k}h_{il}h_{j}^{l}-\frac{1}{2}a_{i}|A|^{2}H^{k}g_{ij}+a_{i}H^{k+1}h_{ij}+kH^{k-1}|A|^{2}h_{ij}.$

By assumption of lemma $M_{ij}\geq 0$ at $t=0$. We now show that $N_{ij}$ is
nonnegative on the null-eigenvectors of $M_{ij}$. Assume that, For some vector
$X=\\{X^{i}\\}$, we have $h_{ij}X^{j}=\frac{1}{2}a_{i}HX_{i}$. Thus, we get

$\displaystyle N_{ij}X^{i}X^{j}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}(k+1)H^{k+2}a_{i}a_{j}g^{ij}X_{i}X_{j}-\frac{1}{2}a_{i}|A|^{2}H^{k}g_{ij}X^{i}X^{j}+\frac{1}{2}a_{i}^{2}H^{k+2}X^{i}X_{i}$
$\displaystyle+\frac{1}{2}ka_{i}H^{k}|A|^{2}X^{i}X_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{4}H^{k+2}(-(k+1)a_{i}a_{j}g^{ij}X_{i}X_{j}+2a_{i}^{2}X^{i}X_{i})$
$\displaystyle+\frac{1}{2}H^{k}|A|^{2}(-a_{i}g_{ij}X^{i}X^{j}+ka_{i}X^{i}X_{i})$
$\displaystyle=$
$\displaystyle\frac{1}{4}(k-1)H^{k}a_{i}(2|A|^{2}-a_{i}H^{2})X_{i}X^{i}.$

Since $|A|^{2}\geq\frac{H^{2}}{n}$, we have $N_{ij}X^{i}X^{j}\geq 0$. Not that

(3.19) $\nabla
H=\nabla(h_{ij}g^{ij})=\nabla[(M_{ij}+\frac{1}{2}a_{i}Hg_{ij})g^{ij}]=g^{ij}\nabla
M_{ij}+\frac{1}{2}\nabla H$

which implies $\nabla H=2g^{ij}\nabla M_{ij}$. Now, Hamilton’s maximum
principle to tensor implies that $M_{ij}\geq 0$ on $M_{t}$ for any
$t\in[0,T)$, that is $a_{i}Hg_{ij}\leq h_{ij}$. Similarly, on $M_{t}$ we have
$a_{i}Hg_{ij}\geq h_{ij}$ for any $t\in[0,T)$. So we get $h_{ij}=a_{i}Hg_{ij}$
on $M\times[0,T)$. ∎

######  Theorem 3.7.

Let $(M^{n},g(t))$ be a solution of the unnormalized $H^{k}$-flow on the
smooth closed oriented manifold $(M^{n},g_{0})$ and $\lambda(t)$ denotes the
evolution of a first eigenvalue under the unnormalized mean curvature flow. If
initial hypersurface $M_{0}$ satisfies (3.14) then under the unnormalized m
$H^{k}$-flow $\lambda(t)$ is nondecreasing.

###### Proof.

From Lemma 3.6 we have

(3.20) $h_{ij}=a_{i}Hg_{ij},\,\,\,\,\,\,\text{on}\,\,\,M_{t}.$

So, from (3.8) we conclude that

$\displaystyle\frac{d\lambda}{dt}(f,t)|_{t=t_{0}}$ $\displaystyle=$
$\displaystyle\lambda(t_{0})\int_{M}H^{2}f^{2}d\mu+2\int_{M}H^{2}a_{i}g^{ij}\nabla_{i}f\nabla_{j}fd\mu-\int_{M}|\nabla
f|^{2}H^{2}d\mu$ $\displaystyle\geq$
$\displaystyle\lambda(t_{0})\int_{M}H^{2}f^{2}d\mu+2(\frac{1}{n}-\epsilon-\frac{1}{2})\int_{M}H^{2}|\nabla
f|^{2}d\mu$

so for $\epsilon\leq\frac{1}{n}-\frac{1}{2}$, we have
$\frac{d}{dt}\lambda(f(t),t)>0$ in any sufficiently small neighborhood of
$t_{0}$, then

$\lambda(f(t_{1}),t_{1})<\lambda(f(t_{0}),t_{0})\,\,\,\,\,\,\text{on}\,\,\,\,\,\,[t_{1},t_{0}].$

Since $\lambda(f(t_{0}),t_{0})=\lambda(t_{0})$ and
$\lambda(f(t_{1}),t_{1})\geq\lambda(t_{1})$ we conclude that
$\lambda(t_{1})<\lambda(t_{0})$, hence $\lambda(t)$ is nondecreasing in any
sufficiently small neighborhood of $t_{0}$. Then $\lambda(t)$ is nondecreasing
along $H^{k}$-flow on $[0,T)$ because of $t_{0}$ is arbitrary. ∎

######  Example 3.8.

Let $M_{0}=\mathbb{S}_{R}^{n}(0)$ be the sphere of radius $R$ around the
origin. The mean curvature $\mathbb{S}_{R}^{n}(0)$ is equal to $\frac{n}{R}$.
If we set $F(x,t)=r(t)F_{0}(x)$, where $F_{0}(x)$ is the standard immersion of
$M_{0}$ in $\mathbb{R}^{n+1}$, then we have

$R^{\prime}(t)F_{0}(x)=\frac{\partial}{\partial
t}F(x,t)=-H(x,t)\nu(x,t)=-\frac{n}{r(t)}F_{0}(x),$

therefore the evolution of $M_{0}$ is given by sphere
$M_{t}=\mathbb{S}_{r(t)}^{n}(0)$, where the radius $r(t)$ evolves according to
the ordinary differential equation

$r^{\prime}(t)=-\frac{n}{r(t)},\,\,\,\,\,r(0)=R.$

The solution is given by $r(t)=\sqrt{R^{2}-2nt}$, thus

$H(x,t)=\frac{n}{\sqrt{R^{2}-2nt}},\,\,\,\,\,h^{ij}(x,t)=\frac{1}{\sqrt{R^{2}-2nt}}g^{ij}(t)=\frac{H(x,t)}{n}g^{ij}(t).$

which these and (3.8) imply that

(3.21)
$\frac{d\lambda}{dt}(f,t)|_{t=t_{0}}=2\frac{H^{2}(x,t_{0})}{n}\int_{M}|\nabla
f|^{2}d\mu=2\frac{H^{2}(x,t_{0})}{n}\lambda(t_{0}).$

we arrive at $\frac{d}{dt}\lambda(f(t),t)>0$ in any sufficiently small
neighborhood of $t_{0}$, therefore

$\lambda(f(t_{1}),t_{1})<\lambda(f(t_{0}),t_{0})\,\,\,\,\,\,\text{on}\,\,\,\,\,\,[t_{1},t_{0}].$

Since $\lambda(f(t_{0}),t_{0})=\lambda(t_{0})$ and
$\lambda(f(t_{1}),t_{1})\geq\lambda(t_{1})$ we get
$\lambda(t_{1})<\lambda(t_{0})$ which show that $\lambda(t)$ is strictly
monotone increasing in any enough small neighborhood of $t_{0}$. Then
$\lambda(t)$ is strictly increasing along the mean curvature flow.

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