Eigenvaluues monotonicity of Witten-Laplacian along the mean curvature flow

TL;DR

This paper investigates how the first eigenvalue of the Witten-Laplace operator evolves along the mean curvature flow, revealing monotonic properties that could inform geometric analysis.

Contribution

It derives the evolution equation for the eigenvalue and identifies monotonic quantities during the mean curvature flow.

Findings

Derived the evolution equation for the first eigenvalue.

Identified monotonic quantities under the flow.

Provided insights into spectral properties during geometric evolution.

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.

Equations

$$F={\int }_M(R+|\nabla f{|}^2)e^{-f}d\nu$$

$$\frac{\partial F}{\partial t}(x,t)=-H(x,t)\nu (x,t),F(.,0)=F_0,x\in M,t\ge 0,$$

$$\frac{\partial F}{\partial t}(x,t)=(r(t)-H(x,t))\nu (x,t),F(.,0)=F_0,x\in M,t\ge 0,$$

$$\frac{\partial F}{\partial t}(x,t)=-H^k(x,t)\nu (x,t),F(.,0)=F_0,x\in M,t\ge 0,$$

$$\frac{\partial F}{\partial t}(x,t)=-\mathcal{S}(x,t)\nu (x,t),F(.,0)=F_0,x\in M,t\ge 0,$$

$$\frac{\partial \mathcal{S}}{\partial k_i}(k_1(x),...,k_n(x))>0,i=1,2,...n,$$

$$\Delta f=div(\nabla f)=g^{ij}({\partial }_i{\partial }_{j}f-{\Gamma }_{ij}^k{\partial }_{k}f).$$

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

$${\Delta }_{\varphi }=\Delta -\nabla \varphi .\nabla$$

$${\int }_M<\nabla u,\nabla v>d\mu =-{\int }_{M}v{\Delta }_{\varphi }ud\mu =-{\int }_{M}u{\Delta }_{\varphi }vd\mu \forall u,v\in C^{\infty }(M),$$

$$-{\Delta }_{\varphi }f={\lambda }_1(t)f,$$

$${\int }_M<\nabla f,\nabla h>d\mu ={\lambda }_1{\int }_{M}fhd\mu ,\forall h\in C^{\infty }(M).$$

$${\lambda }_1=\frac{{\int }_M|\nabla f{|}^{2}d\mu }{{\int }_M{f}^{2}d\mu }.$$

$$\lambda =\underset{f\ne 0}{\min }\left\{{\int }_M|\nabla f{|}^{2}d\mu :f\in C^{\infty }(M),{\int }_M{f}^{2}d\mu =1\right\}.$$

$$\frac{1}{1+ϵ}{g}_1\le g_2\le (1+ϵ)g_1,$$

$$\lambda (g_2)-\lambda (g_1)\le \left((1+ϵ{)}^{\frac{n}{2}+1}-(1+ϵ{)}^{-\frac{n}{2}}\right)(1+ϵ{)}^{\frac{n}{2}}\lambda (g_1).$$

$$l(t)=f_0{\left[\frac{\det (g_{ij}(t_0))}{\det (g_{ij}(t))}\right]}^{\frac{1}{2}}$$

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

$$\lambda (f,t):={\int }_M|\nabla f{|}^{2}d\mu$$

$${\int }_M{f}^{2}d\mu =1.$$

$$\frac{d}{dt}\lambda (f,t){|}_{t=t_0}=\lambda (t_0){\int }_M\mathcal{S}H{f}^{2}d\mu +2{\int }_M\mathcal{S}{h}^{ij}{\nabla }_{i}f{\nabla }_{j}fd\mu -{\int }_M|\nabla f{|}^2\mathcal{S}Hd\mu$$

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

$$\frac{d}{dt}(d{\mu }_t)=-\mathcal{S}Hd\mu$$

$$\displaystyle\frac{d}{dt}\bigg{(}|\nabla f|^{2}\bigg{)}$$

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

$${\int }_M{f}^{2}d\mu =1,$$

$$2{\int }_M{f}^{\prime }fd\mu ={\int }_M{f}^2\mathcal{S}Hd\mu ,$$

$${\int }_M<\nabla f^{\prime },\nabla f>d\mu =\lambda (t_0){\int }_M{f}^{\prime }fd\mu =\frac{\lambda (t_0)}{2}{\int }_M{f}^2\mathcal{S}Hd\mu .$$

$$\frac{d\lambda }{dt}(f,t){|}_{t=t_0}=\lambda (t_0){\int }_M{H}^2{f}^{2}d\mu +2{\int }_{M}H{h}^{ij}{\nabla }_{i}f{\nabla }_{j}fd\mu -{\int }_M|\nabla f{|}^p{H}^{2}d\mu .$$

$$\frac{d\lambda }{dt}(f,t){|}_{t=t_0}$$

Full text

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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 $x\in M$,

[TABLE]

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

[TABLE]

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.

[TABLE]

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

[TABLE]

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

[TABLE]

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 $f\in W^{1,2}(M)$,

[TABLE]

or equivalently

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

then

[TABLE]

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

[TABLE]

along the evolution equation (2.4). We assume that

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

Proof.

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

[TABLE]

On the other hand, we have

[TABLE]

and

[TABLE]

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

[TABLE]

Now, using (2.11), from the condition

[TABLE]

and the time derivative, we can get

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

Replacing this into (3.8) yields

[TABLE]

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

[TABLE]

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.

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

and

[TABLE]

Now (3.1) results that

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

and

[TABLE]

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

[TABLE]

Suppose that

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

So, from (3.8) we conclude that

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

which these and (3.8) imply that

[TABLE]

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

[TABLE]

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.