Parabolic Omori-Yau maximum principle for mean curvature flow and some applications

TL;DR

This paper develops a parabolic Omori-Yau maximum principle for mean curvature flow in spaces with curvature bounds, and applies it to preserve Gauss map images and analyze self-shrinkers.

Contribution

It introduces a new parabolic maximum principle for mean curvature flow and extends existing results to non-compact cases and self-shrinkers.

Findings

Preservation of Gauss map image under mean curvature flow.

Extension of Omori-Yau maximum principle to self-shrinkers.

Generalization of Wang's result to non-compact immersions.

Abstract

We derive a parabolic version of Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on $\ell$-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in euclidean spaces with uniform bounded second fundamental forms. This generalizes the result of Wang \cite{Wang} for compact immersions. We also prove a Omori-Yau maximum principle for properly immersed self-shrinkers, which improves a result in \cite{CJQ}.

Equations

$$u(x)=\sup u,{\nabla }^{M}u(x)=0,{\Delta }^{M}u(x)\le 0.$$

$$u(x_i)\to \sup u,|\nabla u|(x_i)\to 0,\underset{i\to \infty }{\mathrm{limsup}}{\Delta }^{M}u(x_i)\le 0.$$

$$u(x,t)\le B(1+d_{\overline{M}}(y_0,F(x,t){)}^{\alpha }),\forall (x,t)\in M\times [0,T].$$

$$u(x_i,t_i)\to \sup u,|{\nabla }^{M_{t_i}}u(x_i,t_i)|\to 0,\underset{i\to \infty }{\mathrm{liminf}}\left(\frac{\partial }{\partial t}-{\Delta }^{M_{t_i}}\right)u(x_i,t_i)\ge 0.$$

$$f(x)\le C(1+|\tilde{F}(x){|}^{\alpha })$$

$$f(x_i)\to \underset{M}{\sup }f,|\nabla f|(x_i)\to 0,\underset{i\to \infty }{\mathrm{limsup}}\mathcal{L}f(x_i)\le 0.$$

$$\frac{\partial F}{\partial t}(x,t)=\vec{H}(x,t).$$

$$\vec{H}=\text{tr}A$$

$$K_{\overline{M}}^{\ell }(w,V)=\sum _{i=1}^{\ell }⟨R(w,e_i)w,e_i⟩,$$

$$K_{\overline{M}}^{\ell }(w,V)\ge \ell C$$

$$\sum _{i=1}^{\ell }{\nabla }^{2}r(e_i,e_i)\le \ell \sqrt{C}\coth (\sqrt{C}r),$$

$$\underset{x\in M}{\sup }u(x,0)=0.$$

$${ϵ}_i\rho ({\bar{x}}_i,s_i{)}^2\le \frac{1}{i},i=1,2,\cdots .$$

$$u_i(x,t)=u(x,t)-{ϵ}_i\rho (x,t{)}^2.$$

$$u(x_i,t_i)\ge u_i(x_i,t_i)\ge u_i({\bar{x}}_i,s_i)\ge u({\bar{x}}_i,s_i)-\frac{1}{i}.$$

$$u(x_i,t_i)\to \sup u.$$

$${\nabla }^{M_{t_i}}{u}_i=0\text{ and }\left(\frac{\partial }{\partial t}-{\Delta }^{M_{t_i}}\right)u_i\ge 0\text{ at }(x_i,t_i).$$

$${\nabla }^{M_{t_i}}u={ϵ}_i{\nabla }^{M_{t_i}}{\rho }^2=2{ϵ}_i\rho (\nabla r{)}^{\top }$$

$${\Delta }^{M_{t_i}}{\rho }^2=2\sum _{i=1}^n|{\nabla }_{e_i}^{M_{t_i}}r{|}^2+2\rho \sum _{i=1}^n{\nabla }^{2}r(e_i,e_i)+2\rho \bar{g}(\nabla r,\vec{H}).$$

$$\sum _{i=1}^n{\nabla }^{2}r(e_i,e_i)\le C_1\rho .$$

$$\sum _{i=1}^n{\nabla }^{2}r(e_i,e_i)=\frac{1}{n-1}\sum _{j=1}^n\sum _{i\ne j}{\nabla }^{2}r(e_i,e_i).$$

$$\begin{array}{rl}\sum _{i=1}^n{\nabla }^{2}r(e_i,e_i)& \le \frac{n}{n-1}\sum _{j=1}^{n-1}\sqrt{C}\rho \coth (\sqrt{C}\rho )\\ & =n\sqrt{C}\rho \coth (\sqrt{C}\rho ).\end{array}$$

$$e_1=e_1^{\perp }+a{\gamma }^{\prime },$$

$$\begin{array}{rl}{\nabla }^{2}r(e_1,e_1)& =({\nabla }_{e_1}\nabla r)(e_1)\\ & =e_1⟨{\gamma }^{\prime },e_1⟩-⟨{\gamma }^{\prime },{\nabla }_{e_1}{e}_1⟩\\ & =⟨{\nabla }_{e_1}{\gamma }^{\prime },e_1⟩\\ & =⟨{\nabla }_{e_1^{\perp }+a{\gamma }^{\prime }}{\gamma }^{\prime },e_1^{\perp }+a{\gamma }^{\prime }⟩\\ & =⟨{\nabla }_{e_1^{\perp }}{\gamma }^{\prime },e_1^{\perp }⟩+a⟨{\nabla }_{e_1^{\perp }}{\gamma }^{\prime },{\gamma }^{\prime }⟩\\ & ={\nabla }^{2}r(e_1^{\perp },e_1^{\perp }).\end{array}$$

$$\begin{array}{rl}\sum _{i=1}^n{\nabla }^{2}r(e_i,e_i)& =\sum _{i=2}^n{\nabla }^{2}r(e_i,e_i)\\ & \le (n-1)\sqrt{C}\rho \coth (\sqrt{C}\rho )\end{array}$$

$$\begin{array}{rl}\sum _{i=1}^n{\nabla }^{2}r(e_i,e_i)& ={\nabla }^{2}r(e_1^{\perp },e_1^{\perp })+\sum _{i=2}^n{\nabla }^{2}r(e_i,e_i)\\ & =b^2{\nabla }^{2}r(f_1,f_1)+\sum _{i=2}^n{\nabla }^{2}r(e_i,e_i)\\ & =b^2\left({\nabla }^{2}r(f_1,f_1)+\sum _{i=2}^n{\nabla }^{2}r(e_i,e_i)\right)+(1-b^2)\sum _{i=2}^n{\nabla }^{2}r(e_i,e_i).\end{array}$$

$$\begin{array}{rl}\sum _{i=1}^n{\nabla }^{2}r(e_i,e_i)& \le b^{2}n\sqrt{C}\rho \coth (\sqrt{C}\rho )+(1-b^2)(n-1)\sqrt{C}\rho \coth (\sqrt{C}\rho )\\ & \le n\sqrt{C}\rho \coth (\sqrt{C}\rho ).\end{array}$$

$$\sum _{i=1}^n{\nabla }^{2}r(e_i,e_i)\le n\sqrt{C}\rho \coth (\sqrt{C}\rho )\le C_1\rho$$

$$\begin{array}{rl}\left(\frac{\partial }{\partial t}-{\Delta }^{M_{t_i}}\right){\rho }^2& =-2\sum _{i=1}^n|{\nabla }_{e_i}^{M_{t_i}}r{|}^2-2\rho \sum _{i=1}^n{\nabla }^{2}r(e_i,e_i)\\ & \ge -2n-2C_1\rho \end{array}$$

$$|\nabla u|\le 2{ϵ}_i\rho$$

Full text

1. Introduction

Let $(M,g)$ be a Riemannian manifold and let $u:M\to\mathbb{R}$ be a twice differentiable function. If $M$ is compact, $u$ is maximized at some point $x\in M$. At this point, basic advanced calculus implies

[TABLE]

Here $\nabla^{M}$ and $\Delta^{M}$ are respectively the gradient and Laplace operator with respect to the metric $g$. When $M$ is noncompact, a bounded function might not attain a maximum. In this situation, Omori [9] and later Yau [13] provide some noncompact versions of maximum principles. We recall the statement in [13]:

Theorem 1.1**.**

Let $(M,g)$ be a complete noncompact Riemannian manifold with bounded below Ricci curvature. Let $u:M\to\mathbb{R}$ be a bounded above twice differentiable function. Then there is a sequence $\{x_{i}\}$ in $M$ such that

[TABLE]

Maximum principles of this form are called Omori-Yau maximum principles. The assumption on the lower bound on Ricci curvature in Theorem 1.1 has been weaken in (e.g.) [3], [10]. On the other hand, various Omori-Yau type maximum principles have been proved for other elliptic operators and on solitons in geometric flows, such as Ricci solition [2] and self-shrinkers in mean curvature flows [4]. The Omori-Yau maximum principles are powerful tools in studying noncompact manifolds and have a lot of geometric applications. We refer the reader to the book [1] and the reference therein for more information.

In this paper, we derive the following parabolic version of Omori-Yau maximum principle for mean curvature flow.

Theorem 1.2** (Parabolic Omori-Yau Maximum Principle).**

Let $n\geq 2$ and $m\geq 1$. Let $(\overline{M}^{n+m},\bar{g})$ be a $n+m$-dimensional noncompact complete Riemannian manifold such that the $(n-1)$-sectional curvature of $\overline{M}$ is bounded below by $-C$ for some positive constant $C$. Let $M^{n}$ be a $n$-dimensional noncompact manifold and let $F:M^{n}\times[0,T]\to\overline{M}$ be a proper mean curvature flow. Let $u:M\times[0,T]\to\mathbb{R}$ be a continuous function which satisfies

1. (1)

$\sup_{(x,t)\in M\times[0,T]}u>\sup_{x\in M}u(\cdot,0)$, 2. (2)

$u$* is twice differentiable in $M\times(0,T]$, and* 3. (3)

(sublinear growth condition) There are $B>0$, $\alpha\in[0,1)$ and some $y_{0}\in\overline{M}$ so that

[TABLE]

Then there is a sequence of points $(x_{i},t_{i})\in M\times(0,T]$ so that

[TABLE]

We remark that the above theorem makes no assumption on the curvature of the immersion $F_{t}$. See section 2 for the definition of $\ell$-sectional curvature.

With this parabolic Omori-Yau maximum principle, we derive the following results.

In [12], the author studies the gauss map along the mean curvature flow in the euclidean space. He shows that if the image of the gauss map stays inside a geodesic submanifold in the Grassmanians, the same is also true along the flow when the initial immersion is compact. As a first application, we extend Wang’s theorem to the noncompact situation.

Theorem 1.3**.**

Let $F_{0}:M^{n}\to\mathbb{R}^{n+m}$ be a proper immersion and let $F:M^{n}\times[0,T]\to\mathbb{R}^{n+m}$ be a mean curvature flow of $F_{0}$ with uniformly bounded second fundamental form. Let $\Sigma$ be a compact totally geodesic submanifold of the Grassmanians of $n$-planes in $\mathbb{R}^{n+m}$. If the image of the Gauss map $\gamma$ satisfies $\gamma(\cdot,0)\subset\Sigma$, then $\gamma(\cdot,t)\subset\Sigma$ for all $t\in[0,T]$.

As a corollary, we have the following:

Corollary 1.1**.**

Let $F_{0}:M^{n}\to\mathbb{R}^{2n}$ be a proper Lagrangian immersion and let $F:M\times[0,T]\to\mathbb{R}^{2n}$ be a mean curvature flow with uniformly bounded second fundamental form. Then $F_{t}$ is Lagrangian for all $t\in[0,T]$.

The above result is well-known when $M$ is compact [11], [12]. Various forms of Corollary 1.1 are known to the experts (see remark 2 below).

The second application is to derive a Omori-Yau maximum principle for the $\mathcal{L}$-operator of a proper self-shrinker. The $\mathcal{L}$ operator is introduced in [5] when the authors study the entropy stability of a self-shrinker. Since then it proves to be an important operator in mean curvature flow. Using Theorem 1.2, we prove

Theorem 1.4**.**

Let $\widetilde{F}:M^{n}\to\mathbb{R}^{n+m}$ be a properly immersed self-shrinker and let $f:M^{n}\to\mathbb{R}$ be a twice differentiable function so that

[TABLE]

for some $C>0$ and $\alpha\in[0,1)$. Then there exists a sequence $\{x_{i}\}$ in $M$ so that

[TABLE]

The above theorem is a generalization of Theorem 5 in [2] since we assume weaker conditions on $f$.

In section 2, we prove the parabolic Omori-Yau maximum principle. In section 3 we prove Theorem 1.3 and in section 4 we prove Theorem 1.4. The author would like to thank Jingyi Chen for the discussion on Omori-Yau maximum principle and Kwok Kun Kwong for suggesting the work of Li and Wang [7].

2. Proof of the parabolic Omori-Yau maximum principle

Let $(\overline{M}^{n+m},\overline{g})$ be an $n+m$ dimensional complete noncompact Riemannian manifold. Let $F:M\times[0,T]\to\overline{M}$, where $M$ is an $n$-dimensional noncompact manifold, be a family of immersions $\{F(\cdot,t):M\to\overline{M}\}$ which satisfies the mean curvature flow equation

[TABLE]

Here $\vec{H}(x,t)$ is the mean curvature vector given by

[TABLE]

and $A(X,Y)=(\overline{\nabla}_{X}Y)^{\perp}$ is the second fundamental form of the immersion $F(\cdot,t)$.

Next we recall the definition of $\ell$-sectional curvature in [7]. Let $\overline{M}^{N}$ be an $N$-dimensional Riemannian manifold. Let $p\in\overline{M}$, $1\leq\ell\leq N-1$. Consider a pair $\{w,V\}$, where $w\in T_{p}\overline{M}$ and $V\subset T_{p}\overline{M}$ is a $\ell$-dimensional subspace so that $w$ is perpendicular to $V$.

Definition 2.1**.**

The $\ell$-sectional curvature of $\{w,V\}$ is given by

[TABLE]

where $R$ is the Riemann Curvature tensor on $\overline{M}$ and $\{e_{1},\cdots,e_{\ell}\}$ is any orthonormal basis of $V$.

We say that $\overline{M}$ has $\ell$-sectional curvature bounded from below by a constant $C$ if

[TABLE]

for all pairs $\{w,V\}$ at any point $p\in M$. In [7], the authors prove the following comparison theorem for the distance function $r$ on manifolds with lower bound on $\ell$-sectional curvatures.

Theorem 2.1**.**

[Theorem 1.2 in [7]] Assume that $\overline{M}$ has $\ell$-sectional curvature bounded from below by $-C$ for some $C>0$. Let $p\in M$ and $r(x)=d_{\overline{g}}(x,p)$. If $x$ is not in the cut locus of $p$ and $V\subset T_{x}\overline{M}$ is perpendicular to $\nabla r(x)$, then

[TABLE]

where $\{e_{1},\cdots,e_{\ell}\}$ is any orthonormal basis of $V$.

Now we prove Theorem 1.2. We recall that $F$ is assumed to be proper, and $u$ satisfies condition (1)-(3) in the statement of Theorem 1.2.

Proof of Theorem 1.2.

Adding a constant to $u$ if necessary, we assume

[TABLE]

By condition (1) in Theorem 1.2, we have $u(y,s)>0$ for some $(y,s)$. Note that $s>0$. Let $y_{0}\in\overline{M}$ and $r(y)=d_{\bar{g}}(y,y_{0})$ be the distance to $y_{0}$ in $\overline{M}$. Let $\rho(x,t)=r(F(x,t))$. Note that $u(y,s)-\epsilon\rho(y,s)^{2}>0$ whenever $\epsilon$ is small. Let $(\bar{x}_{i},s_{i})$ be a sequence so that $u(\bar{x}_{i},s_{i})\to\sup u\in(0,\infty]$. Let $\{\epsilon_{i}\}$ be a sequence in $(0,\epsilon)$ converging to [math] which satisfies

[TABLE]

Define

[TABLE]

Note that $u_{i}(y,s)>0$ and $u_{i}(\cdot,0)\leq 0$. Using condition (3) in Theorem 1.2, there is $R>0$ so that $u_{i}(x,t)\leq 0$ when $F(x,t)\notin B_{R}(y_{0})$, the closed ball in $\overline{M}$ centered at $y_{0}$ with radius $R$. Since $\overline{M}$ is complete, $B_{R}(y_{0})$ is a compact subset. Furthermore, $F$ is proper and thus $u_{i}$ attains a maximum at some $(x_{i},t_{i})\in M\times(0,T]$. From the choice of $(\bar{x}_{i},s_{i})$ and $\epsilon_{i}$ in (2.5),

[TABLE]

Thus we have

[TABLE]

Now we consider the derivatives of $u$ at $(x_{i},t_{i})$. If $F(x_{i},t_{i})$ is not in the cut locus of $y_{0}$, then $\rho$ is differentiable at $(x_{i},t_{i})$. Then so is $u_{i}$ and we have

[TABLE]

(The inequality holds since $t_{i}>0$). The first equality implies

[TABLE]

at $(x_{i},t_{i})$, where $(\cdot)^{\top}$ denotes the projection onto $T_{x_{i}}M_{t_{i}}$. Let $\{e_{1},\cdots,e_{n}\}$ be any orthonormal basis at $T_{x_{i}}M_{t_{i}}$ with respect to $g_{t_{i}}$. Then

[TABLE]

Next we use the lower bound on $(n-1)$-sectional curvature of $\overline{M}$ to obtain the following lemma.

Lemma 2.1**.**

There is $C_{1}=C_{1}(n,C)>0$ so that

[TABLE]

Proof of lemma.

: We consider two cases. First, if $\gamma^{\prime}$ is perpendicular to $T_{x_{i}}M_{t_{i}}$, write

[TABLE]

Since $\overline{M}$ has $(n-1)$-sectional curvature bounded from below by $-C$, we apply Theorem 2.1 to the plane $V$ spanned by $\{e_{1},\cdots,e_{n}\}\setminus\{e_{i}\}$ for each $i$. Thus

[TABLE]

Second, if $\gamma^{\prime}$ is not perpendicular to $T_{x_{i}}M_{t_{i}}$, since the right hand side of (2.9) is independent of the orthonormal basis chosen, we can assume that $e_{1}$ is parallel to the projection of $\gamma^{\prime}$ onto $T_{x_{i}}M_{t_{i}}$. Write

[TABLE]

where $e_{1}^{\perp}$ lies in the orthogonal complement of $\gamma^{\prime}$ and $a=\langle e_{1},\gamma^{\prime}\rangle$. By a direct calculation,

[TABLE]

We further split into two situations. If $e_{1}^{\perp}=0$, then the above shows $\nabla^{2}r(e_{1},e_{1})=0$. Using Theorem 2.1 we conclude

[TABLE]

If $e_{1}^{\perp}\neq 0$, write $b=\|e_{1}^{\perp}\|$ and $f_{1}=b^{-1}e_{1}^{\perp}$. Then $\{f_{1},e_{2},\cdots,e_{n}\}$ is an orthonormal basis of a $n$-dimensional plane in $T_{F(x_{i},t_{i})}\overline{M}$ orthogonal to $\gamma^{\prime}$. Using (2.11),

[TABLE]

Now we apply Theorem 2.1 again (note that the first term can be dealt with as in (2.10))

[TABLE]

Summarizing (2.10), (2.12) and (2.13), we have

[TABLE]

for some $C_{1}=C_{1}(n,C)>0$. Thus the lemma is proved. ∎

Using Lemma 2.1, (2.8) and $\frac{\partial{\rho}}{\partial t}^{2}=2\rho\bar{g}(\nabla r,\vec{H})$,

[TABLE]

(2.7) and (2.14) imply that at $(x_{i},t_{i})$ we have respectively

[TABLE]

and

[TABLE]

Note

[TABLE]

This implies

[TABLE]

Using the sub-linear growth condition (3) of $u$ and Young’s inequality, we have

[TABLE]

Thus we get

[TABLE]

Together with (2.15), (2.16) and that $\epsilon_{i}\to 0$,

[TABLE]

This proves the theorem if $\rho$ is smooth at $(x_{i},t_{i})$ for all $i$. When $\rho$ is not differentiable at some $(x_{i},t_{i})$, one applies the Calabi’s trick by considering $r_{\epsilon}(y)=d_{\bar{g}}(y,y_{\epsilon})$ instead of $r$, where $y_{\epsilon}$ is a point closed to $y_{0}$. The method is standard and thus is skipped. ∎

Remark 1*.*

Condition (1) in the above theorem is used solely to exclude the case that $u_{i}$ is maximized at $(x_{i},0)$ for some $x_{i}\in M$. The condition can be dropped if that does not happen (see the proof of Theorem 1.4).

3. Preservation of Gauss image

In this section we assume that $F_{0}:M^{n}\to\mathbb{R}^{n+m}$ is a proper immersion. Let $F:M\times[0,T]\to\mathbb{R}^{n+m}$ be a mean curvature flow starting at $F_{0}$. We further assume that the second fundamental form are uniformly bounded: there is $C_{0}>0$ so that

[TABLE]

Lemma 3.1**.**

The mapping $F$ is proper.

Proof.

Let $B_{0}(r)$ be the closed ball in $\mathbb{R}^{n+m}$ centered at the origin with radius $r$. Then by (2.1) and (3.1) we have

[TABLE]

Thus if $(x,t)\in F^{-1}(B_{0}(r))$, then $x$ is in $F_{0}^{-1}(B_{0}(r+C_{0}\sqrt{n}T))$. Let $(x_{n},t_{n})\in F^{-1}(B_{0}(r))$. Since $F_{0}$ is proper, a subsequence of $\{x_{n}\}$ converges to $x\in M$. Since $[0,T]$ is compact, a subsequence of $(x_{n},t_{n})$ converges to $(x,t)$, which must be in $F^{-1}(B_{0}(r))$ since $F$ is continuous. As $r>0$ is arbitrary, $F$ is proper. ∎

In particular, the parabolic Omori-Yau maximum principle (Theorem 1.2) can be applied in this case.

Let $G(n,m)$ be the real Grassmanians of $n$-planes in $\mathbb{R}^{n+m}$ and let

[TABLE]

be the Gauss map of $F$.

Now we prove Theorem 1.3, which is a generalization of a Theorem of Wang [12] to the noncompact situation with bounded second fundamental form.

Proof of Theorem 1.3.

Let $d:G(n,m)\to\mathbb{R}$ be the distance to $\Sigma$. That is $d(\ell)=\inf_{L\in\Sigma}d(L,\ell)$. Since $\gamma(\cdot,0)\subset\Sigma$, we have $d\circ\gamma=0$ when $t=0$. Using chain rule and (3.1), as $d\gamma=A$,

[TABLE]

Since $\Sigma\subset G(n,m)$ is compact, there is $\epsilon_{0}>0$ so that the open set

[TABLE]

lies in a small tubular neighborhood of $\Sigma$ and the function $d^{2}$ is smooth on this neighborhood. Let $T^{\prime}=\epsilon_{0}/2C_{0}$. Then the image of $f:=d^{2}\circ\gamma$ lies in this tubular neighborhood if $t\in[0,T^{\prime}]$ and $f$ is a smooth function on $M\times[0,T^{\prime}]$.

The calculation in Wang [12] shows that

[TABLE]

where $C>0$ depends on $\epsilon_{0}$ and $\Sigma$. Together with (3.1) this shows that

[TABLE]

for some positive constant $C_{1}$.

Let $g=e^{-(C_{1}+1)t}f$. Then $g$ is bounded, nonnegative and $g(\cdot,0)\equiv 0$. On the other hand,

[TABLE]

If $g$ is positive at some point, Theorem 1.2 implies the existence of a sequence $(x_{i},t_{i})$ so that

[TABLE]

Take $i\to\infty$ in (3.4) gives $0\leq-\sup g$, which contradicts that $g$ is positive somewhere. Thus $g$ and so $f$ is identically zero. This is the same as saying that $\gamma(x,t)\in\Sigma$ for all $(x,t)\in[0,T^{\prime}]$. Note that $T^{\prime}$ depends only on $C_{0}$, so we can repeat the same argument finitely many time to conclude that $\gamma(x,t)\in\Sigma$ for all $(x,t)\in M\times[0,T]$. ∎

Proof of Corollary 1.1.

An immersion is Lagrangian if and only if its Gauss map has image in the Lagrangian Grassmanians $LG(n)$, which is a totally geodesic submanifold of $G(n,n)$. The Corollary follows immediately from Theorem 1.3. ∎

Remark 2*.*

Various forms of Corollary 1.1 are known to the experts. In [8], the author comments that the argument used in [11] can be generalized to the complete noncompact case, if one assumes the following volume growth condition:

[TABLE]

The above condition is needed to apply the non-compact maximum principle in [6].

4. Omori-Yau maximum principle for self-shrinkers

In this section, we improve Theorem 5 in [2] using Theorem 1.2. The proof is more intuitive in the sense that we use essentially the fact that a self-shrinker is a self-similar solution to the mean curvature flow (possibly after reparametrization on each time slice).

First we recall some facts about self-shrinker. A self-shrinker to the mean curvature flow is an immersion $\widetilde{F}:M^{n}\to\mathbb{R}^{n+m}$ which satisfies

[TABLE]

Fix $T_{0}\in(-1,0)$. Let $\phi_{t}:M\to M$ be a family of diffeomorphisms on $M$ so that

[TABLE]

Let

[TABLE]

Then $F$ satisfies the MCF equation since by (4.1),

[TABLE]

Lastly, recall the $\mathcal{L}$ operator defined in [5]:

[TABLE]

We are now ready to prove Theorem 1.4:

Proof of Theorem 1.4.

Recall $T_{0}\in(-1,0)$. Let $u:M\times[-1,T_{0}]\to\mathbb{R}$ be given by

[TABLE]

Then

[TABLE]

Thus we can apply Theorem 1.2 (The condition that $u(\cdot,0)\equiv 0$ in Theorem 1.2 is used only to exclude the case $t_{i}=-1$. But since

[TABLE]

in order that $u_{i}$ is maximized at $(x_{i},t_{i})$ we must have $t_{i}=T_{0}$. In particular $t_{i}\neq-1$). Thus there is a sequence $(x_{i},T_{0})$ so that

[TABLE]

Using $\phi_{T_{0}}=\text{Id}$ and the definition of $u$, the first condition gives

[TABLE]

Since $\nabla^{M_{T_{0}}}=\frac{1}{\sqrt{-T_{0}}}\nabla^{M}$, the second condition gives

[TABLE]

Lastly,

[TABLE]

and

[TABLE]

Thus

[TABLE]

and the result follows. ∎

Remark 3*.*

Note that the above theorem is stronger than Theorem 5 in [2], where they assume that $f$ is bounded above (which corresponds to our case when $\alpha=0$).

Remark 4*.*

Our growth condition on $f$ is optimal: the function $f(x)=\sqrt{|x|^{2}+1}$ defined on $\mathbb{R}^{n}$ (as a self-shrinker) has linear growth, but the gradient of $f$

[TABLE]

does not tend to [math] as $f(x)\to\sup f=\infty$.

Remark 5*.*

In Theorem 4 of [2], the authors also derive a Omori-Yau maximum principle on a properly immersed self-shrinker for the Laplace operator. There they assume $u:M\to\mathbb{R}$ satisfies the growth condition

[TABLE]

We remark that the condition can be weaken to

[TABLE]

since the Laplacian of the function $|\widetilde{F}|^{2}$ satisfies better estimates: $\Delta|\widetilde{F}|^{2}\leq 2n$. Thus one can argue as in p.79 in [1] to conclude.