数理解析研究所講究録
第 1817 巻 2012 年 23-37

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Complete self-shrinkers in Euclidean
space
Yejuan Peng
Graduate School of Science and Engineering, Saga University

1. Introduction
1.1. Self-shrinkers.
Let $X$ :be an -dimensional submanifold in the $(n+p)-$. If the position vector $X$ evolves indimensional Euclidean space
the direction of the mean curvature $H$ , then it gives rise to a solution
to the mean curvature flow:$M^{n}arrow \mathbb{R}^{n+p}$$n$$\mathbb{R}^{n+p}$$F(\cdot, t):M^{n}arrow \mathbb{R}^{n+p}$satisfying $F(\cdot, 0)=X(\cdot)$ and(1.1)$( \frac{\partial F(p,t)}{\partial t})^{N}=H(p, t) , p\in M,$

where $H(p, t)$ denotes the mean curvature vector of submanifold $M_{t}=$$F(M^{n}, t)$ at point $F(p, t)$ .

One of the most important problems in mean curvature flow is tounderstand the possible singularities that the flow goes through. Singularities are unavoidable as the flow contracts any closed embedded
submanifold in Euclidean space eventually leading to extinction of theevolving submanifolds.
key starting point for singularity analysis is
Huisken’s monotonicity formula.
$A$$t<t_{0}$ at. Huisken
Let
proved the following monotonicity formula for the mean curvature flow:$\rho(F, t)=(4\pi(t_{0}-t))^{-n/2}e^{-\frac{|F-p_{0}|^{2}}{4(t_{0}-t)}},$$(p_{0}, t_{0})$$\frac{d}{dt}\int_{M_{t}}\rho(F, t)dv_{t}=-\int_{M_{t}}\rho(F, t)|H+\frac{(F(p,t)-p_{0})^{N}}{2(t_{0}-t)}|^{2}dv_{t}$where$dv_{t}$is the measure on$M_{t}.$

A solution of (1.1) is called self-shrinking about$H=- \frac{(F(p,t)-p_{0})^{N}}{2(t_{0}-t)}.$$(p_{0}, t_{0})$if it satisfies

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A submanifold is said to be a self-shrinker if it is the time $t=0$ slice. That is, we call theof solution which is self-shrinking aboutsatisfying
submanifold $X$ :$(p_{0}, t_{0})$$M^{n}arrow \mathbb{R}^{n+p}$(1.2)$H=- \frac{(X-p_{0})^{N}}{2t_{0}}$a self-shrinker.
Define the functional$\mathcal{F}=\frac{1}{(4\pi t_{0})^{n/2}}\int_{M}e^{-\frac{|X-p_{0}|^{2}}{4t_{0}}}dv$

Obviously, we know that $\mathcal{F}=\int_{M}\rho(X, 0)dv.$

By computing the first variation formula, we can prove thata critical point of if and only if$M^{n}$

is$\mathcal{F}$$H=- \frac{(X-p_{0})^{N}}{2t_{0}}.$Without loss of generality, we only consider the case $p_{0}=0$ and. The self-shrinker equation (1.2) is equivalent to$t_{0}= \frac{1}{2}$(1.3)$H=-X^{N}$

Notice thatis a critical point of
can use it to characterize self-shrinkers.

$\mathcal{F}=(2\pi)^{-n/2}\int_{M}e^{-\frac{|X|^{2}}{2}}dv$$M^{n}$. So we1.2. Some typical examples.
Example 1.1. $X$ :$\mathbb{R}^{n}\hookrightarrow \mathbb{R}^{n+1}$

is a complete self-shrinker with $|A|^{2}=$$0.$Example 1.2. For any positive integers. . . $+m_{p}=n$ , the submanifold$m_{1},$$\cdots,$ $m_{p}$such that $m_{1}+$$\mathbb{S}^{m1}(\sqrt{m_{1}})\cross\cdots\cross \mathbb{S}^{m_{p}}(\sqrt{m_{p}})\subset \mathbb{R}^{n+p}$

is an -dimensional compact self-shrinker in$n$$\mathbb{R}^{n+p}$with$H=-X, |A|^{2}=p.$Here, $\mathbb{S}^{m_{t}}(r_{i})=\{X_{i}\in \mathbb{R}^{m_{t}+1} : |X_{i}|^{2}=r_{i}^{2}, i=1, \cdots,p\}$ is adimensional round sphere with mdius$m_{i^{-}}$$r_{i}.$

In particular, we consider the product submanifold, and give a simple comof two submanifolds
andputation.

$\mathbb{S}^{n_{1}}(\sqrt{n_{1}})\cross \mathbb{S}^{n_{2}}(\sqrt{n_{2}})$$\mathbb{S}^{n_{1}}(\sqrt{n_{1}})$

Example 1.3.

$\mathbb{S}^{n_{2}}(\sqrt{n_{2}})$$X:\mathbb{S}^{n_{1}}(\sqrt{n_{1}})\cross \mathbb{S}^{n}2(\sqrt{n_{2}})\hookrightarrow \mathbb{R}^{n_{1}+n+2}2.$

25
Choose local orthonormal vectors
the tangent spaces of$\mathbb{S}^{n_{1}}(\sqrt{n_{1}})$

and

and
, respectively. Put$\{e_{1}, \cdots, e_{n_{1}}\}$$\mathbb{S}^{n_{2}}(\sqrt{n_{2}})$$\{\delta_{1}, \cdots, \delta_{n_{2}}\}$

in$E_{1}=(e_{1},0),$ $\cdots,$ $E_{n_{1}}=(e_{n_{1}},0),$ $E_{n_{1}+1}=(0, \delta_{1}),$ $\cdots,$ $E_{n_{1}+n}2=(0, \delta_{n_{2}})$

Suppose that$v_{1}= \frac{1}{r_{1}}X_{1},$$v_{2}= \frac{1}{r_{2}}X_{2}$.

are the unit normal vectors.

Write$V_{1}=(v_{1},0) , V_{2}=(0, v_{2})$Obviously,
see thatbasis in$V_{1},$$V_{2}$

are the unit normal vectors in$\{E_{1}, \cdots, E_{n_{1}}, E_{n_{1}+1}, \cdots, E_{n_{1}+n_{2}}, V_{1}, V_{2}\}$.

It is easy toforms an orthonormal$\mathbb{R}^{n_{1}+n_{2}+2}$$\mathbb{R}^{n_{1}+n_{2}+2}.$

Denote byandthe second fundamental form forspectively.
By a direct calculation, we obtain that$h_{ij}^{1}$$h_{ij}^{2}$$V_{1},$$V_{2}$, re-$h_{i_{J}}^{1}=\{\begin{array}{ll}-\frac{1}{r_{1}}\delta_{ij}, 1\leq i, j\leq n_{1},0, othercases,\end{array}$$h_{ij}^{2}=\{\begin{array}{ll}-\frac{1}{r_{2}}\delta_{ij}, n_{1}+1\leq i, j\leq n_{1}+n_{2},0, othercases.\end{array}$Hence, the mean curvature vector $H=-X$ , and $|A|^{2}=2$ . Therefore,$H=-X^{N}$ and $X:S^{n1}(\sqrt{n_{1}})\cross S^{n2}(\sqrt{n_{2}})\hookrightarrow \mathbb{R}^{n1+n2+2}$ is an $(n_{1}+n_{2})-$dimensional compact self-shrinker in

Example 1.4. For positive integers$m_{p}+q=n$ , the submanifold$\mathbb{R}^{n1+n_{2}+2}$$m_{1},$$\cdots,$ $m_{p},$$q\geq 1$, with $m_{1}+\cdots+$$M^{n}=\mathbb{S}^{m_{1}}(\sqrt{m_{1}})\cross\cdots\cross \mathbb{S}^{m_{p}}(\sqrt{m_{p}})\cross \mathbb{R}^{q}\subset \mathbb{R}^{n+p}$

is an -dimensional complete noncompact self-shrinker inpolynomial volume growth which satisfies$n$$\mathbb{R}^{n+p}$with$H=-X^{N}, |A|^{2}=p.$

Example 1.5. Let $X$ :minimal surface in$\mathbb{S}^{2}(\sqrt{m(m+1)})\hookrightarrow \mathbb{S}^{2m}(\sqrt{2})\subset \mathbb{R}^{2m+1}$$\mathbb{S}^{2m}(\sqrt{2})$$\mathcal{S}elf-\mathcal{S}$hrinker with.

$c_{on\mathcal{S}}ider$

it as a surface in$H=-X, |A|^{2}=2- \frac{2}{m(m+1)}.$$\mathbb{R}^{2m+1}$

, it

be a
$i_{\mathcal{S}}a$26

2. Some known results
2.1. Complete self-shrinkers in
In this section, would like to introduce some known results. The
classification of mean convex $(i.e. H\geq 0)$ self-shrinkers began with
Huisken’s classification which can be stated as follows:$\mathbb{R}^{n+1}.$$I$

Theorem 2.1. ([8]) If $X$ : $M^{n}arrow \mathbb{R}^{n+1}(n\geq 2)$ is a mooth comThen
pact self-shrinker with nonnegative mean curvature $H$ in$X(M^{n})=S^{n}(\sqrt{n})$ .
$\mathcal{S}$$\mathbb{R}^{n+1}$

When $n=1$ , Abresch and Langer [1] had already classified all sand showed that the embedded
mooth closed self-shrinker curves inones are round circles.
In a second paper, Huisken dealt with the complete case.
$\mathbb{R}^{2}$

be a smooth complete emTheorem 2.2. ([8]) Let $X$ :$H\geq0$, polynomial volume growth.
withbedded self-shrinker inisis bounded. Then
Suppose that the second fundamental form
one of the followin9;
in(1) a round sphere$1\leq k\leq n-1$ in(2) a cylinder(3) a hyperplane in$M^{n}arrow \mathbb{R}^{n+1}$$\mathbb{R}^{n+1}$$M^{n}$$|A|$$S^{n}(\sqrt{n})$$\mathbb{R}^{n+1},$$S^{k}(\sqrt{k})\cross \mathbb{R}^{n-k},$$\mathbb{R}^{n+1},$$\mathbb{R}^{n+1}$

Then, Colding-Minicozzi showed that Huisken’s classification holds
even without the assumption that $|A|$ is bounded.

be a smooth complete emTheorem 2.3. ([3]) Let $X$ :$H\geq0$, polynomial volume growth.
withbedded self-shrinker inis one of the following;Thenin(1) a round sphere$1\leq k\leq n-1$ in(2) a cylinder(3) a hyperplane in$M^{n}arrow \mathbb{R}^{n+1}$$\mathbb{R}^{n+1}$$M^{n}$$S^{n}(\sqrt{n})$$\mathbb{R}^{n+1},$$S^{k}(\sqrt{k})\cross \mathbb{R}^{n-k},$$\mathbb{R}^{n+1},$$\mathbb{R}^{n+1}$

In Huisken and Colding-Minicozzi’s theorems, they all assumed thecondition polynomial volume growth. In fact, in order to prove their
theorems, they need to use integral formulas which are similar to Stokes
formula for compact manifolds. To guarantee that integral formulas
hold, the condition of polynomial volume growth plays a very importantrole.

2.2. Polynomial volume growth.

Definition 2.1. We say that a submanifold
andvolume growth if there exist constants$M^{n}$$C$in

polynomialthat for all $r\geq 1,$$\mathbb{R}^{n+p}ha\mathcal{S}$$d_{\mathcal{S}}uch$

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there

$hold_{\mathcal{S}}$$Vol(B_{r}(0)\cap M)\leq Cr^{d},$wherethe origin.

$B_{r}(0)$is an Euclidean ball in$\mathbb{R}^{n+p}$with radius$r$

and centered at

In 2011, Ding-Xin, Cheng-Zhou proved the following theorem:
Theorem 2.4. ([6],[5]) Any complete non-compact properly immersedin
Euclidean volume growth at most. Preself-shrinker$M^{n}$$\mathbb{R}^{n+p}ha\mathcal{S}$$ci_{\mathcal{S}}ely,$$Vol(B_{r}(0)\cap M)\leq Cr^{n}$for$r\geq 1.$

Proof. From $H=-X^{N}$ , we have$\triangle|X|^{2}=2n-2|X^{N}|^{2}, \nabla|X|^{2}=2X^{T}$

Since $X$ is a proper immersion, it is well defined for$I(t)=t^{-\frac{n}{2}} \int_{D_{r}}e^{-\frac{|X|^{2}}{2t}}dv,$

is a standard
where $D_{r}=M\cap B_{r}(0)$ for the submanifold inball inwith radius and centered at the origin. Then$\mathbb{R}^{n+p},$$\mathbb{R}^{n+p}$$B_{r}(0)$$r$$I’(t)=t^{-\frac{n}{2}-1} \int_{D_{r}}(-\frac{n}{2}+\frac{|X|^{2}}{2t})e^{-\frac{|X|^{2}}{2t}}dv.$

On the other hand,$div(e^{-\frac{|X|^{2}}{2t}} \nabla(\frac{|X|^{2}}{2}))=e^{-\frac{|X|^{2}}{2t}}(\triangle\frac{|X|^{2}}{2}-\frac{1}{4t}|\nabla|X|^{2}|^{2})$$=e^{-\frac{|X|^{2}}{2t}}(n-|X^{N}|^{2}- \frac{1}{t}|X^{T}|^{2})$$=e^{-\frac{|X|^{2}}{2t}}(n-|X^{N}|^{2}- \frac{1}{t}|X|^{2}+\frac{|X^{N}|^{2}}{t})$$=e^{-\frac{|X|^{2}}{2t}}(n- \frac{|X|^{2}}{t}+(\frac{1}{t}-1)|X^{N}|^{2})$$\leq e^{-\frac{|X|^{2}}{2t}}(n-\frac{|X|^{2}}{t})$

,

$($

for

$t\geq 1)$.

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For $t\geq 1,$$I’(t)=t^{-\frac{n}{2}-1} \int_{D_{r}}(-\frac{n}{2}+\frac{|X|^{2}}{2t})e^{-\frac{|X|^{2}}{2t}}dv$$\leq-\frac{1}{2}t^{-\frac{n}{2}-1}\int_{D_{r}}div(e^{-\frac{|X|^{2}}{2t}}\nabla(\frac{|X|^{2}}{2}))dv$$=- \frac{1}{2}t^{-\frac{n}{2}-1}\int_{\partial D_{r}}\langle e^{-\frac{|X|^{2}}{2t}}\nabla\frac{|X|^{2}}{2}, \frac{\nabla\frac{|X|^{2}}{2}}{|\nabla\frac{|X|^{2}}{2}|}\rangle dv$$=- \frac{1}{2}t^{-\frac{n}{2}-1}\int_{\partial D_{r}}e^{-\frac{|X|^{2}}{2t}}|\nabla\frac{|X|^{2}}{2}|dv\leq 0,$

then we get $I(r^{2})\leq I(1)$ for $r\geq 1$ , i.e.
$r^{-n} \int_{D_{r}}e^{-\frac{|X|^{2}}{2r^{l}}}dv\leq\int_{D_{r}}e^{-\frac{|X|^{2}}{2}}dv,$

but on the other hand,(2.1)$\frac{|X|^{2}}{r^{2}}\leq 1$

holds in$D_{r}$. Therefore,$e^{-\frac{1}{2}}r^{-n} \int_{D_{r}}dv\leq r^{-n}\int_{D_{r}}e^{-\frac{|X|^{2}}{2r^{l}}}dv\leq\int_{D_{r}}e^{-\frac{|X|^{2}}{2}}dv.$

Note that$\int_{D_{r}\backslash D_{r-1}}e^{-\frac{|X|^{2}}{2}}dv\leq e^{-\frac{(r-1)^{2}}{2}}e^{\frac{1}{2}}r^{n}\int_{D_{r}}e^{-\frac{|X|^{2}}{2}}dv\leq e^{-r}\int_{D_{r}}e^{-\frac{|X|^{2}}{2}}dv,$

the last inequality holds forabove inequality implies$r\geq r_{0}$with$r_{0}$

sufficiently large. Then the$\cdot$$\int_{D_{r}}e^{-\frac{|X|^{2}}{2}}dv\leq\frac{1}{1-e^{-r}}\int_{D_{r-1}}e^{-\frac{|X|^{2}}{2}}dv.$

Then for any $N,$$\int_{D_{r}}e^{-\frac{|X|^{2}}{2}}dv\leq\prod_{io+N=0}^{N}\frac{1}{1-e^{-(r_{0}+i)}}\int_{D_{r_{0}-1}}e^{-\frac{|X|^{2}}{2}}dv.$

This implies that$\int_{M}e^{-\frac{|X|^{2}}{2}}dv\leq C_{1}\int_{D_{r_{0}}-1}e^{-\frac{|X|^{2}}{2}}dv<C_{2}.$From (2.1), we have$\int_{D_{r}}dv\leq e^{\frac{1}{2}}r^{n}\int_{M}e^{-\frac{|X|^{2}}{2}}dv\leq Cr^{n}$$\square$29
Remark 2.1. Take

$self_{-\mathcal{S}}hrinker$$\mathbb{S}^{k}(\sqrt{k})\cross \mathbb{R}^{n-k}, 0\leq k\leq n, |H|=\sqrt{k},$then the above$e\mathcal{S}$timate implies that
$Vol(B_{r}(0)\cap M)=Cr^{n-k},$which is sharp.

By Theorem 2.4, it’s not hard to prove the following lemma:

Lemma 2.1. If self-shrinker
then for any $m\geq 0$ , we have$M^{n}arrow \mathbb{R}^{n+p}ha\mathcal{S}$polynomial volume growth,$\int_{M}|X|^{m}e^{-\frac{|X|^{2}}{2}}dv<\infty.$

Proof. We can prove our lemma by the following$\int_{M}|X|^{m}e^{-\frac{|X|^{2}}{2}}dv=\sum_{j=0}^{+\infty}\int_{D_{j}+1\backslash D_{j}}|X|^{m}e^{-\frac{|X|^{2}}{2}}dv$$\leq\sum_{j=0}^{+\infty}(j+1)^{m}\int_{D_{j+1}}e^{-\frac{j^{2}}{2}}dv$$= \sum_{j=0}^{+\infty}e^{-L^{2}}$$\leq C\sum_{j=0}^{+\infty}(j+1)^{m+n}e^{-\frac{j^{2}}{2}}$$\square$2.3. Outlines of proofs.
The following linear operator which was introduced and studied firstly on self-shrinker by Colding-Minicozzi (see (3.7) in [3]):(2.2)$\mathcal{L}=\triangle-\langle X, \nabla(\cdot)\rangle=e^{\frac{|X|^{2}}{2}}div(e^{-\frac{|X|^{2}}{2}}\nabla\cdot)$

where and denote the Laplacian and the gradient operator on thedenotes the standard inner productself-shrinker, respectively andof$\triangle$$\nabla$$\langle\cdot,$$\cdot\rangle$$\mathbb{R}^{n+p}.$

The operator$\mathcal{L}$is self-adjoint in a weighted$L^{2}$-space:30
Lemma 2.2. If $X$ :with compact upport, andis a submanifold,-function, thenis a$M^{n}arrow \mathbb{R}^{n+p}$$\mathcal{S}$

(2.3)

$v$

$u$

is a

$C^{1}$-function$C^{2}$$\int_{M}u(\mathcal{L}v)e^{-\frac{|X|^{2}}{2}}=-\int_{M}\langle\nabla v, \nablau\rangle e^{-\frac{|X|^{2}}{2}}$Outline of proof of Theorem 2.1
Step 1. From $\mathcal{L}H=H-|A|^{2}H$ , and by maximum principle, weknow that $H$ satisfies the strict inequality $H>0.$

Step 2. By lemma 2.2, we can get a key inequality:$\int_{M}|\nabla|A|-|A|\nabla logH|^{2}e^{-\frac{|X|^{2}}{2}}\leq 0.$

equivalently, $|A|=\beta H$ , where$\beta$is some positive constant.

Step 3. By $|A|=\beta H$ , we can prove theorem 2.1.
The following corollary is an extension of lemma 2.1, used later tojustify computations when $M$ is not closed.
is a complete submaniCorollary 2.1. Suppose that $X$ :are -functions satisfying
fold, if$M^{n}arrow \mathbb{R}^{n+p}$$C^{2}$$u,$ $v$$\int_{M}(|u\nabla v|+|Vu||Vv|+|u\mathcal{L}v|)e^{-\frac{|X|^{2}}{2}}<+\infty$

then we get(2.4)

Proof. Let$\int_{M}u(\mathcal{L}v)e^{-\frac{|X|^{2}}{2}}=-\int_{M}\langle\nabla v, \nablau\rangle e^{-\frac{|X|^{2}}{2}}$$\eta$

be a cut-off function with compact support on $M$ . Then$\int_{M}\eta u(\mathcal{L}v)e^{-\frac{|X|^{2}}{2}}=\int_{M}\eta udiv(e^{-\frac{|X|^{2}}{2}}\nabla v)$$=- \int_{M}\langle\nabla(\eta u), \nabla v\rangle e^{-\frac{|X|^{2}}{2}}$$=- \int_{M}u\langle\nabla\eta, \nabla v\rangle e^{-\frac{|X|^{2}}{2}}-\int_{M}\eta\langle\nabla u, \nabla v\rangle e^{-\frac{|X|^{2}}{2}}$tobe a cut-off fuction linearly to zero from
$B_{j}=M\cap B_{j}(0)$ withis the Euclidean ball of radius
are bounded by one,andat at the origin. SinceLet$B_{j}$$\eta=\eta_{j}$$B_{j}(0)$$|\eta_{j}|$$|\nabla\eta_{j}|$, wherecenteredand$B_{j+1}$$j$$\eta_{j}arrow 1$$|\nabla\eta_{j}|arrow 0.$$\nabla v\rangle e^{-\frac{|X|^{2}}{2}}$$\int_{M}\eta_{j}u(\mathcal{L}v)e^{-\frac{|X|^{2}}{2}}=-\int_{M}u\langle\nabla\eta_{j},$ $\nabla v\rangle e^{-\frac{|X|^{2}}{2}}-\int_{M}\eta_{j}\langle\nabla u,$

By dominated convergence theorem, we can complete the $pro$ of ofCorollary 2. 1.
$\square$31
Outline of proofs of Theorem 2.2 and Theorem 2.3
Step 1. From $\mathcal{L}H=H-|A|^{2}H$ , and by maximum principle, weknow that $H$ satisfies the strict inequality $H>0.$
Step 2. We show that $|A|=\beta H$ for a constant $\beta>0$ . This geometricidentity is a key for proving the classification.
$\int_{M}|\nabla|A|-|A|\nabla logH|^{2}e^{-\frac{|X|^{2}}{2}}$$= \int_{M}|\nabla|A||^{2}e^{-\frac{|X|^{2}}{2}}-\int_{M}\langle\nabla|A|^{2},$ $\nabla logH\rangle e^{-\frac{|X|^{2}}{2}}+\int_{M}|A|^{2}|\nabla logH|^{2}e^{-\frac{|X|^{2}}{2}}$In order to insure the above integrations to make sense, the following
lemma is needed.

Proposition 2.1. Let $X$ :self-shrinker with $H>0$ . If

be an -dimensional complete
has polynomial volume growth, then$M^{n}arrow \mathbb{R}^{n+p}$$M^{n}$$n$$\int_{M}(|A|^{2}|\nabla logH|+|\nabla|A|^{2}||\nabla logH|+|A|^{2}|\mathcal{L}logH|)e^{-\frac{|X|^{2}}{2}}<\infty,$$\int_{M}(|A||\nabla|A||+|\nabla|A||^{2}+|A||\mathcal{L}|A||)e^{-\frac{|X|^{2}}{2}}<\infty.$

Let $u=|A|,$ $v=|A|$ in corollary 2.1, then we obtain that by proposition 2. 1$\int_{M}(|A||\nabla|A||+|\nabla|A||^{2}+|A||\mathcal{L}|A||)e^{-\frac{|X|^{2}}{2}}<\infty,$

which satisfies the condition of corollary 2.1, so we have$\int_{M}|\nabla|A||^{2}e^{-\frac{|X|^{2}}{2}}=-\int_{M}|A|(\mathcal{L}|A|)e^{-\frac{|X|^{2}}{2}}$Let $u=|A|^{2},$ $v=logH$ in corollary 2.1, then we obtain that byproposition 2. 1$\int_{M}(|A|^{2}|\nabla logH|+|\nabla|A|^{2}||\nabla logH|+|A|^{2}|\mathcal{L}logH|)e^{-\frac{|X|^{2}}{2}}<\infty,$

which satisfies the condition of corollary 2.1, so we have$\int_{M}\langle\nabla|A|^{2}, \nabla logH\rangle e^{-\frac{|X|^{2}}{2}}=-\int_{M}|A|^{2}(\mathcal{L}logH)e^{-\frac{|X|^{2}}{2}}$Step 3. From $|A|=\beta H$ , we can get theorem 2.2 and theorem 2.3 bya series of discussions.

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2.4. Complete self-shrinkers in$\mathbb{R}^{n+p}.$

defined by (2.2), $N.$
Recently, based on the differential operator
Q. Le and N. Sesum [10] proved a gap theorem for self-shrinkers of
is a smooth complete
codimension one: if a hypersurface
embedded self-shrinker and with polynomial volume growth, and satis a hyperplane.
isfies $|A|^{2}<1$ , then$\mathcal{L}$$M^{n}\subset \mathbb{R}^{n+1}$$M^{n}$In 2012, Cao-Li extended Le-Sesum’s result to any codimension:
Theorem 2.5. ([2]) Let $X$ :shrinker with polynomimal volume growth inof the second fundamental form satisfies

be a complete self. If the squared norm$M^{n}arrow \mathbb{R}^{n+p}(p\geq 1)$$\mathbb{R}^{n+p}$$|A|^{2}\leq 1.$

Then $M$ is one of the following:in(1) a round phere S(2) a cylinder(3) a hyperplane in$(\sqrt{n})$$\mathcal{S}$$S^{k}(\sqrt{k})\cross \mathbb{R}^{n-k},$$\mathbb{R}^{n+1},$$1\leq k\leq n-1$in$\mathbb{R}^{n+1},$$\mathbb{R}^{n+1}$

Question 2.1: Is it possible to remove the assumption on polynomial
volume growth in Theorem 2.5?

3. Classifications of self-shrinkers in$\mathbb{R}^{n+p}$

We study complete self-shrinkers without the assumption on polynomial volume growth and obtain the following results by extending thegeneralized maximum principle of Yau to -operator. Our main results
can be stated as follows:
be an -dimensional
Theorem 3.1. ([4]) Let $X$ :, then one of the following holds:
complete self-shrinker in(1) $\sup|A|\geq 1,$
is a hyperplane in(2) $|A|\equiv 0,$ $i.e.$$\mathcal{L}$$M^{n}arrow \mathbb{R}^{n+p}(p\geq 1)$$n$$\mathbb{R}^{n+p}$$\mathbb{R}^{n+1}$$M^{n}$

Corollary 3.1. Let $X$ :shrinker, and satisfy$M^{n}arrow \mathbb{R}^{n+p}$$(p\geq 1)$

be a complete self-$\sup|A|^{2}<1.$

Then $M$ is a hyperplane in$\mathbb{R}^{n+1}$

and the cylinder
Remark 3.1. The round sphere$1\leq k\leq n-1$ are complete self-shrinkers inwith $|A|=1.$
Thus, our result is sharp.
$S^{k}(\sqrt{k})\cross$$S^{n}(\sqrt{n})$$\mathbb{R}^{n-k},$$\mathbb{R}^{n+1}$

Remark 3.2. From corollary 3.1 and remark 3.1, we can easily knowthat theorem 2.5 holds without assumption on polynomial volume growth,which gives an affirmative answer to Question 2.1.

33
In order to prove our results, first of all, we extend the generalized
maximum principle of Yau to -operator on complete self-shrinkers.
We call the following theorem as genemlized maximum principle for-opemtor which can be stated as follows:$\mathcal{L}$$\mathcal{L}$Theorem 3.2. ([4]) Let $X$ :be a complete self-shrinker
with Ricci curvature bounded from below. Let
be any-functionbounded from above on this self-shrinker. Then, there exists a sequence, such thatof points$M^{n}arrow \mathbb{R}^{n+p}$$C^{2}$$f$$\{p_{k}\}\subset M^{n}$$\lim_{karrow\infty}f(X(p_{k}))=supf,\lim_{karrow\infty}|\nabla f|(X(p_{k}))=0,$

Since$\lim_{karrow}\sup_{\infty}\mathcal{L}f(X(p_{k}))\leq 0.$

is a complete self-shrinker, the self-shrinker equation (1.3)
is equivalent to(3.1)$M^{n}$$H^{\alpha}=-\langle X, e_{\alpha}\rangle, n+1\leq\alpha\leq n+p.$

Taking covariant derivative of (3.1) with respect to(3.2)$e_{j}$$e_{i}$, we have$H_{i}^{\alpha}= \sum_{k}h_{ik}^{\alpha}\langle X, e_{k}\rangle, 1\leq i\leq n, n+1\leq\alpha\leq n+p.$

Furthermore, by taking covariant derivative of (3.2) with respect to, we have$H_{i_{J}}^{\alpha}= \sum_{k}h_{ikj}^{\alpha}\langle X, e_{k}\rangle+h_{i_{J}}^{\alpha}+\sum_{\beta,k}h_{ik}^{\alpha}h_{kj}^{\beta}\langle X, e_{\beta}\rangle$(3.3)$= \sum_{k}h_{ik_{J}}^{\alpha}\langle X, e_{k}\rangle+h_{tJ}^{\alpha}-\sum_{\beta,k}H^{\beta}h_{ik}^{\alpha}h_{k_{J}}^{\beta},$

According to (3.3), we obtain(3.4)$\mathcal{L}|H|^{2}=2|\nabla H|^{2}+2|H|^{2}-2\sum_{\alpha,\beta,i,k}H^{\alpha}H^{\beta}h_{ik}^{\alpha}h_{ik}^{\beta}.$

Outline of proof of Theorem 3.1
Step 1. By using Cauchy-Schwarz inequality, we can obtain that$\mathcal{L}|H|^{2}=2|\nabla H|^{2}+2|H|^{2}-2\sum_{\alpha,\beta,i,k}H^{\alpha}H^{\beta}h_{ik}^{\alpha}h_{ik}^{\beta}$$\geq 2|\nabla H|^{2}+2(1-|A|^{2})|H|^{2}$

Step 2. If $\sup|A|^{2}\geq 1$ , there is nothing to do.
If$\sup|A|^{2}<1,\sum_{\alpha,i,j}(h_{ij}^{\alpha})^{2}<1$

is bounded from below and. We can know that the Ricci curvature$|H|^{2}$is bounded from above.

34
Step 3. By applying the generalized maximum principle for -operator, we haveto the function
lim $sup\mathcal{L}|H|^{2}\geq 2(1-\sup|A|^{2})\sup|H|^{2}$
From $\sup|A|^{2}<1$ , we have $\sup|H|^{2}=0$ , that is $H\equiv 0$ . Fromis a smooththe self-shrinker equation (1.3), we can know thatminimal cone. Hence, it follows that the only smooth cone throughis a hyperplane.
hrinker.
be a complete
Theorem 3.3. ([4]) Let $X$ :$inf|A|^{2}\leq
1.$
is bounded, thenIf $\inf H^{2}>0$ and$\mathcal{L}$$H^{2}$$0\geq$$M^{n}$$0$$M^{n}arrow \mathbb{R}^{n+1}$$self-\mathcal{S}$$|A|^{2}$Corollary 3.2. ([4]) Let $X$ :is constant, thenIf $\inf H^{2}>0$ andor the cylinder
sphere$M^{n}arrow \mathbb{R}^{n+1}$$|A|^{2}$be a complete self-shrinker.
is the roundand$|A|^{2}\equiv 1$$S^{k}(\sqrt{k})\cross \mathbb{R}^{n-k},$$S^{n}(\sqrt{n})$$M^{n}$$1\leq k\leq n-1.$

Remark 3.3. Notice that condition $\inf H^{2}>0$ is necessary since Angenent has proved that there exist embedded self-shrinkers fromwith $\inf H^{2}=0.$
intoOutline of proof of Theorem 3.3
Step 1. By a direct calculation, we can get$S^{1}\cross$$S^{n-1}$$\mathbb{R}^{n+1}$$\mathcal{L}H=(1-|A|^{2})H.$

Step 2. Since
is bounded, we know that $H$ is bounded and the
Ricci curvature is bounded from below.
Since $\inf H^{2}>0$ , we can divide it into two cases: $\inf H>0$ or$|A|^{2}$$\sup H<0.$

If $\inf H>0$ , applying the generalized maximum principle for-operator to $-H$ , we obtain(1– $inf|A|^{2}$ ) $\inf H.$$1^{o}$$\mathcal{L}$$0\leq$Hence, $inf|A|^{2}\leq 1.$
If $\sup H<0$ , applying the generalized maximum principle for-operator to $H$ , we obtain$2^{o}$$\mathcal{L}$$0\geq$lim $sup\mathcal{L}H\geq(1-\sup|A|^{2})supH.$

Then, $inf|A|^{2}\leq\sup|A|^{2}\leq 1$ . This finishes the proof of Theorem

3.3.
Outline of proof of Corollary 3.3
Step 1.
$0,$$|A|^{2}\equiv 1.$

In fact, from Theorem 3.3, we know that $inf|A|^{2}\leq 1$ . Since $inf|H|^{2}>$$H\neq 0$ , that is
is not totally geodesic. According to Theorem$M^{n}$

35
3.1, we have $\sup|A|^{2}\geq 1$ . From our assumption that
so we get$|A|^{2}$is a constant,$|A|^{2}\equiv 1.$

Step 2. By a direct calculation, we can obtain that$\frac{1}{2}\mathcal{L}|A|^{2}=|\nabla A|^{2}+|A|^{2}(1-|A|^{2})$.

Step 3.

Substituting $|A|^{2}=1$ into the above equation, we know
. According to the theorem of Lawson [9], we know thatis isometric to the round sphereor the cylinder$|\nabla A|^{2}\equiv 0$$M^{n}$$S^{k}(\sqrt{k})\cross$$S^{n}(\sqrt{n})$$\mathbb{R}^{n-k},$$1\leq k\leq n-1.$

4. Complete proper self-shrinkers of dimension 2
and 3
4.1. Complete proper self-shrinkers of dimension 3.
First, we study complete proper self-shrinkers of dimension 3 inwith constant squared norm of the second fundamental form, and then,obtain a complete classification which can be stated as follows:
Theorem 4 $.1.([4])LetX:M^{3}arrow \mathbb{R}^{5}bea3-$$properself-$ shrinkerwithH $>0$
parallel in the normal bundle ofand the squared norm of the second
isfundamental form constant, then $M$ is one of the following:$1\leq k\leq 3$ with $|A|^{2}=1,$(1)(2)
with $|A|^{2}=2,$(3)with $|A|^{2}=2,$(4) the three dimensional minimal isoparametric Cartan hypersurface
with $|A|^{2}=3.$$\mathbb{R}^{5}$dimensionalcomleteIftheprincipalnormalv$= \frac{pH}{H}is$$M^{3}$$S^{k}(\sqrt{k})\cross \mathbb{R}^{3-k},$$S^{1}(1)\cross S^{1}(1)\cross \mathbb{R}$$S^{1}(1)\cross S^{2}(\sqrt{2})$Outline of proof of Theorem 4.1
Step 1. Since
is a complete proper self-shrinker, we know thathas polynomial volume growth. By a result of Li-Wei [11], we know
thatis isometric toor, where is an AbreschLanger curve andis a minimal hypersurface in sphere.
$M^{3}$$M^{3}$$M^{3}$$\Gamma\cross \mathbb{R}^{2}$$\tilde{M}^{r}\cross \mathbb{R}^{3-r}$$\Gamma$$\tilde{M}$$S^{r+1}(\sqrt{r})$Step 2. Since
is constant, then the Abresch-Langer curve
be a circle. In this case,is isometric to$|A|^{2}$$M^{3}$$\Gamma$must$S^{1}(1)\cross \mathbb{R}^{2}$

Step 3. If $|A|^{2}\leq 1$ , by Cao-Li’s results (see Theorem 2.5), we have
$1\leq k\leq 3.$$|A|^{2}=1$ andis$M^{3}$$S^{k}(\sqrt{k})\cross \mathbb{R}^{3-k},$

Step 4. We consider the case of $|A|^{2}>1.$
When $r=2,\tilde{M}$ is the Clifford torus $S^{1}(1)\cross S^{1}(1)$ in$\cdot$$S^{3}(\sqrt{2})$

;

36
in
When $r=3,\tilde{M}$ is the Clifford torus
with $|A|^{2}=2$ or the three dimensional minimal isoparametric Cartan$S^{1}(1)\cross S^{2}(\sqrt{2})$hypersurface in$S^{4}(\sqrt{3})$$S^{4}(\sqrt{3})$with $|A|^{2}=3.$

4.2. Complete proper self-shrinkers of dimension 2.
Furthermore, we study complete proper self-shrinker of dimension 2,
and give a complete classification theorem for arbitrary codimension.
$>0.Iftheprincipalnor \gamma nal\nu=\frac{pH}{H}isTheorem4.2.([4])LetX:M^{2}arrow \mathbb{R}^{2+p}bea2-dimensionalcomlete$$properself-shr’$inkerwithH

and the squared norm of the second
pamllel in the normal bundle of, then $M$ is one of the following:
fundamental form is$1\leq k\leq 2$ with $|A|^{2}=1,$(1)with $p=2m-1$(2) the Boruvka sphere $S^{2}(\sqrt{m(m+1)})$ in
and $|A|^{2}=2- \frac{2}{m(m+1)},$$M^{2}$$con\mathcal{S}tant$$S^{k}(\sqrt{k})\cross \mathbb{R}^{2-k},$$S^{2m}(\sqrt{2})$(3) a compact flat minimal surface in $S^{2m+1}(\sqrt{2})$ with $p=2m$ and$|A|^{2}=2.$$O$

utline of proof of Theorem 4.2

Step 1. Since
is a complete proper self-shrinker, we know thathas polynomial volume growth. By a result of Li-Wei [11], we know
, where is an Abresch-Langeror
is isometric tothat.
is a compact minimal hypersurface in spherecurve and$M^{2}$$M^{2}$$M^{2}$$\Gamma\cross \mathbb{R}^{1}$$\tilde{M}^{2}$$\Gamma$$S^{p+1}(\sqrt{2})$$\tilde{M}$

Step 2. Since
is constant, then the Abresch-Langer curveis isometric to
be a circle. In this case,$|A|^{2}$$M^{2}$

Step 3. If$|A|^{2}\leq 1$$\Gamma$must$S^{1}(1)\cross \mathbb{R}.$, we have $|A|^{2}=1$ and$M^{2}$

is$S^{1}(1)\cross \mathbb{R}.$Step 4. Consider the case of $|A|^{2}>1$ . We can prove thatwithis isometric to a Boruvka sphere $S^{2}(\sqrt{m(m+1)})$ in$p=2m-1$ and $|A|^{2}=2- \frac{2}{m(m+1)}$ or a compact flat minimal surface$M^{2}$$S^{2m}(\sqrt{2})$in $S^{2m+1}(\sqrt{2})$ with $p=2m$ and $|A|^{2}=2.$

References[1] U. Abresch and J. Langer, The normahzed curve shortening flow and homothetic solutions, J. Diff. Geom., 23(1986), 175-196.
[2] H.-D. Cao and H. Li, A Gap Theorem for Self-shrinkers of the Mean Curvature
Flow in Arbitrary Codimension, Calc. Var. 2012, DOI: 10.1007/s00526-Ol2-0508-1.

[3] Tobias H. Colding and William P. Minicozzi II, Generic Mean Curvature Flow
I; Generic Singularities, Ann. of Math., 175 (2012), 755-833.
[4] Q. -M. Cheng and Y. Peng, Complete self-shrinkers of the mean curvature
flow, arXiv: 1202. 1053, 2012.

37[5] X. Cheng and D. Zhou, Volume estimate about shrinkers, arXiv:1106.4950,

2011.

[6] Q. Ding, Y. L. Xin, Volume growth, eigenvalue and compactness for selfshrinkers, arXiv: 1101.1411, 2011.
[7] G. Huisken, Asymptotic Behavior for Singularities of the Mean Curvature
Flow, J. Differential Geom., 31 (1990), 285-299.
[8] G. Huisken, Local and global behaviour of hypersurfaces moving by mean
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[9] H. B. Lawson, Local rigidity theorems for minimal hypersurfaces, Ann. of
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[10] Nam Q. Le and N. Sesum, Blow-up rate of the mean curvature during themean curvature flow and a gap theorem for self-shrinkers, arXiv:1011.5245,

2010.
[11] H. Li and Y. Wei, Classification and rigidity of self-shrinkers in the mean
curvature flow, arXiv:1201.4623, 2012.
Department of Mathematics
Graduate School of Science and Engineering

Saga University
Saga 840-8502
Japan-mail address: yejuan666@gmail.com$E$