2-dimensional complete self-shrinkers in R 3
aa r X i v : . [ m a t h . DG ] A p r QING-MING CHENG* AND SHIHO OGATA
Dedicated to Professor Yoshihiko Suyama for his 70th birthday
Abstract.
It is our purpose to study complete self-shrinkers in Euclidean space.First of all, we show some examples of complete self-shrinkers without poly-nomial volume growth. By making use of the generalized maximum principlefor L -operator, we give a complete classification for 2-dimensional complete self-shrinkers with constant squared norm of the second fundamental form in R . In[8], Ding and Xin have proved this result under the assumption of polynomialvolume growth, which is removed in our theorem. introduction Let X : M n → R n +1 be an n -dimensional hypersurface in the n + 1-dimensionalEuclidean space R n +1 . If the position vector X evolves in the direction of the meancurvature H , then it gives rise to a solution to mean curvature flow: X ( · , t ) : M n → R n +1 satisfying X ( · ,
0) = X ( · ) and(1.1) ∂X ( p, t ) ∂t = H ( p, t ) , ( p, t ) ∈ M × [0 , T ) , where H ( p, t ) denotes the mean curvature vector of hypersurface M t = X ( M n , t )at point X ( p, t ). The equation (1.1) is called the mean curvature flow equation.The study of the mean curvature flow from the perspective of partial differentialequations commenced with Huisken’s paper [11] on the flow of convex hypersurfaces(cf. [9]).One of the most important problems in the mean curvature flow is to understand thepossible singularities that the flow goes through. A key starting point for singularityanalysis is Huisken’s monotonicity formula because the monotonicity implies thatthe flow is asymptotically self-similar near a given singularity and thus, is modeledby self-shrinking solutions of the flow.An n -dimensional hypersurface X : M → R n +1 in the ( n + 1)-dimensional Euclideanspace R n +1 is called a self-shrinker if it satisfies H + h X, N i = 0 , Key words and phrases : mean curvature flow, complete self-shrinkers, the generalized maximumprinciple2010
Mathematics Subject Classification : 53C44, 53C40.* Research partially Supported by JSPS Grant-in-Aid for Scientific Research (B) No. 24340013and Challenging Exploratory Research No. 25610016. where H and N denote the mean curvature and the unit normal vector of the hy-persurface, respectively. It is known that self-shrinkers play an important role inthe study of the mean curvature flow because they describe all possible blow upat a given singularity of the mean curvature flow. For classifications of completeself-shrinkers, Abresch and Langer [1], Huisken [12, 13] and Colding and Minicozzi[6] have obtained very important results. In fact, Abresch and Langer [1] classifiedclosed self-shrinker curves in R and showed that the round circle is the only em-bedded self-shrinkers. Huisken [12, 13] and Colding and Minicozzi [6] have provedthat if X : M → R n +1 is an n -dimensional complete embedded self-shrinker in R n +1 with H ≥ X : M → R n +1 isisometric to either R n , the round sphere S n ( √ n ), or a cylinder S m ( √ m ) × R n − m ,1 ≤ m ≤ n −
1. In [2], Cao has conjectured that complete self-shrinkers must havepolynomial volume growth. Furthermore, Ding and Xin [7] and X. Cheng and Zhou[5] have proved that a complete self-shrinker has polynomial volume growth if andonly if it is proper. From the following proposition, we know that there are manycomplete self-shrinkers without polynomial volume growth.
Proposition 1.1.
For any integer n > , there exist n -dimensional complete self-shrinkers without polynomial volume growth in R n +1 . In fact, in [10], Halldorsson has proved there exist complete self-shrinker curves Γin R , which is contained in an annulus around the origin and whose image is densein the annulus. Hence, these complete self-shrinker curves Γ are not proper. Thus,for any integer n >
0, Γ × R n − is a complete self-shrinker in R n +1 , which does nothave polynomial volume growth.In [2], Cao and Li have proved that if an n -dimensional complete self-shrinker X : M → R n +1 with polynomial volume growth satisfies S ≤
1, then X : M → R n +1 isisometric to either R n , the round sphere S n ( √ n ), or a cylinder S m ( √ m ) × R n − m ,1 ≤ m ≤ n − Theorem DX . Let X : M → R be a -dimensional complete self-shrinker withpolynomial volume growth in R . If the squared norm S of the second fundamentalform is constant, then X : M → R is isometric to one of the following: (1) R , (2) a cylinder S (1) × R (3) the round sphere S ( √ . In this paper, we want to remove the assumption of polynomial volume growth inthe above theorem of Ding and Xin and to prove that the above result of Ding andXin holds by making use of a different method.
Theorem 1.1.
Let X : M → R be a -dimensional complete self-shrinker in R . Ifthe squared norm S of the second fundamental form is constant, then X : M → R is isometric to one of the following: (1) R , (2) a cylinder S (1) × R -DIMENSIONAL COMPLETE SELF-SHRINKERS 3 (3) the round sphere S ( √ . Proof of theorem 1.1
Let X : M → R be a 2-dimensional surface in R . We choose a local orthonormalframe field { e A } A =1 in R with dual co-frame field { ω A } A =1 , such that, restricted to M , e , e are tangent to M . Hence, we have dX = X i =1 ω i e i , de i = X j =1 ω ij e j + ω i e . We restrict these forms to M , then(2.1) ω = 0and ω i = X j =1 h ij ω j , h ij = h ji , where h ij denote components of the second fundamental form of X : M → R .Take e , e such that, at any fixed point, h ij = λ i δ ij , where λ and λ are the principal curvatures of X : M → R . Thus, the Gausscurvature K and the mean curvature H are given by K = λ λ , H = λ + λ . For a smooth function f , the L -operator is defined by(2.2) L f = ∆ f − h X, ∇ f i where ∆ and ∇ denote the Laplacian and the gradient operator on the self-shrinker,respectively. In order to prove our results, the following generalized maximum prin-ciple for L -operator on self-shrinkers is very important, which is proved by Chengand Peng in [3]: Lemma 2.1. ( Generalized maximum principle for L -operator ) Let X : M n → R n + p ( p ≥ be a complete self-shrinker with Ricci curvature bounded from below. Let f be any C -function bounded from above on this self-shrinker. Then, there exists asequence of points { p k } ⊂ M n , such that lim k →∞ f ( X ( p k )) = sup f, lim k →∞ |∇ f | ( X ( p k )) = 0 , lim sup k →∞ L f ( X ( p k )) ≤ . Proof of theorem 1.1 . Since X : M → R is a complete self-shrinker, we have(2.3) H + h X, N i = 0 . By a simple calculation, we have12 L S = X i,j,k h ijk + S (1 − S ) , QING-MING CHENG AND SHIHO OGATA where S = P i,j =1 h ij is the squared norm of the second fundamental form and h ijk denote components of the first covariant derivative of the second fundamental form.Since S is constant, we have(2.4) X i,j,k h ijk + S (1 − S ) = 0 . If S = 1, then we know h ijk ≡
0. Hence, X : M → R is isometric to the roundsphere S ( √
2) or the cylinder S (1) × R from the results of Lawson [15]. If S < X : M → R is isometricto R .Next, we prove that S ≤ L| X | = 2 − | X | . Since S is constant, we know that the Gauss curvature satisfies K = λ λ ≥ − λ + λ − S . Therefore, the Gauss curvature is bounded from below. Since −| X | ≤ L -operator to thefunction −| X | . Thus, there exists a sequence { p k } in M such that(2.6) lim k →∞ | X | ( p k ) = inf | X | , lim k →∞ |∇| X | ( p k ) | = 0 , lim inf k →∞ L| X | ( p k ) ≥ . From (2.5) and (2.6), we have(2.7) inf | X | ≤ . Since |∇| X | | = P i =1 h X, e i i holds, we have from (2.6)lim k →∞ |∇| X | ( p k ) | = lim k →∞ X i =1 h X, e i i ( p k ) = 0 . Hence, we get from (2.3)(2.8) inf | X | = lim k →∞ H ( p k ) , lim k →∞ |∇ H | ( p k ) = 0 . Since S is constant, from the definition of the mean curvature H and (2.3), weobtain, for j = 1 , k →∞ (cid:0) h j ( p k ) + h j ( p k ) (cid:1) = 0 , lim k →∞ (cid:0) λ ( p k ) h j ( p k ) + λ ( p k ) h j ( p k ) (cid:1) = 0 . Since S is constant, we know that { λ j ( p k ) } and { h iij ( p k ) } are bounded sequences.Thus, we can assume lim k →∞ h iij ( p k ) = ¯ h iij , lim k →∞ λ j ( p k ) = ¯ λ j , for i, j = 1 ,
2. From (2.9), we obtain(2.10) ( ¯ h j + ¯ h j = 0 , ¯ λ ¯ h j + ¯ λ ¯ h j = 0 . -DIMENSIONAL COMPLETE SELF-SHRINKERS 5 If ¯ λ = ¯ λ is satisfies, according to (2.10), we infer¯ h iij = 0for i, j = 1 ,
2. According to Codazzi equations, we have X i,j,k ¯ h ijk = 0 . From (2.4), we have S = 1 or S = 0. Hence S ≤ λ = ¯ λ holds, we have S = ¯ λ + ¯ λ = (cid:0) ¯ λ + ¯ λ (cid:1) k →∞ H ( p k )2 . According to (2.7) and (2.8), we have S ≤ . Hence, S = 0 or S = 1. According to the theorem of Lawson [15], we know that M n is isometric to the round sphere S ( √ S (1) × R or R . (cid:3) Acknowledgement . Authors would like to thank professor Wei Guoxin for fruitfuldiscussions.
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