几何发展方程中的若干问题研究
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摘要
本论文研究了平均曲率流的自相似解和一类逆平均曲率流,以及球面中极小超曲面的第二空隙。几何发展方程是研究数学问题的强有力工具,近几十年来受到了越来越多的关注。平均曲率流差不多是子流形几何中最为重要的几何流。平均曲率流中最为重要的问题之一是研究这个流可能出现的奇点。自相似解不仅与平均曲率流的奇点有密切的联系,而且是一类重要的子流形。我们广泛研究了平均曲率流的自收缩解,得到了它们的一些几何和分析的性质。通过给定无穷远处的边界值,我们在闵可夫斯基空间中构造了很多个整体光滑凸的严格类空平移解。球面中的极小子流形是微分几何中一个优美而重要的课题,它们自然的联系着欧氏空间中的极小锥。由Chern-do Carmo-Kobayashi提出的一个公开问题是研究球面中极小超曲面数量曲率的空隙。近来,我们在不要求常数量曲率的假设下,对所有维数证明了第二空隙的存在性。
本文的结构安排如下。
在第一章,首先我们回忆了平均曲率流的历史和现状,以及自收缩解是如何由平均曲率流的奇点所引起的。自相似解的研究对平均曲率流的奇点的了解起到关键的作用。其次,我们讨论了逆平均曲率流,它是研究微分几何和广义相对论中数学问题的重要工具。最后,我们探讨了陈省身关于极小超曲面的刚性问题。
在第二章,为了便于讨论,我们引入了子流形几何的基本语言和符号。我们陈述了自收缩解以及平移解方面已知的工作和我们的研究结果。总共花了3章(第三章到第五章)来叙述我们在自收缩解方面的系列工作。我们给出了关于子流形的第二基本型方面的一些熟知的公式。特别的,对在欧氏空间以及伪欧氏空间中可以表示为图的包括拉格朗日情形在内的自收缩解作了仔细的讨论。
在第三章,我们研究了任意余维数的自收缩解。同欧氏空间中极小子流形完全不一样的是,逆紧的非紧的自收缩解有下面最优的体积增长估计。
定理1.([50])任意一个浸入在Rn+m中的完备非紧的逆紧的自收缩解M都有至多欧氏体积增长.即存在一个仅依赖于n和M∩B8n的体积的常数C,使得对任意r≥1有
假定自收缩解可以表示成向量值函数u的图,我们能够证明u有线性增长。
定理2.(48)设M={(x,u(x))|x∈Rn}是Rn+m中可表示为整体图的自收缩解且u(x)=(u1(x),…,um(x)),则其中x∈Rn及
我们导出了一个自收缩解版本的Ruh-Vilms型定理：自收缩解的高斯映照是带权的调和映照。通过对第二基本型仔细的分析,可以得到自收缩解的Bernstein型定理,其所要求的条件比极小的情形要稍弱一点。借助Sobolev不等式,我们证明了一个关于高余维的自收缩解的第二基本型的模长范数的刚性定理。
在第四章,我们重点研究了拉格朗日的自收缩解,即它既是拉格朗日子流形又是自收缩解。通过积分的办法,我们证明了具有不定度量的伪欧氏空间Rn2n中拉格朗日平均曲率流的可表示为整图的类空自收缩解一定是平坦的。
定理3.([52])方程在Rn上的任意整体光滑凸解是二次多项式u(0)+{(D2u(0)x,x).
这个结果去掉了[78]和[22]中额外的条件,从而完善了他们的结果。利用相同的方式,我们再次证明了[22]中建立的欧氏空间的对应情形。
在第五章,我们探讨了欧氏空间中的自收缩超曲面。我们利用[49]中类似的想法,研究了自收缩解的第二基本型的模长范数的第二空隙。
定理4.([51])假设Mn是浸入在Rn+1中的完备逆紧的自收缩解,其第二基本型记为B,那么存在一个正数δ=0.011使得如果1/2≤|B|2≤1/2+δ,那么|B|2≡1/2
通过比较Hoffeman-Osserman-Schoen[76]关于R3中常平均曲率曲面的著名结果,我们对自收缩解证明了相应的部分。定理5.([53])设M是浸入在Rn+1中的完备逆紧的自收缩超曲面.如果Gauss映照的像被包含在一个开半球里面,那么M是超平面.如果Gauss映照的像被包含在一个闭半球里面,那么M是超平面或者是Rn中(n-1)-维的自收缩解与R的乘积.
记n-1维的上半开球s+n-1={(x1,…,xn)∈Rn|x12+…+xn2=1,xn>0}.使用由Jost-Xin-Yang[92]所研究的球面凸几何的技术,我们得到了下面这个关于高斯像的范围的最优的刚性定理。
定理6.([53])设Mn浸入在Rn+1中的完备逆紧的自收缩超曲面.如果Gauss映照的像包含在Sn\S+n-1中,那么M不得不是超平面.
给定一个非负的整数9和常数D>0,令Sg,D表示所有具有亏格不超过g直径不超过D的紧嵌入在R3中的自收缩解.通过估计自收缩解L算子的第一特征值的下界,在没有有界熵条件下,2维的紧嵌入自收缩解也存在紧性定理。
定理7.([50])对固定的g和D,空间Sg,D是紧的.即Sg,D中任意序列有子序列在Ck(任意k≥0)拓扑意义下一致收敛到Sg,D中某个曲面.
在R3里面,第二基本型的模长范数是常数时,自收缩曲面可以被分类。
在第六章,我们研究了闵可夫斯基空间中平均曲率流的整体类空平移解。通过计算第二基本型,我们能够懂得平移解的有界下水平集的凸性。那么在光滑有界凸区域上的一类狄利克雷问题是可解的。设Q是包含所有的齐一次的凸函数且该函数的梯度存在时其模长为1的集合。通过在不同的凸区域上构造一系列凸解,我们能够得到：
定理8.([47])果V∈Q且V不是一个线性函数,那么方程在Rn上一定有一个光滑凸的严格类空整体解u,使得u blow down到V,即对Vx∈Rn成立
在第七章,我们探讨了某些旋转对称空间中从闭的星型超曲面出发的逆平均曲率流。
定理9.([46])设N是一个(n+1)-维的具有非正截面曲率的旋转对称空间,考虑嵌入映照X0:Sn→N,且M0=X0(Sn)是一张光滑闭具有正平均曲率的星型超曲面.那么从M0出发的逆平均曲率流X=1/Hν有长时间光滑解.此外
情形1.如果N有欧氏体积增长,那么拉伸后的曲面X(t)=e-t/nX(t)收敛到一个唯一的球.
情形2.如果N是双曲空间,那么拉伸后的曲面X(t)=n/tX(t)收敛到一个唯一的具有半径为1的球.
在第八章,我们考虑了球面中极小超曲面的数量曲率的第二空隙问题,它由Chern-do Carmo-Kobayashi [35]所提出。Peng-Terng在[118]和[119]中首先研究了这个问题。他们证明了球面中具有常数量曲率的极小超曲面在任意维数都存在第二空隙,以及对不需要有常数量曲率假设的极小超曲面在低维存在第二空隙。此后关于这个问题有很多的工作。最近,我们肯定了球面中不需要有常数量曲率假设的极小超曲面在任意维数都存在第二空隙[49],并且当维数n≥6时得到了具体的拼挤常数,这些拼挤常数当n≥7时优于之前的所有结果。
定理10.([49])设M是单位球面Sn+1中的紧致的极小超曲面,s是它的第二基本型模长的平方.那么存在一个仅依赖于n的正常数δ(n),使得如果n≤s≤n+δ(n),那么S≡n,也就是说M是一个Clifford极小超曲面。当维数n≥6时,拼挤常数δ(n)=n/(23).
In the thesis, we shall study self-similar solutions for mean curvature flow and a class of inverse mean curvature flows, as well as the second gap for minimal hypersurfaces in spheres. Geometric evolution equations are powerful tools in studying mathematical prob-lems and receive more and more attentions in the past few decades. Mean curvature flow (abbreviated by MCF in what follows) perhaps is the most important geometric flow in the geometry of submanifolds. One of the most important problems in MCF is to under-stand the possible singularities that the flow goes through. Self-similar solutions are not only closely related to the singularities of MCF, but also an important class of subman-ifolds. We study the self-shrinkers of MCF extensively and obtain some geometric and analytic properties of them. By prescribing boundary data at infinity we construct many entire smooth convex strictly spacelike translating solitons in Minkowski space. Minimal submanifolds in the sphere are elegant and important subject in differential geometry, which relate to minimal cones in Euclidean space naturally. An open question proposed by Chern-do Carmo-Kobayashi is studying the gaps of scalar curvature of minimal hy-persurfaces in the sphere. Recently, we establish the existence of the second gap in any dimension without constant scalar curvature assumption.
The present thesis is organized as follows.
In Chapter1, firstly we recall the history and current state of the mean curvature flow and how are self-similar solutions raising from the singularities of MCF. The research of self-similar solutions plays a key role in studying the singularities of MCF. Secondly, we discuss the inverse mean curvature flow which is an important tool in differential geometry and the mathematical problem in general relativity. At last, it is devoted to explore Chern's problem for rigidity of minimal hypersurfaces.
In Chapter2, the basic language and notations of geometry of submanifolds are introduced for the convenience of talking about our topics. We specify the known works and our studies on self-shrinkers as well as translating solitons. It takes three Chapters (Chapter3-5) to present our series of works on self-shrinkers. Some familiar formulas on the second fundamental form of self-shrinkers are given. Particularly, the graphic self-shrinkers including the Lagrangian case are discussed carefully both in Euclidean space and pseudo-Euclidean space.
In Chapter3, we investigate the self-shrinkers of arbitrary codimension. We show that every proper noncompact self-shrinkers in Euclidean space has optimal volume growth, which is in a sharp contrast to complete minimal submanifolds in Euclidean space.
Theorem1.([50])Any complete non-compact properly immersed self-shrinker Mn in Rn+m has Euclidean volume growth at most. Precisely,∫M∩gBr ldμ≤Crn for r≥1, where C is a constant depending only on n and the volume of M n B8n.
If a self-shrinker could be written as a graph of some vector-valued function u, we could verify the linear growth of u.
Theorem2.([48]) Let M={(x,u(x))|x∈Rn} be an entire graphic self-shrinker in Rn+m with u(x)=(u1(x),…,um(x)), then where x∈Rn and
We derive a Ruh-Vilms type theorem for self-shrinkers. Precisely, the Gauss map of a self-shrinker is a weighted harmonic map. By careful analysis on second fundamental form, a Bernstein type theorem is deduced for self-shrinkers, whose condition is a litter weaker than minimal submanifolds. Using Sobolev inequality we get a rigidity result on squared norm of the second fundamental form for self-shrinkers of high codimensions.
In Chapter4, we focus on the Lagrangian self-shrinkers, which are both Lagrangian submanifolds and self-shrinkers. By the integral method we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in Rn2n with the indefinite metric∑i dxidyi is flat.
Theorem3.([52]) Any entire smooth convex solution u(x) to the equation is the quadratic polynomial u(0)+1/2(D2u(0)x,x).
This result improves the previous ones in [78] and [22] by removing the additional assumption in their results. In a similar manner, we reprove its Euclidean counterpart which is established in [22].
In Chapter5, we explore the self-shrinking hypersurfaces in Euclidean space. We employ the similar idea in our work [49] to study the second gap of the squared norm of the second fundamental form for self-shrinkers.
Theorem A.([51]) Let Mn be a complete properly immersed self-shrinker in Rn+1with second fundamental form B, then there exists a positive number δ=0.011such that if1/2≤|B|2≤1/2+δ, then|B|2=1/2.
Compared with the well-known results of Hoffeman-Osserman-Schoen [76] on con-stant mean curvature surfaces in R3, we obtain a counterpart for self-shrinkers.
Theorem5.([53]) Let M be a complete self-shrinker hypersurface properly immersed in Rn+1. If the image under the Gauss map is contained in an open hemisphere, then M has to be a hyperplane. If the image under the Gauss map is contained in a closed hemisphere, then M is a hyperplane or a cylinder whose cross section is an (n-1)-dimensional self-shrinker in Rn.
Let S+n-1={(x1,…,xn)∈Rn|x12+…+xn2=1, xn>0} be an n-dimensional open semisphere. Using technique of the convex geometry of the sphere studied by Jost-Xin-Yang [92] a rigidity theorem for the range of the Gauss image is obtained which is the best possible.
Theorem6.([53]) Let Mn be a complete self-shrinker hypersurface properly immersed in Rn+1.If the image under Gauss map is contained in Sn\S+-n-1, then M has to be a hyperplane.
Let Sg,D denote the space of all compact embedded self-shrinkers in R3with genus at most g, and diameter at most D for any non-negative integer g and constant D>0.By estimating the lower bound of the first eigenvalues of self-shrinkers for the operator L, a compactness theorem for two dimensional compact embedded self-shrinkers holds without bounded entropy.
Theorem7.([50]) For each fixed g and D, the space Sg,D is compact. Namely, any sequence in Sg,D has a subsequence that converges uniformly in the Ck topology (for any k≥0) to a surface in Sg,D.
In R3, we shall classify self-shrinking surfaces with constant squared norm of the second fundamental form.
In Chapter6, we study entire spacelike translating solitons of mean curvature flow in Minkowski space. By a calculation for the second fundamental form, we could understand convexity of the bounded sublevel sets for any solutions to translating solitons equation. Then it is able to solve a class of Dirichlet problems for smooth convex bounded domains. Let Q be a set defined by all convex homogeneous of degree one functions whose gradient has norm one whenever defined. By constructing a sequence of convex solutions in different bounded convex domains, we obtain
Theorem8.([4.7]) For any function V in Q except linear functions there is an entire smooth convex strictly spacelike solution u to the equation such that u blows down to V, namely, limr→∞u(rx)/r=V(x),(?)x∈Rn.
In Chapter7, We discuss the motion of inverse mean curvature flow which starts from a closed star-shaped hypersurface in some rotationally symmetric spaces.
Theorem9.([46]) Let N be an (n+1)-dimensional rotationally symmetric space with nonpositive sectional curvature, M0be a smooth closed, star-shaped hypersurface with pos-itive mean curvature in N, which is given as an embedding X0:Sn→Nn+1. Then the inverse mean curvature has a unique smooth solution for all times. Moreover
Case1. If N has Euclidean volume growth, then the rescaled surfaces X(t)=e-t/nX(t) converge to a uniquely determinate sphere.
Case2. If N is hyperbolic space, then the rescaled surfaces X(t)=n/tX(t) converge to a uniquely determinate sphere of radius1.
In Chapter8, we consider the second gap for the scalar curvature of minimal hyper-surfaces in the spheres, which is proposed by Chern-do Carmo-Kobayashi [35]. Peng-Terng in [118] and [119] firstly studied this problem and obtained pinching results for minimal hypersurfaces of constant scalar curvature in any dimension and that without the constant scalar curvature assumption in lower dimensions. After that, there are many works on this problem. Recently, we confirm the second gap in any dimension without constant scalar curvature assumption [49] and obtain the concrete pinching constant for dimension n≥6where they are better than all previous results for dimension n≥7.
Theorem10.([49]) Let M be a compact minimal hypersurface in Sn+1with the squared length of the second fundamental form S. Then there exists a positive constant δ(n) de-pending only on n, such that if n≤S≤n+δ(n), then S≡n. i.e.,M is a Clifford minimal hypersurface. Moreover, if the dimension is n≥6, then the pinching constant δ(n)=n/23.
本文的结构安排如下。
在第一章,首先我们回忆了平均曲率流的历史和现状,以及自收缩解是如何由平均曲率流的奇点所引起的。自相似解的研究对平均曲率流的奇点的了解起到关键的作用。其次,我们讨论了逆平均曲率流,它是研究微分几何和广义相对论中数学问题的重要工具。最后,我们探讨了陈省身关于极小超曲面的刚性问题。
在第二章,为了便于讨论,我们引入了子流形几何的基本语言和符号。我们陈述了自收缩解以及平移解方面已知的工作和我们的研究结果。总共花了3章(第三章到第五章)来叙述我们在自收缩解方面的系列工作。我们给出了关于子流形的第二基本型方面的一些熟知的公式。特别的,对在欧氏空间以及伪欧氏空间中可以表示为图的包括拉格朗日情形在内的自收缩解作了仔细的讨论。
在第三章,我们研究了任意余维数的自收缩解。同欧氏空间中极小子流形完全不一样的是,逆紧的非紧的自收缩解有下面最优的体积增长估计。
定理1.([50])任意一个浸入在Rn+m中的完备非紧的逆紧的自收缩解M都有至多欧氏体积增长.即存在一个仅依赖于n和M∩B8n的体积的常数C,使得对任意r≥1有
假定自收缩解可以表示成向量值函数u的图,我们能够证明u有线性增长。
定理2.(48)设M={(x,u(x))|x∈Rn}是Rn+m中可表示为整体图的自收缩解且u(x)=(u1(x),…,um(x)),则其中x∈Rn及
我们导出了一个自收缩解版本的Ruh-Vilms型定理：自收缩解的高斯映照是带权的调和映照。通过对第二基本型仔细的分析,可以得到自收缩解的Bernstein型定理,其所要求的条件比极小的情形要稍弱一点。借助Sobolev不等式,我们证明了一个关于高余维的自收缩解的第二基本型的模长范数的刚性定理。
在第四章,我们重点研究了拉格朗日的自收缩解,即它既是拉格朗日子流形又是自收缩解。通过积分的办法,我们证明了具有不定度量的伪欧氏空间Rn2n中拉格朗日平均曲率流的可表示为整图的类空自收缩解一定是平坦的。
定理3.([52])方程在Rn上的任意整体光滑凸解是二次多项式u(0)+{(D2u(0)x,x).
这个结果去掉了[78]和[22]中额外的条件,从而完善了他们的结果。利用相同的方式,我们再次证明了[22]中建立的欧氏空间的对应情形。
在第五章,我们探讨了欧氏空间中的自收缩超曲面。我们利用[49]中类似的想法,研究了自收缩解的第二基本型的模长范数的第二空隙。
定理4.([51])假设Mn是浸入在Rn+1中的完备逆紧的自收缩解,其第二基本型记为B,那么存在一个正数δ=0.011使得如果1/2≤|B|2≤1/2+δ,那么|B|2≡1/2
通过比较Hoffeman-Osserman-Schoen[76]关于R3中常平均曲率曲面的著名结果,我们对自收缩解证明了相应的部分。定理5.([53])设M是浸入在Rn+1中的完备逆紧的自收缩超曲面.如果Gauss映照的像被包含在一个开半球里面,那么M是超平面.如果Gauss映照的像被包含在一个闭半球里面,那么M是超平面或者是Rn中(n-1)-维的自收缩解与R的乘积.
记n-1维的上半开球s+n-1={(x1,…,xn)∈Rn|x12+…+xn2=1,xn>0}.使用由Jost-Xin-Yang[92]所研究的球面凸几何的技术,我们得到了下面这个关于高斯像的范围的最优的刚性定理。
定理6.([53])设Mn浸入在Rn+1中的完备逆紧的自收缩超曲面.如果Gauss映照的像包含在Sn\S+n-1中,那么M不得不是超平面.
给定一个非负的整数9和常数D>0,令Sg,D表示所有具有亏格不超过g直径不超过D的紧嵌入在R3中的自收缩解.通过估计自收缩解L算子的第一特征值的下界,在没有有界熵条件下,2维的紧嵌入自收缩解也存在紧性定理。
定理7.([50])对固定的g和D,空间Sg,D是紧的.即Sg,D中任意序列有子序列在Ck(任意k≥0)拓扑意义下一致收敛到Sg,D中某个曲面.
在R3里面,第二基本型的模长范数是常数时,自收缩曲面可以被分类。
在第六章,我们研究了闵可夫斯基空间中平均曲率流的整体类空平移解。通过计算第二基本型,我们能够懂得平移解的有界下水平集的凸性。那么在光滑有界凸区域上的一类狄利克雷问题是可解的。设Q是包含所有的齐一次的凸函数且该函数的梯度存在时其模长为1的集合。通过在不同的凸区域上构造一系列凸解,我们能够得到：
定理8.([47])果V∈Q且V不是一个线性函数,那么方程在Rn上一定有一个光滑凸的严格类空整体解u,使得u blow down到V,即对Vx∈Rn成立
在第七章,我们探讨了某些旋转对称空间中从闭的星型超曲面出发的逆平均曲率流。
定理9.([46])设N是一个(n+1)-维的具有非正截面曲率的旋转对称空间,考虑嵌入映照X0:Sn→N,且M0=X0(Sn)是一张光滑闭具有正平均曲率的星型超曲面.那么从M0出发的逆平均曲率流X=1/Hν有长时间光滑解.此外
情形1.如果N有欧氏体积增长,那么拉伸后的曲面X(t)=e-t/nX(t)收敛到一个唯一的球.
情形2.如果N是双曲空间,那么拉伸后的曲面X(t)=n/tX(t)收敛到一个唯一的具有半径为1的球.
在第八章,我们考虑了球面中极小超曲面的数量曲率的第二空隙问题,它由Chern-do Carmo-Kobayashi [35]所提出。Peng-Terng在[118]和[119]中首先研究了这个问题。他们证明了球面中具有常数量曲率的极小超曲面在任意维数都存在第二空隙,以及对不需要有常数量曲率假设的极小超曲面在低维存在第二空隙。此后关于这个问题有很多的工作。最近,我们肯定了球面中不需要有常数量曲率假设的极小超曲面在任意维数都存在第二空隙[49],并且当维数n≥6时得到了具体的拼挤常数,这些拼挤常数当n≥7时优于之前的所有结果。
定理10.([49])设M是单位球面Sn+1中的紧致的极小超曲面,s是它的第二基本型模长的平方.那么存在一个仅依赖于n的正常数δ(n),使得如果n≤s≤n+δ(n),那么S≡n,也就是说M是一个Clifford极小超曲面。当维数n≥6时,拼挤常数δ(n)=n/(23).
In the thesis, we shall study self-similar solutions for mean curvature flow and a class of inverse mean curvature flows, as well as the second gap for minimal hypersurfaces in spheres. Geometric evolution equations are powerful tools in studying mathematical prob-lems and receive more and more attentions in the past few decades. Mean curvature flow (abbreviated by MCF in what follows) perhaps is the most important geometric flow in the geometry of submanifolds. One of the most important problems in MCF is to under-stand the possible singularities that the flow goes through. Self-similar solutions are not only closely related to the singularities of MCF, but also an important class of subman-ifolds. We study the self-shrinkers of MCF extensively and obtain some geometric and analytic properties of them. By prescribing boundary data at infinity we construct many entire smooth convex strictly spacelike translating solitons in Minkowski space. Minimal submanifolds in the sphere are elegant and important subject in differential geometry, which relate to minimal cones in Euclidean space naturally. An open question proposed by Chern-do Carmo-Kobayashi is studying the gaps of scalar curvature of minimal hy-persurfaces in the sphere. Recently, we establish the existence of the second gap in any dimension without constant scalar curvature assumption.
The present thesis is organized as follows.
In Chapter1, firstly we recall the history and current state of the mean curvature flow and how are self-similar solutions raising from the singularities of MCF. The research of self-similar solutions plays a key role in studying the singularities of MCF. Secondly, we discuss the inverse mean curvature flow which is an important tool in differential geometry and the mathematical problem in general relativity. At last, it is devoted to explore Chern's problem for rigidity of minimal hypersurfaces.
In Chapter2, the basic language and notations of geometry of submanifolds are introduced for the convenience of talking about our topics. We specify the known works and our studies on self-shrinkers as well as translating solitons. It takes three Chapters (Chapter3-5) to present our series of works on self-shrinkers. Some familiar formulas on the second fundamental form of self-shrinkers are given. Particularly, the graphic self-shrinkers including the Lagrangian case are discussed carefully both in Euclidean space and pseudo-Euclidean space.
In Chapter3, we investigate the self-shrinkers of arbitrary codimension. We show that every proper noncompact self-shrinkers in Euclidean space has optimal volume growth, which is in a sharp contrast to complete minimal submanifolds in Euclidean space.
Theorem1.([50])Any complete non-compact properly immersed self-shrinker Mn in Rn+m has Euclidean volume growth at most. Precisely,∫M∩gBr ldμ≤Crn for r≥1, where C is a constant depending only on n and the volume of M n B8n.
If a self-shrinker could be written as a graph of some vector-valued function u, we could verify the linear growth of u.
Theorem2.([48]) Let M={(x,u(x))|x∈Rn} be an entire graphic self-shrinker in Rn+m with u(x)=(u1(x),…,um(x)), then where x∈Rn and
We derive a Ruh-Vilms type theorem for self-shrinkers. Precisely, the Gauss map of a self-shrinker is a weighted harmonic map. By careful analysis on second fundamental form, a Bernstein type theorem is deduced for self-shrinkers, whose condition is a litter weaker than minimal submanifolds. Using Sobolev inequality we get a rigidity result on squared norm of the second fundamental form for self-shrinkers of high codimensions.
In Chapter4, we focus on the Lagrangian self-shrinkers, which are both Lagrangian submanifolds and self-shrinkers. By the integral method we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in Rn2n with the indefinite metric∑i dxidyi is flat.
Theorem3.([52]) Any entire smooth convex solution u(x) to the equation is the quadratic polynomial u(0)+1/2(D2u(0)x,x).
This result improves the previous ones in [78] and [22] by removing the additional assumption in their results. In a similar manner, we reprove its Euclidean counterpart which is established in [22].
In Chapter5, we explore the self-shrinking hypersurfaces in Euclidean space. We employ the similar idea in our work [49] to study the second gap of the squared norm of the second fundamental form for self-shrinkers.
Theorem A.([51]) Let Mn be a complete properly immersed self-shrinker in Rn+1with second fundamental form B, then there exists a positive number δ=0.011such that if1/2≤|B|2≤1/2+δ, then|B|2=1/2.
Compared with the well-known results of Hoffeman-Osserman-Schoen [76] on con-stant mean curvature surfaces in R3, we obtain a counterpart for self-shrinkers.
Theorem5.([53]) Let M be a complete self-shrinker hypersurface properly immersed in Rn+1. If the image under the Gauss map is contained in an open hemisphere, then M has to be a hyperplane. If the image under the Gauss map is contained in a closed hemisphere, then M is a hyperplane or a cylinder whose cross section is an (n-1)-dimensional self-shrinker in Rn.
Let S+n-1={(x1,…,xn)∈Rn|x12+…+xn2=1, xn>0} be an n-dimensional open semisphere. Using technique of the convex geometry of the sphere studied by Jost-Xin-Yang [92] a rigidity theorem for the range of the Gauss image is obtained which is the best possible.
Theorem6.([53]) Let Mn be a complete self-shrinker hypersurface properly immersed in Rn+1.If the image under Gauss map is contained in Sn\S+-n-1, then M has to be a hyperplane.
Let Sg,D denote the space of all compact embedded self-shrinkers in R3with genus at most g, and diameter at most D for any non-negative integer g and constant D>0.By estimating the lower bound of the first eigenvalues of self-shrinkers for the operator L, a compactness theorem for two dimensional compact embedded self-shrinkers holds without bounded entropy.
Theorem7.([50]) For each fixed g and D, the space Sg,D is compact. Namely, any sequence in Sg,D has a subsequence that converges uniformly in the Ck topology (for any k≥0) to a surface in Sg,D.
In R3, we shall classify self-shrinking surfaces with constant squared norm of the second fundamental form.
In Chapter6, we study entire spacelike translating solitons of mean curvature flow in Minkowski space. By a calculation for the second fundamental form, we could understand convexity of the bounded sublevel sets for any solutions to translating solitons equation. Then it is able to solve a class of Dirichlet problems for smooth convex bounded domains. Let Q be a set defined by all convex homogeneous of degree one functions whose gradient has norm one whenever defined. By constructing a sequence of convex solutions in different bounded convex domains, we obtain
Theorem8.([4.7]) For any function V in Q except linear functions there is an entire smooth convex strictly spacelike solution u to the equation such that u blows down to V, namely, limr→∞u(rx)/r=V(x),(?)x∈Rn.
In Chapter7, We discuss the motion of inverse mean curvature flow which starts from a closed star-shaped hypersurface in some rotationally symmetric spaces.
Theorem9.([46]) Let N be an (n+1)-dimensional rotationally symmetric space with nonpositive sectional curvature, M0be a smooth closed, star-shaped hypersurface with pos-itive mean curvature in N, which is given as an embedding X0:Sn→Nn+1. Then the inverse mean curvature has a unique smooth solution for all times. Moreover
Case1. If N has Euclidean volume growth, then the rescaled surfaces X(t)=e-t/nX(t) converge to a uniquely determinate sphere.
Case2. If N is hyperbolic space, then the rescaled surfaces X(t)=n/tX(t) converge to a uniquely determinate sphere of radius1.
In Chapter8, we consider the second gap for the scalar curvature of minimal hyper-surfaces in the spheres, which is proposed by Chern-do Carmo-Kobayashi [35]. Peng-Terng in [118] and [119] firstly studied this problem and obtained pinching results for minimal hypersurfaces of constant scalar curvature in any dimension and that without the constant scalar curvature assumption in lower dimensions. After that, there are many works on this problem. Recently, we confirm the second gap in any dimension without constant scalar curvature assumption [49] and obtain the concrete pinching constant for dimension n≥6where they are better than all previous results for dimension n≥7.
Theorem10.([49]) Let M be a compact minimal hypersurface in Sn+1with the squared length of the second fundamental form S. Then there exists a positive constant δ(n) de-pending only on n, such that if n≤S≤n+δ(n), then S≡n. i.e.,M is a Clifford minimal hypersurface. Moreover, if the dimension is n≥6, then the pinching constant δ(n)=n/23.
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