Some Problems On Geometric Evolution Equations

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 andWe 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 obtainTheorem8.([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. MoreoverCase1. 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.