Geometric Flow Topology, A Number Of Issues

This paper mainly include two parts. The second and the third chapters belong to geometric flow part, and the last two chapters are topology part.Many of the most exciting recent developments in geometric analysis have arisen from the study of geometric flow equation. Among the most prominent examples, one is the proof of the Riemannian Penrose inequality given by Huisken and Ilmanen by using the inverse mean curvature flow, and the other is the proof of Poincare conjecture given by Huaidong Cao and Xiping Zhu by using Ricci flow.In the second chapter, we mainly study the first eigenvalue of Laplace op-erator along Hk-flow. Firstly, we derive the evolution equation for the first eigenvalue along Hk-flow.Let F0:M→Rn+1 be a smooth immersion of an n-dimensional convex hypersurface in Euclidean space, where n≥2 and H(Fo(Mn))> 0. The evolution of M0=F0(M) by powers of mean curvature flow is the one-parameter family of smooth immersions F:M x [0,T)→Rn+1 satisfying where k>0, H is the mean curvature and v is the outer unit normal. We call the above system unnormalized Hk-flow. Note that for k=1, this flow is the mean curvature flow.Theorem A:Assume Mn(t) is the solution of the Hk-flow. Letλ(t) be the first eigenvalue of Laplacian on Mn(t), and u(t) be the eigenfunction correspond-ing toλ,i.e.,-Δu=λu. If the metric evolves by the normalized Hk-flow, then we have Similarly, along the unnormalized Hk—low,Therefore, we can get the following corollaryCorollary B:Letλ=λ(t) be the first eigenvalue of the Laplace operator on Mn(t) which evolves by the normalized Hk—flow. If the initial surface M is totally umbilical, then the eigenvalue is nondecreasing along the unnormalized Hk-flow.Recently, hyperbolic geometric flow has received considerable attention.In the third chapter, we consider the hyperbolic Yamabe flow introduced by Professor Kefeng Liu and Dexing Kong.Let M be n-dimensional complete Riemannian manifold with Riemannian metric gij. Considering the following geometric flowwe derive some solutions for hyperbolic Yamabe flow. Firstly, the solutions of Einstein initial metric are given. Secondly, we investigate the solutions with axial symmetry. At last, as the special solution of the flow, the steady hyperbolic Yamabe soliton is defined and we get the equation satisfied by the soliton solution.In the next two chapters, we will consider the topology part.In the fourth chapter, we deal with the ends of the complete oriented minimal surface. We give an explicit upper bound for the number of the ends and derive Hoffman and Meeks'conjecture is true in special cases.Hoffman and Meeks'conjecture:For complete minimal embedded surface S, assume S has finite total curvature, then g≥r-2, that is to say r≤g+2. In this theorem, r is the number of the ends of surface S and g is the genus of compactification of S.Theorem C:Let M be a complete oriented minimal embedded surfaces in R3, which satisfies∫S|K|<∞, and it is not a plane, then the number of the ends r satisfies 2≤r≤2K-λ/4K+λ(g-1),g is the 9enus of compactification of M. For A in the above theorem is the following eigenvalue form. where D is the compact domain sequence and K represents the Gauss curvature on M.The topic on rigidity theorems of compact hypersurfaces has been studied extensively. One remarkable result of them is about compact hypersurfaces in a unit sphere with scalar curvature proportional to mean curvature.In the last chapter, we have studied the rigidity theorems of compact hyper-surfaces in real space forms and compact spacelike hypersurfaces in Lorentzian space forms.Theorem E:Let M be an n-dimensional compact hypersurface with non-negative sectional curvature in space forms Nn+1(c)(c≥0). If the normalized scalar curvature r and the mean curvature H satisfies r=f(H), where the function f satisfies (n-1)(f')2-4nHf'+4nf-4nc≥0, then M is either totally umbilical, or c>0, M=Sn-k×Sk.Moreover, we consider the compact spacelike hypersurface in Lorentzian space forms. Similarly, we get the following rigidity theorems.Theorem F:Let M be an n-dimensional compact spacelike hypersur-face with nonnegative sectional curvature in (n+1)-dimensional Lorentzian space forms R1n+1(c)(c> 0). Suppose the normalized scalar curvature r and mean curvature H satisfies r=f(H), where the function f satisfies then M is totally umbilical.