Several Types Of Geometric Flows And Their Application

In this dissertation,we mainly study several types of geometric flow and their application.In Chapter 3,we consider the flow by powers of Gauss curvature.We prove that convex hypersurfaces under the flow by powers α>0 of the Gauss curvature in space forms Nn+1(κ)of constant sectional curvature κ(j=±1)contract to a point in finite time T*.Moreover,convex hypersurfaces under the flow by power α>1/n+2 of the Gauss curvature converge(after rescaling)to a limit which is the geodesic sphere in Nn+1(κ).This extends the known results in Euclidean space to space forms.In Chapter 4,we apply curvature flows to prove the geometric inequalities.We proved the Alexandrov-Fenchel inequalities for embedded,closed,connected and convex C2-hypersurface in Sn+1,Ak≥ξk,k-2(Ak-2)for any 1 ≤k≤ n-1,where Ak is the quermassintegral(see Definition 1.1)and ξk,k-2 is the unique positive function such that the equality holds when M is a geodesic sphere.The equality holds if and only if M is a geodesic sphere.In Chapter 5,we study the regularity of harmonic maps.We attempt to weaken the assumption of minimizing maps in Theorem 2,3,4 and corollary 5 in[1].We prove these theorems still hold for stationary maps.We obtain the regularity for stationary maps(Theorem 1.1,1.2).Since we can construct nonconstant stationary maps from Rk to Sn which are bounded away from a totally geodesic subsphere of codimension two(Example 1.4),we need stability assumption to establish a Liouville theorem for stationary maps.More generally,we deduce the Liouville theorem for stationary pharmonic maps(Theorem 1.7).In Chapter 6,we focus on the singularity analysis of the heat flow of harmonic maps.We mainly study the eigenfunction of Quasi-Laplacian Δg=|x|2/e2(m-2)(Δg0-▽g0h·▽g0)=|x|/e2(m-2)Δh for h=|x|2/4.We study the analytic property and asymptotic behavior(at the infinity)of the eigenfunctions of the Laplace operator of(Rm,g),denoted byΔg=|x|2/e2(m-2)(Δg-▽g0h·▽g0),where h=|x|2/4.