Translating Solutions Of The Nonparametric Mean Curvature Flow With Nonzero Neumann Boundary Data In Product Manifold M~n×R

Mean curvature flow is a class of important extrinsic curvature flows,which has important research significance.Whether the mean curvature flow exists for all the time or not?Under what conditions,does the mean curvature flow have the long-time existence,and then how to accurately describe the asymptotic behavior of the flow?These problems attract geometers' attention.For the mean curvature curvature flow equation,it has an important class of special solutions-translating solutions.Translating solutions play an important role in the study of type ? singularities of mean curvature flow.Naturally,it is also meaningful to know when the mean curvature flow equation has translating solutions.Given the product manifold Mn×R with the metric g:=?ijd?i(?)d?j+ds(?)ds,where Mn is a complete Riemannian manifold of nonnegative Ricci curvature.R is the 1-dimensional Euclidean space.Using Mn×R as the ambient space.Assume that ?(?)Mn is a compact strictly convex domain with smooth boundary(?)Q,and G is a smooth graphic hyper surface defined over ?.We would like to investigate the evolution of G(?)Mn×R under the nonparametric mean curvature flow with nonzero Neumann boundary data,can prove that this flow exists for all the time,and has translating solutions.This thesis is based on the previous joint-work[6]with my collaborators.The main body of this thesis is organized as follows.In Chapter 1,we briefly introduce the research background and application of mean curvature flow,as well as the main conclusions of this thesis.In Chapter 2,we give the fundamental knowledge,some formulas and notations.In Chapter 3,we firstly get the short time existence of the flow,and then derive the time derivative estimate and the gradient estimate of the flow equation.Finally,together with the extremum principle and Schauder theory,we can give the proof of our main conclusions.