Existence And Uniqueness Of Solutions To The Constant Mean Curvature Equation With Nonzero Neumann Boundary Data In Product Manifold M~n × R

Submanifolds of constant curvature are important in the study of submanifolds,and attract geometers' attention a lot.The study of constant curvature submainfolds(especially minimal submanifolds)in space forms is relatively mature,and many beau-tiful results have been obtained.Naturally,people would try to extend those conclu-sions hold in the Euclidean space to more general geometric spaces.Recent years,the study of constant curvature submanifolds in warped product manifolds attracts more and more geometers'attention.In this paper,we can prove the existence and uniqueness of solutions to the con-stant mean curvature equation with nonzero Neumann boundary data in product mani-fold Mn × R,where Mn is an n-dimensional(n? 2)complete Riemannian manifold with nonnegative Ricci curvature,and R is the Euclidean 1-space.The conclusion ex-tends the relevant result in Euclidean space obtain by Professor Xi-Nan Ma et al..This thesis is based on the previous joint-work[7]with my collaborators.The main body of this thesis consists of three parts.More precisely,they are:In the first part,we give the research background,significance and the main con-clusions of the thesis.In the second part,some preliminaries have been introduced.For example,Ricci identity,and the strong(or weak)maximum principlie of second-order elliptic equa-tions,and the existence theorem of the supersolution and subsolution in the approxima-tion problem.In the third part,we first derive the gradient estimate of the equation,and then,together with the maximum principle and the Schauder theory,can prove the main conclusion of the thesis.