A Class Of Prescribed Weingarten Curvature Equations In Warped Product Manifolds

Prescribed Weingarten curvature problem is one of the most important geometric problems,its research is of great significance in the field of convex geometry,differ-ential geometry and partial differential equation.Many domestic and foreign scholars have engaged in the study of prescribed Weingarten curvature problem,and have ob-tained a series of important results.In this paper,we investigate a class of prescribed Weingarten curvature equations on hypersurfaces in warped product manifolds M~n×_fI,where M~nis an n-dimensional compact Riemannian manifold,I is an interval of R,f is a function defined on I.Under appropriate conditions,we obtain the prior estimates for the solution to the prescribed Weingarten curvature equation.And on this basis,we obtain the existence of the solu-tion of this equation.In other words,we obtain the existence of hypersurfaces where its principal curvature satisfies certain constraints in a given warped product manifold.This dissertation is mainly based on the previous joint-work with my collaborators.This thesis is organized as follows.In Chapter 1,we briefly introduce the research background of the prescribed Weingarten curvature equation we studied,as well as the main conclusions of this thesis.In Chapter 2,we give some preliminaries.For instance,several geometric properties of hypersurfaces in warped product manifolds,the Ricci identity,Newton-Maclaurin inequality,and so on.In Chapter 3,we derived C~0,C~1and C~2estimates for the solution to the equation we studied in turn.Then we can get the existence of the solution to this equation according to degree theory,and we can also obtain the existence of k-convex,closed hypersurfaces.In particular,we can obtain the existence of k-convex,star-shaped closed hypersurfaces in some special cases.