A Study On A Class Of Prescribed Curvature Problems In Hyperbolic Space

Prescribed Weingarten curvature problem is a very important geometric problem.The study of this problem can promote the development of partial differential equations,differential geometry and convex geometry.In this paper,we consider a kind of the prescribed shifted Gauss curvature problem of h-convex hypersurfaces in hyperbolic spaces.By a priori estimate of the solution of the equation and the standard degree theory,we obtain the existence of the solution of the problem.This article is structured as follows.In the first chapter,we first briefly introduce some geometric concepts and their backgrounds in hyperbolic spaces,and the research history and current situation of the prescribed Weingarten curvature equation.Then in the second section,we introduce the main research contents and conclusions of this paper.In the second chapter,we give the preliminary knowledge related to this paper,including the basic concepts in Riemannian geometry,immersed hypersurfaces and star hypersurfaces in hyperbolic space,and the lemmas needed for later calculations.In the third chapter,we obtain a priori estimate of the solution of the equation until the second derivative,and then use the degree theory to prove the existence of the solution of the equation according to the Evans-Krylov estimate and Schauder estimate.