Researches On Fully Nonlinear Hessian Equations Of Elliptic And Parabolic Type On Riemannian Manifolds

Many Problems in differential geometry can be reduced to solving fully nonlinear Hessian equations. For example, the Calabi conjecture is equivalent to solve a complex Monge-Ampere equation on a compact Kahler manifold. Minkowski problem is equiva-lent to solve a Monge-Ampere equation on a sphere. The k-Yamabe problem in conformal geometry is equivalent to solve the fully nonlinear Hessian equations on closed Rieman-nian manifolds. With help of the k-Yamabe flow, it also can be resolved by solving the fully nonlinear Hessian equations of parabolic type. Besides, the motion of strict convex hypersurfaces by their Gauss curvature can be transformed to the Monge-Ampere equa-tion of parabolic type of its support function with the Gauss map. All these show that fully nonlinear Hessian equations are closely related to the research of differential geometry. Therefore, the study of fully nonlinear Hessian equations is extremely important in the aspect of both theory and application.In this dissertation, we study some fully nonlinear Hessian equations on Riemannian manifolds with boundary. We proved the C2 a priori estimates for admissible solution-s. Higher order estimates are obtained by the Evans-Krylov theorem and the classical Schauder theory. Hence, using the method of continuity and degree theory, we obtained the existence of solutions. Specifically, we have the following results:Firstly, we proved the existence of smooth solutions for the Dirichlet problem of some fully nonlinear Hessian equations of elliptic type. As a conclusion, the prescribed negative curvature problem arising in conformal geometry has a solution, i.e. there exists a conformal metric g on Riemannian manifold (M,g) with boundary which coincides with g on (?)M and the modified Schouten tensor Agt with respect to g satisfies the prescribed equation.Secondly, we proved the existence of smooth solutions for the first-initial boundary value problem of fully nonlinear Hessian equations of parabolic type on MT:= M × (0, T]. As in the elliptic case, the existence of smooth solutions is obtained by deriving the C2 a priori estimates for admissible solutions. We assume there exists a sub-solution to avoid introducing geometric assumptions on the boundary. Besides the fundamental conditions, we do not need any other assumptions for the fully nonlinear operator in deriving the C2 a priori estimates. Only when deriving a priori estimates for the gradient, we need a technical assumption.Finally, we proved the existence of C1,1 solutions to the obstacle problem of ful-ly nonlinear Hessian equations. It often appears in the problem of finding the greatest hypersurface with Gauss curvature constraints. As an application, we proved the exis-tence of solutions for the obstacle problem of prescribed negative curvature equations in conformal geometry with the same method.