Some Nonlinear Problems On Manifolds Arising From Conformal Geometry

This thesis focuses on the the fully nonlinear Yamabe-type problem on manifolds with boundary of admissible negative curvature, which is an important problem in Geometric Analysis and has been extensively studied.This problem is essentially an elliptic problem with Neumann boundary condi-tion. Firstly, we use Maximum Principle and perturbation method to derive C0bound. Then, we creatively utilize the tubular neighborhood normal coordinates to derive C1and C2boundary estimates. When the C2estimate is established, this problem turn-s out to be uniformly elliptic. So, by the theory of Lieberman and Trudinger’s and the concave condition, the C2,α estimate can be obtained. Furthermore, using the s-tandard Schauder estimate, we can get C4,α estimate. Finally, the existence result is gained through method of continuity and the uniqueness of this problem is derived by Maximum Principle.