| In this thesis, we consider boundary value problems for a class of fully nonlinear partial differential equations in conformal geometry. This class of equations arises from the study of the integrand in the Gauss-Bonnet formula under conformal deformations. A key step in achieving the existence of solutions for such equations is to derive a priori estimates. The main advantage of our method in achieving these estimates is to get C 2 estimates directly from C0 estimates, and obtain C1 estimates as a consequence. The method is also applicable to a large class of fully nonlinear partial differential equations. We will prove the existence theorems and apply them to studies of conformal geometry on manifolds with boundary and conformally compact Einstein manifolds. |
