Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds

We derive a priori estimates for second order derivatives of solutions to a wide class of fully nonlinear elliptic equations on Riemannian manifolds. There had been significant work in this direction, especially in connection with important geometric problems and other applications, but one had to make use of the special structures or needed extra assumptions which are more technical in nature to overcome various difficulties. In this paper we are able to remove most of the technical assumptions and derive the estimates under conditions which are close to optimal. These estimates enable one to prove existence results which are new even for bounded domains in Euclidean space.