On the fully nonlinear Yamabe problem

This dissertation consists of three parts. In the first part, we broaden the scope of the fully nonlinear Yamabe problem by establishing optimal Liouville-type theorems, local gradient estimates, and new existence and compactness results for conformal metrics on a closed Riemannian manifold with prescribed symmetric functions of the Schouten (Ricci) tensor. Our results allow the scalar curvature of conformal metrics to have varying signs. A crucial new ingredient in our proofs is our enhanced understanding of solution behavior near isolated singularities of the associated equations. In the second part, we extend these results in a parallel way to manifolds with boundary, incorporating the boundary mean curvature. In the last part, we establish a Liouville-type theorem as well as a local gradient estimate for the σₖ—Yamabe problem with boundary curvature arising from the Chern—Gauss—Bonnet formula.