Title: Some recent developments on the fully nonlinear Yamabe problems
Abstract: In recent joint work with YanYan Li and Zongyuan Li, we broaden the scope of fully nonlinear Yamabe problems 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 accommodate conformal metrics with scalar curvature of varying signs. A crucial new ingredient in our proofs is our enhanced understanding of solution behavior near isolated singularities of the associated equations. In addition to above results, I will also describe our developments on the fully nonlinear Yamabe problems on manifolds with boundary, discussing both boundary mean curvature and boundary curvatures arising from the Chern–Gauss–Bonnet formula.