On Fully Nonlinear Yamabe Problems: Existence, Compactness, and Singularities

This proposal focuses on fully nonlinear Yamabe problems and related fully nonlinear elliptic equations. The classical Yamabe problem, formulated as a semilinear elliptic equation, seeks a conformal metric with constant scalar curvature. The fully nonlinear Yamabe problem generalizes this by considering more general curvature quantities, often resulting in fully nonlinear, and possibly non-uniformly, elliptic equations. Building on my previous works on Liouville theorems, the first part of the proposal targets the existence and compactness of solutions to fully nonlinear Yamabe problems. Our research emphasizes the development of viscosity solution theories for those fully nonlinear elliptic equations arising from conformal geometry. In the second part, we investigate the removability of non-isolated singularities in these equations. Ultimately, we aim to develop a comprehensive viscosity theory that can advance the study of fully nonlinear Yamabe problems.