The Dirichlet problem for Fully Nonlinear Equations Arising from Conformal Geometry

We study the Dirichlet problem for a class of curvature equations arising from conformal geometry on Riemannian manifolds $(M^n, g)$ with boundary where $n \geq 3$. We prove there exists a unique solution using the continuity method which is based on \emph{a priori} estimates for admissible solutions. In deriving the estimates, a crucial step is to derive a lower bound for the gradient on the boundary. This is overcome by constructing a cluster of subsolutions.