Conformal Deformation of a Riemannian Metric to a Scalar Flat Metric with Constant mean Curvature on the Boundary

One of the most celebrated theorems in mathematics is the Riemann mapping theorem. It says that an open, simply connected, proper subset of the plane is conformally diffeomorphic to the disk. In higher dimensions, very few regions are conformally diffeomorphic to the ball. However we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways, namely, it has zero scalar curvature and its boundary has constant mean curvature. In this paper we generalize the Riemann mapping theorem to higher dimensions in that sense.