A Riemannian mapping type Theorem in higher dimensions, Part I: the   conformally flat case with umbilic boundary
  In this paper we prove that every Riemannian metric on a locally conformally flat manifold with umbilic boundary can be conformally deformed to a scalar flat metric having constant mean curvature. This result can be seen as a generalization to higher dimensions of the well known Riemann mapping Theorem in the plane. 
Mohameden Ould Ahmedou