The principal investigator will study applications of linear and nonlinear partial differential equations to the geometry of Riemannian and CR manifolds. Specifically, he will find Einstein metrics of the ball with prescribed conformal structure at infinity on the sphere, and solve the CR Yamabe problem in dimension three by developing a theory of "CR minimal surfaces." In two related projects he will prove the existence of spherical CR structures on certain 3-manifolds, and determine whether the Dirichlet-to-Neumann map of a Riemannian manifold with boundary determines the metric up to isometry. Geometers consider spheres to be positively curved. The torus, or skin of an inner tube, is positively curved in some parts but negatively curved in others. By smoothly deforming the concept of distance, the torus can be made flat or have zero curvature. Two-holed tori can be deformed until they have constant negative curvature. The principal investigator will find a solution to the three-dimensional Yamabe problem. Thus he will deform 3-manifolds with negative Euler characteristic until they have constant negative curvature.
