The primary goal of the work described in this proposal is to understand some nonlinear elliptic equations which arise from problems in geometry and analysis. The research will be focused on geometric problems in four areas: Plateau-type problems for hypersurfaces in Euclidean and hyperbolic spaces, complete conformal metrics determined by functions of Ricci tensor on Riemannian manifolds with boundary, problems in isometric embedding of 2-dimensional Riemannian manifolds of positive or negative curvature, and complex Monge-Ampre equations. These problems involve solving some fully nonlinear partial differential equations. Because of their origin in geometry, these equations are closely related and therefore share many common technical issues. Progress in this project may lead to solutions to others problems in the theory of nonlinear partial differential equations on manifolds and geometric analysis.
The theory and methods of geometric analysis and fully nonlinear equations play important roles in understanding difficult problems in pure and applied mathematics, such as the Poincare conjecture, the positive mass theorem and Penrose inequality in general relativity, and applications in image processing, optical reflector designs, optimal mass transport, mathematical biology and physics. The proposal concerns some very typical problems in differential geometry and highly nonlinear partial differential equations. The research will advance the understanding of these problems and develop techniques that will be useful in studying fully nonlinear equations which arise from other problems in mathematics and physical sciences.
