Fully Nonlinear Partial Differential Equations in Geometry

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

Abstract Award: DMS-0505632 Principal Investigator: Bo Guan The primary goal of this project is to understand some nonlinear elliptic and parabolic equations which arise from problems in differential geometry. The research will be focused on three types of geometric problems: Plateau-type problems for hypersurfaces in Euclidean space and in more general manifolds; the Minkowski-type problems of finding closed convex hypersurfaces of prescribed Weingarten curvature, and problems in isometric embedding. Each of these problems involves solving some fully nonlinear partial differential equations. Because of their origins in geometry, these equations are closely related and therefore share many common technical issues. Thus, progress in one of the problems will very likely lead to progress in the others, and more generally in the theory of nonlinear partial differential equations (on manifolds) and geometric analysis. Fully nonlinear equations also arise in other problems in geometry and physics as well as in science and engineering. Results from our research may have impact and applications in areas such as string theory in physics, and optimal control theory.