Fully Nonlinear Elliptic and Parabolic Equations in Differential Geometry

University Of Tennessee Knoxville, Knoxville TN

Investigators

Abstract

NSF proposal DMS - 0204590 Principal Investigator: Bo Guan, University of Tennessee Title: FULLY NONLINEAR ELLIPTIC AND PARABOLIC EQUATIONS IN DIFFERENTIAL GEOMETRY Abstract: Fully nonlinear elliptic and parabolic equations arise from many problems in differential geometry. In recent years these equations have attracted a lot of attention and significant progresses have been made to understand these equations and related geometric problems. In this project, the principal investigator will continue his research in this direction. The problems to be investigated in this project include isometric embeddings of metrics of nonnegative curvature; questions about hypersurfaces of nonnegative constant Gauss curvature with boundary in Euclidean space and more general Riemannian manifolds, including existence, uniqueness and regularity; spacelike entire graphs of constant Gauss curvature in Minkowski space; Minkowski type problems of finding closed convex hypersurfaces of prescribed Weingarten curvatures; regularity of solutions to degenerate Monge-Ampere equations in non-convex domains; regularity of pluricomplex Green functions and existence of holomorphic functions in Kahler manifolds; Hessian equations on Riemannian manifolds and applications in geometric problems; hypersurfaces in hyperbolic space of constant mean curvature (or Weingarten hypersurfaces) with prescribed asymptotic boundary at infinity; and evolution of hypersurfaces by curvature functions. Equations arising from most of these problems are highly nonlinear. These equations also model various phenomena in sciences and engineering. Solving such equations heavily depends on establishing apriori estimates up to second order derivatives. For many of the proposed problems in this project, these estimates alone are often not enough to lead to existence of solutions; there are other obstructions from geometry and analysis. These all impose challenging questions. Research on these problems may also develop methods of numerical approximations to the solutions that are useful in engineering and science.