Nonlinear Problems in Geometry

Johns Hopkins University, Baltimore MD

Investigators

Abstract

Abstract Award: DMS-0072242 Principal Investigator: Joel Spruck The principal investigator proposes to study a number of classical problems in Riemannian geometry that are related in that they may be described by, or have a strong connection with, fully nonlinear elliptic equations such as Monge-Ampere equations or mean curvature equations in some novel way. These include extensions of the classical sharp isoperimetric inequality to negatively curved Riemannian manifolds, hypersurfaces of constant mean curvature in hyperbolic space with prescribed boundary at infinity, and the problem of deciding if any local Riemannian metric can be realized by an embedding into Euclidean three dimensional space. This last problem is interesting not only for its geometric content but also for important concrete problems in computer vision. Our research is motivated by concrete geometric and physical problems that are of basic interest to pure and applied scientists. What are the basic geometric design principles (variational principles) that govern the structure of the protein in our genes and can we find a fast computational algorithm to predict this structure. How can we make better color computer screens and smart cameras. These questions and numerous others depend upon a deep understanding of the geometry of surfaces and the complicated nonlinear equations that describe how they twist and turn and move about in space. The effective solution of these problems involves the study of geometry, partial differential equations and high speed computation.