Nonlinear Problems in Geometry

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The principal investigator proposes to study a number of current problems in Lorentzian and Riemannian geometry that have a strong connection with fully nonlinear elliptic equations such as Monge-Ampere equations or mean curvature equations. In particular the PI will attempt to extend his study of complete hypersurfaces of constant curvature in hyperbolic space, especially in the convex case, to the constant curvature simply connected Lorentz space forms, namely Minkowski space, de Sitter space and anti-de Sitter space. The PI will also study boundedness and monotonicity properties of positive solutions of semilinear elliptic pde's, especially in the so-called supercritical case. Global boundedness fails in general but the PI will attempt to show monotonicity of solutions in a uniform neighborhood of the boundary. He will also explore the closely related problem of uniform boundedness of semi-stable solutions. There is a close connection of this problem with the open Bernstein problem for stable minimal hypersurfaces. Finally the PI is interested in the totally degenerate complex Monge-Ampere equation described by Mabuchi, Semmes and Donaldson, which arises in the study of geodesics in the space of Kahler potentials in a fixed Kahler class.This project is aimed at developing new analytic techniques to solve problems in geometry, physics, biology and astronomy where the underlying physical and geometrical equations can be described by elliptic partial differential equations. Such techniques have been extremely successful in problems which model curvature phenomena or use curvature flows as an analytic tool. These methods have broad applications in pure and applied mathematics, especially quantum physics and cosmology, image processing, optimal design and computational biology.