The principal investigator proposes to study a number of problems in Differential and 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 optimal domain for the fundamental tone of a clamped plate.
The aim of the Principal investigator is to develop fundamental geometric and analytic methods to study highly nonlinear problems that are of importance in several fields of pure and applied mathematics, especially in Differential Geometry, image processing, optimal design, magnetohydrodynamics and mathematical physics. These problems are formulated in terms of highly nonlinear PDE's involving implicitly defined functions of curvature (or dynamic curvature flows such as mean curvature flow) and are often variational in nature involving free boundaries.
