Nonlinear Problems in Geometry

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The principal investigator will study a number of classical problems in Differential and Riemannian geometry that are related to mean curvature equations or have a strong connection with fully nonlinear elliptic equations such as Monge-Ampere equations in some novel way. These include the equations of prescribed mean curvature and more general curvature functions of graphs in Riemannian manifolds MxR and hypersurfaces of constant mean curvature in hyperbolic space with prescribed boundary at infinity. We also apply the methods of fully nonlinear elliptic pde to study the geometric problem of estimating from below, the total absolute Gauss curvature for convex hypersurfaces in a Cartan-Hadamard manifold.We continue to study the geometric aspects of the theory of fully nonlinear elliptic equations that arise naturally in problems involving curvature. These equations are the most useful and the most difficult to study and are broadly applicable in pure and applied mathematics, especially in image processing, optimal design, computational biology and mathematical physics. For example, the use of mean curvature flow and other curvature flows in image processing and problems of brain registration is now quite standard. We similarly expect that our current research will become a standard tool in the future.