Non-linear Problems in Geometry

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9703154 Spruck The proposed research deals with a number of geometric problems in the area of Riemannian and differential geometry. Asymptotic shape of curvature flows, local two dimensional isometric embedding problem, complex Green's function and existence of analytic functions on Stein manifolds, and isoperimetric inequality on Cartan-Hadamard manifolds are among the problems to be pursued. The main tool to be used is the theory of Monge-Ampere equations. The use of nonlinear elliptic partial differential equations to solve fundamental problems in differential geometry has been one of the great achievements of modern mathematics. Perhaps the most important nonlinear elliptic equation is the Monge-Ampere equation and there have been significant advances in our understanding of this equation in the last several years. Partial differential equations model various natural phenomena where several parameters vary simultaneously and their rates of change over a short time interval can be observed. Thus far, a general theory of such equations is very much lacking, especially when the equation is fully nonlinear.