Mathematical Sciences: Monge-Ampere Type Equations and Related Problems in Differential Geometry

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9626722 Guan This project pursues geometric partial differential equations. Analysis of many problems in differential geometry often leads to questions concerning nonlinear partial differential equations. The investigator plans to study Monge-Ampere type equations with an emphasis on the regularity of solutions in the degenerate case; the Christoffel-Minkowski problems and related Hessian equations; the Dirichlet problem for hypersurfaces of prescribed curvatures; curvature evolution equations for hypersurfaces. Unlike linear partial differential equations there is little general theory regarding nonlinear partial differential equations. Yet many problems arising in various geometric settings can only be formulated as nonlinear differential equations. And the proposed research represents an attempt to pursue these equations from a geometric perspective - the main advantage of the geometric approach seems to be that one is able to make fairy strong, but geometrically meaningful, assumptions thereby simplifying the equations.