Mathematical Sciences: Global Differential Geometry and Nonlinear Partial Differential Equations

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Vladimir Oliker will continue his research in the area of global differential geometry and related problems in nonlinear partial differential equations. He will study existence, uniqueness and regularity problems for compact and noncompact complete hypersurfaces with prescribed curvature functions. The emphasis is placed on the analytic approach based on a study of corresponding questions for Monge-Ampere equations. As a part of this research variational problems for nonlinear elliptic equations will also be investigated. Earlier results on the realization of a given function as a curvature function have been based on pointwise growth conditions. Oliker will attempt to replace these with integral estimates. He will also look for corresponding results in Lorenz space. These investigations will encounter the added difficulty that the unit sphere is not compact.