Local Solvability of a Class of Degenerate Monge-Ampere Equations and Applications to Geometry

Marcus A. Khuri

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arXiv · 1003.2243

Local Solvability of a Class of Degenerate Monge-Ampere Equations and Applications to Geometry

Marcus A. Khuri

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Abstract

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampere type. These are: the problem of locally prescribed Gaussian curvature for surfaces in R^3, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove a general local existence result for a large class of Monge-Ampere equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature possesses a nondegenerate critical point.

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Marcus A. Khuri. 2010-03-11. Local Solvability of a Class of Degenerate Monge-Ampere Equations and Applications to Geometry. https://arxiv.org/abs/1003.2243

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