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

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

2010-03-11

arxiv.org/abs/1003.2243v1

Electron. J. Diff. Eqns., 2007 (2007), No. 65, 1-37

Mathematics

Analysis of PDEs

Scientific paper

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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