Local solvability of degenerate Monge-Ampere equations and applications to geometry

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 $mathbb{R}^{3}$, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove a general local existence result for a large class of degenerate 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 vanishes and possesses a nonvanishing Hessian matrix at a critical point.

Keywords

• Local solvability
• Monge-Ampere equations
• isometric embeddings.