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

TL;DR

This paper proves local solvability results for a class of degenerate Monge-Ampere equations, with applications to geometric problems like prescribed Gaussian curvature and isometric embedding of surfaces.

Contribution

It establishes a general local existence theorem for degenerate Monge-Ampere equations, enabling solutions to geometric problems under specific curvature conditions.

Findings

Existence of regular solutions when Gaussian curvature has a nondegenerate critical point

Application to local prescribed Gaussian curvature problem in R^3

Application to local isometric embedding problem for Riemannian surfaces

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.