Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 65, pp. 1-37.

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

Author: Marcus A. Khuri (Stony Brook Univ., NY, USA)

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.
  
Submitted February 28, 2007. Published May 9, 2007.
Math Subject Classifications: 53B20, 53A05, 35M10.
Key Words: Local solvability; Monge-Ampere equations; isometric embeddings.