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.