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

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

Marcus A. Khuri

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampère 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-Ampère 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-Ampère equations; isometric embeddings.

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Marcus A. Khuri
Department of Mathematics, Stony Brook University
Stony Brook, NY 11794, USA
email: khuri@math.sunysb.edu

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