Journal of Differential Geometry

The local isometric embedding in $\mathbb{R}^3$ of two-dimensional Riemannian manifolds with Gaussian curvature changing sign to finite order on a curve

Marcus A. Khuri

Abstract

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampère type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in $\mathbb{R}^3$. We prove a general local existence result for a large class of 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 to arbitrary finite order on a single smooth curve.

Article information

Source
J. Differential Geom., Volume 76, Number 2 (2007), 249-291.

Dates
First available in Project Euclid: 25 May 2007

https://projecteuclid.org/euclid.jdg/1180135679

Digital Object Identifier
doi:10.4310/jdg/1180135679

Mathematical Reviews number (MathSciNet)
MR2330415

Zentralblatt MATH identifier
1130.53017

Citation

Khuri, Marcus A. The local isometric embedding in $\mathbb{R}^3$ of two-dimensional Riemannian manifolds with Gaussian curvature changing sign to finite order on a curve. J. Differential Geom. 76 (2007), no. 2, 249--291. doi:10.4310/jdg/1180135679. https://projecteuclid.org/euclid.jdg/1180135679