The Local Isometric Embedding in R^3 of Two-Dimensional Riemannian Manifolds With Gaussian Curvature Changing Sign to Finite Order on a Curve

摘要

We consider two natural problems arising in geometry which are equivalent tothe local solvability of specific equations of Monge-Ampere type. These twoproblems are: the local isometric embedding problem for two-dimensionalRiemannian manifolds, and the problem of locally prescribed Gaussian curvaturefor surfaces in R^3. We prove a general local existence result for a largeclass of Monge-Ampere equations in the plane, and obtain as corollaries theexistence of regular solutions to both problems, in the case that the Gaussiancurvature vanishes to arbitrary finite order on a single smooth curve.