Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature

Abstract

We prove that the image of an isometric embedding into R 3 of a two dimensionnal complete Riemannian manifold (Σ, g) without boundary is a convex surface provided both the embedding and the metric g enjoy a C 1,α regularity for some α > 2/3 and the distributional Gaussian curvature of g is nonnegative and nonzero. The analysis must pass through some key observations regarding solutions to the very weak Monge-Ampère equation.