Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature

Abstract

We prove that the image of an isometric embedding into ${\mathbb{R}}^3$ of a two dimensionnal complete Riemannian manifold $(\Sigma ,g)$ without boundary is a convex surface provided both the embedding and the metric $g$ enjoy a $C^{1,\alpha }$ regularity for some $\alpha >2/3$ 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.

new version with minor corrections and improvements, 18 pages

References in corpus (2)

• Global Nash-Kuiper theorem for compact manifolds
• $W^{s,\frac{n}{s}}$-maps with positive distributional Jacobians