Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature

TL;DR

This paper proves that certain two-dimensional Riemannian manifolds with nonnegative, nonzero distributional Gaussian curvature can be isometrically embedded into three-dimensional space as convex surfaces, under specific regularity conditions.

Contribution

It establishes convexity of isometric embeddings for manifolds with low regularity and distributional curvature, advancing understanding of weak solutions to the Monge-Ampère equation.

Findings

Embedding surfaces are convex under specified regularity and curvature conditions.

Key insights into solutions of the very weak Monge-Ampère equation.

Extension of convexity results to low-regularity Riemannian manifolds.

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-Amp\`ere equation.