Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature

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