Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature
It is a well-known fact of differential geometry that any \(C^2\)-smooth complete surface in \(\mathbb{R}^3\), whose Riemannian metric has non-negative and non-zero intrinsic Gaussian curvature, is a convex surface. The statement is a consequence of the fact that, by the Theorema Egregium, the intrinsic and extrinsic notions of curvatures coincide for such a surface.\N\NIn continuation of a series of papers on generalisations of this result, the author formulates a general research question: Under which regularity assumptions on a surface and its metric do the intrinsic and the extrinsic notions of the Gaussian curvature coincide? The present paper makes a fine contribution to this question.\N\NThe results obtained are adequately described in the author's abstract:\N\N``We prove that the image of an isometric embedding into \(\mathbb{R}^3\) 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.''
scientific article; zbMATH DE number 7919641
scientific article