Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature
type: Item
instance of: scholarly article
title: Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature
author: Mohammad Reza Pakzad
published in: Journal of Functional Analysis
publication date: 2024-09-27T00:00:00Z
review text: 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.''
reviewed by: Vagn Lundsgaard Hansen
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 53C24
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35J96
zbMATH DE Number: 7919641
zbMATH Keywords: non-negative Gauss curvature
zbMATH Keywords: differential geometry at low regularity
zbMATH Keywords: convex surfaces
MaRDI profile type: Publication
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DOI: 10.1016/J.JFA.2024.110616
description: scientific article; zbMATH DE number 7919641