Sphere theorems via Alexandrov for constant Weingarten curvature hypersurfaces. Appendix to a note of A. Ros
type: Item
instance of: scholarly article
title: Sphere theorems via Alexandrov for constant Weingarten curvature hypersurfaces. Appendix to a note of A. Ros
zbMATH Open document ID: 0638.53052
author: Nicholas J. Korevaar
published in: Journal of Differential Geometry
publication date: 1988-01-01T00:00:00Z
review text: We use the classical Alexandrov reflection technique to prove that compact embedded hypersurfaces without boundary (in \({\mathbb{R}}^{n+1})\) must be Euclidean spheres, if they have constant intermediate curvature \(H_ r\) (the rth symmetric function of the n principal curvatures, \(1\leq r\leq n)\). The proof follows from a simple observation related to work by \textit{R. Ros} (see the preceding review), the curvature function \(H_ r\) is necessarily ``elliptic'' on any candidate surface, so the a priori convexity assumptions made in previous works are unnecessary.
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 4040767
zbMATH Keywords: Alexandrov reflection technique
zbMATH Keywords: hypersurfaces
zbMATH Keywords: constant intermediate curvature
zbMATH Keywords: principal curvatures
MaRDI profile type: Publication
Wikidata QID: Q115183996
full work available at URL: https://doi.org/10.4310/jdg/1214441780
OpenAlex ID: W1479891047
DOI: 10.4310/JDG/1214441780
description: scientific article; zbMATH DE number 4040767