On stable complete hypersurfaces with vanishing \(r\)-mean curvature
type: Item
instance of: scholarly article
title: On stable complete hypersurfaces with vanishing \(r\)-mean curvature
zbMATH Open document ID: 1062.53052
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2004-11-05T00:00:00Z
review text: The \(r\)-mean curvature of a hypersurface \(x: M^n\to\mathbb{R}^{n+1}\) is \[ H_r= {n\choose r}^{-1} \sum_{i_1<\cdots< i_r} k_{i_1}\cdots k_{i_r}, \] where \(k_{i_1}\cdots k_{i_r}\) are the principal curvatures of \(x\). It is proven that there exists no complete orientable, proper, stable hypersurface \(x: M^n\to \mathbb{R}^{n+1}\), of finite total curvature \(n\geq 3\), with \(H_{n-1}= 0\) and \(H_n\neq 0\) everywhere.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 2114240
zbMATH Keywords: stable hypersurface
zbMATH Keywords: \(r\)-mean curvature
zbMATH Keywords: finite total curvature
author: Manfredo P. do Carmo
author: Maria Fernanda Elbert
reviewed by: Ülo Lumiste
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113246548
OpenAlex ID: W2033740414
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DOI: 10.2748/TMJ/1113246548
description: scientific article; zbMATH DE number 2114240