An inequality of ``Reilly''-type for submanifolds of the hyperbolic space
type: Item
instance of: scholarly article
title: An inequality of ``Reilly''-type for submanifolds of the hyperbolic space
zbMATH Open document ID: 0758.53029
DOI: 10.1007/BF02566494
published in: Commentarii Mathematici Helvetici
publication date: 1993-01-16T00:00:00Z
full work available at URL: https://eudml.org/doc/140263
review text: Given a compact submanifold in the Euclidean space, \(M^ n\subset \mathbb{R}^ m\), \textit{R. C. Reilly} [ibid. 52, 525-533 (1977; Zbl 0382.53038)] proved the inequality \(n\int_ M H^ 2\geq \lambda_ 1 V\), \(\lambda_ 1\) and \(V\) being the first eigenvalue of the Laplacian and the volume of \(M\), and \(H\) being the mean curvature vector of the immersion. If the equality holds, then \(M\) is minimally contained in an umbilical hypersurface of \(\mathbb{R}^ m\). The authors obtain an analogous result for submanifolds of the hyperbolic space. In this case they prove the inequality \(n\int_ M H^ 2\geq(\lambda_ 1+n)V\). Finally, the authors give applications to an eigenvalue estimate for \(\Delta+W\) on submanifolds and to the study of stability of constant mean curvature hypersurfaces.
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 93802
zbMATH Keywords: total mean curvature
zbMATH Keywords: conformal volume
zbMATH Keywords: eigenvalue estimate
zbMATH Keywords: constant mean curvature hypersurfaces
author: Ahmad El Soufi
author: Said Ilias
MaRDI profile type: Publication
OpenAlex ID: W1993537380
description: scientific article; zbMATH DE number 93802