Upper bounds of the first eigenvalue of closed hypersurfaces by the quotient area/volume
type: Item
instance of: scholarly article
title: Upper bounds of the first eigenvalue of closed hypersurfaces by the quotient area/volume
zbMATH Open document ID: 1068.53045
author: Fernando Giménez
author: Vicente Miquel
author: J. Javier Orengo
published in: Archiv der Mathematik
publication date: 2004-11-05T00:00:00Z
review text: \textit{D. D. Bleecker} and \textit{J. L. Weiner} [Comment. Math. Helv. 51, 601--609 (1976; Zbl 0341.53034)] evaluated an upper bound of the first eigenvalue \(\lambda_1(M)\) of a closed submanifold \(M\) of \(\mathbb{R}^n\) which depends only on the volume, the dimension of \(M\) and the integral along \(M\) of the square of the norm of the second fundamental form. In this paper the authors deal with the case of a closed hypersurface \(M\) of dimension \(n\) and of non-positive sectional curvature \(K_{\text{sec}}\) and evaluate \(\lambda_1(M)\) which is given by \[ \lambda_1(M)\leq {n-1\over n^2}\,\Biggl({\text{volume}(M)\over \text{volume}(\Omega)}\Biggr)^2, \] where \(\Omega\) is the compact domain bounded by \(M\). This result is extended to the case that \(K_{\text{sec}}\) is less than some certain number \(\lambda\), positive or negative, and also to the case that the ambient space is \(\mathbb{C} H^n(\lambda)\) or \(\mathbb{Q} H^n(\lambda)\).
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 2113847
zbMATH Keywords: global differential geometry
zbMATH Keywords: immersions (prescribed curvature)
reviewed by: Tanjiro Okubo
MaRDI profile type: Publication
full work available at URL: s00013-004-4954-7
OpenAlex ID: W2066655344
DOI: 10.1007/S00013-004-4954-7
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description: scientific article; zbMATH DE number 2113847