On the spectrum of non-compact manifolds with finite volume
type: Item
instance of: scholarly article
title: On the spectrum of non-compact manifolds with finite volume
zbMATH Open document ID: 0537.58040
DOI: 10.1007/BF01161957
author: Robert Brooks
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173501
review text: Let M be a complete non-compact manifold with finite volume, and \(\lambda_ 0^{ess}\) the lower bound of the essential spectrum of the Laplacian on M \((\lambda_ 0^{ess}=\infty\) if M has discrete spectrum). We prove the estimates \((1/4)h^{ess^ 2}\leq \lambda_ 0^{ess}\leq(1/4)\mu^ 2_ f\) where \(h^{ess}\) is the essential Cheeger's isoperimetric constant, and \[ \mu_ f=\overline{\lim}_{r\to \infty}(-1/r) \log(Vol(M)-Vol(Br)) \] measures how quickly the volume of a ball of radius r approaches the volume of M. Techniques are presented for estimating these constants, and examples are given.
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35P15
zbMATH DE Number: 3854076
zbMATH Keywords: Laplacian on Riemannian manifold
zbMATH Keywords: essential spectrum
zbMATH Keywords: discrete spectrum
zbMATH Keywords: Cheeger's isoperimetric constant
MaRDI profile type: Publication
cites work: Q3969238
cites work: Noncompact Riemannian manifolds with discrete spectra
cites work: On the essential spectrum of a complete Riemannian manifold
cites work: Pure point spectrum and negative curvature for noncompact manifolds
OpenAlex ID: W2094939467
description: scientific article; zbMATH DE number 3854076