The \(p\)-spectrum of the Laplacian on compact hyperbolic three manifolds
type: Item
instance of: scholarly article
title: The \(p\)-spectrum of the Laplacian on compact hyperbolic three manifolds
zbMATH Open document ID: 0801.53034
DOI: 10.1007/BF01459527
author: Jeffrey McGowan
published in: Mathematische Annalen
publication date: 1994-11-24T00:00:00Z
full work available at URL: https://eudml.org/doc/165156
review text: Let \(M\) be a compact hyperbolic 3-dimensional manifold with volume \(V\) and diameter \(R\). Let \(N(0,\lambda)\) denote the number of eigenvalues of the Laplacian on \(p\)-forms for \(p = 1,2\) which are less than or equal to \(\lambda\). The author shows there are universal constants so that \[ N\left(0,{1\over c_ 1 V(1+R^ 2)}\right) \leq c_ 2 V; \] this bounds the number of small eigenvalues for the Laplacian. Furthermore, this estimate is qualitatively sharp; the dimension of the space of coexact eigenforms of degree 1 with eigenvalues in the interval \((0,R^{-2})\) is at least \(c_ 3 V\) if \(R\) is sufficiently large.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 537341
zbMATH Keywords: hyperbolic 3-dimensional manifold
zbMATH Keywords: eigenvalues
zbMATH Keywords: Laplacian
zbMATH Keywords: eigenforms
reviewed by: Peter B. Gilkey
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 537341