Riemannian foliations and eigenvalue comparison
type: Item
instance of: scholarly article
title: Riemannian foliations and eigenvalue comparison
zbMATH Open document ID: 0915.53015
DOI: 10.1023/A:1006573301591
author: Jeffrey M. Lee
author: Ken Richardson
published in: Annals of Global Analysis and Geometry
publication date: 1999-06-28T00:00:00Z
review text: The authors provide several lower and upper bounds for eigenvalues of the basic Laplacian \(\Delta_B\) on a Riemannian manifold \(M\) equipped with a Riemannian foliation \(\mathcal F\) (and a bundle-like metric). Here are some of their results: (1) \(\lambda^B_j\geq\lambda_j\) for all \(j > 0\) (here, \(\lambda^B_j\) are eigenvalues of \(\Delta_B\) while \(\lambda_j\) eigenvalues of \(\Delta\), the standard Laplace operator on \(M\)), (2) \(\lambda^B\geq\frac{1}{4}h_B^2\), where \(\lambda_B\) is the smallest positive eigenvalue of \(\Delta_B\) and \(h_B\) is the Cheeger constant obtained while considering splittings of \(M\) into two components bounded by \(\mathcal F\)-saturated submanifolds only, (3) if the sectional curvature \(K\) of \(M\) satisfies \(K\geq\alpha\), then \(\lambda^B(L, R)\leq\lambda^B_{\alpha , h}\), where \(\lambda^B(L, R)\) is the first Dirichlet eigenvalue of \(\Delta_B\) on the tubular neighbourhood of a compact leaf closure \(L\), \(R\) is the radius of the tube, \(h\) is a sufficiently large constant and \(\lambda^R_{\alpha , h}\) is the first eigenvalue of a suitable (described explicitly in the paper) 1-dimensional eigenvalue problem with parameters \(\alpha\), \(h\) and \(R\).
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1220811
zbMATH Keywords: Riemannian foliation
zbMATH Keywords: Laplacian
zbMATH Keywords: eigenvalue
reviewed by: Paweł Walczak
MaRDI profile type: Publication
full work available at URL: a:1006573301591
OpenAlex ID: W192883556
description: scientific article; zbMATH DE number 1220811