A harmonic mean bound for the spectral gap of the Laplacian on Riemannian manifolds
type: Item
instance of: scholarly article
title: A harmonic mean bound for the spectral gap of the Laplacian on Riemannian manifolds
zbMATH Open document ID: 1207.58026
author: Laurent Veysseire
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2011-01-12T00:00:00Z
review text: According to the author, most known lower bounds on the spectral gap of the Laplacian using Ricci curvature are based on the infimum of the Ricci curvature, so they can be poor when the Ricci curvature is large everywhere but on a small subset on which it is small. He, however, shows that the harmonic mean of the Ricci curvature also is a lower bound on the spectral gap of the Laplacian, if the Ricci curvature is everywhere nonnegative. In fact, his main theorem states as follows: Let \(M\) be a compact connected Riemannian manifold without boundary, with nonnegative Ricci curvature. Then the following inequality holds: \[ \frac{1}{\lambda_1}\leq\int_M\frac{d\mu(x)}{\kappa(x)} \] where \(\mu\) is the renormalized volume measure \((d\mu =\frac{\,\text{vol}}{\text{vol}(M)})\), and \(\kappa(x)=\inf_{v\in\text{T}_x M,\,\| v\| =1}\text{Ric}(v,v)\). In the last section, the author generalizes this result for a generator \(L\) of a Markovian process under the condition that \(L\) satisfies a certain so-called curvature-dimension inequality.
reviewed by: Eleutherius Symeonidis
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 5835325
zbMATH Keywords: Laplacian
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: spectral gap
zbMATH Keywords: Ricci curvature
zbMATH Keywords: reversible diffusion
MaRDI profile type: Publication
full work available at URL: j.crma.2010.10.015
OpenAlex ID: W2052489962
cites work: Finiteness of π1π1 and geometric inequalities in almost positive Ricci curvature
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Wikidata QID: Q115358256
DOI: 10.1016/J.CRMA.2010.10.015
description: scientific article; zbMATH DE number 5835325