Upper bounds on the first eigenvalue for a diffusion operator via Bakry-Émery Ricci curvature
type: Item
instance of: scholarly article
title: Upper bounds on the first eigenvalue for a diffusion operator via Bakry-Émery Ricci curvature
zbMATH Open document ID: 1183.58028
author: Jia-Yong Wu
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-11-05T00:00:00Z
review text: Let \((M,g)\) be an \(n\)-dimensional complete Riemannian manifold and let \(\phi\) be a \(C^2\) function. Consider the symmetric diffusion operator \(L=\Delta-\nabla\phi\cdot\phi\). This is the infinitesimal generator of the Dirichlet form associated to \(d\mu:=e^{-\phi}dx\). It is self-adjoint with respect to the weighted measure \(d\mu\). Let \(Ric_{m,n}\) be the associated Bakry-Emery Ricci curvature of the diffusion operator \(L\) where \(m\) is the Bakry-Emery dimension of \(L\) [\textit{D. Bakry} and \textit{M. Emery}, Proc., Lect. Notes Math. 1123, 177--206 (1985; Zbl 0561.60080)] -- this is not necessarily an integer. Assume that \(Ric_{m,n}\geq-(n-1)\). Then the authors establish an upper bound estimate on the first nontrivial eigenvalue of the diffusion operator \(L\) of the form \[ \lambda_1\leq(m-1)(n-1)/4\,. \] This generalizes a previous result of \textit{S.-Y. Cheng} [Math. Z. 143, 289--297 (1975; Zbl 0329.53035)] for the Laplacian to the case of the diffusion operator.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 5625382
zbMATH Keywords: Bakry-Emery Ricci curvature
zbMATH Keywords: diffusion operator
zbMATH Keywords: eigenvalue estimate
zbMATH Keywords: Dirichlet form
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.09.019
OpenAlex ID: W2059392945
cites work: Q4309937
cites work: Q5186515
cites work: Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator
cites work: Harnack inequalities on a manifold with positive or negative Ricci curvature
cites work: Some new results on eigenvectors via dimension, diameter, and Ricci curvature
cites work: Q3392348
cites work: An extension of E. Hopf's maximum principle with an application to Riemannian geometry
cites work: Eigenvalue comparison theorems and its geometric applications
cites work: Differential equations on riemannian manifolds and their geometric applications
cites work: Q3883196
cites work: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds
cites work: Some geometric properties of the Bakry-Émery-Ricci tensor
cites work: Q5485637
cites work: Harnack inequality and Liouville properties for \(L\)-harmonic operator
cites work: On the geometry of metric measure spaces. I
cites work: On the geometry of metric measure spaces. II
cites work: Q5186325
DOI: 10.1016/J.JMAA.2009.09.019
description: scientific article; zbMATH DE number 5625382