Upper bounds on the first eigenvalue for a diffusion operator via Bakry–Émery Ricci curvature

Let L = Δ − ∇ φ ⋅ ∇ be a symmetric diffusion operator with an invariant measure d μ = e − φ d x on a complete Riemannian manifold. In this paper we give an upper bound estimate on the first eigenvalue of the diffusion operator L on the complete manifold with the m-dimensional Bakry–Émery Ricci curva...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2010, Vol.361 (1), p.10-18
Main Author: Wu, Jia-Yong
Format: Article
Language:English
Subjects:
Bakry–Émery Ricci curvature
Diffusion operator
Eigenvalue estimate
Exact sciences and technology
Mathematical analysis
Mathematics
Measure and integration
Sciences and techniques of general use
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Summary:Let L = Δ − ∇ φ ⋅ ∇ be a symmetric diffusion operator with an invariant measure d μ = e − φ d x on a complete Riemannian manifold. In this paper we give an upper bound estimate on the first eigenvalue of the diffusion operator L on the complete manifold with the m-dimensional Bakry–Émery Ricci curvature satisfying Ric m , n ( L ) ⩾ − ( n − 1 ) , and therefore generalize a Cheng's result on the Laplacian (S.-Y. Cheng (1975) [8]) to the case of the diffusion operator.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2009.09.019