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Upper Bounds on the First Eigenvalue for a Diffusion Operator via Bakry–Émery Ricci Curvature II

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Abstract

Let \({L=\Delta-\nabla\varphi\cdot\nabla}\) be a symmetric diffusion operator with an invariant measure \({d\mu=e^{-\varphi}dx}\) on a complete Riemannian manifold. In this paper we prove Li–Yau gradient estimates for weighted elliptic equations on the complete manifold with \({|\nabla \varphi| \leq \theta}\) and ∞-dimensional Bakry–Émery Ricci curvature bounded below by some negative constant. Based on this, we give an upper bound on the first eigenvalue of the diffusion operator L on this kind manifold, and thereby generalize a Cheng’s result on the Laplacian case (Math Z, 143:289–297, 1975).

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  1. Department of Mathematics, Shanghai Maritime University, Haigang Avenue 1550, 201306, Shanghai, People’s Republic of China

    Jia-Yong Wu

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  1. Jia-Yong Wu
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Correspondence to Jia-Yong Wu.

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This work is partially supported by the NSFC (No. 11101267) and the Science and Technology Program of Shanghai Maritime University (No. 20120061).

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Wu, JY. Upper Bounds on the First Eigenvalue for a Diffusion Operator via Bakry–Émery Ricci Curvature II. Results. Math. 63, 1079–1094 (2013). https://doi.org/10.1007/s00025-012-0254-x

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