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arXiv · 1010.4175

Upper bounds on the first eigenvalue for a diffusion operator via Bakry-\'{E}mery Ricci curvature II

Jia-Yong Wu
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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 $\infty$-dimensional Bakry-\'{E}mery 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 (1975) 289-297).

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Jia-Yong Wu. 2010-10-20. Upper bounds on the first eigenvalue for a diffusion operator via Bakry-\'{E}mery Ricci curvature II. https://doi.org/10.1007/s00025-012-0254-x

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