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

Abstract

Let $L=\Delta -\nabla \phi \cdot \nabla$ be a symmetric diffusion operator with an invariant measure $d\mu =e^{-\phi }dx$ on a complete Riemannian manifold. In this paper we prove Li-Yau gradient estimates for weighted elliptic equations on the complete manifold with $|\nabla \phi |\le \theta$ and $\infty$-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 (1975) 289-297).

Final version. The original proof of Theorem 2.1 using Li-Yau gradient estimate method has been moved to the appendix. The new proof is simple and direct

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