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

TL;DR

This paper establishes upper bounds for the first eigenvalue of a diffusion operator on complete Riemannian manifolds using Bakry-Émery Ricci curvature, extending classical results to weighted elliptic operators.

Contribution

It provides new upper bounds on the first eigenvalue for diffusion operators with weighted measures, generalizing Cheng's classical results to the Bakry-Émery setting.

Findings

Li-Yau gradient estimates for weighted elliptic equations

Upper bounds on the first eigenvalue of diffusion operators

Extension of Cheng's eigenvalue bounds to weighted manifolds

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).