Global Lower Bounds on the First Eigenvalue for a Diffusion Operator

We derive global lower bounds for the first eigenvalue of a symmetric diffusion ΔX:=Δ-∇X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _X:=\Delta -\nabla _X$$\end{document} on Riemannian manifolds with the Bakry–Émery–Ricci curvature bounded from below.