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

Let L=Δ−∇φ⋅∇ be a symmetric diffusion operator with an invariant measure on a complete Riemannian manifold. In this paper we give an upper bound estimate on the first eigenvalue of the diffusion operator L on the complete manifold with the m-dimensional Bakry–Émery Ricci curvature satisfying Ricm,n(L)⩾−(n−1), and therefore generalize a Cheng's result on the Laplacian (S.-Y. Cheng (1975) [8]) to the case of the diffusion operator.