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

Let $L=Δ-\nablaφ\cdot\nabla$ be a symmetric diffusion operator with an invariant measure $dμ=e^{-φ}dx$ on a complete Riemannian manifold. In this paper we prove Li-Yau gradient estimates for weighted elliptic equations on the complete manifold with $|\nabla φ|\leqθ$ 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).

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