No abstract

Abstract. We derive a local Gaussian upper bound for the f -heat kernel on complete smooth metric measure space (M, g, e −f dv) with nonnegative Bakry-Émery Ricci curvature. As applications, we obtain a sharp L 1 f -Liouville theorem for f -subharmonic functions and an L 1 f -uniqueness property for nonnegative solutions of the f -heat equation, assuming f is of at most quadratic growth. In particular, any L 1 f -integrable f -subharmonic function on gradient shrinking and steady Ricci solitons must be constant. We also provide explicit f -heat kernel for Gaussian solitons.

Let L = − ∇ϕ · ∇ be a symmetric diffusion operator with an invariant measure dμ = e −ϕ dx 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 mdimensional Bakry-Émery Ricci curvature satisfying Ric m,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.