Smooth metric measure spaces with non-negative curvature

In this paper, we study both function theoretic and spectral properties on complete non-compact smooth metric measure space (M, g, e -f dv) with non-negative Bakry-mery Ricci curvature. Among other things, we derive a gradient estimate for positive f -harmonic functions and obtain as a consequence the strong Liouville property under the optimal sublinear growth assumption on f. We also establish a sharp upper bound of the bottom spectrum of the f -Laplacian in terms of the linear growth rate of f. Moreover, we show that if equality holds and M is not connected at infinity, then M must be a cylinder. As an application, we conclude steady Ricci solitons must be connected at infinity.