We study both function theoretic and spectral properties of the weighted Laplacian Δ f on complete smooth metric measure space (M, g, e −f dv) with its Bakry-Émery curvature Ric f bounded from below by a constant. In particular, we establish a gradient estimate for positive f -harmonic functions and a sharp upper bound of the bottom spectrum of Δ f in terms of the lower bound of Ric f and the linear growth rate of f. We also address the rigidity issue when the bottom spectrum achieves its

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... l upper bound under a slightly stronger assumption that the gradient of f is bounded. Applications to the study of the geometry and topology of gradient Ricci solitons are also considered. Among other things, it is shown that the volume of a noncompact shrinking Ricci soliton must be of at least linear growth. It is also shown that a nontrivial expanding Ricci soliton must be connected at infinity provided its scalar curvature satisfies a suitable lower bound. where J f (x, r, ξ) := e −f J (x, r, ξ) is the f -volume form in the geodesic polar coordinates. For a set Ω we will denote by V (Ω) the volume of Ω with respect to the usual volume form dv, and V f (Ω) the f -volume of Ω. Proof of Lemma 2.1. As discussed above, we write dV | exp p (rξ) = J (r, ξ) drdξ for ξ ∈ S p M. Let J f (r, ξ) = e −f (r,ξ) J (r, ξ) be the corresponding weighted volume form. In the following, we will omit the dependence of these quantities on ξ. Along a minimizing geodesic starting from p, we have for some constant C 0 > 0 independent of r. Let us denote where c(n) is a constant depending only on n and a + := max{a, 0}. In the following, we will denote by c a constant depending only on n, which may change from line to line. Plugging this into (6.2) results in J J