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We study both function theoretic and spectral properties of the weightedLaplacian $\Delta_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$with its Bakry-\'{E}mery curvature $Ric_f$ bounded from below by a constant. Inparticular, we establish a gradient estimate for positive $f-$harmonicfunctions and a sharp upper bound of the bottom spectrum of $\Delta_f$ in termsof the lower bound of $Ric_{f}$ and the linear growth rate of $f.$ We alsoaddress the rigidity issue when the bottom spectrum achieves its optimal upperbound under a slightly stronger assumption that the gradient of $f$ is bounded.Applications to the study of the geometry and topology of gradient Riccisolitons are also considered. Among other things, it is shown that the volumeof a noncompact shrinking Ricci soliton must be of at least linear growth. Itis also shown that a nontrivial expanding Ricci soliton must be connected atinfinity provided its scalar curvature satisfies a suitable lower bound.

Analysis of weighted Laplacian and applications to Ricci solitons