Analysis of weighted Laplacian and applications to Ricci solitons

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 Ricf 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 optimal 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.