Analysis of weighted Laplacian and applications to Ricci solitons

TLDR: In this paper, the authors studied both function theoretic and spectral properties of the weighted Laplacian ∆f on complete smooth metric measure space (M, g, e dv) with its Bakry-Emery curvature bounded from below by a constant.

Abstract: We study both function theoretic and spectral properties of the weighted Laplacian ∆f on complete smooth metric measure space (M, g, e dv) with its Bakry-Emery 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 ∆f in terms of the lower bound of Ricf 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.