Analysis of weighted Laplacian and applications to Ricci solitons

Abstract

We study both function theoretic and spectral properties of the weighted Laplacian f on complete smooth metric measure space (M, g, e -f dv) with its Bakry-mery 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 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.