### Analysis of weighted Laplacian and applications to Ricci solitons

Ovidiu Munteanu, Jiaping Wang
2012 Communications in analysis and geometry
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
more » ... 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