Analysis of weighted Laplacian and applications to Ricci solitons

Abstract

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 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.

Heat Kernel Estimates and the Essential Spectrum on Weighted Manifolds

“…We recall that Munteanu and Wang in [23,25] have demonstrated a similar upper bound for λ 1 (M) under the assumption that f has linear growth at a point and Ric f has a lower bound.…”

“…We would like to further point out that only assuming Ric f bounded below and f of linear growth at a point as in [25] would not be sufficient to obtain the global heat kernel bound this way. In order to get the heat kernel estimates, an assumption on the uniform linear growth of f is needed, which is almost equivalent to assuming that the gradient of f is bounded.…”

Estimates for the Eigenvalues of the Drifting Laplacian on Some Complete Ricci Solitons

“…As an important elliptic operator, estimates for eigenvalues of the drifting Laplacian have been attracting more and more attention from many mathematicians in recent years. On the one hand, upper bounds for the first eigenvalue of the drifting Laplacian on complete Riemannian manifolds have been studied in [27,28,31,34]. On the other hand, many mathematicians have also considered the lower bounds for the eigenvalues of the drifting Laplacian on the compact Riemannian manifolds.…”

“…where Ric and Hess f denote the Ricci tensor of M n and the Hessian of f , respectively (see [2,23])-have been obtained, for example, in [24,25,27,28,33]. Furthermore, the socalled drifting Laplacian (also called the f -Laplacian or Witten-Laplacian) with respect to the metric measure space plays an important role in geometric analysis, which is defined by…”

“…During the last two decades, the metric measure space has received a lot of attention in geometric analysis (cf. [1,3,4,7,13,14,27,28,32,35]). In a celebrated work [29], Perelman introduced a functional that involves an integral of the scalar curvature in the sense of the weighted measure, such that the Ricci flow becomes a gradient flow of such a functional.…”