Smooth metric measure spaces with non-negative curvature

Abstract

In this paper we study both function theoretic and spectral properties on complete noncompact smooth metric measure space (M, g, e −f dv) with nonnegative Bakry-Émery Ricci curvature. Among other things, we derive a gradient estimate for positive f -harmonic functions and obtain as a consequence the strong Liouville property under the optimal sublinear growth assumption on f. We also establish a sharp upper bound of the bottom spectrum of the f -Laplacian in terms of the linear growth rate of f. Moreover, we show that if equality holds and M is not connected at infinity, then M must be a cylinder. As an application, we conclude steady Ricci solitons must be connected at infinity. ∆ f := ∆ − ∇f · ∇ is more naturally associated with such a smooth metric measure space than the classical Laplacian as it is symmetric with respect to the measure e −f dv. That is,for any ϕ, ψ ∈ C ∞ 0 (M ) . Again, we point out that the operator f -Laplacian is very much related to the Laplacian of a suitable conformal change of the background Riemannian metric. It also appears as the generator of a class of stochastic diffusion processes, the Brownian motion with drifts.The Bakry-Émery Ricci tensor [1] of the metric measure space (M, g, e −f dv) is defined byRic f := Ric + Hess(f ),

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.…”

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.…”

On Ricci-harmonic metrics

“…If (N, h) = (R, dr 2 ) and φ : M → R is a constant function, then the gradient Ricci-harmonic soliton metric defined in (1.3) is a gradient Ricci soliton metric. The works on the gradient Ricci soliton metric can be referred to [2,3,11,26,27,30] and the references therein. When the potential function f is a constant function, the gradient Ricci-harmonic soliton metric is called harmonic Einstein, which satisfies the following coupled system…”