Smooth metric measure spaces with non-negative curvature

TL;DR

This paper investigates geometric and spectral properties of complete noncompact smooth metric measure spaces with nonnegative Bakry-Émery Ricci curvature, deriving key estimates and characterizing the structure of such spaces.

Contribution

It provides new gradient estimates for positive $f$-harmonic functions, establishes a sharp spectral bound, and characterizes the structure of spaces where equality holds, including applications to Ricci solitons.

Findings

Gradient estimate for positive $f$-harmonic functions

Sharp upper bound for the bottom spectrum of the $f$-Laplacian

Characterization of spaces with equality as cylinders and implications for Ricci solitons

Abstract

We study both function theoretic and spectral properties on complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-\'{E}mery 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.