A Liouville-type Theorem for Smooth Metric Measure Spaces

TL;DR

This paper proves a Liouville-type theorem for smooth metric measure spaces with nonnegative Bakry-Emery Ricci tensor, extending Yau's classical result to a broader geometric setting.

Contribution

It generalizes Yau's Liouville theorem to smooth metric measure spaces with nonnegative Bakry-Emery Ricci curvature, using gradient estimates for f-harmonic functions.

Findings

Liouville-type theorem established for spaces with nonnegative Bakry-Emery Ricci tensor

Gradient estimates derived for f-harmonic functions in this setting

Result recovers Yau's theorem when the weight function f is constant

Abstract

For smooth metric measure spaces $(M, g, e^{-f} dvol)$ we prove a Liuoville-type theorem when the Bakry-Emery Ricci tensor is nonnegative. This generalizes a result of Yau, which is recovered in the case $f$ is constant. This result follows from a gradient estimate for f-harmonic functions on smooth metric measure spaces with Bakry-Emery Ricci tensor bounded from below.