A Liouville-type Theorem for Smooth Metric Measure Spaces

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.

7 pages. The proofs are modified to remove the assumption that f is bounded. An example is included demonstrating necessity of the remaining assumptions and the exposition is revised to correct typos and increase readability