A Liouville-type Theorem for Smooth Metric Measure Spaces

Kevin Brighton

A Liouville-type Theorem for Smooth Metric Measure Spaces

Differential Geometry 2011-01-17 v2

Authors: Kevin Brighton

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.

Keywords

Liouville-type theorem smooth metric measure spaces Bakry-Emery Ricci tensor f-harmonic functions gradient estimate

Cite

@article{arxiv.1006.0751,
  title  = {A Liouville-type Theorem for Smooth Metric Measure Spaces},
  author = {Kevin Brighton},
  journal= {arXiv preprint arXiv:1006.0751},
  year   = {2011}
}

Comments

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

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