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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2010
- DOI:
- arXiv:
- arXiv:1006.0751
- Bibcode:
- 2010arXiv1006.0751B
- Keywords:
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- Mathematics - Differential Geometry
- E-Print:
- 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