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Complete shrinking Ricci solitons have finite fundamental group


Author: William Wylie
Journal: Proc. Amer. Math. Soc. 136 (2008), 1803-1806
MSC (2000): Primary 53C20
DOI: https://doi.org/10.1090/S0002-9939-07-09174-5
Published electronically: October 18, 2007
MathSciNet review: 2373611
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Abstract: We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor have finite fundamental group. The method of proof is to generalize arguments of García-Río and Fernández-López in the compact case.


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Additional Information

William Wylie
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095
Email: wylie@math.ucla.edu

DOI: https://doi.org/10.1090/S0002-9939-07-09174-5
Keywords: Ricci soliton, noncompact manifold, fundamental group
Received by editor(s): March 29, 2007
Published electronically: October 18, 2007
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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