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 | References | Similar Articles | Additional Information

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.