|
Complete shrinking Ricci solitons have finite fundamental group
Author(s):
William
Wylie
Journal:
Proc. Amer. Math. Soc.
136
(2008),
1803-1806.
MSC (2000):
Primary 53C20
Posted:
October 18, 2007
Retrieve article in:
PDF
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.
References:
-
- [A]
- W. Ambrose, A theorem of Myers, Duke Math. J. 24 (1957), 345-348. MR 0089464 (19,680c)
- [CLN]
- B. Chow, P. Lu, and L. Ni, Hamilton's Ricci flow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI, 2006. MR 2274812
- [FG]
- M. Fernández-López and E. García-Río, A remark on compact Ricci solitons, preprint.
- [H]
- R. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), Int. Press, Cambridge, MA, 1995, pp. 7-136. MR 1375255 (97e:53075)
- [L]
- J. Lott, Some geometric properties of the Bakry-Émery-Ricci tensor, Comment. Math. Helv. 78 (2003), no. 4, 865-883. MR 2016700 (2004i:53044)
- [N]
- A. Naber, Some geometry and analysis on Ricci solitons, arXiv:math.DG/0612532.
- [P]
- G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/0211159.
- [Q]
- Z. Qian, Estimates for weighted volumes and applications, Quart. J. Math. Oxford Ser. (2) 48 (1997), no. 190, 235-242. MR 1458581 (98e:53058)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
53C20
Retrieve articles in all Journals with MSC
(2000):
53C20
Additional Information:
William
Wylie
Affiliation:
Department of Mathematics, University of California, Los Angeles, California 90095
Email:
wylie@math.ucla.edu
DOI:
10.1090/S0002-9939-07-09174-5
PII:
S 0002-9939(07)09174-5
Keywords:
Ricci soliton,
noncompact manifold,
fundamental group
Received by editor(s):
March 29, 2007
Posted:
October 18, 2007
Communicated by:
Jon G. Wolfson
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org