A bound for the order of the fundamental group of a complete noncompact Ricci shrinker
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- by Bennett Chow and Peng Lu
- Proc. Amer. Math. Soc. 144 (2016), 2623-2625
- DOI: https://doi.org/10.1090/proc/12983
- Published electronically: December 15, 2015
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Abstract:
In the case of a complete noncompact shrinking gradient Ricci soliton, building upon the works of Derdzinski, Fernández-López and García-Rio, Lott, Naber, and Wylie, we obtain a bound for the order of its fundamental group $\pi _1$ in terms of the dimension $n$ and the logarithmic Sobolev constant. Under the additional assumption of being strongly $\kappa$-noncollapsed, a bound for the order of $\pi _1$ is $\kappa ^{-1}$ times a function of $n$.References
- Huai-Dong Cao and Detang Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010), no. 2, 175–185. MR 2732975
- José A. Carrillo and Lei Ni, Sharp logarithmic Sobolev inequalities on gradient solitons and applications, Comm. Anal. Geom. 17 (2009), no. 4, 721–753. MR 3010626, DOI 10.4310/CAG.2009.v17.n4.a7
- Andrzej Derdzinski, A Myers-type theorem and compact Ricci solitons, Proc. Amer. Math. Soc. 134 (2006), no. 12, 3645–3648. MR 2240678, DOI 10.1090/S0002-9939-06-08422-X
- M. Fernández-López and E. García-Río, A remark on compact Ricci solitons, Math. Ann. 340 (2008), no. 4, 893–896. MR 2372742, DOI 10.1007/s00208-007-0173-4
- Robert Haslhofer and Reto Müller, A compactness theorem for complete Ricci shrinkers, Geom. Funct. Anal. 21 (2011), no. 5, 1091–1116. MR 2846384, DOI 10.1007/s00039-011-0137-4
- John Lott, Some geometric properties of the Bakry-Émery-Ricci tensor, Comment. Math. Helv. 78 (2003), no. 4, 865–883. MR 2016700, DOI 10.1007/s00014-003-0775-8
- Ovidiu Munteanu and Jiaping Wang, Analysis of weighted Laplacian and applications to Ricci solitons, Comm. Anal. Geom. 20 (2012), no. 1, 55–94. MR 2903101, DOI 10.4310/CAG.2012.v20.n1.a3
- Ovidiu Munteanu and Jiaping Wang, Geometry of manifolds with densities, Adv. Math. 259 (2014), 269–305. MR 3197658, DOI 10.1016/j.aim.2014.03.023
- Aaron Naber, Some Geometry and Analysis on Ricci Solitons, arXiv:math/0612532.
- Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:0211159.
- William Wylie, Complete shrinking Ricci solitons have finite fundamental group, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1803–1806. MR 2373611, DOI 10.1090/S0002-9939-07-09174-5
- Takumi Yokota, Perelman’s reduced volume and a gap theorem for the Ricci flow, Comm. Anal. Geom. 17 (2009), no. 2, 227–263. MR 2520908, DOI 10.4310/CAG.2009.v17.n2.a3
- Zhenlei Zhang, On the finiteness of the fundamental group of a compact shrinking Ricci soliton, Colloq. Math. 107 (2007), no. 2, 297–299. MR 2284167, DOI 10.4064/cm107-2-9
Bibliographic Information
- Bennett Chow
- Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093
- Peng Lu
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
- Received by editor(s): July 14, 2015
- Published electronically: December 15, 2015
- Additional Notes: The second author was partially supported by a grant from the Simons Foundation.
- Communicated by: Guofang Wei
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2623-2625
- MSC (2010): Primary 53C44
- DOI: https://doi.org/10.1090/proc/12983
- MathSciNet review: 3477080