Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A Myers-type theorem and compact Ricci solitons

Author: Andrzej Derdzinski
Journal: Proc. Amer. Math. Soc. 134 (2006), 3645-3648
MSC (2000): Primary 53C25; Secondary 53C20
Published electronically: June 13, 2006
MathSciNet review: 2240678
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Abstract: Let the Ricci curvature of a compact Riemannian manifold be greater, at every point, than the Lie derivative of the metric with respect to some fixed smooth vector field. It is shown that the fundamental group then has only finitely many conjugacy classes. This applies, in particular, to all compact shrinking Ricci solitons.

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  • 1. Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684
  • 2. Huai-Dong Cao, Existence of gradient Kähler-Ricci solitons, Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, MA, 1996, pp. 1–16. MR 1417944
  • 3. A. Derdzinski, Compact Ricci solitons, in preparation.
  • 4. Mikhail Feldman, Tom Ilmanen, and Dan Knopf, Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons, J. Differential Geom. 65 (2003), no. 2, 169–209. MR 2058261
  • 5. M. Fernández-López and E. García-Río, A remark on compact Ricci solitons, preprint.
  • 6. Daniel Harry Friedan, Nonlinear models in 2+𝜖 dimensions, Ann. Physics 163 (1985), no. 2, 318–419. MR 811072, 10.1016/0003-4916(85)90384-7
  • 7. Richard S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237–262. MR 954419, 10.1090/conm/071/954419
  • 8. Tom Ilmanen and Dan Knopf, A lower bound for the diameter of solutions to the Ricci flow with nonzero 𝐻¹(𝑀ⁿ;ℝ), Math. Res. Lett. 10 (2003), no. 2-3, 161–168. MR 1981893, 10.4310/MRL.2003.v10.n2.a3
  • 9. Thomas Ivey, Ricci solitons on compact three-manifolds, Differential Geom. Appl. 3 (1993), no. 4, 301–307. MR 1249376, 10.1016/0926-2245(93)90008-O
  • 10. Norihito Koiso, On rotationally symmetric Hamilton’s equation for Kähler-Einstein metrics, Recent topics in differential and analytic geometry, Adv. Stud. Pure Math., vol. 18, Academic Press, Boston, MA, 1990, pp. 327–337. MR 1145263
  • 11. J. A. Makowsky, On some conjectures connected with complete sentences, Fund. Math. 81 (1974), 193–202. Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, III. MR 0366647
  • 12. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint,
  • 13. Xu-Jia Wang and Xiaohua Zhu, Kähler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math. 188 (2004), no. 1, 87–103. MR 2084775, 10.1016/j.aim.2003.09.009

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Additional Information

Andrzej Derdzinski
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210

Received by editor(s): December 8, 2004
Received by editor(s) in revised form: July 11, 2005
Published electronically: June 13, 2006
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.