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A Myers-type theorem and compact Ricci solitons
Author(s):
Andrzej
Derdzinski
Journal:
Proc. Amer. Math. Soc.
134
(2006),
3645-3648.
MSC (2000):
Primary 53C25;
Secondary 53C20
Posted:
June 13, 2006
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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.
References:
-
- 1.
- A. L. Besse, Einstein Manifolds, Ergebnisse (3) 10, Springer-Verlag, Berlin, 1987. MR 0867684 (88f:53087)
- 2.
- H.-D. Cao, Existence of gradient Kähler-Ricci solitons, Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), 1-16, A. K. Peters, Wellesley, MA, 1996. MR 1417944 (98a:53058)
- 3.
- A. Derdzinski, Compact Ricci solitons, in preparation.
- 4.
- M. Feldman, T. Ilmanen and D. Knopf, Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons, J. Differential Geom. 65 (2003), 169-209. MR 2058261 (2005e:53102)
- 5.
- M. Fernández-López and E. García-Río, A remark on compact Ricci solitons, preprint.
- 6.
- D. H. Friedan, Nonlinear models in
 dimensions, Ann. Physics 163 (1985), 318-419. MR 0811072 (87f:81130) - 7.
- R. S. Hamilton, The Ricci flow on surfaces, in: Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., Vol. 71, Amer. Math. Soc., Providence, RI, 1988. MR 0954419 (89i:53029)
- 8.
- T. Ilmanen and D. Knopf, A lower bound for the diameter of solutions to the Ricci flow with nonzero
 , Math. Res. Lett. 10 (2003), 161-168. MR 1981893 (2004a:53085) - 9.
- T. Ivey, Ricci solitons on compact three-manifolds, Differential Geom. Appl. 3 (1993), 301-307. MR 1249376 (94j:53048)
- 10.
- N. Koiso, On rotationally symmetric Hamilton's equation for Kähler-Einstein metrics, Recent topics in differential and analytic geometry, 327-337, Adv. Stud. Pure Math., 18-I, Academic Press, Boston, MA, 1990.MR 1145263 (93d:53057)
- 11.
- J. Makowsky, On some conjectures connected with complete sentences, Fund. Math. 81 (1974), 193-202. MR 0366647 (51:2894)
- 12.
- G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, http://arXiv.org/abs/math.DG/0211159
- 13.
- X. J. Wang and X. Zhu, Kähler-Ricci solitons on toric manifolds with positive first Chern class, Adv. in Math. 188 (2004), 87-103.MR 2084775 (2005d:53074)
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Additional Information:
Andrzej
Derdzinski
Affiliation:
Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Email:
andrzej@math.ohio-state.edu
DOI:
10.1090/S0002-9939-06-08422-X
PII:
S 0002-9939(06)08422-X
Received by editor(s):
December 8, 2004
Received by editor(s) in revised form:
July 11, 2005
Posted:
June 13, 2006
Communicated by:
Jon G. Wolfson
Copyright of article:
Copyright
2006,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
Forward Citation(s): Information for authors on submitting citations The following works have cited this article Fu-quan Fang, Jian-wen Man, Zhen-lei Zhang, Complete gradient shrinking Ricci solitons have finite topological type, C. R. Math. Acad. Sci. Paris 346 (2008), 10.1016/j.crma.2008.03.021, posted on 04/29/2008, 653-656. (English) MR 2423272
M. Fernandez-Lopez, E. Garcia-Rio, A remark on compact Ricci solitons, Math. Ann. 340 (2008), 10.1007/s00208-007-0173-4, posted on 09/18/2007, 893-896. (English) MR 2372742 (2008j:53077)
Manolo Eminenti, Gabriele La Nave, Carlo Mantegazza, Ricci solitons: the equation point of view, Manuscripta Math. 127 (2008), 10.1007/s00229-008-0210-y, posted on 09/23/2008, 345-367. (English)
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