A Myers-type theorem and compact Ricci solitons
HTML articles powered by AMS MathViewer
- by Andrzej Derdzinski PDF
- Proc. Amer. Math. Soc. 134 (2006), 3645-3648 Request permission
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
- 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, DOI 10.1007/978-3-540-74311-8
- 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
- A. Derdzinski, Compact Ricci solitons, in preparation.
- 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
- M. Fernández-López and E. García-Río, A remark on compact Ricci solitons, preprint.
- Daniel Harry Friedan, Nonlinear models in $2+\varepsilon$ dimensions, Ann. Physics 163 (1985), no. 2, 318–419. MR 811072, DOI 10.1016/0003-4916(85)90384-7
- 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, DOI 10.1090/conm/071/954419
- Tom Ilmanen and Dan Knopf, A lower bound for the diameter of solutions to the Ricci flow with nonzero $H^1(M^n;\Bbb R)$, Math. Res. Lett. 10 (2003), no. 2-3, 161–168. MR 1981893, DOI 10.4310/MRL.2003.v10.n2.a3
- Thomas Ivey, Ricci solitons on compact three-manifolds, Differential Geom. Appl. 3 (1993), no. 4, 301–307. MR 1249376, DOI 10.1016/0926-2245(93)90008-O
- 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, DOI 10.2969/aspm/01810327
- J. A. Makowsky, On some conjectures connected with complete sentences, Fund. Math. 81 (1974), 193–202. MR 366647, DOI 10.4064/fm-81-3-193-202
- G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, http://arXiv.org/abs/math.DG/0211159
- 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, DOI 10.1016/j.aim.2003.09.009
Additional Information
- Andrzej Derdzinski
- Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
- Email: andrzej@math.ohio-state.edu
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 3645-3648
- MSC (2000): Primary 53C25; Secondary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-06-08422-X
- MathSciNet review: 2240678