On gradient Ricci solitons with symmetry
HTML articles powered by AMS MathViewer
- by Peter Petersen and William Wylie PDF
- Proc. Amer. Math. Soc. 137 (2009), 2085-2092 Request permission
Abstract:
We study gradient Ricci solitons with maximal symmetry. First we show that there are no nontrivial homogeneous gradient Ricci solitons. Thus, the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. However, we apply the main result in our paper “Rigidity of gradient Ricci solitons” to show that there are no noncompact cohomogeneity one shrinking gradient solitons with nonnegative curvature.References
- Paul Baird and Laurent Danielo, Three-dimensional Ricci solitons which project to surfaces, J. Reine Angew. Math. 608 (2007), 65–91. MR 2339469, DOI 10.1515/CRELLE.2007.053
- Shigetoshi Bando, Real analyticity of solutions of Hamilton’s equation, Math. Z. 195 (1987), no. 1, 93–97. MR 888130, DOI 10.1007/BF01161602
- Christoph Böhm and Burkhard Wilking. Manifolds with positive curvature operators are space forms. Ann. of Math. (2) 167(3): 1079-1097, 2008.
- 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
- Huai-Dong Cao, Limits of solutions to the Kähler-Ricci flow, J. Differential Geom. 45 (1997), no. 2, 257–272. MR 1449972
- Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI; Science Press Beijing, New York, 2006. MR 2274812, DOI 10.1090/gsm/077
- Andrew Dancer and Mckenzie Wang. On Ricci Solitons of cohomogeneity one. arXiv:math/0802.0759.
- Manolo Eminenti, Gabriele La Nave, and Carlo Mantegazza. Ricci Solitons - the Equation Point of View. Manuscripta Math. 127(3): 345-367, 2008.
- 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
- 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
- Thomas Ivey, New examples of complete Ricci solitons, Proc. Amer. Math. Soc. 122 (1994), no. 1, 241–245. MR 1207538, DOI 10.1090/S0002-9939-1994-1207538-5
- 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
- Brett Kotschwar. On rotationally invariant shrinking gradient Ricci solitons. Pacific J. Math. 236(1): 73-88, 2008.
- Jorge Lauret, Ricci soliton homogeneous nilmanifolds, Math. Ann. 319 (2001), no. 4, 715–733. MR 1825405, DOI 10.1007/PL00004456
- John Lott, On the long-time behavior of type-III Ricci flow solutions, Math. Ann. 339 (2007), no. 3, 627–666. MR 2336062, DOI 10.1007/s00208-007-0127-x
- Aaron Naber. Noncompact shrinking $4$-solitons with nonnegative curvature. arXiv:math.DG/0710.5579.
- Lei Ni, Ancient solutions to Kähler-Ricci flow, Math. Res. Lett. 12 (2005), no. 5-6, 633–653. MR 2189227, DOI 10.4310/MRL.2005.v12.n5.a3
- Lei Ni and Nolan Wallach. On a classification of the gradient shrinking solitons. Math. Research Letters 15(5): 941-955, 2008.
- Lei Ni and Nolan Wallach. On 4-dimensional gradient shrinking solitons. Internat. Math. Res. Notices, vol. 2008, rnm152, 13 pp.
- G. Ya. Perelman. Ricci flow with surgery on three manifolds. arXiv: math.DG/0303109.
- Peter Petersen and William Wylie. Rigidity of gradient Ricci solitons. arXiv:math.DG/0710.3174. To appear in Pacific J. Math.
- Peter Petersen and William Wylie. On the classification of gradient Ricci solitons. arXiv:math.DG/0712.1298.
- 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
- Brian Weber. Convergence of compact Ricci solitons. arXiv:math.DG/0804.1158.
- Guofang Wei and William Wylie. Comparison geometry for the Bakry-Emery Ricci tensor. arXiv:math.DG/0706.1120.
- 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
- Bo Yang. A characterization of Koiso’s typed solitons. arXiv:math.DG/0802.0300.
Additional Information
- Peter Petersen
- Affiliation: Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, California 90095
- Email: petersen@math.ucla.edu
- William Wylie
- Affiliation: Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, California 90095
- Address at time of publication: Department of Mathematics, David Rittenhouse Laboratory, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
- Email: wylie@math.upenn.edu
- Received by editor(s): October 12, 2007
- Received by editor(s) in revised form: August 5, 2008
- Published electronically: January 22, 2009
- Communicated by: Richard A. Wentworth
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 2085-2092
- MSC (2000): Primary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-09-09723-8
- MathSciNet review: 2480290