|
On gradient Ricci solitons with symmetry
Author(s):
Peter
Petersen;
William
Wylie
Journal:
Proc. Amer. Math. Soc.
137
(2009),
2085-2092.
MSC (2000):
Primary 53C20
Posted:
January 22, 2009
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
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:
-
- 1.
- Paul Baird and Laurent Danielo.
Three-dimensional Ricci solitons which project to surfaces.
J. Reine Angew. Math. 608: 65-91, 2007. MR 2339469 (2008f:53086) - 2.
- Shigetoshi Bando.
Real analyticity of solutions of Hamilton's equation.
Math Z. 195: 93-97, 1987. MR 888130 (88i:53073) - 3.
- Christoph Böhm and Burkhard Wilking.
Manifolds with positive curvature operators are space forms.
Ann. of Math. (2) 167(3): 1079-1097, 2008. - 4.
- Huai-Dong Cao.
Existence of gradient Kä hler-Ricci solitons.
In Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), pages 1-16. A K Peters, Wellesley, MA, 1996. MR 1417944 (98a:53058) - 5.
- Huai-Dong Cao.
Limits of solutions to the Kä hler-Ricci flow.
J. Differential Geom. 45(2): 257-272, 1997. MR 1449972 (99g:53042) - 6.
- Bennet Chow, Peng Lu, and Lei Ni.
Hamilton's Ricci flow.
Graduate Studies in Mathematics 77, Amer. Math. Soc., Providence, RI; Science Press, New York, 2006. MR 2274812 (2008:53068) - 7.
- Andrew Dancer and Mckenzie Wang.
On Ricci Solitons of cohomogeneity one.
arXiv:math/0802.0759. - 8.
- Manolo Eminenti, Gabriele La Nave, and Carlo Mantegazza.
Ricci Solitons - the Equation Point of View. Manuscripta Math. 127(3): 345-367, 2008. - 9.
- Mikhail Feldman, Tom Ilmanen, and Dan Knopf.
Rotationally symmetric shrinking and expanding gradient K ähler-Ricci solitons.
J. Differential Geom. 65(2): 169-209, 2003. MR 2058261 (2005e:53102) - 10.
- Richard Hamilton.
The Ricci flow on surfaces.
In Mathematics and General Relativity, Contemporary Mathematics 71, pages 237-262. Amer. Math. Soc., Providence, RI, 1988. MR 954419 (89i:53029) - 11.
- Thomas Ivey.
New examples of complete Ricci solitons.
Proc. Amer. Math. Soc. 122(1): 241-245, 1994. MR 1207538 (94k:53057) - 12.
- Norihito Koiso.
On rotationally symmetric H amilton's equation for Kähler-Einstein metrics.
In Recent Topics in Differential and Analytic Geometry, Adv. Stud. Pure Math. 18, pages 327-337. Academic Press, Boston, MA, 1990. MR 1145263 (93d:53057) - 13.
- Brett Kotschwar.
On rotationally invariant shrinking gradient Ricci solitons. Pacific J. Math. 236(1): 73-88, 2008. - 14.
- Jorge Lauret.
Ricci soliton homogeneous nilmanifolds.
Math. Ann. 319: 715-733, 2001. MR 1825405 (2002k:53083) - 15.
- John Lott.
On the long-time behavior of type-III Ricci flow solutions.
Math. Ann. 339(3): 627-666, 2007. MR 2336062 (2008i:53093) - 16.
- Aaron Naber.
Noncompact shrinking  -solitons with nonnegative curvature.
arXiv:math.DG/0710.5579. - 17.
- Lei Ni.
Ancient solutions to Kähler Ricci flow.
Math. Research Letters 12: 633-653, 2005. MR 2189227 (2006i:53097) - 18.
- Lei Ni and Nolan Wallach.
On a classification of the gradient shrinking solitons. Math. Research Letters 15(5): 941-955, 2008. - 19.
- Lei Ni and Nolan Wallach.
On 4-dimensional gradient shrinking solitons. Internat. Math. Res. Notices, vol. 2008, rnm152, 13 pp. - 20.
- G. Ya. Perelman.
Ricci flow with surgery on three manifolds.
arXiv: math.DG/0303109. - 21.
- Peter Petersen and William Wylie.
Rigidity of gradient Ricci solitons.
arXiv:math.DG/0710.3174. To appear in Pacific J. Math. - 22.
- Peter Petersen and William Wylie.
On the classification of gradient Ricci solitons.
arXiv:math.DG/0712.1298. - 23.
- X.J. Wang and X.H. Zhu.
Kähler-Ricci solitons on toric manifolds with positive first Chern class.
Adv. Math. 188(1): 87-103, 2004. MR 2084775 (2005d:53074) - 24.
- Brian Weber.
Convergence of compact Ricci solitons.
arXiv:math.DG/0804.1158. - 25.
- Guofang Wei and William Wylie.
Comparison geometry for the Bakry-Emery Ricci tensor.
arXiv:math.DG /0706.1120. - 26.
- William Wylie.
Complete shrinking Ricci solitons have finite fundamental group.
Proc. Amer. Math. Soc. 136(5): 1803-1806, 2008. MR 2373611 - 27.
- Bo Yang.
A characterization of Koiso's typed solitons.
arXiv:math.DG/0802.0300.
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
53C20
Retrieve articles in all Journals with MSC
(2000):
53C20
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
DOI:
10.1090/S0002-9939-09-09723-8
PII:
S 0002-9939(09)09723-8
Received by editor(s):
October 12, 2007,
Received by editor(s) in revised form:
August 5, 2008
Posted:
January 22, 2009
Communicated by:
Richard A. Wentworth
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org