On locally conformally flat gradient steady Ricci solitons
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- by Huai-Dong Cao and Qiang Chen PDF
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Abstract:
In this paper, we classify $n$-dimensional ($n\geq 3$) complete noncompact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact nonflat locally conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.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
- Bryant, R., unpublished work.
- 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, Recent progress on Ricci solitons, Recent advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 11, Int. Press, Somerville, MA, 2010, pp. 1–38. MR 2648937
- Huai-Dong Cao and Xi-Ping Zhu, A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math. 10 (2006), no. 2, 165–492. MR 2233789, DOI 10.4310/AJM.2006.v10.n2.a2
- Catino, G. and Mantegazza, C., Evolution of the Weyl tensor under the Ricci flow, arXiv: 0910.4761.
- Bing-Long Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009), no. 2, 363–382. MR 2520796
- Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: techniques and applications. Part I, Mathematical Surveys and Monographs, vol. 135, American Mathematical Society, Providence, RI, 2007. Geometric aspects. MR 2302600, DOI 10.1090/surv/135
- Sun-Chin Chu, Geometry of 3-dimensional gradient Ricci solitons with positive curvature, Comm. Anal. Geom. 13 (2005), no. 1, 129–150. MR 2154669
- Fernández-López, M. and García-Río, E., Rigidity of shrinking Ricci solitons, preprint (2009).
- Detlef Gromoll and Wolfgang Meyer, On complete open manifolds of positive curvature, Ann. of Math. (2) 90 (1969), 75–90. MR 247590, DOI 10.2307/1970682
- Hongxin Guo, Area growth rate of the level surface of the potential function on the 3-dimensional steady gradient Ricci soliton, Proc. Amer. Math. Soc. 137 (2009), no. 6, 2093–2097. MR 2480291, DOI 10.1090/S0002-9939-09-09792-5
- Richard S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153–179. MR 862046
- 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
- Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7–136. MR 1375255
- Israel, W., Event horizons in static vacuum space-times, Phys. Rev., 164, no. 5 (1967), 1776-1779.
- 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
- Brett Kotschwar, On rotationally invariant shrinking Ricci solitons, Pacific J. Math. 236 (2008), no. 1, 73–88. MR 2398988, DOI 10.2140/pjm.2008.236.73
- Munteanu, O. and Sesum, N., On gradient Ricci solitons, arXiv:0910.1105.
- Perelman, G., The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159.
- Robinson, D. C., A simple proof of the generalization of Israel’s theorem, General Relativity and Gravitation, 8, no. 8 (1977), 695-698.
- Zhu-Hong Zhang, On the completeness of gradient Ricci solitons, Proc. Amer. Math. Soc. 137 (2009), no. 8, 2755–2759. MR 2497489, DOI 10.1090/S0002-9939-09-09866-9
- Zhu-Hong Zhang, Gradient shrinking solitons with vanishing Weyl tensor, Pacific J. Math. 242 (2009), no. 1, 189–200. MR 2525510, DOI 10.2140/pjm.2009.242.189
Additional Information
- Huai-Dong Cao
- Affiliation: Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015
- MR Author ID: 224609
- ORCID: 0000-0002-4956-4849
- Email: huc2@lehigh.edu
- Qiang Chen
- Affiliation: Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015
- Email: qic208@lehigh.edu
- Received by editor(s): March 9, 2010
- Received by editor(s) in revised form: March 15, 2010
- Published electronically: December 13, 2011
- Additional Notes: The first author was partially supported by NSF Grants DMS-0506084 and DMS-0909581.
The second author was partially supported by NSF Grant DMS-0354621 and a Dean’s Fellowship of the School of Arts and Sciences at Lehigh University. - © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 2377-2391
- MSC (2010): Primary 53C21, 53C25
- DOI: https://doi.org/10.1090/S0002-9947-2011-05446-2
- MathSciNet review: 2888210