On locally conformally flat gradient steady Ricci solitons

Cao, Huai-Dong; Chen, Qiang

doi:10.1090/S0002-9947-2011-05446-2

On locally conformally flat gradient steady Ricci solitons
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Trans. Amer. Math. Soc. 364 (2012), 2377-2391 Request permission

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Bryant, R., unpublished work.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