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Generalization of the Hamilton-Ivey estimate to the higher dimensional Ricci flow with a vanishing Weyl tensor. (English) Zbl 1314.53126

Summary: In this study, we consider curvature estimates of the Ricci flow and we generalize the Hamilton-Ivey pinching estimate to a higher dimensional Ricci flow with a vanishing Weyl tensor. We also prove a pinching estimate for the Ricci curvature under the Ricci flow.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K55 Nonlinear parabolic equations
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[1] Angenent, S.; Knopf, D., An example of neckpinching for Ricci flow on \(S^{n + 1}\), Math. Res. Lett., 11, 4, 493-518 (2004) · Zbl 1064.58022
[2] Böhm, C.; Wilking, B., Manifolds with positive curvature operators are space forms, Ann. of Math. (2), 167, 3, 1079-1097 (2008) · Zbl 1185.53073
[3] Brendle, S.; Schoen, R., Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc., 22, 1, 287-307 (2009) · Zbl 1251.53021
[4] Cao, H.-D., Existence of gradient Kähler-Ricci solitons, (Elliptic and Parabolic Methods in Geometry. Elliptic and Parabolic Methods in Geometry, Minneapolis, MN, 1994 (1996), A K Peters: A K Peters Wellesley, MA), 1-16 · Zbl 0868.58047
[5] Cao, H. D.; Zhu, X. P., A complete proof of the Poincaré and geometrization conjecture - application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math., 10, 2, 165-492 (2006) · Zbl 1200.53057
[6] Chen, B. L., Strong uniqueness of the Ricci flow, J. Differential Geom., 82, 2, 363-382 (2009) · Zbl 1177.53036
[7] Chen, B. L.; Tang, S. H.; Zhu, X. P., Complete classification of compact four-manifolds with positive isotropic curvature, J. Differential Geom., 91, 1-169 (2012)
[8] Chen, B. L.; Xu, G. Y.; Zhang, Z. H., Local pinching estimates in 3-dim Ricci flow, Math. Res. Lett., 20, 5, 845-855 (2015) · Zbl 1304.35149
[9] Chen, B. L.; Zhu, X. P., Ricci flow with surgery on four-manifolds with positive isotropic curvature, J. Differential Geom., 74, 177-264 (2006) · Zbl 1103.53036
[10] Hamilton, R. S., Three manifolds with positive Ricci curvature, J. Differential Geom., 17, 255-306 (1982) · Zbl 0504.53034
[11] Hamilton, R. S., Four-manifolds with positive curvature operator, J. Differential Geom., 24, 153-179 (1986) · Zbl 0628.53042
[12] Hamilton, R. S., The formation of singularities in the Ricci flow, (Cambridge, MA, 1993. Cambridge, MA, 1993, Surv. Differ. Geom., vol. 2 (1995), International Press: International Press Cambridge), 7-136 · Zbl 0867.53030
[13] Hamilton, R. S., Non-singular solutions to the Ricci flow on three manifolds, Comm. Anal. Geom., 7, 695-729 (1999) · Zbl 0939.53024
[14] Ivey, T., Ricci solitons on compact three-manifolds, Diff. Geom. Appl., 3, 301-307 (1993) · Zbl 0788.53034
[15] Koiso, N., On rotationally symmetric Hamilton’s equation for Kähler-Einstein metrics, (Recent Topics in Differential and Analytic Geometry. Recent Topics in Differential and Analytic Geometry, Adv. Stud. Pure Math., vol. 18:I (1990), Academic Press: Academic Press Boston, MA), 327-337 · Zbl 0739.53052
[16] Perelman, G., The entropy formula for the Ricci flow and its geometric applications (November 11, 2002), preprint
[17] Perelman, G., Ricci flow with surgery on three manifolds (March 10, 2003), preprint
[18] Zhang, Z. H., Gradient shrinking solitons with vanishing Weyl tensor, Pacific J. Math., 242, 1, 189-200 (2009), MR2525510 (2010f:53116) · Zbl 1171.53332
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