Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere
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We prove a comparison theorem for the isoperimetric profiles of solutions of the normalized Ricci flow on the two-sphere: If the isoperimetric profile of the initial metric is greater than that of some positively curved axisymmetric metric, then the inequality remains true for the isoperimetric profiles of the evolved metrics. We apply this using the Rosenau solution as the model metric to deduce sharp time-dependent curvature bounds for arbitrary solutions of the normalized Ricci flow on the two-sphere. This gives a simple and direct proof of convergence to a constant curvature metric without use of any blowup or compactness arguments, Harnack estimates, or any classification of behaviour near singularities.
Mathematics Subject Classification (2000)35K55 35K45 58J35
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- 1.Aubin, T.: Some nonlinear problems in Riemannian geometry. In: Springer monographs in mathematics. Springer-Verlag, Berlin, 1998 (MR1636569 (99i:58001))Google Scholar
- 6.Chavel, I.: Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol.115. Academic Press Inc., Orlando, FL, (1984) (Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk, MR768584) (86g:58140)Google Scholar
- 8.Gray A.: The volume of a small geodesic ball of a Riemannian manifold. Michigan Math. J. 20(1973), 329–344 (1974) (MR0339002 (49 #3765))Google Scholar
- 10.Hamilton, R.S.: The Ricci flow on surfaces, mathematics and general relativity, Santa Cruz, CA,1986 Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, 237–262, MR954419 (89i:53029)Google Scholar
- 12.Hamilton, R.S.: The formation of singularities in the Ricci flow. In: Surveys in differential geometry, vol. II (Cambridge, MA, 1993)pp. 7–136. Int. Press, Cambridge, MA (1995) (MR1375255 (97e:53075))Google Scholar
- 13.Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. (2002). Available at arXiv:math/0211159v1[math.DG]
- 14.Perelman, G.: Ricci flow with surgery on three-manifolds. (2003). Available at arXiv:math/0303109v1[math.DG]
- 15.Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. (2003). Available at arXiv:math/0307245v1[math.DG]