Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere

Article

Abstract

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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References

  1. 1.
    Aubin, T.: Some nonlinear problems in Riemannian geometry. In: Springer monographs in mathematics. Springer-Verlag, Berlin, 1998 (MR1636569 (99i:58001))Google Scholar
  2. 2.
    Böhm C., Wilking B.: Manifolds with positive curvature operators are space forms. Ann. Math. 167(3), 1079–1097 (2008) (MR2415394)MATHCrossRefGoogle Scholar
  3. 3.
    Brendle S., Schoen R.: Manifolds with 1/4-pinched curvature are space forms. J. Amer. Math. Soc. 22(1), 287–307 (2009) (MR2449060)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Brendle S., Schoen R.M.: Classification of manifolds with weakly 1/4-pinched curvatures. Acta Math. 200(1), 1–13 (2008) (MR2386107)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Brendle S.: A general convergence result for the Ricci flow in higher dimensions. Duke Math. J. 145(3), 585–601 (2008) (MR2462114)MATHCrossRefMathSciNetGoogle Scholar
  6. 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
  7. 7.
    Chow B.: The Ricci flow on the 2-sphere J. Differ. Geom. 33(2), 325–334 (1991) (MR1094458 (92d:53036))MATHGoogle Scholar
  8. 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
  9. 9.
    Hamilton R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982) (MR664497 (84a:53050))MATHMathSciNetGoogle Scholar
  10. 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
  11. 11.
    Hamilton R.S.: An isoperimetric estimate for the Ricci flow on the two-sphere, modern methods in complex analysis (Princeton, NJ,1992). Ann. Math. Stud. 137, 191–200 (1995) (MR1369139 (96k:53059))MathSciNetGoogle Scholar
  12. 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. 13.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. (2002). Available at arXiv:math/0211159v1[math.DG]
  14. 14.
    Perelman, G.: Ricci flow with surgery on three-manifolds. (2003). Available at arXiv:math/0303109v1[math.DG]
  15. 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]
  16. 16.
    Ritoré M.: Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces. Comm. Anal. Geom. 9(5), 1093–1138 (2001) (MR1883725 (2003a:53018))MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Centre for Mathematics and Its ApplicationsAustralian National UniversityCanberraAustralia
  2. 2.Department of MathematicsAustralian National UniversityCanberraAustralia

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