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



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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© 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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