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

Ben Andrews*, Paul Bryan

*Corresponding author for this work

Research output: Contribution to journalArticle

4 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)419-428
Number of pages10
JournalCalculus of Variations and Partial Differential Equations
Volume39
Issue number3
DOIs
Publication statusPublished - 2010
Externally publishedYes

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