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

TL;DR

This paper establishes a comparison theorem for isoperimetric profiles under normalized Ricci flow on the two-sphere, leading to sharp curvature bounds and a straightforward proof of convergence to constant curvature.

Contribution

It introduces a new comparison theorem for isoperimetric profiles that simplifies proving convergence and curvature bounds without complex singularity analysis.

Findings

Comparison theorem for isoperimetric profiles under Ricci flow

Sharp time-dependent curvature bounds derived

Simplified proof of convergence to constant curvature

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.