The Ricci flow on the sphere with marked points

TL;DR

This paper proves the convergence of Ricci flow on the 2-sphere with marked points across all stability cases, revealing new behaviors in semi-stable and unstable scenarios and characterizing the limiting geometries.

Contribution

It extends Ricci flow convergence results to semi-stable and unstable cases, providing new proofs and describing the limiting spaces as conical metrics or Ricci solitons.

Findings

Flow converges to a 2-sphere with different marked points in semi-stable and unstable cases.

Limiting metrics are unique conical constant curvature or Ricci solitons.

Convergence is established in the Gromov-Hausdorff topology.

Abstract

The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is given. The semi-stable and unstable cases are new, and it is shown that the flow converges in the Gromov-Hausdorff topology to a limiting metric space which is also a 2-sphere, but with different marked points and hence a different complex structure. The limiting metric space carries a unique conical constant curvature metric in the semi-stable case, and a unique conical shrinking gradient Ricci soliton in the unstable case.