The Ricci flow on the sphere with marked points

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 semistable 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 is the unique conical constant curvature metric in the semi-stable case, and the unique conical shrinking gradient Ricci soliton metric in the unstable case.