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.
Funding Statement
Work supported in part by National Science Foundation grants DMS-12-66033, DMS-1406124 and DMS-0905873 and a Collaboration Grants for Mathematicians from this Simons Foundation.
Citation
D. H. Phong. Jian Song. Jacob Sturm. Xiaowei Wang. "The Ricci flow on the sphere with marked points." J. Differential Geom. 114 (1) 117 - 170, January 2020. https://doi.org/10.4310/jdg/1577502023