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January 2020 The Ricci flow on the sphere with marked points
D. H. Phong, Jian Song, Jacob Sturm, Xiaowei Wang
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J. Differential Geom. 114(1): 117-170 (January 2020). DOI: 10.4310/jdg/1577502023


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.


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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.


Received: 21 December 2016; Published: January 2020
First available in Project Euclid: 28 December 2019

zbMATH: 07147344
MathSciNet: MR4047553
Digital Object Identifier: 10.4310/jdg/1577502023

Rights: Copyright © 2020 Lehigh University


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Vol.114 • No. 1 • January 2020
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