The Ricci flow on the sphere with marked points

2014/07/04 by D. H. Phong, Phong, D. H., Jian Song +5 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.CV #math.DG

paper · pdf · doi:10.48550/arxiv.1407.1118

arxiv created 2014/07/04 · openalex publication_date 2014/07/04 · arxiv updated 2014/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Connecting toric manifolds by conical Kahler-Einstein metrics
• K-stability and Kähler-Einstein metrics
• Space of Ricci flows (II)
• Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 π

Cited by

• Convergence of the conical Ricci flow on S2 to a soliton

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