Convergence of the conical Ricci flow on S2 to a soliton

2015/03/15 by D. H. Phong, Phong, D. H., Jian Song +5
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #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.1503.04488

arxiv created 2015/03/15 · openalex publication_date 2015/03/15 · arxiv updated 2015/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In our previous work [PSSW], we showed that the Ricci flow on S² whose initial metric has conical singularities ∑_j=1^k β_j[p_j] converges to a constant curvature metric with conic singularities (in the stable and semi-stable cases) or to a gradient shrinking soliton with conical singularities (in the unstable case). The purpose of this note is to show that in the unstable case, that is, the case where β_k>β_k'=\s_j<kβ_j, that the limiting metric is the unique shrinking soliton with cone singularity β_k[p_∞]+β_k'[q_∞]. This verifies the prediction made in [PSSW].

Citations

• The Ricci flow on the sphere with marked points
• Connecting toric manifolds by conical Kahler-Einstein metrics

Related

• The Ricci flow on the sphere with marked points
• Ricci flows which terminate in cones
• Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers
• Conical structure for shrinking Ricci solitons
• Ricci-mean curvature flows in gradient shrinking Ricci solitons