Convergence of the conical Ricci flow on S2 to a soliton

Abstract

In our previous work [PSSW], we showed that the Ricci flow on S^2 whose initial metric has conical singularities \sum_{j=1}^k \beta_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 \beta_k>\beta_k'=\s_{j<k}\beta_j, that the limiting metric is the unique shrinking soliton with cone singularity \beta_k[p_\infty]+\beta_k'[q_\infty]. This verifies the prediction made in [PSSW].