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Convergence of the conical Ricci flow on S2 to a soliton

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