Ricci Solitons, Conical Singularities, and Nonuniqueness

In dimension $$n=3$$ , there is a complete theory of weak solutions of Ricci flow—the singular Ricci flows introduced by Kleiner and Lott (Acta Math 219(1):65–134, 2017, in: Chen, Lu, Lu, Zhang (eds) Geometric analysis. Progress in mathematics, vol 333. Birkhäuser, Cham, 2018)—which Bamler and Kleiner (Uniqueness and stability of Ricci flow through singularities, arXiv:1709.04122v1 , 2017) proved are unique across singularities. In this paper, we show that uniqueness should not be expected to hold for Ricci flow weak solutions in dimensions $$n\ge 5$$ . Specifically, for any integers $$p_1,p_2\ge 2$$ with $$p_1+p_2\le 8$$ , and any $$K\in {\mathbb N}$$ , we construct a complete shrinking soliton metric $$g_K$$ on $$\mathcal S^{p_1}\times {\mathbb R}^{p_2+1}$$ whose forward evolution $$g_K(t)$$ by Ricci flow starting at $$t=-1$$ forms a singularity at time $$t=0$$ . As $$t\nearrow 0$$ , the metric $$g_K(t)$$ converges to a conical metric on $$\mathcal S^{p_1}\times \mathcal S^{p_2}\times (0,\infty )$$ . Moreover there exist at least K distinct, non-isometric, forward continuations by Ricci flow expanding solitons on $$\mathcal S^{p_1}\times {\mathbb R}^{p_2+1}$$ , and also at least K non-isometric, forward continuations expanding solitons on $${\mathbb R}^{p_1+1}\times \mathcal S^{p_2}$$ . In short, there exist smooth complete initial metrics for Ricci flow whose forward evolutions after a first singularity forms are not unique, and whose topology may change at the singularity for some solutions but not for others.