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 ≥ 5 . Specifically, for any integers p 1 , p 2 ≥ 2 with p 1 + p 2 ≤ 8 , and any K ∈ N , we construct a complete shrinking soliton metric g K on S p 1 × 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 ↗ 0 , the metric g K ( t ) converges to a conical metric on S p 1 × S p 2 × ( 0 , ∞ ) . Moreover there exist at least K distinct, non-isometric, forward continuations by Ricci flow expanding solitons on S p 1 × R p 2 + 1 , and also at least K non-isometric, forward continuations expanding solitons on R p 1 + 1 × 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.