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. Birkhauser, 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 is an element of N, we construct a complete shrinking soliton metric g(K) on S-p1 x Rp2+1 whose forward evolution g(K) (t) by Ricci flow starting at t = -1 forms a singularity at time t = 0. As t NE arrow 0, the metric g(K) (t) converges to a conical metric on S-p1 x S-p2 x (0, infinity). More-over there exist at least K distinct, non-isometric, forward continuations by Ricci flow expanding solitons on S-p1 x Sp2+1, and also at least K non-isometric, forward continuations expanding solitons on R-p1(+1) x S-p2. 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.