Ricci Solitons, Conical Singularities, and Nonuniqueness

TL;DR

This paper demonstrates that in dimensions five and higher, Ricci flow weak solutions can exhibit nonuniqueness, especially near conical singularities, contrasting with the unique solutions in three dimensions.

Contribution

The authors construct explicit examples of asymptotically conical gradient shrinking solitons with non-unique forward continuations, highlighting nonuniqueness phenomena in higher-dimensional Ricci flows.

Findings

Existence of non-unique Ricci flow continuations in dimensions ≥5.

Construction of asymptotically conical gradient shrinking solitons.

Topological nonuniqueness in Ricci flow solutions.

Abstract

In dimension $n=3$, there is a complete theory of weak solutions of Ricci flow - the singular Ricci flows introduced by Kleiner and Lott - which are unique across singularities, as was proved by Bamler and Kleiner. We show that uniqueness should not be expected to hold for Ricci flow weak solutions in dimensions $n\geq5$. Specifically, we exhibit a discrete family of asymptotically conical gradient shrinking solitons, each of which admits non-unique forward continuations by gradient expanding solitons. (v2) We recast the Main Theorem in the language of Kleiner and Lott's Ricci Flow spacetimes and in addition show that topological nonuniqueness is possible for the solutions we construct.