On $\kappa$-solutions and canonical neighborhoods in 4d Ricci flow

TL;DR

This paper proposes a classification conjecture for 4d Ricci flow $ppa$-solutions, introduces a new family of symmetric bubble-sheet ovals, and establishes a canonical neighborhood theorem assuming a stronger conjecture, revealing new phenomena in 4d Ricci flow.

Contribution

It introduces a new classification conjecture for 4d Ricci flow solutions, constructs a novel symmetric bubble-sheet family, and proves a canonical neighborhood theorem under a stronger conjecture.

Findings

A new family of symmetric bubble-sheet ovals is constructed.

A stronger classification conjecture implies a canonical neighborhood theorem.

Quotient-necks may cause non-uniqueness in 4d Ricci flow.

Abstract

We introduce a classification conjecture for $\kappa$-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of $\mathbb{Z}_2^2\times \mathrm{O}_3$-symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.