On κ-solutions and\break canonical neighborhoods in 4d Ricci flow

Abstract

We introduce a classification conjecture for κ-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new one-parameter family of ℤ 2 2 × O 3 {\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.