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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>ℤ</m:mi> <m:mn>2</m:mn> <m:mn>2</m:mn> </m:msubsup> <m:mo>×</m:mo> <m:msub> <m:mi mathvariant="normal">O</m:mi> <m:mn>3</m:mn> </m:msub> </m:mrow> </m:math> {Z₂²₃} -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.