$\kappa$-solutions with the round cylinder as an asymptotic shrinker

TL;DR

This paper proves that certain high-dimensional Ricci flow solutions asymptotic to a round cylinder are necessarily uniformly PIC, leading to classification results for these solutions.

Contribution

It establishes that $ abla$-solutions with cylindrical asymptotics in dimensions ≥4 are uniformly PIC, extending classification of Ricci flow solutions.

Findings

Solutions asymptotic to the round cylinder are uniformly PIC.

Classifies noncompact solutions as either the round cylinder or Bryant steady soliton.

Classifies compact solutions as Perelman's ancient solutions.

Abstract

We show that $\kappa$-solutions to the Ricci flow in dimensions $n\geq 4$ whose asymptotic shrinking Ricci soliton is the round cylinder $\mathbb{S}^{n-1}\times\mathbb{R}$ must be uniformly PIC. Combined with earlier classification results, this implies that any such noncompact solution is either the round shrinking cylinder or the Bryant steady soliton, and any such compact solution is Perelman's ancient solution.