Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions

TL;DR

This paper extends the classification of ancient solutions to the Ricci flow in higher dimensions, showing that noncompact solutions with certain curvature conditions are limited to known models like cylinders and the Bryant soliton.

Contribution

It generalizes previous results to dimensions four and higher, establishing uniqueness of ancient $ppa$-solutions under specific curvature and noncollapsing conditions.

Findings

Noncompact ancient ppa-solutions are isometric to shrinking cylinders, quotients, or Bryant solitons.

The classification applies to dimensions  and above with uniform PIC and weak PIC2.

The results extend the uniqueness part of re20 to higher dimensions.

Abstract

We extend the second part of \cite{Bre20} on the uniqueness of ancient $\kappa$-solutions to higher dimensions. In dimensions $n \geq 4$, an ancient $\kappa$-solution is a nonflat, complete, ancient solution of the Ricci flow that is uniformly PIC and weakly PIC2; has bounded curvature; and is $\kappa$-noncollapsed. We show that the only noncompact ancient $\kappa$-solutions up to isometry are a family of shrinking cylinders, a quotient thereof, or the Bryant soliton.