# Ancient solutions to the Ricci flow in higher dimensions

## TL;DR

This paper classifies certain ancient solutions to the Ricci flow in higher dimensions, showing that under specific conditions, they must be the Bryant soliton, thus extending previous results to more complex settings.

## Contribution

It generalizes Brendle's classification of ancient solutions to higher dimensions with additional assumptions on asymptotic behavior.

## Key findings

- Noncompact $
abla$-noncollapsed ancient solutions with positive curvature are Bryant solitons.
- Under the assumptions, solutions asymptotic to the standard cylinder are classified.
- Extension of classification results to higher-dimensional Ricci flows.

## Abstract

In this paper, we study $\kappa$-noncollapsed ancient solutions to the Ricci flow with nonnegative curvature operator in higher dimensions. We impose one further assumption: one of the asymptotic shrinking gradient Ricci solitons is the standard cylinder $\mathbb{S}^{n-1}\times\mathbb{R}$. By making use of the properties of such ancient solutions, we generalize part one of Brendle \cite{brendle2018ancient} to higher dimensions, that is, every noncompact $\kappa$-noncollapsed rotationally symmetric ancient solution to the Ricci flow with bounded positive curvature operator must be the Bryant soliton.