Ancient solutions to the Ricci flow in higher dimensions

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.

arXiv commentFinal version, to appear in Communications in Analysis and Geometry