Communications in Analysis and Geometry

Volume 30 (2022)

Number 9

Ancient solutions to the Ricci flow in higher dimensions

Pages: 2011 – 2048



Xiaolong Li (Department of Mathematics, University of California, Irvine, Calif., U.S.A.; and Department of Mathematics, Statistics and Physics, Wichita State University, Wichita, Kansas, U.S.A.)

Yongjia Zhang (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China)


In this paper, we study $\kappa$-noncollapsed ancient solutions to the Ricci flow with nonnegative curvature operator in higher dimensions $n \geq 4$. We impose one further assumption: one of the asymptotic shrinking gradient Ricci solitons is the standard cylinder $\mathbb{S}^{n-1} \times \mathbb{R}$. First, Perelman’s structure theorem on three-dimensional ancient $\kappa$-solutions is generalized to all higher dimensions. Second, we prove that every noncompact $\kappa$-noncollapsed rotationally symmetric ancient solution to the Ricci flow with bounded positive curvature operator must be the Bryant soliton, thus extending a very recent result of Brendle in three dimensions to all higher dimensions.

Yongjia Zhang’s research is partially supported by Shanghai Sailing Program 23YF1420400, and by Research Start-up Fund of SJTU WH220407110.

Received 7 August 2019

Accepted 11 May 2020

Published 17 August 2023