Communications in Analysis and Geometry

Volume 30 (2022)

Number 9

Ancient solutions to the Ricci flow in higher dimensions

Pages: 2011 – 2048

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n9.a3

Authors

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)

Abstract

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.

Full Text (PDF format)

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