Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

TL;DR

This paper classifies compact ancient solutions to the higher-dimensional Ricci flow on spheres, showing they are either shrinking spheres or Perelman's Type II solutions, extending previous results to higher dimensions.

Contribution

It extends the classification of ancient solutions to n-dimensional Ricci flow on spheres, identifying all such solutions as either shrinking spheres or Perelman's Type II solutions.

Findings

Ancient solutions are either shrinking spheres or Perelman's Type II solutions.

The classification extends previous results to higher dimensions.

Provides a complete characterization of compact ancient solutions on spheres.

Abstract

In this paper, we study the classification of $\kappa$-noncollapsed ancient solutions to n-dimensional Ricci flow on $S^n$, extending the result in [13] to higher dimensions. We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.