On the Ricci Flow of Homogeneous Metrics on Spheres

TL;DR

This paper analyzes the Ricci flow on a family of homogeneous metrics on spheres, classifies ancient solutions, and introduces new solutions with larger symmetry groups, revealing their long-term behaviors and geometric limits.

Contribution

It classifies ancient solutions of Ricci flow on spheres with Sp(n+1)-invariant metrics and introduces a new one-parameter family of ancient solutions with larger isometry groups.

Findings

Identified a new one-parameter family of ancient solutions on spheres.

Classified ancient solutions with larger isometry groups such as Sp(n+1)Sp(1), Sp(n+1)U(1), U(2n+2).

Some solutions converge to Einstein metrics, others collapse to Ziller's Einstein metric.

Abstract

We study the Ricci flow of the four-parameter family of Sp(n+1)-invariant metrics on spheres. We determine their forward behaviour and also classify ancient solutions. In doing so, we exhibit a new one-parameter family of ancient solutions on spheres. These (non-isometric) ancient solutions all have a larger isometry group, namely Sp(n+1)Sp(1), Sp(n+1)U(1), or U(2n+2). Two ancient solutions are non-collapsed and converge, under the backwards flow, to Jensen's second Einstein metric. One solution parametrizes the well known Berger metrics. The rest are new and collapse, under a rescaling of the backwards flow, to Ziller's second homogeneous Einstein metric on complex projective space.