Ancient solutions of the homogeneous Ricci flow on flag manifolds

TL;DR

This paper classifies non-collapsing ancient solutions of the homogeneous Ricci flow on flag manifolds, showing they originate from Einstein metrics and analyzing their stability and singularity development.

Contribution

It provides a detailed description of ancient solutions on flag manifolds, linking them to Einstein metrics and analyzing their dynamical stability using Poincaré compactification.

Findings

Ancient solutions emerge from invariant Einstein metrics.

Solutions develop Type I singularities in finite time.

Fixed points at infinity correspond to Einstein metrics and are studied for stability.

Abstract

For any flag manifold $M=G/K$ of a compact simple Lie group $G$ we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions emerge from an invariant Einstein metric on $M$, and by [B\"oLS17] they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold $M=G/K$ with second Betti number $b_{2}(M)=1$, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose $\alpha$-limit set consists of fixed points at infinity of ${\mathscr{M}}^G$. Based on the Poincar\'{e} compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system's phase space.