The Ricci flow of left invariant metrics on full flag manifold SU(3)/T from a dynamical systems point of view

TL;DR

This paper analyzes the long-term behavior of the Ricci flow on the full flag manifold SU(3)/T using dynamical systems techniques, revealing invariant lines and convergence to Einstein metrics.

Contribution

It introduces a dynamical systems approach to study Ricci flow on SU(3)/T, identifying invariant lines and convergence properties to Einstein metrics.

Findings

Four invariant lines for Ricci flow correspond to Einstein metrics.

The bi-invariant metric is an Einstein metric on SU(3)/T.

Left-invariant metrics converge to bi-invariant Einstein metric under Ricci flow.

Abstract

In this paper we study the behavior of the Ricci flow at infinity for the full flag manifold $SU(3)/T$ using techniques of the qualitative theory of differential equations, in special the Poincar\'e Compactification and Lyapunov exponents. We prove that there are four invariant lines for the Ricci flow equation, each one associated with a singularity corresponding to a Einstein metric. In such manifold, the bi-invariant normal metric is Einstein. Moreover, around each invariant line there is a cylinder of initial conditions such that the limit metric under the Ricci flow is the corresponding Einstein metric; in particular we obtain the convergence of left-invariant metrics to a bi-invariant metric under the Ricci flow.