The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

TL;DR

This paper analyzes the normalized Ricci flow on certain flag manifolds with two or three isotropy summands, using dynamical systems to understand invariant Einstein metrics and aid classification.

Contribution

It provides a global phase portrait analysis of the Ricci flow on these manifolds, revealing insights into the existence and form of Einstein metrics.

Findings

Identification of invariant Einstein metrics on flag manifolds.

Qualitative phase portrait analysis of Ricci flow dynamics.

Insights into classification of Einstein metrics on homogeneous spaces.

Abstract

We give the global picture of the normalized Ricci flow on generalized flag manifolds with two or three isotropy summands. The normalized Ricci flow for these spaces descents to a parameter depending system of two or three ordinary differential equations, respectively. We present here the qualitative study of these system's global phase portrait, by using techniques of Dynamical Systems theory. This study allows us to draw conclusions about the existence and the analytical form of invariant Einstein metrics on such manifolds, and seems to offer a better insight to the classification problem of invariant Einstein metrics on compact homogeneous spaces.