Global study of the Ricci flow on flag manifolds with second Betti number equal to 1

Abstract: For any flag manifold $M = G/K$ of a compact simple Lie group $G$ we describe some qualitative properties of the homogeneous un-normalized Ricci flow. We engage ourselves with the global study of the dynamical system induced by this flow on any flag manifold $M = G/K$ with second Betti number $b_2(M) = 1$, and present non-collapsed ancient solutions, whose $\alpha$-limit set consists of fixed points at infinity of $\mathscr{M}^G$ . Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics, which we classify according to their stability properties, illuminating thus the structure of the system¢s phase space. This is a joint work with Ioannis Chrysikos.

differential geometry

Audience: researchers in the topic

Workshop on compact homogeneous Einstein manifolds

Series comments: The workshop aims at bringing together mathematicians that have contributed and are contributing to the study of Einstein metrics on the challenging class of all compact homogeneous spaces.