Einstein metrics on the full flag $F(n)$

TL;DR

This paper studies Einstein metrics on full flag manifolds, especially F(5), identifying new metrics and analyzing their stability, which advances understanding of their classification and stability properties.

Contribution

It introduces four new Einstein metrics on F(5) and provides detailed stability analysis, including coindices and Hessian spectra, enhancing the classification of Einstein metrics on flag manifolds.

Findings

Four new Einstein metrics on F(5) identified

Stability analysis confirms metrics are pairwise non-homothetic

Provides data supporting the finiteness conjecture for Einstein metrics

Abstract

Let $ M = G/K $ be a full flag manifold. In this work, we investigate the $ G$-stability of Einstein metrics on $M$ and analyze their stability types, including coindices, for several cases. We specifically focus on $F(n) = \mathrm{SU}(n)/T$, emphasizing $n = 5$, where we identify four new Einstein metrics in addition to known ones. Stability data, including coindex and Hessian spectrum, confirms that these metrics on $F(5)$ are pairwise non-homothetic, providing new insights into the finiteness conjecture.