A numerical treatment to the problem of the quantity of Einstein metrics on flag manifolds

TL;DR

This paper uses numerical methods to estimate and analyze Einstein metrics on full flag manifolds, providing improved bounds and examining isometric properties, with applicability to any dimension n.

Contribution

Introduces a numerical approach to study Einstein metrics on flag manifolds, improving classical estimates and analyzing isometry for any fixed n.

Findings

Estimated the number of Einstein metrics on $SU(n+1)/T^n$ for n=4,5

Improved classical bounds on the count of such metrics

Analyzed isometric properties of these Einstein metrics

Abstract

In this paper we employ numerical methods to study the Einstein equation \[ Ric(g)=\lambda\, g, \] where $Ric$ is the Ricci tensor and $\lambda$ is the Einstein constant, restricted to a class of full flag manifolds. These metrics describe the gravitational field of a vacuum with cosmological constant (vacuum is the case $\lambda=0$). In particular, we give estimates to the number of such metrics on the full flag manifolds $SU(n+1)/T^n$ for $n=4,5$, improving some classical estimatives. We also examine the isometric problem for these Einstein metrics. Our method can be applied for any fixed $n$.