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

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 $n$.