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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 theEinstein constant, restricted to a class of full flag manifolds. These metricsdescribe the gravitational field of a vacuum with cosmological constant (vacuumis the case $\lambda=0$). In particular, we give estimates to the number ofsuch metrics on the full flag manifolds $SU(n+1)/T^n$ for $n=4,5$, improvingsome classical estimatives. We also examine the isometric problem for theseEinstein metrics. Our method can be applied for any fixed $n$.

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