New Homogeneous Einstein Metrics from Detection

Abstract

We use a detection method for invariant Einstein equations to construct and distinguish asymmetric homogeneous Einstein metrics on several families of compact homogeneous spaces. The method retains a geometrically meaningful symmetry-breaking parameter, eliminates the remaining Einstein equations, and then reconstructs positive metrics from the resulting detector. We prove a sharp threshold in a two-factor symplectic family, construct two asymmetric Einstein metrics on each admissible space $\SU(n)^3/Δ\SO(n)$, obtain a sharp threshold for $\SU(2n)^3/Δ\Sp(n)$, and construct asymmetric metrics on three exceptional three-factor families. The four-factor case exhibits a further phenomenon. On \[ M_9=\SU(18)^4/Δ\Sp(9) \] the two symmetry types $3+1$ and $2+2$ are both realized: up to homothety and internal block permutations, the corresponding fixed families contain exactly three asymmetric Einstein metrics, two with three equal horizontal scales and one distinct scale (type $3+1$), and one with two equal pairs of horizontal scales (type $2+2$). They are pairwise non-isometric, Riemannian irreducible, and not naturally reductive. The $2+2$ metric is globally unique in the full positive $2+2$ fixed family up to block interchange.