The Alekseevskii conjecture in 9 and 10 dimensions

Abstract

We show that non-compact homogeneous spaces not diffeomorphic to Euclidean space of dimension 9 or 10 admit no homogeneous Einstein metrics of negative Ricci curvature, with only three potential exceptions. The main ingredient in the proof is to show, via a cohomogeneity-one approach, that non-compact homogeneous spaces admitting an ideal isomorphic to sl2(R) admit no homogeneous Einstein metrics of negative Ricci curvature.