The Alekseevskii Conjecture in 9 and 10 dimensions

We show that noncompact 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 noncompact homogeneous spaces admitting an ideal isomorphic to $\mathfrak{sl}_2(\mathbb{R})$ admit no homogeneous Einstein metrics of negative Ricci curvature.