The Alekseevskii Conjecture in 9 and 10 dimensions

TL;DR

This paper proves that certain high-dimensional noncompact homogeneous spaces do not support negative Ricci curvature Einstein metrics, advancing understanding of the Alekseevskii conjecture in dimensions 9 and 10.

Contribution

It establishes new nonexistence results for homogeneous Einstein metrics in dimensions 9 and 10, using a cohomogeneity-one approach and Lie algebra analysis.

Findings

No negative Ricci curvature Einstein metrics on most noncompact homogeneous spaces in 9 and 10 dimensions.

Identification of three potential exceptions to the nonexistence results.

Application of cohomogeneity-one methods to analyze Einstein metrics.

Abstract

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.