The Alekseevskii conjecture in low dimensions

TL;DR

This paper extends the verification of the Alekseevskii conjecture for connected homogeneous Einstein spaces with negative scalar curvature to dimensions up to 10 when G is not semisimple, and up to dimension 8 in general, advancing understanding in geometric analysis.

Contribution

It proves the Alekseevskii conjecture in dimensions up to 10 for non-semisimple G and up to 8 with few exceptions for arbitrary G, extending previous results.

Findings

Confirmed the conjecture in dimensions up to 10 for non-semisimple G.

Extended validity of the conjecture up to dimension 8 with five possible exceptions.

Provided new classifications for homogeneous Einstein spaces in low dimensions.

Abstract

The long-standing Alekseevskii conjecture states that a connected homogeneous Einstein space G/K of negative scalar curvature must be diffeomorphic to R^n. This was known to be true only in dimensions up to 5, and in dimension 6 for non-semisimple G. In this work we prove that this is also the case in dimensions up to 10 when G is not semisimple. For arbitrary G, besides 5 possible exceptions, we show that the conjecture holds up to dimension 8.