All known examples of homogeneous Einstein metrics of negative Ricci curvature can be realized as left-invariant Riemannian metrics on solvable Lie groups. After defining a notion of maximal symmetry among left-invariant Riemannian metrics on a Lie group, we prove that any left-invariant Einstein metric of negative Ricci curvature on a solvable Lie group is maximally symmetric. This theorem is motivated both by the Alekseevskii Conjecture and by the question of stability of Einstein metrics under the Ricci flow. We also address questions of existence of maximally symmetric left-invariant Riemannian metrics more generally.
The first author’s research was partially supported by National Science Foundation grant DMS-0906168.
The second author’s research was partially supported by National Science Foundation grant DMS-1612357 and a grant from the Simons Foundation (#360562, Michael Jablonski).
"Einstein solvmanifolds have maximal symmetry." J. Differential Geom. 111 (1) 1 - 38, January 2019. https://doi.org/10.4310/jdg/1547607686