Seminar

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 show that any left-invariant Einstein metric of negative Ricci curvature on a solvable Lie group is maximally symmetric.   This theorem is philosophically motivated  both by the Alekseevskii Conjecture, which states that every homogeneous Einstein manifold of negative Ricci curvature is diffeomorphic to Euclidean space, 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.  (This is joint work with Michael Jablonski.)