Einstein solvmanifolds: existence and non-existence questions

Abstract

The aim of this paper is to study the problem of which solvable Lie groups admit an Einstein left invariant metric. The space N of all nilpotent Lie brackets on Rn parametrizes a set of (n + 1)-dimensional rank-one solvmanifolds {Sμ : μ ∈ N }, containing the set of all those which are Einstein in that dimension. The moment map for the natural GLn-action on N , evaluated at μ ∈ N , encodes geometric information on Sμ and suggests the use of strong results from geometric invariant theory. For instance, the functional on N whose critical points are precisely the Einstein Sμ’s, is the square norm of this moment map. We use a GLn-invariant stratification for the space N and show that there is a strong interplay between the strata and the Einstein condition on the solvmanifolds Sμ. As an application, we obtain criteria to decide whether a given nilpotent Lie algebra can be the nilradical of a rank-one Einstein solvmanifold or not. We find several examples of N-graded (even 2-step) nilpotent Lie algebras which are not. A classification in the 7-dimensional, 6-step case and an existence result for certain 2-step algebras associated to graphs are also given.