Einstein solvmanifolds and the pre-Einstein derivation

Abstract

An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable extension. For every nilpotent Lie algebra, we construct an (essentially unique) derivation, the pre- Einstein derivation, the solvable extension by which may carry an Einstein inner product. Using the pre-Einstein derivation, we then give a variational characterization of Einstein nilradicals. As an application, we prove an easy-to-check convex geometry condition for a nilpotent Lie algebra with a basis to be an Einstein nilradical and also show that a typical two-step nilpotent Lie algebra is an Einstein nilradical. The theory of Riemannian homogeneous spaces with an Einstein metric splits into three very different cases depending on the sign of the Einstein constant, the scalar curvature. Among them, the picture is complete only the Ricci-flat case: by the result of (AK), every Ricci-flat homogeneous space is flat. The major open conjecture the case of negative scalar curvature is the Alekseevski Conjecture (Al1) asserting that a noncompact Einstein homogeneous space admits a simply transitive solvable isometry group. This is equivalent to saying that any such space is a solvmanifold, a solvable Lie group with a left-invariant Riemannian metric satisfying the Einstein condition. By a deep result of J.Lauret (La5), any Einstein solvmanifold is standard. This means that the metric solvable Lie algebra s of such a solvmanifold has the following property: the orthogonal complement to the derived algebra of s is abelian. The systematic study of standard Einstein solvmanifolds (and the term standard) originated from the paper of J.Heber (Heb). On the Lie algebra level, all the metric Einstein solvable Lie algebras can be obtained as the result of the following construction (Heb, La1, La5, LW). One starts with the three pieces of data: a nilpotent Lie algebra n, a semisimple derivationof n, and an inner product h� , �i n on n, with respect to which � is symmetric. An extension of n byis a solvable Lie algebra s = RH ⊕ n (as a linear space) with (adH)|n := �. The inner product on s is defined by h H, ni = 0, k Hk 2 = Tr � (and coincides with the existing one on n). The resulting metric solvable Lie algebra (s, h� , �i ) is Einstein provided n is nice and the derivationand the inner product h� , �i n are chosen in the correct way (note, however, that these conditions are expressed by a system of algebraic equations, which could hardly be analyzed directly, see Section 2). Metric Einstein solvable Lie algebras of higher rank (with the codimension of the nilradical greater than one) having the same nilradical n can be obtained from s via a known procedure, by further adjoining to n semisimple derivation commuting with �. It turns out that the structure of an Einstein metric solvable Lie algebra is completely encoded its nilradical the following sense: given a nilpotent Lie algebra n, there is no more than one (possibly none) choice ofand of h� , �i n, up to conjugation by Aut(n) and scaling, which may result an Einstein metric solvable Lie algebra (s, h� , �i ). Definition 1. A nilpotent Lie algebra is called an Einstein nilradical, if it is the nilradical of an Einstein metric solvable Lie algebra. A derivationof an Einstein nilradical n and an inner product h� , �i n, for which the metric solvable Lie algebra (s, h� , �i ) is Einstein are called an Einstein derivation and a nilsoliton inner product respectively.