Einstein solvmanifolds: existence and non-existence questions

JORGE LAURET, CYNTHIA WILL

MATHEMATISCHE ANNALEN

SPRINGER

Año: 2010 vol. 350 p. 199 - 225

0025-5831

The aim of this paper is to study the problem of which solvable Lie groups admit anEinstein left invariant metric. The space $ ca$ of all nilpotent Lie brackets on$RR^n$ parametrizes a set of $(n+1)$-dimensional rank-one solvmanifolds ${S_{mu}:muin ca}$, containing the set of all those which are Einstein in thatdimension. The moment map for the natural $G$-action on $ ca$, evaluated at$muin ca$, encodes geometric information on $S_{mu}$ and suggests the use ofstrong results from geometric invariant theory. For instance, the functional on$ ca$ whose critical points are precisely the Einstein $S_{mu}$´s, is the squarenorm of this moment map. We use a $G$-invariant stratification for the space$ ca$ and show that there is a strong interplay between the strata and the Einsteincondition on the solvmanifolds $S_{mu}$.  As an application, we obtain criteria todecide whether a given nilpotent Lie algebra can be the nilradical of a rank-oneEinstein solvmanifold or not.  We find several examples of $NN$-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 algebrasassociated to graphs are also given.