Mathematics – Differential Geometry
Scientific paper
2006-02-22
Mathematics
Differential Geometry
Final version to appear in Math. Annalen
Scientific paper
The general aim of this paper is to study which are the solvable Lie groups admitting an Einstein left invariant metric. The space N of all nilpotent Lie brackets on R^n parametrizes a set of (n+1)-dimensional rank-one solvmanifolds, containing the set of all those which are Einstein in that dimension. The moment map for the natural GL(n)-action on N evaluated in a point of N encodes geometric information on the corresponding solvmanifold, allowing us to use strong and well-known results from geometric invariant theory. For instance, the functional on N whose critical points are precisely the Einstein solvmanifolds is the square norm of this moment map. We also use a GL(n)-invariant stratification for the space N following essentially a construction given by F. Kirwan and show that there is a strong interplay between the strata and the Einstein condition on the solvmanifolds. As applications, we obtain several examples of graded (even 2-step) nilpotent Lie algebras which are not the nilradicals of any standard Einstein solvmanifold, as well as a classification in the 7-dimensional 6-step case and an existence result for certain 2-step algebras associated to graphs.
Lauret Jorge
Will Cynthia
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