Abstract
The aim of this paper is to study the problem of which solvable Lie groups admit an Einstein left invariant metric. The space \({\mathcal{N}}\) of all nilpotent Lie brackets on \({\mathbb{R}^n}\) parametrizes a set of (n + 1)-dimensional rank-one solvmanifolds \({\{S_{\mu}:\mu\in\mathcal{N}\}}\), containing the set of all those which are Einstein in that dimension. The moment map for the natural GL n -action on \({\mathcal{N}}\), evaluated at \({\mu\in\mathcal{N}}\), encodes geometric information on S μ and suggests the use of strong results from geometric invariant theory. For instance, the functional on \({\mathcal{N}}\) whose critical points are precisely the Einstein S μ ’s, is the square norm of this moment map. We use a GL n -invariant stratification for the space \({\mathcal{N}}\) and show that there is a strong interplay between the strata and the Einstein condition on the solvmanifolds S μ . 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 \({\mathbb{N}}\)-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.
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Supported by grants from Fundación Antorchas, CONICET, FONCyT and SeCyT (Univ. Nac. de Córdoba).
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Lauret, J., Will, C. Einstein solvmanifolds: existence and non-existence questions. Math. Ann. 350, 199–225 (2011). https://doi.org/10.1007/s00208-010-0552-0
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DOI: https://doi.org/10.1007/s00208-010-0552-0