Einstein solvmanifolds: existence and non-existence questions

By Jorge Lauret, C. Will. 2006

100 citations recorded by the source corpus.

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 GLn-action on $${\mathcal{N}}$$, evaluated at $${\mu\in\mathcal{N}}$$, encodes geometric information on Sμ 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 Sμ’s, is the square norm of this moment map. We use a GLn-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 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 $${\mathbb{N}}$$-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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4a9ada77bf536fa9d65acf119df4d6f28aec51b1

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The existence of soliton metrics for nilpotent Lie groups (2008, background): “The example in the theorem was also considered in [ 19 ]a nd [22].”
Structure of homogeneous Ricci solitons and the Alekseevskii conjecture (2012, background): “[23, 25] There exists a finite subset B ⊂ t+, and for each β ∈ B a GLn(R)invariant subset Sβ ⊂ V (a stratum) such that”