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Finding Einstein solvmanifolds by a variational method. (English) Zbl 1015.53028

A solvmanifold is a solvable Lie group $S$ endowed with a left invariant Riemannian metric; $S$ is called standard if $𝔞={𝔫}^{\perp }$ is abelian, where if $𝔰$ is the Lie algebra of $S$ then $𝔫=\left[𝔰,𝔰\right]$. The Einstein solvmanifolds have been investigated by J. Heber in [Invent. Math. 133, 279-352 (1998; Zbl 0906.53032)]. If $S$ is Einstein, then for some distinguished element $H\in 𝔞$, the eigenvalues of ad${H|}_{𝔫}$ are all positive integers without common divisors, say ${k}_{1}<\cdots <{k}_{r}$. If ${d}_{1},\cdots ,{d}_{r}$ denote the corresponding multiplicities, then the tuple $\left(k;d\right)=\left({k}_{1}<\cdots <{k}_{r};{d}_{1},\cdots ,{d}_{r}\right)$ is called the eigenvalue type of $S$. In every dimension, only finitely many eigenvalue types occur. Most of the known examples of Einstein solvmanifolds are given of eigenvalue type $\left(1<2;{d}_{1},{d}_{2}\right)$.

The aim of this paper is to approach the construction of new families of explicit examples of Einstein solvmanifolds of several different eigenvalue types by using the variational method given by the author in [Q. J. Math. 52, 463-470 (2001; Zbl 1015.53025)]: the $\left(n+1\right)$-dimensional rank-one $\left(dim𝔞=1\right)$ Einstein solvmanifolds are critical points of a certain polynomial of degree 4 restricted to the sphere of a vector space which contains the set of all $n$-dimensional nilpotent Lie algebras as a real algebraic subset.

The author proves that any nilpotent Lie algebra having a codimension-one abelian ideal admits a rank-one solvable extension which can be endowed with an Einstein left-invariant metric. Also it is proved that any nilpotent Lie algebra of dimension $\le 5$ is the nilradical of rank-one Einstein solvmanifold and their eigenvalue types are determined. In the last section the author presents a curve of pairwise non-isometric 8-dimensional Einstein solvmanifolds, which is the lowest possible for the existence of such a curve.

Reviewer: I.Pop (Iaşi)
##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Homogeneous manifolds (differential geometry) 22E25 Nilpotent and solvable Lie groups