A solvmanifold is a solvable Lie group endowed with a left invariant Riemannian metric; is called standard if is abelian, where if is the Lie algebra of then . The Einstein solvmanifolds have been investigated by J. Heber in [Invent. Math. 133, 279-352 (1998; Zbl 0906.53032)]. If is Einstein, then for some distinguished element , the eigenvalues of ad are all positive integers without common divisors, say . If denote the corresponding multiplicities, then the tuple is called the eigenvalue type of . In every dimension, only finitely many eigenvalue types occur. Most of the known examples of Einstein solvmanifolds are given of eigenvalue type .
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 -dimensional rank-one 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 -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 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.