Ricci soliton homogeneous nilmanifolds

We study a notion weakening the Einstein condition on a left invariant Riemannian metric g on a nilpotent Lie group N. We consider those metrics satisfying Ric(g) = cl + D for some c is an element of R and some derivation D of the Lie algebra n of N, where Ric(g) denotes the Ricci operator of (N. g). This condition is equivalent to the metric g to be a Ricci soliton. We prove that a Ricci soliton left invariant metric on N is unique up to isometry and scaling. The following characterization is also given: (N. R) is a Ricci soliton if and only if (N. g) admits a metric standard solvable extension whose corresponding standard solvmanifold (S. (g) over tilde) is Einstein. This gives several families of new tramples of Ricci solitons. By a variational approach, we furthermore show that the Ricci soliton homogeneous nilmanifolds (N. g) are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set.