Ricci soliton homogeneous nilmanifolds

Abstract.

We study a notion weakening the Einstein condition on a left invariant Riemannian metric g on a nilpotent Lie groupN. We consider those metrics satisfying Ric$_g=cI+D$ for some $c\in{mathbb R}$ and some derivationD of the Lie algebra ${\mathfrak n}$ ofN, 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,g) is a Ricci soliton if and only if (N,g) admits a metric standard solvable extension whose corresponding standard solvmanifold $(S,\tilde{g})$ is Einstein. This gives several families of new examples 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.