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.