Recall that the fixed points of this flow are precisely the Einstein metrics, and the general philosophy is that the initial metric g(0) is "improved" in some sense by the flow. Thus the metrics g(0) for which the flow g(t) is isometric to g(0) for any t, called Ricci solitons, may be considered as very special ones.
Searching for a distinguished left invariant metric g on a given nilpotent Lie group N, we have found that g is a Ricci soliton if and only if (N,g) admits a solvable extension (S,g) which is Einstein. We also characterize the Ricci solitons (N.g) as the critical points of a natural functional defined on a real algebraic variety parametrizing all the left invariant Riemannian metrics on all the nilpotent Lie groups of a given dimension. This variational approach has proven to be a useful tool to find explicit examples of Ricci solitons and consequently of Einstein solvmanifolds.