The Ricci flow for simply connected nilmanifolds

We prove that the Ricci flow g(t) starting at any metric on R-n that is invariant by a transitive nilpotent Lie group N can be obtained by solving an ordinary differential equation (ODE) for a curve of nilpotent Lie brackets on R-n. By using that this ODE is the negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-III, and, up to pull-back by time-dependent diffeomorphisms, that g(t) converges to the flat metric, and the rescaling vertical bar R(g(t))vertical bar g(t) converges to a Ricci soliton in C-infinity, uniformly on compact sets in R-n. The Ricci soliton limit is also invariant by some transitive nilpotent Lie group, though possibly nonisomorphic to N.