The Ricci flow for simply connected nilmanifolds

We prove that the Ricci flow $g(t)$ starting at any metricon $\RR^n$ that is invariant by a transitive nilpotent Liegroup $N$ can be obtained by solving an ordinary differentialequation (ODE) for a curve ofnilpotent Lie brackets on $\RR^n$. By using that this ODEis the negative gradient flow of a homogeneous polynomial,we obtain that $g(t)$ is type-III, and, up to pull-back bytime-dependent diffeomorphisms, that $g(t)$ converges tothe flat metric, and the rescaling$|\!\scalar(g(t))|\,g(t)$ converges to a Ricci soliton in$C^\infty$, uniformly on compact sets in $\RR^n$. TheRicci soliton limit is also invariant by some transitivenilpotent Lie group, though possibly nonisomorphic to~$N$.

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Published 3 February 2012