The Ricci flow for simply connected nilmanifolds

JORGE LAURET

COMMUNICATIONS IN ANALYSIS AND GEOMETRY

INT PRESS BOSTON, INC

Año: 2011 vol. 19 p. 831 - 854

1019-8385

We prove that the Ricci flow $g(t)$ starting at any metric on $RR^n$ that is invariant by a transitive nilpotent Lie group $N$ can be obtained by solving an ODE for a curve  of nilpotent Lie brackets on $RR^n$.  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 $|!scalar(g(t))|,g(t)$ converges to a Ricci soliton in $C^infty$, uniformly on compact sets in $RR^n$.  The Ricci soliton limit is also invariant by some transitive nilpotent Lie group, though possibly nonisomorphic to $N$.