The Ricci flow for simply connected nilmanifolds

Details The Ricci flow for simply connected nilmanifolds The Ricci flow for simply connected nilmanifolds

2010-04-06

arxiv.org/abs/1004.0946v2

Mathematics

Differential Geometry

15 pages, final version to appear in Comm. Anal. Geom

Scientific paper

We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. 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 |R(g(t))|g(t) converges smoothly to a Ricci soliton, uniformly on compact sets. The Ricci soliton limit is also invariant by some transitive nilpotent Lie group, though possibly non-isomorphic to N.

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