The Ricci flow for simply connected nilmanifolds

TL;DR

This paper demonstrates that Ricci flow on simply connected nilmanifolds can be described by an ODE for Lie brackets, leading to convergence results and classification of Ricci solitons in this setting.

Contribution

It introduces an ODE approach for Ricci flow on nilmanifolds and proves convergence to Ricci solitons, extending understanding of geometric evolution on these spaces.

Findings

Ricci flow can be described by an ODE for nilpotent Lie brackets.

Flow solutions are type-III and converge to flat metrics.

Rescaled metrics converge to Ricci solitons, possibly with different symmetry groups.

Abstract

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.