The Ricci flow in a class of solvmanifolds

TL;DR

This paper analyzes the Ricci flow on a specific class of solvmanifolds, showing solutions are immortal, converge to flat manifolds, and that negative curvature metrics lead to finite-time negative curvature, using bracket flow techniques.

Contribution

It introduces a bracket flow approach to study Ricci flow on solvmanifolds with an abelian ideal, proving convergence to algebraic solitons and flat limits, and characterizing curvature evolution.

Findings

Solutions are immortal under Ricci flow.

Sequences of solutions converge to flat manifolds.

Negative curvature metrics become negatively curved in finite time.

Abstract

In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the omega-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence in which the Ricci flow converges, in the pointed topology, to a manifold which is locally isometric to a flat manifold. We give a functional which is non-increasing along a normalized bracket flow that will allow us to prove that given a sequence of times, one can extract a subsequence converging to an algebraic soliton, and to determine which of these limits are flat. Finally, we use these results to prove that if a Lie group in this class admits a Riemannian metric of negative sectional curvature, then the curvature of any Ricci flow solution will become negative in finite time.