Geometric flows and their solitons on homogeneous spaces

TL;DR

This paper introduces a unified method using the bracket flow to analyze geometric flows on homogeneous spaces, enabling better visualization of limits and solitons, with applications to G2-structures and Laplacian flow.

Contribution

It extends the bracket flow approach to general geometric flows on homogeneous spaces, providing new insights into solitons and limits, including a novel G2-structure example.

Findings

Identified a new expanding soliton G2-structure on a nilpotent Lie group.

Demonstrated the effectiveness of the bracket flow in visualizing geometric flow limits.

Extended the method to a broad class of flows on homogeneous spaces.

Abstract

We develop a general approach to study geometric flows on homogeneous spaces. Our main tool will be a dynamical system defined on the variety of Lie algebras called the bracket flow, which coincides with the original geometric flow after a natural change of variables. The advantage of using this method relies on the fact that the possible pointed (or Cheeger-Gromov) limits of solutions, as well as self-similar solutions or soliton structures, can be much better visualized. The approach has already been worked out in the Ricci flow case and for general curvature flows of almost-hermitian structures on Lie groups. This paper is intended as an attempt to motivate the use of the method on homogeneous spaces for any flow of geometric structures under minimal natural assumptions. As a novel application, we find a closed G2-structure on a nilpotent Lie group which is an expanding soliton for the Laplacian flow and is not an eigenvector.