Curvature flows for almost-hermitian Lie groups

TL;DR

This paper introduces a unified approach using the bracket flow ODE system to analyze curvature flows on almost-hermitian Lie groups, revealing insights into solution behaviors and special solutions.

Contribution

It develops a novel framework applying the bracket flow to study curvature flows on homogeneous manifolds, enhancing understanding of their limits and regularity.

Findings

Bracket flow simplifies analysis of curvature flows.

Identification of conditions for special solutions like solitons.

Application to Chern-Ricci and symplectic curvature flows.

Abstract

We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way, by considering a generic flow under just a few natural conditions on the broad class of almost-hermitian structures. As a main tool, we use an ODE system defined on the variety of 2n-dimensional Lie algebras, called the bracket flow, whose solutions differ from those to the original curvature flow by only pull-back by time-dependent diffeomorphisms. The approach, which has already been used to study the Ricci flow on homogeneous manifolds, is useful to better visualize the possible pointed limits of solutions, under diverse rescalings, as well as to address regularity issues. Immortal, ancient and self-similar solutions arise naturally from the qualitative analysis of the bracket flow. The Chern-Ricci flow and the symplectic curvature flow are considered in more detail.