The long-time behavior of the homogeneous pluriclosed flow

TL;DR

This paper investigates the long-term behavior of the pluriclosed flow on homogeneous spaces, showing convergence to self-similar solutions and revealing new phenomena like shrinking solitons and finite-time extinction.

Contribution

It provides the first analysis of the pluriclosed flow on homogeneous manifolds, demonstrating convergence to solitons and discovering novel flow behaviors.

Findings

Solutions on certain Lie groups converge to self-similar solutions.

Some homogeneous manifolds exhibit shrinking solitons as limits.

Existence of solutions with finite extinction time and eternal solutions.

Abstract

We study the asymptotic behavior of the pluriclosed flow in the case of left-invariant Hermitian structures on Lie groups. We prove that solutions on 2-step nilpotent Lie groups and on almost-abelian Lie groups converge, after a suitable normalization, to self-similar solutions of the flow. Given that the spaces are solvmanifolds, an unexpected feature is that some of the limits are shrinking solitons. We also exhibit the first example of a homogeneous manifold on which a geometric flow has some solutions with finite extinction time and some that exist for all positive times.