Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups

TL;DR

This paper investigates the long-term behavior of the positive Hermitian curvature flow on complex Lie groups, demonstrating convergence to solitons in nilpotent and almost-abelian cases, and classifying all such solitons.

Contribution

It provides the first comprehensive analysis of the flow on complex Lie groups, proving convergence to solitons and classifying all almost-abelian solitons.

Findings

Flow exists for all positive times on nilpotent groups.

Flow converges to a soliton in both nilpotent and almost-abelian cases.

Complete classification of almost-abelian solitons with construction methods.

Abstract

We study the positive Hermitian curvature flow on the space of left-invariant metrics on complex Lie groups. We show that in the nilpotent case, the flow exists for all positive times and subconverges in the Cheeger-Gromov sense to a soliton. We also show convergence to a soliton when the complex Lie group is almost abelian. That is, when its Lie algebra admits a (complex) co-dimension one abelian ideal. Finally, we study solitons in the almost-abelian setting. We prove uniqueness and completely classify all left-invariant, almost-abelian solitons, giving a method to construct examples in arbitrary dimensions, many of which admit co-compact lattices.