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.

我们研究了复杂李群的左不变度量空间上的正厄米曲率流。我们表明，在幂零情况下，流存在于所有正时间，并在 Cheeger-Gromov 意义上子收敛到孤子。当复李群几乎是阿贝尔群时，我们还展示了对孤子的收敛。也就是说，当它的李代数承认一个（复）余维一个阿贝尔理想时。最后，我们在几乎阿贝尔环境中研究孤子。我们证明了唯一性并对所有左不变的、几乎阿贝尔的孤子进行了完全分类，给出了一种在任意维度上构造示例的方法，其中许多维度都允许共紧格。