Positive Hermitian curvature flow on 2-step nilpotent Lie groups

We study the positive Hermitian curvature flow for left-invariant metrics on $2$-step nilpotent Lie groups with a left-invariant complex structure $J$. We describe the long-time behavior of the flow under the assumption that $J[\mathfrak{g}, \mathfrak{g}]$ is contained in the center of $\mathfrak{g}$. We show that under our assumption the flow $g_{t}$ exists for all positive $t$ and $(G,(1+t)^{-1}g_{t})$ converges, in the Cheeger-Gromov topology, to a $2$-step nilpotent Lie group with a non flat semi-algebraic soliton. Moreover, we prove that, in our class of Lie groups, there exists at most one semi-algebraic soliton solution, up to homothety. Similar results were proved by M. Pujia and J. Stanfield for nilpotent complex Lie groups \cite{P2021, S2021}. In the last part of the paper we study the Hermitian curvature flow for the same class of Lie groups.