Positive Hermitian curvature flow on 2-step nilpotent Lie groups
Ettore Lo Giudice
TL;DR
This work analyzes the positive Hermitian curvature flow $HCF_{+}$ on simply-connected $2$-step nilpotent Lie groups with a left-invariant complex structure $J$, under the condition $Jigl[ rak g, rak gigr] o Z( rak g)$. Using Lauret’s bracket-flow framework, it proves global existence and shows that the rescaled metrics $(G,(1+t)^{-1}g_t)$ converge in Cheeger–Gromov topology to a non-flat semi-algebraic soliton, which, in this setting, is unique up to homothety and expanding. The paper also relates $HCF_{+}$ to the type IIB flow via conformal changes and provides an explicit example illustrating the soliton structure; it further studies the analogous flow $K(g)$ in the same class and discusses long-time behavior for the $ ext{Ric}^{1,1}$-flow under additional hypotheses. Overall, it characterizes the asymptotic self-similar behavior of $HCF_{+}$ on 2-step nilpotent Hermitian Lie groups and clarifies the nature and uniqueness of semi-algebraic solitons in this context.
Abstract
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.