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 G with a left-invariant complex structure J . We describe the long-time behavior of the flow under the assumption that J[g, g] J [ g , g ] is contained in the center of g g . We show that under our assumption the flow gₜ g t exists for all positive t and (G,(1+t)⁻¹gₜ) ( 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 [21, 24]. In the last part of the paper we study the Hermitian curvature flow for the same class of Lie groups.