Hermitian curvature flow on complex locally homogeneous surfaces

Abstract

We study the Hermitian curvature flow of locally homogeneous non-Kähler metrics on compact complex surfaces. In particular, we characterize the long-time behaviour of the solutions to the flow. Finally, we compute the Gromov-Hausdorff limit of immortal solutions after a suitable normalization. Our results follow by a case-by-case analysis of the flow on each complex model geometry.

Positive Hermitian curvature flow on complex 2-step nilpotent Lie groups

“…Moreover, in [19] it has been shown that expanding algebraic solitons on complex Lie groups lead to strong constrains on the algebraic structure. Finally, the behaviour of the 'original' HCF of locally homogeneous non-Kähler metrics on compact complex surfaces, together with a strong convergence result, was investigated in [13].…”

“…Proof. Let us suppose that there exists a left-invariant metric g on G satisfying (13). Let {Z 1 , .…”