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 behavior of the solutions to the flow. We also provide the first example of a compact complex non-Kähler manifold admitting a finite time singularity for the Hermitian curvature 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.

arXiv commentSome minor changes. A new appendix containing explicit tensor components. To appear on Ann. Mat. Pura Appl

References in corpus (13)

• The Kähler-Ricci flow on surfaces of positive Kodaira dimension
• On the evolution of a Hermitian metric by its Chern-Ricci form
• Ricci flow on Kähler-Einstein manifolds
• The Chern-Ricci flow on complex surfaces
• The long-time behavior of the homogeneous pluriclosed flow
• Collapsing of the Chern-Ricci flow on elliptic surfaces
• Inoue surfaces and the Chern-Ricci flow
• The Hermitian curvature flow on manifolds with non-negative Griffiths curvature
• Homogeneous Solutions of Pluriclosed Flow on Closed Complex Surfaces
• Hermitian Curvature flow on unimodular Lie groups and static invariant metrics
• A remark on the Bismut-Ricci form on 2-step nilmanifolds
• Expanding solitons to the Hermitian curvature flow on complex Lie groups
• Geometric formalities along the Chern-Ricci flow

Cited by in corpus (3)

• The Hull-Strominger system and the Anomaly flow on a class of solvmanifolds
• The Anomaly flow on nilmanifolds
• Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups