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.

中文翻译：

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复杂局部均匀表面上的厄米曲率流

我们研究紧致复杂表面上局部均匀非Kähler度量的Hermitian曲率流。特别是，我们描述了流解决方案的长期行为。我们还提供了一个紧凑的复杂非Kähler流形的第一个示例，该流形允许Hermitian曲率流具有有限的时间奇点。最后，我们通过适当的归一化计算永生解的Gromov–Hausdorff极限。我们的结果是对每个复杂模型几何体上的流进行了个案分析。