The Hopf surfaces provide a family of minimal non-Kähler surfaces of class VII on which little is known about the Chern–Ricci flow. We use a construction of Gauduchon–Ornea for locally conformally Kähler metrics on primary Hopf surfaces of class 1 to study solutions of the Chern–Ricci flow. These solutions reach a volume collapsing singularity in finite time, and we show that the metric tensor satisfies a uniform upper bound, supporting the conjecture that the Gromov-Hausdorff limit is isometric to a round\(S^1\). Uniform \(C^{1+\beta }\) estimates are also established for the potential. Previous results had only been known for the simplest examples of Hopf surfaces.

Hopf曲面提供了最小的VII类非Kähler曲面族，在该曲面上对Chern-Ricci流的了解很少。我们使用Gauduchon-Ornea的构造对1类主要Hopf曲面上的局部保形Kähler度量进行研究，以研究Chern-Ricci流的解。这些解在有限时间内达到了体积崩溃的奇点，并且我们证明了度量张量满足一个统一的上限，这支持了Gromov-Hausdorff极限等距到圆\（S ^ 1 \）的猜想。还建立了潜力的统一\（C ^ {1+ \ beta} \）估计。以前的结果仅是关于Hopf表面的最简单示例才知道的。