Abstract:We construct a family of Hermitian metrics on the Hopf surface $\mathbb {S}^3\times \mathbb {S}^1$, whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group. These metrics are locally conformally Kähler. Among the toric fibres of $\pi :\mathbb {S}^3\times \mathbb {S}^1\to \mathbb {C} P^1$ two of them are stable minimal surfaces and each of the two has a neighbourhood so that fibres therein are given by stable harmonic maps from 2-torus and outside, far away from the two tori, there are unstable harmonic ones that are also unstable minimal surfaces. A similar result is true for $\mathbb {S}^{2n-1}\times \mathbb {S}^{1}$.

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中文翻译：

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作为调和映射和极小曲面的 Hopf 曲面纤维的稳定性

摘要：我们在 Hopf 曲面 $\mathbb {S}^3\times \mathbb {S}^1$ 上构建了一系列 Hermitian 度量，其基本类代表 Aeppli 上同调群中的不同上同调类。这些度量是局部一致的 Kähler。在 $\pi :\mathbb {S}^3\times \mathbb {S}^1\to \mathbb {C} P^1$ 的复曲面纤维中，其中两个是稳定的最小曲面，并且两个中的每一个都有一个邻域，因此其中的纤维由来自 2-环面的稳定谐波映射给出，并且在远离两个环面的外部，存在不稳定的谐波，它们也是不稳定的最小表面。$\mathbb {S}^{2n-1}\times \mathbb {S}^{1}$ 也有类似的结果。

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