On stability of the fibres of Hopf surfaces as harmonic maps and minimal surfaces

Chen, Jingyi; Huang, Liding

doi:10.1090/tran/8520

On stability of the fibres of Hopf surfaces as harmonic maps and minimal surfaces
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Trans. Amer. Math. Soc. 374 (2021), 8661-8690 Request permission

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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