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

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.

We generalize Theorem 1.2 to ${\mathbb{S}}^{2n-1}\times {\mathbb{S}}^1$ in section 4