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

TL;DR

This paper investigates the stability of fibers in Hopf surfaces, constructing specific Hermitian metrics and analyzing the stability of certain minimal and harmonic surfaces within this geometric context.

Contribution

It introduces a family of locally conformally Kähler Hermitian metrics on Hopf surfaces and examines the stability of their toric fibers as minimal surfaces and harmonic maps.

Findings

Two toric fibers are stable minimal surfaces.

Nearby fibers are stable harmonic maps from 2-torus.

Outside the stable fibers, harmonic maps are unstable.

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 K\"ahler. 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.