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arXiv · 2410.20396

Unstable minimal spheres with degree-1 Gauss lift in hyperk\"ahler 4-manifolds

Lorenzo Foscolo·Federico Trinca
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Abstract

We exhibit new minimal 2-spheres in hyperk\"ahler 4-manifolds arising from the Gibbons--Hawking ansatz and in the K3 manifold endowed with a hyperk\"ahler metric. These minimal surfaces are obtained via a gluing construction using the Scherk surface in flat space and the holomorphic cigar in the Taub-NUT space as building blocks. As for the stable minimal 2-sphere in the Atiyah--Hitchin manifold, the minimal surfaces we construct are not holomorphic with respect to any complex structure compatible with the metric, have degree-1 positive Gauss lift so they can be parametrised by a harmonic map that satisfies a first-order Fueter-type PDE, and yet are unstable. This shows that there is no characterisation of stable minimal surfaces in hyperk\"ahler 4-manifolds in terms of topological data.

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Lorenzo Foscolo, Federico Trinca. 2024-10-27. Unstable minimal spheres with degree-1 Gauss lift in hyperk\"ahler 4-manifolds. https://arxiv.org/abs/2410.20396

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