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

TL;DR

This paper constructs new unstable minimal 2-spheres in hyperk"ahler 4-manifolds, demonstrating that stability cannot be characterized solely by topological data, using advanced gluing techniques and harmonic map parametrizations.

Contribution

It introduces novel minimal surfaces in hyperk"ahler 4-manifolds via a gluing construction, revealing their instability and properties related to Gauss lift and harmonic maps.

Findings

Constructed new minimal 2-spheres in hyperk"ahler 4-manifolds.

Showed these minimal surfaces are unstable despite positive Gauss lift.

Demonstrated no topological characterization of stability for such surfaces.

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.