Unstable minimal spheres with degree-1 Gauss lift in hyperkähler 4-manifolds
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909367565025280 |
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| author | Foscolo, Lorenzo Trinca, Federico |
| author_facet | Foscolo, Lorenzo Trinca, Federico |
| contents | We exhibit new minimal 2-spheres in hyperkähler 4-manifolds arising from the Gibbons--Hawking ansatz and in the K3 manifold endowed with a hyperkähler 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 hyperkähler 4-manifolds in terms of topological data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20396 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unstable minimal spheres with degree-1 Gauss lift in hyperkähler 4-manifolds Foscolo, Lorenzo Trinca, Federico Differential Geometry 53C40, 53C42 We exhibit new minimal 2-spheres in hyperkähler 4-manifolds arising from the Gibbons--Hawking ansatz and in the K3 manifold endowed with a hyperkähler 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 hyperkähler 4-manifolds in terms of topological data. |
| title | Unstable minimal spheres with degree-1 Gauss lift in hyperkähler 4-manifolds |
| topic | Differential Geometry 53C40, 53C42 |
| url | https://arxiv.org/abs/2410.20396 |