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Unstable minimal spheres with degree-1 Gauss lift in hyperkähler 4-manifolds

Lorenzo Foscolo, Federico Trinca

Abstract

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.

Unstable minimal spheres with degree-1 Gauss lift in hyperkähler 4-manifolds

Abstract

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.

Paper Structure

This paper contains 25 sections, 22 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The ambient space
  3. The ansatz
  4. Minimal surfaces
  5. Symmetries
  6. Metric estimates
  7. Approximate minimal surfaces
  8. Scherk's desingularization
  9. Construction of the initial surface
  10. Mean curvature estimates
  11. Perturbation to minimal surfaces
  12. The linear problem on the building blocks
  13. Scherk's surface
  14. Cigar in the Taub--NUT space
  15. Totally geodesic cylinder
  16. ...and 10 more sections

Key Result

Theorem 1

On the K3 manifold there exist hyperkähler metrics that contain stable minimal $2$-spheres with positive Gauss lift of degree $1$ (equivalently, with self-intersection number $-4$) and hyperkähler metrics that contain unstable minimal 2-spheres with the same topological properties.

Figures (1)

  • Figure 1: Schematic representation of the initial surface $\mathcal{S}_{T_1,T_2}^d$ in $\mathbb R^3$.

Theorems & Definitions (51)

  • Theorem
  • Remark
  • Remark
  • Remark
  • Remark
  • Remark
  • Lemma 2.1: Equivariant trivialization near the origin
  • proof
  • Remark 2.2
  • Lemma 2.3
  • ...and 41 more