### Abstract

We construct large families of new collapsing hyperkähler metrics on the K3 surface. The limit space is a flat Riemannian 3-orbifold T
^{3}/Z
_{2}. Away from finitely many exceptional points the collapse occurs with bounded curvature. There are at most 24 exceptional points where the curvature concentrates, which always contains the 8 fixed points of the involution on T
^{3}. The geometry around these points is modelled by ALF gravitational instantons: of dihedral type (D
_{k}) for the fixed points of the involution on T
^{3} and of cyclic type (A
_{k}) otherwise. The collapsing metrics are constructed by deforming approximately hyperkähler metrics obtained by gluing ALF gravitational instantons to a background (incomplete) S
^{1}–invariant hyperkähler metric arising from the Gibbons–Hawking ansatz over a punctured 3-torus. As an immediate application to submanifold geometry, we exhibit hyperkähler metrics on the K3 surface that admit a strictly stable minimal sphere which cannot be holomorphic with respect to any complex structure compatible with the metric.

Original language | English |
---|---|

Pages (from-to) | 79-120 |

Number of pages | 42 |

Journal | Journal of Differential Geometry |

Volume | 112 |

Issue number | 1 |

DOIs | |

Publication status | Published - 8 May 2019 |

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### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Cite this

**surface.**

*K*3*Journal of Differential Geometry*,

*112*(1), 79-120. https://doi.org/10.4310/jdg/1557281007