### 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 |

### Fingerprint

### 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

}

**surface',**

*K*3*Journal of Differential Geometry*, vol. 112, no. 1, pp. 79-120. https://doi.org/10.4310/jdg/1557281007

**ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface.** / Foscolo, Lorenzo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface

AU - Foscolo, Lorenzo

PY - 2019/5/8

Y1 - 2019/5/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85071910782&partnerID=8YFLogxK

U2 - 10.4310/jdg/1557281007

DO - 10.4310/jdg/1557281007

M3 - Article

VL - 112

SP - 79

EP - 120

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 1

ER -

**surface. Journal of Differential Geometry. 2019 May 8;112(1):79-120. https://doi.org/10.4310/jdg/1557281007**

*K*3