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

Lorenzo Foscolo

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)79-120
Number of pages42
JournalJournal of Differential Geometry
Volume112
Issue number1
DOIs
Publication statusPublished - 8 May 2019

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K3 Surfaces
Collapsing
Instantons
Metric
Involution
Fixed point
Curvature
Gluing
Orbifold
Complex Structure
Submanifolds
Torus
Strictly

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

Cite this

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abstract = "We construct large families of new collapsing hyperk{\"a}hler 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 hyperk{\"a}hler metrics obtained by gluing ALF gravitational instantons to a background (incomplete) S 1–invariant hyperk{\"a}hler metric arising from the Gibbons–Hawking ansatz over a punctured 3-torus. As an immediate application to submanifold geometry, we exhibit hyperk{\"a}hler 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.",
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ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface. / Foscolo, Lorenzo.

In: Journal of Differential Geometry, Vol. 112, No. 1, 08.05.2019, p. 79-120.

Research output: Contribution to journalArticle

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