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

Research output: Contribution to journalArticle

Abstract

We construct large families of new collapsing hyperkähler metrics on the K3 surface. The limit space is the quotient of a flat 3-torus by an involution. 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 the 3-torus. The geometry around these points is modelled by ALF gravitational instantons: of dihedral type (Dk) for the fixed points of the involution on the 3-torus and of cyclic type (Ak) otherwise. The collapsing metrics are constructed by deforming approximately hyperkähler metrics obtained by gluing ALF gravitational instantons to a background (incomplete) 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.
LanguageEnglish
Pages79-120
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
Torus
Involution
Fixed point
Curvature
Gluing
Complex Structure
Submanifolds
Quotient
Strictly

Cite this

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title = "ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface",
abstract = "We construct large families of new collapsing hyperk{\"a}hler metrics on the K3 surface. The limit space is the quotient of a flat 3-torus by an involution. 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 the 3-torus. The geometry around these points is modelled by ALF gravitational instantons: of dihedral type (Dk) for the fixed points of the involution on the 3-torus and of cyclic type (Ak) otherwise. The collapsing metrics are constructed by deforming approximately hyperk{\"a}hler metrics obtained by gluing ALF gravitational instantons to a background (incomplete) 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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