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

TL;DR

This paper constructs new collapsing hyperk"ahler metrics on the K3 surface, with the collapse described by ALF gravitational instantons and a limit space related to a quotient of a 3-torus, revealing novel geometric phenomena.

Contribution

It introduces a method to build collapsing hyperk"ahler metrics on K3 using ALF gravitational instantons and analyzes their geometric and curvature properties.

Findings

Collapse occurs with bounded curvature away from 24 exceptional points.

At exceptional points, curvature concentrates modeled by ALF gravitational instantons.

Existence of hyperk"ahler metrics with stable minimal spheres not holomorphic in any compatible complex structure.

Abstract

We construct large families of new collapsing hyperk\"ahler 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\"ahler metrics obtained by gluing ALF gravitational instantons to a background (incomplete) hyperk\"ahler metric arising from the Gibbons-Hawking ansatz over a punctured 3-torus. As an immediate application to submanifold geometry, we exhibit hyperk\"ahler 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.