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

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.