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

Abstract

We construct large families of new collapsing hyperkahler metrics on the $K3$ surface. The limit space is a flat Riemannian $3$-orbifold $T^3 / \mathbb{Z}_2$. 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 T3 and of cyclic type ($A_k$) otherwise. The collapsing metrics are constructed by deforming approximately hyperkahler metrics obtained by gluing ALF gravitational instantons to a background (incomplete) $S^1$–invariant hyperkahler metric arising from the Gibbons–Hawking ansatz over a punctured $3$-torus. As an immediate application to submanifold geometry, we exhibit hyperkahler 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.