Foscolo, L; (2019) ALF gravitational instantons and collapsing Ricci-flat metrics on the K3 surface. Journal of Differential Geometry , 112 (1) pp. 79-120. 10.4310/jdg/1557281007.

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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.

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