Journal of Differential Geometry

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

Lorenzo Foscolo

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We construct large families of new collapsing hyperkähler 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 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.

Article information

J. Differential Geom., Volume 112, Number 1 (2019), 79-120.

Received: 20 March 2016
First available in Project Euclid: 8 May 2019

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Foscolo, Lorenzo. ALF gravitational instantons and collapsing Ricci-flat metrics on the $K3$ surface. J. Differential Geom. 112 (2019), no. 1, 79--120. doi:10.4310/jdg/1557281007.

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