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May 2019 ALF gravitational instantons and collapsing Ricci-flat metrics on the $K3$ surface
Lorenzo Foscolo
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J. Differential Geom. 112(1): 79-120 (May 2019). DOI: 10.4310/jdg/1557281007


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.


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Lorenzo Foscolo. "ALF gravitational instantons and collapsing Ricci-flat metrics on the $K3$ surface." J. Differential Geom. 112 (1) 79 - 120, May 2019.


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

zbMATH: 07054920
MathSciNet: MR3948228
Digital Object Identifier: 10.4310/jdg/1557281007

Rights: Copyright © 2019 Lehigh University


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Vol.112 • No. 1 • May 2019
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