As we have already mentioned, in this paper we will need to glue hyperkähler structures rather than solving a complex Monge–Ampère equation on a given complex manifold. In Section 2, following Donaldson [15], we explain how to set up the problem of deforming approximately hyperkähler metrics based on the notion of definite triples.

Section 3 is a detailed summary of the theory of ALF spaces: we give precise definitions, describe detailed asymptotics for such metrics and recall the construction and classification of examples.

In Section 4 we use the Gibbons–Hawking ansatz to construct (incomplete) hyperkähler metrics on circle bundles over a punctured $3$–torus. In Section 5 we use ALF spaces of cyclic and dihedral type together with the metrics constructed in Section 4 to produce families of approximately hyperkähler metrics. In Section 6 we use analysis to deform these approximate solutions into exact hyperkähler metrics. This is done by means of an Implicit Function Theorem in weighted Hölder spaces. As usual in gluing problems, most of the work goes into showing that the relevant linear operator has no small eigenvalues as $\epsilon\rightarrow 0$ and the geometry degenerates.

Finally, Section 7 contains the proof of Theorem 7.1 about the existence of non-holomorphic strictly stable minimal spheres.