In this thesis we focus on the following question: when a family of manifolds with special metrics degenerates, what does the metric look like near the singularities?
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A natural setting in which to study this question is that of manifolds with so-called special holonomy. We provide the background in Riemannian and complex geometry needed to discuss Yau’s theorem, which provides the existence theory for Calabi–Yau metrics. In real dimension four the Calabi–Yau condition is equivalent to the existence of a closed triple of two-forms, and we give a complete proof of this equivalence, following an outline of Donaldson. The Gibbons–Hawking construction generates such triples from positive harmonic functions on open subsets of ℝ3; importantly for this thesis it produces the Eguchi–Hanson metric, which can be understood to model the simplest kind of singularity, known as an ordinary double point. We compute the curvature of an arbitrary Gibbons–Hawking metric in closed form, following Donaldson, and for the Eguchi–Hanson metric determine the behaviour at infinity. We then show that the curvature energy density of a Gibbons–Hawking metric is the divergence of an explicit vector field on ℝ3, and compute the energy of the Ak spaces as a flux. Finally, we outline the use of the Eguchi–Hanson metric in gluing constructions: the Kummer construction, and Joyce’s construction of the first compact G2-manifolds.