Gravitational Instantons and Degenerations of Ricci-flat Metrics on the K3 Surface

The study of degenerations of metrics with special holonomy is an important theme unifying the study of convergence of Einstein metrics, the study of complete non-compact manifolds with special holonomy and the construction of spaces with special holonomy by singular perturbation methods. We survey three constructions of degenerating sequences of hyperkähler metrics on the (smooth 4-manifold underlying a complex) K3 surface—the classical Kummer construction, Gross–Wilson’s work on collapse along the fibres of an elliptic fibration, and the author’s construction of sequences collapsing to a 3-dimensional limit—describing how they fit into the general theory and highlighting the role played in each construction by gravitational instantons, i.e. complete non-compact hyperkähler 4-manifolds with decaying curvature at infinity.