Abstract
We investigate the collapsing geometry of hyperkähler 4-manifolds. As applications we prove two well-known conjectures in the field. (1) Any collapsed limit of unit-diameter hyperkähler metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3-torus by an involution, a singular special Kähler metric on the 2-sphere, or the unit interval. (2) Any complete hyperkähler 4-manifold with finite energy (i.e., gravitational instanton) is asymptotic to a model end at infinity.
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