The Energy of Maps Accompanying the Collapsing of the K3 Surface to a Flat 3-Dimensional Orbifold

Abstract We study the Dirichlet energy of some smooth maps appearing in a collapsing family of hyper-Kähler metrics on the K 3 surface constructed by Foscolo. We introduce an invariant for homotopy classes of smooth maps from the K 3 surface with a hyper-Kähler metric to a flat Riemannian orbifold of dimension 3, then show that it gives a lower bound of the energy. Moreover, we show that the ratio of the energy to the invariant converges to 1 for Foscolo’s collapsing families.