The energy of maps accompanying the collapsing of the $K3$ surface to a flat 3-dimensional orbifold

We study the Dirichlet energy of some smooth maps appearing in a collapsing family of hyper-Kähler metrics on the $K3$ surface constructed by Foscolo. We introduce an invariant for homotopy classes of smooth maps from the $K3$ 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.