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

TL;DR

This paper investigates the Dirichlet energy of smooth maps from collapsing hyper-K"ahler $K3$ surfaces to flat 3-orbifolds, introducing an invariant that bounds energy and analyzing its behavior in collapsing families.

Contribution

It introduces a new invariant for homotopy classes of maps from $K3$ surfaces and demonstrates its role in understanding energy during collapsing hyper-K"ahler metrics.

Findings

Invariant provides a lower bound for energy.

Energy-to-invariant ratio approaches 1 in collapsing families.

Results connect geometric collapse with energy estimates.

Abstract

We study the Dirichlet energy of some smooth maps appearing in a collapsing family of hyper-K\"ahler 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-K\"ahler 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.