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

Kota Hattori

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arXiv · 2410.15545

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

Kota Hattori

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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.

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Kota Hattori. 2024-10-21. The energy of maps accompanying the collapsing of the $K3$ surface to a flat 3-dimensional orbifold. https://arxiv.org/abs/2410.15545

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