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<p>We study the Dirichlet energy of some smooth maps appearing in a collapsing family of hyper-Kähler metrics on the <i>K</i>3 surface constructed by Foscolo. We introduce an invariant for homotopy classes of smooth maps from the <i>K</i>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.</p>

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

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  • 发表日期：

    2025年3月18日

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  • Kota Hattori

摘要

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.