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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.15545 |
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Table of Contents:
- 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.