The energy of maps accompanying the collapsing of the $K3$ surface to a flat 3-dimensional orbifold
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909357202997248 |
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| author | Hattori, Kota |
| author_facet | Hattori, Kota |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15545 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The energy of maps accompanying the collapsing of the $K3$ surface to a flat 3-dimensional orbifold Hattori, Kota Differential Geometry 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. |
| title | The energy of maps accompanying the collapsing of the $K3$ surface to a flat 3-dimensional orbifold |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2410.15545 |