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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.06967 |
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| _version_ | 1866918489054248960 |
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| author | Gong, Xun Lin, Zuo Sanchez, Anthony |
| author_facet | Gong, Xun Lin, Zuo Sanchez, Anthony |
| contents | We study flat metrics arising from right regular $n$-prisms by viewing them as $n$-differentials and analyzing their associated unfoldings. We show that the unfolding of a right regular $n$-prism is never a lattice surface unless $n=4$, in contrast with the case of Platonic solids. Despite this, we prove that these surfaces admit translation coverings to hyperelliptic surfaces, allowing us to determine their $\mathrm{GL}(2,\mathbb{R})$-orbit closures using the classification of hyperelliptic components of strata.
As a consequence, we obtain exact quadratic asymptotics for a certain average of the number of saddle connections on the base surfaces, their unfoldings, and the original prisms, including their Siegel--Veech constants. This provides a natural infinite family of non-lattice surfaces for which orbit closures and counting problems can be computed explicitly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06967 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Translation Surfaces arising from Right Regular Prisms Gong, Xun Lin, Zuo Sanchez, Anthony Geometric Topology We study flat metrics arising from right regular $n$-prisms by viewing them as $n$-differentials and analyzing their associated unfoldings. We show that the unfolding of a right regular $n$-prism is never a lattice surface unless $n=4$, in contrast with the case of Platonic solids. Despite this, we prove that these surfaces admit translation coverings to hyperelliptic surfaces, allowing us to determine their $\mathrm{GL}(2,\mathbb{R})$-orbit closures using the classification of hyperelliptic components of strata. As a consequence, we obtain exact quadratic asymptotics for a certain average of the number of saddle connections on the base surfaces, their unfoldings, and the original prisms, including their Siegel--Veech constants. This provides a natural infinite family of non-lattice surfaces for which orbit closures and counting problems can be computed explicitly. |
| title | Translation Surfaces arising from Right Regular Prisms |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2605.06967 |