Decomposing Centrally Symmetric Convex Polyhedral Surfaces into Parallelograms
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| Format: | Preprint |
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2026
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| _version_ | 1866918416332357632 |
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| author | Wang, Zili Wu, Cong |
| author_facet | Wang, Zili Wu, Cong |
| contents | Let $\mathcal{M}_{2N}(δ_1, δ_2,\dots, δ_N)$ be the moduli space of centrally symmetric convex polyhedral surfaces with $2N$ labeled vertices and prescribed cone-deficits $δ_1$, $δ_2$, $\dots$, $δ_N$. We show that $\mathcal{M}_{2N}(δ_1, δ_2,\dots, δ_N)$ has the structure of a real hyperbolic manifold of dimension $2N-3$. When $N=4$ and $5$, we show that every surface in $\mathcal{M}_{2N}(δ_1, δ_2,\dots, δ_N)$ can be decomposed into at most $2\binom{2N-2}{2}$ parallelograms, and the decomposition is invariant under the antipodal map. Using the edge-lengths of these parallelograms as coordinates, we show that the moduli space of centrally symmetric polyhedral surfaces with $8$ unlabeled vertices and cone-deficits $\fracπ{2}$ is isometric to the quotient of a real hyperbolic regular ideal $5$-simplex by the dihedral group $D_6$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_21199 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Decomposing Centrally Symmetric Convex Polyhedral Surfaces into Parallelograms Wang, Zili Wu, Cong Geometric Topology Combinatorics Let $\mathcal{M}_{2N}(δ_1, δ_2,\dots, δ_N)$ be the moduli space of centrally symmetric convex polyhedral surfaces with $2N$ labeled vertices and prescribed cone-deficits $δ_1$, $δ_2$, $\dots$, $δ_N$. We show that $\mathcal{M}_{2N}(δ_1, δ_2,\dots, δ_N)$ has the structure of a real hyperbolic manifold of dimension $2N-3$. When $N=4$ and $5$, we show that every surface in $\mathcal{M}_{2N}(δ_1, δ_2,\dots, δ_N)$ can be decomposed into at most $2\binom{2N-2}{2}$ parallelograms, and the decomposition is invariant under the antipodal map. Using the edge-lengths of these parallelograms as coordinates, we show that the moduli space of centrally symmetric polyhedral surfaces with $8$ unlabeled vertices and cone-deficits $\fracπ{2}$ is isometric to the quotient of a real hyperbolic regular ideal $5$-simplex by the dihedral group $D_6$. |
| title | Decomposing Centrally Symmetric Convex Polyhedral Surfaces into Parallelograms |
| topic | Geometric Topology Combinatorics |
| url | https://arxiv.org/abs/2603.21199 |