Decomposing Centrally Symmetric Convex Polyhedral Surfaces into Parallelograms

TL;DR

This paper studies the structure of moduli spaces of centrally symmetric convex polyhedral surfaces, showing they form real hyperbolic manifolds and can be decomposed into parallelograms with specific geometric properties.

Contribution

It establishes the hyperbolic structure of these moduli spaces and provides explicit decompositions into parallelograms for certain cases, linking geometry and topology.

Findings

Moduli space $ ext{M}_{2N}$ has a real hyperbolic manifold structure.

Surfaces with 8 vertices can be decomposed into parallelograms invariant under antipodal map.

The space of 8-vertex surfaces with specific cone-deficits is isometric to a quotient of a hyperbolic simplex.

Abstract

Let $\mathcal{M}_{2N}(\delta_1, \delta_2,\dots, \delta_N)$ be the moduli space of centrally symmetric convex polyhedral surfaces with $2N$ labeled vertices and prescribed cone-deficits $\delta_1$, $\delta_2$, $\dots$, $\delta_N$. We show that $\mathcal{M}_{2N}(\delta_1, \delta_2,\dots, \delta_N)$ has the structure of a real hyperbolic manifold of dimension $2N-3$. When $N=4$ and $5$, we show that every surface in $\mathcal{M}_{2N}(\delta_1, \delta_2,\dots, \delta_N)$ 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 $\frac{\pi}{2}$ is isometric to the quotient of a real hyperbolic regular ideal $5$-simplex by the dihedral group $D_6$.