# A remark on spaces of flat metrics with cone singularities of constant sign curvatures

## TL;DR

This paper explores the geometric structures of moduli spaces of flat metrics with cone singularities of constant sign curvatures, revealing decompositions into hyperbolic and spherical polyhedral regions.

## Contribution

It provides a new geometric perspective on these moduli spaces, connecting complex hyperbolic, real hyperbolic, and spherical decompositions using polyhedral geometry.

## Key findings

- Moduli space with positive curvatures decomposes into real hyperbolic convex polyhedra.
- Moduli space with negative curvatures decomposes into spherical convex polyhedra.
- The structure depends on the prescribed curvatures and involves complex, real, and spherical geometries.

## Abstract

By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with $n$ cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension $n-3$. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a natural decomposition into real hyperbolic convex polyhedra of dimensions $n-3$ and $\leq \frac{1}{2}(n-1)$. By a result of W.~Veech, the moduli space of flat metrics on a compact surface with cone singularities of prescribed negative curvatures has a foliation whose leaves have a local structure of complex pseudo-spheres. The complex structure comes again from the area of the metric. The form can be degenerate; its signature depends on the curvatures prescribed. Using polyhedral surfaces in Minkowski space, we show that this moduli space has a natural decomposition into spherical convex polyhedra.