Polyhedral hyperbolic metrics on surfaces

TL;DR

This paper proves the connectedness of a space of polyhedral hyperbolic metrics on surfaces, completing a previous argument that established a homeomorphism between this space and a moduli space.

Contribution

It provides a missing proof of the connectedness of the space of polyhedral hyperbolic metrics, solidifying the topological relationship with the moduli space.

Findings

Confirmed the connectedness of the space of polyhedral hyperbolic metrics

Established the homeomorphism between the metric space and the moduli space

Filled a gap in the previous proof by providing a missing step

Abstract

In the last section of \cite{CompHyp} it is proved that the map $\mathcal{I}$ is a finite-sheeted covering map between $\mathcal{P}$ and $\mathcal{M}$. As $\mathcal{M}$ is simply connected it is deduced that $\mathcal{I}$ is a homeomorphism. The fact that $\mathcal{P}$ is connected is missing. Here we provide a proof.