Pseudo-developing maps for ideal triangulations II: Positively oriented ideal triangulations of cone-manifolds

TL;DR

This paper extends the theory of ideal triangulations to cone-manifolds with arbitrary genus vertex links, demonstrating that the space of hyperbolic structures with prescribed cone angles forms a smooth complex manifold and can be locally parameterized holomorphically.

Contribution

It generalizes previous work to include vertex links of any genus and establishes a local holomorphic parameterization of the space of hyperbolic cone-manifold structures.

Findings

The space of hyperbolic cone-manifold structures is a smooth complex manifold.

The dimension of this manifold equals the sum of the genera of the vertex links.

Peripheral complex lengths provide a local holomorphic parameterization.

Abstract

We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescribed cone angles at all edges is (if non-empty) a smooth complex manifold of dimension the sum of the genera of the vertex links. Moreover, we show that the complex lengths of a collection of peripheral elements give a local holomorphic parameterisation of this manifold.