Regular globally hyperbolic maximal anti-de Sitter structures

TL;DR

This paper introduces a new class of anti-de Sitter structures on punctured surfaces, providing two parameterizations of their deformation space related to Teichmüller theory and meromorphic quadratic differentials.

Contribution

It defines regular globally hyperbolic maximal anti-de Sitter structures on punctured surfaces and offers two novel parameterizations of their deformation space.

Findings

Two parameterizations of the deformation space are established.

The deformation space is described as an enhanced product of Fricke spaces.

The second parameterization involves meromorphic quadratic differentials with poles of order at most 2.

Abstract

Let $\Sigma$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti-de Sitter structures on $\Sigma \times \mathbb{R}$ and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of $\Sigma$ and as the bundle over the Teichm\"uller space of $\Sigma$ whose fibre consists of meromorphic quadratic differentials with poles of order at most $2$ at the punctures.