Degeneration of globally hyperbolic maximal anti-de Sitter structures along pinching sequences

TL;DR

This paper investigates how globally hyperbolic maximal anti-de Sitter structures on a surface evolve along pinching sequences, revealing that certain regular structures naturally emerge as limits in this process.

Contribution

It provides a detailed analysis of the degeneration process of anti-de Sitter structures along pinching sequences, connecting to the parameterization via the cotangent bundle over Teichmüller space.

Findings

Regular anti-de Sitter structures appear as limits during degeneration.

The behavior of structures is characterized along pinching sequences.

The study links geometric degeneration to the cotangent bundle parameterization.

Abstract

Let $S$ be a closed oriented surface of genus at least $2$. Using the parameterisation of the deformation space of globally hyperbolic maximal anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle over the Teichm\"uller space of $S$, we study the behaviour of these geometric structures along pinching sequences. We show, in particular, that the regular globally hyperbolic anti-de Sitter structures introduced in arXiv:1806.08176 naturally appear as limiting points.