Degeneration of globally hyperbolic maximal anti-de Sitter structures along rays

TL;DR

This paper investigates how key geometric quantities of globally hyperbolic maximal anti-de Sitter structures on surfaces degenerate along rays in the deformation space, revealing their asymptotic behaviors.

Contribution

It provides a detailed analysis of the degeneration of geometric invariants in anti-de Sitter structures using the cotangent bundle parameterization of the deformation space.

Findings

Lorentzian Hausdorff dimension of the limit set degenerates along rays

Width of the convex core tends to zero or infinity

Hölder exponent exhibits specific degeneration patterns

Abstract

Using the parameterisation of the deformation space of GHMC anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle of the Teichm\"uller space of $S$, we study how some geometric quantities, such as the Lorentzian Hausdorff dimension of the limit set, the width of the convex core and the H\"older exponent, degenerate along rays of quadratic differentials.