AdS manifolds with particles and earthquakes on singular surfaces

TL;DR

This paper establishes an earthquake theorem for hyperbolic surfaces with cone singularities and characterizes the space of AdS manifolds with particles using Teichmüller space, advancing understanding of singular geometric structures.

Contribution

It proves an earthquake theorem for hyperbolic surfaces with cone singularities and parametrizes AdS manifolds with particles via Teichmüller space, linking singular surface theory and Lorentzian geometry.

Findings

Unique left earthquake connects any two cone singularity metrics.

Parametrization of AdS manifolds with particles by Teichmüller space.

Extension of earthquake theory to singular surfaces.

Abstract

We prove two related results. The first is an ``Earthquake Theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than $\pi$: any two such metrics in are connected by a unique left earthquake. The second result is that the space of ``globally hyperbolic'' AdS manifolds with ``particles'' -- cone singularities (of given angle) along time-like lines -- is parametrized by the product of two copies of the Teichm\"uller space with some marked points (corresponding to the cone singularities). The two statements are proved together.