Cauchy-compact flat spacetimes with extreme BTZ

TL;DR

This paper establishes a parametrization of certain 3D Lorentzian manifolds with singularities modeled on extreme BTZ black holes, extending classical hyperbolic surface results to a Lorentzian setting with cusps.

Contribution

It introduces a new parametrization of Cauchy-compact flat spacetimes with extreme BTZ singularities using the tangent bundle of Teichmüller space, extending previous regular case results.

Findings

Parametrization of singular Lorentzian manifolds via Teichmüller space tangent bundle.

Development of a BTZ-extension procedure analogous to hyperbolic cusp compactification.

Extension of classical hyperbolic surface theory to Lorentzian 3-manifolds with singularities.

Abstract

Cauchy-compact flat spacetimes with extreme BTZ are Lorentzian analogue of complete hyperbolic surfaces of finite volume. Indeed, the latter are 2-manifolds locally modeled on the hyperbolic plane, with group of isometries $\mathrm{PSL}_2(\R)$, admitting finitely many cuspidal ends while the regular part of the former are 3-manifolds locally models on 3 dimensionnal Minkowski space, with group of isometries $\mathrm{PSL}_2(\R)\ltimes \RR^3$, admitting finitely many ends whose neighborhoods are foliated by cusps. We prove a Theorem akin to the classical parametrization result for finite volume complete hyperbolic surfaces: the tangent bundle of the Teichm\"uller space of a punctured surface parametrizes globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ. Previous results of Mess, Bonsante and Barbot provide already a satisfactory parametrization of regular parts of such manifolds, the particularity of the present work reside in the consideration of manifolds with a singular geometrical structure with a singularities modeled on extreme BTZ. We present a BTZ-extension procedure akin to the procedure compactifying finite volume complete hyperbolic surface by adding cusp points at infinity