Cauchy-compact flat spacetimes with extreme BTZ

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