Cauchy-compact flat spacetimes with extreme BTZ

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 $$\mathrm {PSL}_2(\mathbb {R})$$ , 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 $$\mathrm {PSL}_2(\mathbb {R})\ltimes \mathbb {R}^3$$ , 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 Teichmuller 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 lies in the consideration of manifolds with a singular geometrical structure with 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.