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 Teichmüller 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.

具有极端BTZ的柯西紧凑平面时空是有限体积的完整双曲表面的洛伦兹模拟。的确，后者是在双曲平面上局部建模的2个流形，具有等距组\（\ mathrm {PSL} _2（\ mathbb {R}）\），允许有限的尖齿末端，而前者的正则部分是3维Minkowski空间上的3个流形局部模型，具有等距组\（\ mathrm {PSL} _2（\ mathbb {R}）\ ltimes \ mathbb {R} ^ 3 \），可以接受许多末端被尖齿剥落的末端。我们证明了一个与经典参数化结果相似的定理，它适用于有限体积完整的双曲曲面：被刺穿的表面参数的Teichmüller空间的切线束具有全局双曲的Cauchy-极大和Cauchy-紧致的局部Minkowski流形，具有极端BTZ。Mess，Bonsante和Barbot的先前结果已经提供了此类歧管的常规零件的令人满意的参数化，当前工作的特殊性在于考虑了具有奇异几何结构的歧管，并以极端BTZ为模型。我们提出了一种BTZ扩展程序，类似于通过在无限远处添加尖点来压缩有限体积的完整双曲曲面的程序。