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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 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.

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This work has been part of a PhD thesis supervised by Thierry Barbot at Université d’Avignon et des Pays de Vaucluse and is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement ERC advanced Grant 740021–ARTHUS, PI: Thomas Buchert).

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Correspondence to Léo Brunswic.

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Brunswic, L. Cauchy-compact flat spacetimes with extreme BTZ. Geom Dedicata 214, 571–608 (2021).

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  • Geometry
  • Geometry topology
  • Minkowski space
  • Singular geometrical structure
  • Lorentzian manifold

Mathematics Subject Classification

  • 57K35; 51H20; 83A05; 83C57; 57K20; 51M10