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The Alexandrov theorem for $2+1$ flat radiant spacetimes

Léo Maxime Brunswic

Algebraic & Geometric Topology 25 (2025) 1321–1375
Bibliography
1 S Alexander, V Kapovitch, A Petrunin, An invitation to Alexandrov geometry : CAT(0) spaces, Springer (2019) MR3930625
2 A Alexandrov, Existence of a convex polyhedron and of a convex surface with a given metric, Mat. Sb. 53 (1942) 15 MR13540
3 A D Alexandrov, Convex polyhedra, Springer (2005) MR2127379
4 T Barbot, F Bonsante, J M Schlenker, Collisions of particles in locally AdS spacetimes, I : Local description and global examples, Comm. Math. Phys. 308 (2011) 147 MR2842974
5 P Bernard, S Suhr, Lyapounov functions of closed cone fields : from Conley theory to time functions, Comm. Math. Phys. 359 (2018) 467 MR3783554
6 A I Bobenko, I Izmestiev, Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes, Ann. Inst. Fourier (Grenoble) 58 (2008) 447 MR2410380
7 F Bonsante, A Seppi, Spacelike convex surfaces with prescribed curvature in (2+1)-Minkowski space, Adv. Math. 304 (2017) 434 MR3558215
8 G E Bredon, Topology and geometry, 139, Springer (1993) MR1224675
9 L Brunswic, Surfaces de Cauchy polyédrales des espaces temps-plats singuliers, PhD thesis, Université d’Avignon (2017)
10 L Brunswic, Cauchy-compact flat spacetimes with extreme BTZ, Geom. Dedicata 214 (2021) 571 MR4308293
11 L Brunswic, On branched coverings of singular (G,X)-manifolds, Geom. Dedicata 218 (2024) 43 MR4706074
12 C Ehresmann, Œuvres complètes et commentées, I-1,2 : Topologie algébrique et géométrie différentielle, 24 (suppl. 1), Cahiers Topologie Géom Différentielle (1983) MR794193
13 D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67 MR918457
14 W Fenchel, Elementary geometry in hyperbolic space, 11, de Gruyter (1989) MR1004006
15 F Fillastre, Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces, Ann. Inst. Fourier (Grenoble) 57 (2007) 163 MR2313089
16 F Fillastre, Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces, from: "Conference on metric geometry of surfaces and polyhedra", Current Prob. Math. Mech. 6, Moscow State Univ. (2010) 208
17 F Fillastre, Fuchsian polyhedra in Lorentzian space-forms, Math. Ann. 350 (2011) 417 MR2794916
18 F Fillastre, I Izmestiev, Hyperbolic cusps with convex polyhedral boundary, Geom. Topol. 13 (2009) 457 MR2469522
19 F Fillastre, I Izmestiev, Gauss images of hyperbolic cusps with convex polyhedral boundary, Trans. Amer. Math. Soc. 363 (2011) 5481 MR2813423
20 W M Goldman, Geometric structures on manifolds, 227, Amer. Math. Soc. (2022) MR4500072
21 C D Hodgson, I Rivin, A characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math. 111 (1993) 77 MR1193599
22 I Izmestiev, A variational proof of Alexandrov’s convex cap theorem, Discrete Comput. Geom. 40 (2008) 561 MR2453328
23 D Kane, G N Price, E D Demaine, A pseudopolynomial algorithm for Alexandrov’s theorem, from: "Algorithms and data structures", Lecture Notes in Comput. Sci. 5664, Springer (2009) 435 MR2550627
24 H Masur, J Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math. 134 (1991) 455 MR1135877
25 G Mess, Lorentz spacetimes of constant curvature, Geom. Dedicata 126 (2007) 3 MR2328921
26 A D Milka, Space-like convex surfaces in pseudo-Euclidean spaces, from: "Some questions of differential geometry in the large", Amer. Math. Soc. Transl. Ser. 2 176, Amer. Math. Soc. (1996) 97 MR1406843
27 R C Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987) 299 MR919235
28 R C Penner, Decorated Teichmüller theory, Eur. Math. Soc. (2012) MR3052157
29 , SageMath, version 9.4 (2022)
30 J Sbierski, On the existence of a maximal Cauchy development for the Einstein equations : a dezornification, Ann. Henri Poincaré 17 (2016) 301 MR3447847
31 J M Schlenker, Hyperbolic manifolds with polyhedral boundary, preprint (2001) arXiv:math/0111136
32 R Souam, The Schläfli formula for polyhedra and piecewise smooth hypersurfaces, Differential Geom. Appl. 20 (2004) 31 MR2030165
33 W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)
34 Y A Volkov, Existence of a polyhedron with prescribed development, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 476 (2018) 50 MR3904042