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The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti-de Sitter geometry

Francesco Bonsante, Jeffrey Danciger, Sara Maloni and Jean-Marc Schlenker

Appendix: Boubacar Diallo

Geometry & Topology 25 (2021) 2827–2911
Bibliography
1 L V Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963) 291 MR154978
2 A D Alexandrov, Convex polyhedra, Springer (2005) MR2127379
3 T Barbot, F Béguin, A Zeghib, Prescribing Gauss curvature of surfaces in 3–dimensional spacetimes : application to the Minkowski problem in the Minkowski space, Ann. Inst. Fourier (Grenoble) 61 (2011) 511 MR2895066
4 M Belraouti, Sur la géométrie de la singularité initiale des espaces-temps plats globalement hyperboliques, Ann. Inst. Fourier (Grenoble) 64 (2014) 457 MR3330911
5 M Belraouti, Asymptotic behavior of Cauchy hypersurfaces in constant curvature space-times, Geom. Dedicata 190 (2017) 103 MR3704814
6 R Benedetti, F Bonsante, Canonical Wick rotations in 3–dimensional gravity, 926, Amer. Math. Soc. (2009) MR2499272
7 Y Benoist, A survey on divisible convex sets, from: "Geometry, analysis and topology of discrete groups" (editors L Ji, K Liu, L Yang, S T Yau), Adv. Lect. Math. 6, International (2008) 1 MR2464391
8 L Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960) 94 MR111834
9 A Beurling, L Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956) 125 MR86869
10 F Bonsante, Constant curvature (2+1)–spacetimes and projective structures, from: "Actes de Séminaire de Théorie Spectrale et Géométrie", Sémin. Théor. Spectr. Géom. 23, Univ. Grenoble I (2005) 9 MR2270222
11 F Bonsante, J Danciger, S Maloni, J M Schlenker, Quasicircles and width of Jordan curves in ℂℙ1, Bull. Lond. Math. Soc. 53 (2021) 507 MR4239192
12 F Bonsante, J M Schlenker, Maximal surfaces and the universal Teichmüller space, Invent. Math. 182 (2010) 279 MR2729269
13 F Bonsante, A Seppi, Spacelike convex surfaces with prescribed curvature in (2+1)–Minkowski space, Adv. Math. 304 (2017) 434 MR3558215
14 F Bonsante, A Seppi, Area-preserving diffeomorphisms of the hyperbolic plane and K–surfaces in anti-de Sitter space, J. Topol. 11 (2018) 420 MR3789829
15 M Bridgeman, R D Canary, The Thurston metric on hyperbolic domains and boundaries of convex hulls, Geom. Funct. Anal. 20 (2010) 1317 MR2738995
16 R D Canary, D B A Epstein, P Green, Notes on notes of Thurston, from: "Analytical and geometric aspects of hyperbolic space" (editor D B A Epstein), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 3 MR903850
17 J Danciger, S Maloni, J M Schlenker, Polyhedra inscribed in a quadric, Invent. Math. 221 (2020) 237 MR4105089
18 B Diallo, Prescribing metrics on the boundary of convex cores of globally hyperbolic maximal compact AdS 3–manifolds, preprint (2013) arXiv:1303.7406
19 B Diallo, Métriques prescrites sur le bord du coeur convexe d’une variété anti-de sitter globalement hyperbolique maximale compacte de dimension trois, PhD thesis, Université de Toulouse (2014)
20 D B A Epstein, A Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, from: "Analytical and geometric aspects of hyperbolic space" (editor D B A Epstein), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 113 MR903852
21 F Fillastre, A Seppi, Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions, from: "Eighteen essays in non-Euclidean geometry" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 29, Eur. Math. Soc. (2019) 321 MR3965127
22 A Fletcher, V Markovic, Quasiconformal maps and Teichmüller theory, 11, Oxford Univ. Press (2007) MR2269887
23 F P Gardiner, J Hu, N Lakic, Earthquake curves, from: "Complex manifolds and hyperbolic geometry" (editors C J Earle, W J Harvey, S Recillas-Pishmish), Contemp. Math. 311, Amer. Math. Soc. (2002) 141 MR1940169
24 D A Herron, W Ma, D Minda, Estimates for conformal metric ratios, Comput. Methods Funct. Theory 5 (2005) 323 MR2205417
25 C D Hodgson, I Rivin, A characterization of compact convex polyhedra in hyperbolic 3–space, Invent. Math. 111 (1993) 77 MR1193599
26 K Krasnov, J M Schlenker, Minimal surfaces and particles in 3–manifolds, Geom. Dedicata 126 (2007) 187 MR2328927
27 F Labourie, Immersions isométriques elliptiques et courbes pseudo-holomorphes, J. Differential Geom. 30 (1989) 395 MR1010166
28 F Labourie, Problème de Minkowski et surfaces à courbure constante dans les variétés hyperboliques, Bull. Soc. Math. France 119 (1991) 307 MR1125669
29 F Labourie, Métriques prescrites sur le bord des variétés hyperboliques de dimension 3, J. Differential Geom. 35 (1992) 609 MR1163450
30 F Labourie, Surfaces convexes dans l’espace hyperbolique et CP1–structures, J. London Math. Soc. 45 (1992) 549 MR1180262
31 O Lehto, K I Virtanen, Quasiconformal mappings in the plane, 126, Springer (1973) MR0344463
32 G Mess, Lorentz spacetimes of constant curvature, Geom. Dedicata 126 (2007) 3 MR2328921
33 H Miyachi, D Šarić, Uniform weak topology and earthquakes in the hyperbolic plane, Proc. Lond. Math. Soc. 105 (2012) 1123 MR3004100
34 A V Pogorelov, Extrinsic geometry of convex surfaces, 35, Amer. Math. Soc. (1973) MR0346714
35 C Pommerenke, Boundary behaviour of conformal maps, 299, Springer (1992) MR1217706
36 I Rivin, Intrinsic geometry of convex ideal polyhedra in hyperbolic 3–space, from: "Analysis, algebra, and computers in mathematical research" (editors M Gyllenberg, L E Persson), Lecture Notes in Pure and Appl. Math. 156, Dekker (1994) 275 MR1280952
37 H Rosenberg, J Spruck, On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolic space, J. Differential Geom. 40 (1994) 379 MR1293658
38 J M Schlenker, Surfaces convexes dans des espaces lorentziens à courbure constante, Comm. Anal. Geom. 4 (1996) 285 MR1393565
39 J M Schlenker, Métriques sur les polyèdres hyperboliques convexes, J. Differential Geom. 48 (1998) 323 MR1630178
40 J M Schlenker, Hypersurfaces in Hn and the space of its horospheres, Geom. Funct. Anal. 12 (2002) 395 MR1911666
41 J M Schlenker, Hyperbolic manifolds with convex boundary, Invent. Math. 163 (2006) 109 MR2208419
42 R M Schoen, The role of harmonic mappings in rigidity and deformation problems, from: "Complex geometry" (editors G Komatsu, Y Sakane), Lecture Notes in Pure and Appl. Math. 143, Dekker (1993) 179 MR1201611
43 D Sullivan, Travaux de Thurston sur les groupes quasi-fuchsiens et les variétés hyperboliques de dimension 3 fibrées sur S1, from: "Bourbaki Seminar, 1979/80", Lecture Notes in Math. 842, Springer (1981) 196 MR636524
44 A Tamburelli, Prescribing metrics on the boundary of anti-de Sitter 3–manifolds, Int. Math. Res. Not. 2018 (2018) 1281 MR3801462
45 W P Thurston, Three-dimensional geometry and topology, I, 35, Princeton Univ. Press (1997) MR1435975
46 W P Thurston, Complete metric on the space of jordan curves?, MathOverflow comment (2010)
47 M Troyanov, The Schwarz lemma for nonpositively curved Riemannian surfaces, Manuscripta Math. 72 (1991) 251 MR1118545
48 D Šarić, Real and complex earthquakes, Trans. Amer. Math. Soc. 358 (2006) 233 MR2171231
49 S T Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978) 197 MR486659