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

Bonsante, Francesco¹ ; Danciger, Jeffrey² ; Maloni, Sara³ ; Schlenker, Jean-Marc⁴

¹ Dipartimento di Matematica, Università degli Studi di Pavia, Pavia, Italy
² Department of Mathematics, University of Texas at Austin, Austin, TX, United States
³ Department of Mathematics, University of Virginia, Charlottesville, VA, United States
⁴ Department of Mathematics, University of Luxembourg, Esch-sur-Alzette, Luxembourg

Geometry & topology, Tome 25 (2021) no. 6, pp. 2827-2911.

Résumé

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces compares near points of the limiting Jordan curve. Restricting further to the case in which the induced metrics on the two bounding surfaces have constant curvature $K\in \left[-1,0\right)$ and the Jordan curve at infinity is a quasicircle, the gluing map is naturally a quasisymmetric homeomorphism of the circle. The main result is that for each value of $K$, every quasisymmetric map is achieved as the gluing map at infinity along some quasicircle. We also prove analogous results in the setting of three-dimensional anti-de Sitter geometry. Our results may be viewed as universal versions of the conjectures of Thurston and Mess about prescribing the induced metric on the boundary of the convex core of quasifuchsian hyperbolic manifolds and globally hyperbolic anti-de Sitter spacetimes.

Détail

DOI : 10.2140/gt.2021.25.2827

Classification : 30F40, 53B30, 53C45, 53C50, 57M50, 30C62, 37F30
Keywords: quasicircles, induced metric, Weyl problem

Bibliographie

[1] L V Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963) 291 | DOI

[2] A D Alexandrov, Convex polyhedra, Springer (2005) | DOI

[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 | DOI

[4] M Belraouti, Sur la géométrie de la singularité initiale des espaces-temps plats globalement hyperboliques, Ann. Inst. Fourier (Grenoble) 64 (2014) 457 | DOI

[5] M Belraouti, Asymptotic behavior of Cauchy hypersurfaces in constant curvature space-times, Geom. Dedicata 190 (2017) 103 | DOI

[6] R Benedetti, F Bonsante, Canonical Wick rotations in 3–dimensional gravity, 926, Amer. Math. Soc. (2009) | DOI

[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

[8] L Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960) 94 | DOI

[9] A Beurling, L Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956) 125 | DOI

[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 | DOI

[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 | DOI

[12] F Bonsante, J M Schlenker, Maximal surfaces and the universal Teichmüller space, Invent. Math. 182 (2010) 279 | DOI

[13] F Bonsante, A Seppi, Spacelike convex surfaces with prescribed curvature in (2+1)–Minkowski space, Adv. Math. 304 (2017) 434 | DOI

[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 | DOI

[15] M Bridgeman, R D Canary, The Thurston metric on hyperbolic domains and boundaries of convex hulls, Geom. Funct. Anal. 20 (2010) 1317 | DOI

[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

[17] J Danciger, S Maloni, J M Schlenker, Polyhedra inscribed in a quadric, Invent. Math. 221 (2020) 237 | DOI

[18] B Diallo, Prescribing metrics on the boundary of convex cores of globally hyperbolic maximal compact AdS 3–manifolds, preprint (2013)

[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

[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 | DOI

[22] A Fletcher, V Markovic, Quasiconformal maps and Teichmüller theory, 11, Oxford Univ. Press (2007)

[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 | DOI

[24] D A Herron, W Ma, D Minda, Estimates for conformal metric ratios, Comput. Methods Funct. Theory 5 (2005) 323 | DOI

[25] C D Hodgson, I Rivin, A characterization of compact convex polyhedra in hyperbolic 3–space, Invent. Math. 111 (1993) 77 | DOI

[26] K Krasnov, J M Schlenker, Minimal surfaces and particles in 3–manifolds, Geom. Dedicata 126 (2007) 187 | DOI

[27] F Labourie, Immersions isométriques elliptiques et courbes pseudo-holomorphes, J. Differential Geom. 30 (1989) 395

[28] F Labourie, Problème de Minkowski et surfaces à courbure constante dans les variétés hyperboliques, Bull. Soc. Math. France 119 (1991) 307 | DOI

[29] F Labourie, Métriques prescrites sur le bord des variétés hyperboliques de dimension 3, J. Differential Geom. 35 (1992) 609

[30] F Labourie, Surfaces convexes dans l’espace hyperbolique et CP1–structures, J. London Math. Soc. 45 (1992) 549 | DOI

[31] O Lehto, K I Virtanen, Quasiconformal mappings in the plane, 126, Springer (1973)

[32] G Mess, Lorentz spacetimes of constant curvature, Geom. Dedicata 126 (2007) 3 | DOI

[33] H Miyachi, D Šarić, Uniform weak∗ topology and earthquakes in the hyperbolic plane, Proc. Lond. Math. Soc. 105 (2012) 1123 | DOI

[34] A V Pogorelov, Extrinsic geometry of convex surfaces, 35, Amer. Math. Soc. (1973)

[35] C Pommerenke, Boundary behaviour of conformal maps, 299, Springer (1992) | DOI

[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

[37] H Rosenberg, J Spruck, On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolic space, J. Differential Geom. 40 (1994) 379

[38] J M Schlenker, Surfaces convexes dans des espaces lorentziens à courbure constante, Comm. Anal. Geom. 4 (1996) 285 | DOI

[39] J M Schlenker, Métriques sur les polyèdres hyperboliques convexes, J. Differential Geom. 48 (1998) 323

[40] J M Schlenker, Hypersurfaces in Hn and the space of its horospheres, Geom. Funct. Anal. 12 (2002) 395 | DOI

[41] J M Schlenker, Hyperbolic manifolds with convex boundary, Invent. Math. 163 (2006) 109 | DOI

[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

[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 | DOI

[44] A Tamburelli, Prescribing metrics on the boundary of anti-de Sitter 3–manifolds, Int. Math. Res. Not. 2018 (2018) 1281 | DOI

[45] W P Thurston, Three-dimensional geometry and topology, I, 35, Princeton Univ. Press (1997)

[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 | DOI

[48] D Šarić, Real and complex earthquakes, Trans. Amer. Math. Soc. 358 (2006) 233 | DOI

[49] S T Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978) 197 | DOI