Geometry & Topology

The convex core of quasifuchsian manifolds with particles

Cyril Lecuire and Jean-Marc Schlenker

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We consider quasifuchsian manifolds with “particles”, ie cone singularities of fixed angle less than π going from one connected component of the boundary at infinity to the other. Each connected component of the boundary at infinity is then endowed with a conformal structure marked by the endpoints of the particles. We prove that this defines a homeomorphism between the space of quasifuchsian metrics with n particles (of fixed angle) and the product of two copies of the Teichmüller space of a surface with n marked points. This extends the Bers double uniformization theorem to quasifuchsian manifolds with “particles”.

Quasifuchsian manifolds with particles also have a convex core. Its boundary has a hyperbolic induced metric, with cone singularities at the intersection with the particles, and is pleated along a measured geodesic lamination. We prove that any two hyperbolic metrics with cone singularities (of prescribed angle) can be obtained, and also that any two measured bending laminations, satisfying some obviously necessary conditions, can be obtained, as in Bonahon and Otal [Ann. of Math. 160 (2004) 1013–1055] in the nonsingular case.

Article information

Geom. Topol., Volume 18, Number 4 (2014), 2309-2373.

Received: 18 February 2013
Accepted: 5 October 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] 57M50: Geometric structures on low-dimensional manifolds

quasifuchsian groups cone singularities conformal structures bending laminations


Lecuire, Cyril; Schlenker, Jean-Marc. The convex core of quasifuchsian manifolds with particles. Geom. Topol. 18 (2014), no. 4, 2309--2373. doi:10.2140/gt.2014.18.2309.

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  • L Ahlfors, L Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. 72 (1960) 385–404
  • A D Aleksandrov, V A Zalgaller, Intrinsic geometry of surfaces, Trans. Math. Monog. 15, Amer. Math. Soc. (1967)
  • L Andersson, T Barbot, R Benedetti, F Bonsante, W M Goldman, F Labourie, K P Scannell, J-M Schlenker, Notes on: “Lorentz spacetimes of constant curvature” by G Mess, Geom. Dedicata 126 (2007) 47–70
  • W Ballmann, M Gromov, V Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics 61, Birkhäuser, Boston (1985)
  • 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) 147–200
  • T Barbot, F Bonsante, J-M Schlenker, Collisions of particles in locally AdS spacetimes, II: Moduli of globally hyperbolic spaces, Comm. Math. Phys. 327 (2014) 691–735
  • R Benedetti, C Petronio, Lectures on hyperbolic geometry, Springer, Berlin (1992)
  • L Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960) 94–97
  • A L Besse, Einstein manifolds, Ergeb. Math. Grenzgeb. 10, Springer, Berlin (1987)
  • M Boileau, B Leeb, J Porti, Geometrization of $3$–dimensional orbifolds, Ann. of Math. 162 (2005) 195–290
  • M Boileau, J Porti, Geometrization of $3$–orbifolds of cyclic type, Astérisque 272, Soc. Math. France, Paris (2001) 208
  • F Bonahon, J-P Otal, Laminations measurées de plissage des variétés hyperboliques de dimension 3, Ann. of Math. 160 (2004) 1013–1055
  • F Bonsante, J-M Schlenker, AdS manifolds with particles and earthquakes on singular surfaces, Geom. Funct. Anal. 19 (2009) 41–82
  • F Bonsante, J-M Schlenker, Fixed points of compositions of earthquakes, Duke Math. J. 161 (2012) 1011–1054
  • M Bridgeman, Average bending of convex pleated planes in hyperbolic three-space, Invent. Math. 132 (1998) 381–391
  • M Bridgeman, R D Canary, From the boundary of the convex core to the conformal boundary, Geom. Dedicata 96 (2003) 211–240
  • J F Brock, R D Canary, Y N Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. 176 (2012) 1–149
  • J F Brock, H Masur, Y N Minsky, Asymptotics of Weil–Petersson geodesic, I: Ending laminations, recurrence, and flows, Geom. Funct. Anal. 19 (2010) 1229–1257
  • K Bromberg, Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives, J. Amer. Math. Soc. 17 (2004) 783–826
  • K Bromberg, Rigidity of geometrically finite hyperbolic cone-manifolds, Geom. Dedicata 105 (2004) 143–170
  • R Brooks, J P Matelski, Collars in Kleinian groups, Duke Math. J. 49 (1982) 163–182
  • A J Casson, S A Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Math. Soc. Student Texts 9, Cambridge Univ. Press (1988)
  • D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5, Math. Soc. Japan, Tokyo (2000)
  • E B Dryden, H Parlier, Collars and partitions of hyperbolic cone-surfaces, Geom. Dedicata 127 (2007) 139–149
  • D Dumas, Complex projective structures, from: “Handbook of Teichmüller theory, Vol. II”, (A Papadopoulos, editor), IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc., Zürich (2009) 455–508
  • 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”, (D B A Epstein, editor), London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press (1987) 113–253
  • D B A Epstein, V Markovic, The logarithmic spiral: A counterexample to the $K=2$ conjecture, Ann. of Math. 161 (2005) 925–957
  • J L Harer, R C Penner, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton Univ. Press (1992)
  • C D Hodgson, S P Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998) 1–59
  • G 't Hooft, The evolution of gravitating point particles in $2+1$ dimensions, Classical Quantum Gravity 10 (1993) 1023–1038
  • G 't Hooft, Quantization of point particles in $(2+1)$–dimensional gravity and spacetime discreteness, Classical Quantum Gravity 13 (1996) 1023–1039
  • S P Kerckhoff, The Nielsen realization problem, Ann. of Math. 117 (1983) 235–265
  • K Krasnov, J-M Schlenker, On the renormalized volume of hyperbolic $3$–manifolds, Comm. Math. Phys. 279 (2008) 637–668
  • K Krasnov, J-M Schlenker, A symplectic map between hyperbolic and complex Teichmüller theory, Duke Math. J. 150 (2009) 331–356
  • K Krasnov, J-M Schlenker, The Weil–Petersson metric and the renormalized volume of hyperbolic $3$–manifolds, from: “Handbook of Teichmüller theory, Vol. III”, (A Papadopoulos, editor), IRMA Lect. Math. Theor. Phys. 17, Eur. Math. Soc., Zürich (2012) 779–819
  • R S Kulkarni, U Pinkall, A canonical metric for Möbius structures and its applications, Math. Z. 216 (1994) 89–129
  • F Labourie, Métriques prescrites sur le bord des variétés hyperboliques de dimension $3$, J. Differential Geom. 35 (1992) 609–626
  • C Lecuire, Plissage des variétés hyperboliques de dimension $3$, Invent. Math. 164 (2006) 85–141
  • C T McMullen, The moduli space of Riemann surfaces is Kähler hyperbolic, Ann. of Math. 151 (2000) 327–357
  • G Mess, Lorentz spacetimes of constant curvature, Geom. Dedicata 126 (2007) 3–45
  • R Meyerhoff, A lower bound for the volume of hyperbolic $3$–manifolds, Canad. J. Math. 39 (1987) 1038–1056
  • Y N Minsky, The classification of punctured-torus groups, Ann. of Math. 149 (1999) 559–626
  • S Moroianu, J-M Schlenker, Quasi-Fuchsian manifolds with particles, J. Differential Geom. 83 (2009) 75–129
  • J-P Otal, Le théorème d'hyperbolisation pour les variétés fibrées de dimension $3$, Astérisque 235, Soc. Math. France, Paris (1996)
  • J-M Schlenker, Métriques sur les polyèdres hyperboliques convexes, J. Differential Geom. 48 (1998) 323–405
  • J-M Schlenker, Convex polyhedra in Lorentzian space-forms, Asian J. Math. 5 (2001) 327–363
  • J-M Schlenker, Des immersions isométriques de surfaces aux variétés hyperboliques à bord convexe, Sémin. Théor. Spectr. Géom. 21, Univ. Grenoble I, Saint-Martin-d'Hères (2003) 165–216
  • J-M Schlenker, Hyperbolic manifolds with convex boundary, Invent. Math. 163 (2006) 109–169
  • G Schumacher, S Trapani, Weil–Petersson geometry for families of hyperbolic conical Riemann surfaces, Michigan Math. J. 60 (2011) 3–33
  • M Spivak, A comprehensive introduction to differential geometry, Vol. III, 2nd edition, Publish or Perish, Wilmington, DE (1979)
  • L A Takhtajan, L-P Teo, Liouville action and Weil–Petersson metric on deformation spaces, global Kleinian reciprocity and holography, Comm. Math. Phys. 239 (2003) 183–240
  • L A Takhtajan, P G Zograf, On the uniformization of Riemann surfaces and on the Weil–Petersson metric on the Teichmüller and Schottky spaces, Mat. Sb. 132(174) (1987) 304–321, 444 In Russian; translated in Math. USSR-Sb. 60 (1988) 297–313
  • L A Takhtajan, P G Zograf, Hyperbolic $2$–spheres with conical singularities, accessory parameters and Kähler metrics on $\mathscr{M} \sb {0,n}$, Trans. Amer. Math. Soc. 355 (2003) 1857–1867
  • W P Thurston, The geometry and topology of $3$–manifolds, lecture notes, Princeton University (1978–1981) Available at \setbox0\makeatletter\@url {\unhbox0
  • W P Thurston, Earthquakes in two-dimensional hyperbolic geometry, from: “Low-dimensional topology and Kleinian groups”, (D B A Epstein, editor), London Math. Soc. Lecture Note Ser. 112, Cambridge Univ. Press (1986) 91–112
  • M Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821
  • F Waldhausen, Eine Verallgemeinerung des Schleifensatzes, Topology 6 (1967) 501–504