Des immersions isométriques de surfaces aux variétés hyperboliques à bord convexe

Jean-Marc Schlenker

Séminaire de théorie spectrale et géométrie (2002-2003)

  • Volume: 21, page 165-216
  • ISSN: 1624-5458

How to cite


Schlenker, Jean-Marc. "Des immersions isométriques de surfaces aux variétés hyperboliques à bord convexe." Séminaire de théorie spectrale et géométrie 21 (2002-2003): 165-216. <>.

author = {Schlenker, Jean-Marc},
journal = {Séminaire de théorie spectrale et géométrie},
keywords = {anti-de Sitter manifold; hyperbolic manifold with convex boundary; isometric immersion},
language = {fre},
pages = {165-216},
publisher = {Institut Fourier},
title = {Des immersions isométriques de surfaces aux variétés hyperboliques à bord convexe},
url = {},
volume = {21},
year = {2002-2003},

AU - Schlenker, Jean-Marc
TI - Des immersions isométriques de surfaces aux variétés hyperboliques à bord convexe
JO - Séminaire de théorie spectrale et géométrie
PY - 2002-2003
PB - Institut Fourier
VL - 21
SP - 165
EP - 216
LA - fre
KW - anti-de Sitter manifold; hyperbolic manifold with convex boundary; isometric immersion
UR -
ER -


  1. [Ale58a] A.D. ALEKSANDROV, Vestnik Leningrad Univ., 13(1), 1958. 
  2. [Ale58b] A.D. ALEXANDROW, Konvexe polyeder. Akademie-Verlag, Berlin, 1958. MR92989
  3. [And70] E.M. ANDREEV, Convex polyhedra in Lobacevskii space. Mat. Sb. (N.S.), 81 (123):445-478, 1970. Zbl0194.23202MR259734
  4. [And71] E.M. ANDREEV, On convex polyhedra of finite volume in Lobacevskii space. Math. USSR Sbornik, 12 (3):225-259, 1971. Zbl0252.52005MR273510
  5. [BB02] X. BAO and F. BONAHON, Hyperideal polyhedra in hyperbolic 3 space. Preprint available at Bull. Soc. Math. France, to appear, 2002. Zbl1033.52009MR1943885
  6. [BC03] M. BRIDGEMAN and R.-D. CANARY, From the boundary of the convex core to the conformal boun dary. Geom. Dedicata, 96:211-240, 2003. Zbl1083.57024MR1956842
  7. [BO01] F. BONAHON and J.-P. OTAL, Laminations mesurées de plissage des variétés hyperboliques de dimension 3., 2001. Zbl1083.57023
  8. [Bon96] F. BONAHON, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form. Ann. Fac. Sci. Toulouse Math. (6), 5(2):233-297, 1996. Zbl0880.57005MR1413855
  9. [Bon01] F. BONAHON, Geodesic laminations on surfaces. In Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998), volume 269 of Contemp. Math., pages 1-37. Amer. Math. Soc, Providence, RI, 2001. Zbl0996.53029MR1810534
  10. [Brä92] W. BRÄGGER, Kreispackungen und Triangulierungen. Enseign. Math. (2), 38(3-4):201-217, 1992. Zbl0770.52007MR1189006
  11. [Cal61] E. CALABI, On compact Riemannian manifolds with constant curvature, I. AMS proceedings of Symposia in Pure Math, 3:155-180, 1961. Zbl0129.14102MR133787
  12. [Cau13] A.L. CAUCHY, Sur les polygones et polyèdres, second mémoire. Journal de l'Ecole Polytechnique, 19:87-98, 1813. 
  13. [CdV91] Y. COLIN DE VERDIÈRE, Un principe variationnel pour les empilements de cercles. Invent. Math., 104(3):655-669, 1991. Zbl0745.52010MR1106755
  14. [EM86] D.B.A. EPSTEIN and A. MARDEN, Convex hulls in hyperbolic spaces, a theorem of Sullivan, and measured pleated surfaces. In D.B.A. Epstein, editor, Analytical and geometric aspects of hyperbolic space, volume 111 of L.M.S. Lecture Note Series. Cambridge University Press, 1986. Zbl0612.57010
  15. [Fil] F. FILLASTRE, Surfaces convexes fuchsiennes. In preparation. 
  16. [Gro85] M. GROMOV, Pseudo-holomorphic curves in symplectic manifolds. Inventiones Math., 82:307-347, 1985. Zbl0592.53025
  17. [Gro86] M. GROMOV, Partial Differential Relations. Springer, 1986. Zbl0651.53001MR864505
  18. [HK98] C.D. HODGSON and S.P. KERCKHOFF, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. J. Differential Geom., 48:1-60, 1998. Zbl0919.57009MR1622600
  19. [Isk00] I. ISKHAKOV, On hyperbolic surface tessellations and equivariant spacelike convex polyhedral surfaces in Minkowski space. PhD thesis, Ohio State University, 2000. 
  20. [Koe36] P. KOEBE, Kontaktprobleme der konformen Abbildung. Abh. Sächs. Akad. Wiss. Leipzig Math.-Natur. Kl., 88:141-164, 1936. JFM62.1217.04
  21. [Lab89] F. LABOURIE, Immersions isométriques elliptiques et courbes pseudo-holomorphes. J. Differential Geom., 30:395-424, 1989. Zbl0682.53063MR1010166
  22. [Lab92] F. LABOURIE, Métriques prescrites sur le bord des variétés hyperboliques de dimension 3. J. Differential Geom., 35:609-626, 1992. Zbl0768.53017MR1163450
  23. [Lab94] F. LABOURIE, Exemples de courbes pseudo-holomorphes en géométrie riemannienne. In Audin and Lafontaine, editors, Pseudo-Holomorphic Curves in Symplectic Geometry, pages 251-270. Birkhauser, 1994. MR1274933
  24. [Lab97] F. LABOURIE, Problèmes de Monge-Ampère, courbes holomorphes et laminations. Geom. Funct. Anal., 7(3):496-534, 1997. Zbl0885.32013MR1466336
  25. [Lab 00] F. LABOURIE, Un lemme de Morse pour les surfaces convexes. Invent. Math., 141 (2):239-297, 2000. Zbl0981.52002MR1775215
  26. [LecO2] C. LECUIRE, Plissage des variétés hyperboliques de dimension 3. Preprint 301, UMPA, ENS Lyon, 2002. MR2207784
  27. [LegII] A.-M. LEGENDRE, Eléments de géométrie. Paris, 1793 (an II). Première édition, note XII, pp. 321-334. 
  28. [LS00] F. LABOURIE and J.-M. SCHLENKER, Surfaces convexes fuchsiennes dans les espaces lorentziens à courbure constante. Math. Annalen, 316:465-483, 2000. Zbl0968.53047MR1752780
  29. [Mes90] G. MESS, Lorentz spacetimes of constant curvature. Preprint I.H.E.S./M/90/28, 1990. MR2328921
  30. [Mou02] G. MOUSSONG, Personal communication. July 2002. 
  31. [Pog73] A.V. POGORELOV, Extrinsic Geometry of Convex Surfaces. American Mathematical Society, 1973. Translations of Mathematical Monographs. Vol.35. Zbl0311.53067MR346714
  32. [RH93] I. RIVIN and C. D. HODGSON, A characterization of compact convex polyhedra in hyperbolic 3-space. Invent.Math., 111:77-111, 1993. Zbl0784.52013MR1193599
  33. [Riv86] I. RIVIN, Thesis. PhD thesis, Princeton University, 1986. 
  34. [Riv92] I. RIVIN, Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space. In M. Gyllenberg and L. E. Persson, editors, Analysis, Algebra, and Computers in Mathematical Research, pages 275-292. Marcel Dekker, 1992. (Proc. of the 21 st Nordic Congress of Mathematicians). Zbl0823.52008MR1280952
  35. [Riv94] I. RIVIN, Euclidean structures on simplicial surfaces and hyperbolic volume. Annals of Math., 139:553-580, 1994. Zbl0823.52009MR1283870
  36. [Riv96] I. RIVIN, A characterization of ideal polyhedra in hyperbolic 3-space. Annals of Math., 143:51-70, 1996. Zbl0874.52006MR1370757
  37. [Rou02] M. ROUSSET, Sur la rigidité de polyèdres hyperboliques en dimension 3 : cas de volume fini, cas hyperidéal, cas fuchsien. math.GT/0211280; Bull. Soc. Math. France, to appear, 2002. Zbl1061.52007MR2075567
  38. [Sch96] J.-M. SCHLENKER, Surfaces convexes dans des espaces lorentziens à courbure constante. Commun. Anal, and Geom., 4:285-331, 1996. Zbl0864.53016MR1393565
  39. [Sch98a] J.-M. SCHLENKER, Métriques sur les polyèdres hyperboliques convexes. J. Differential Geom., 48 (2) :323-405, 1998. Zbl0912.52008MR1630178
  40. [Sch98b] J.-M. SCHLENKER, Représentations de surfaces hyperboliques complètes dans H3. Annales de l'Institut Fourier, 48 (3):837-860, 1998. Zbl0916.53029MR1644101
  41. [Sch00] J.-M. SCHLENKER, Dihedral angles of convex polyhedra. Discrete Comput. Geom., 23 (3):409-417, 2000. Zbl0951.52006MR1744513
  42. [Sch01a] textsc J.-M. Schlenker, Convex polyhedra in Lorentzian space-forms. Asian J. of Math., 5: 327-364, 2001. Zbl1020.53046MR1868937
  43. [Sch01b] J.-M. SCHLENKER, Einstein manifolds with convex boundaries. Commentarii Math. Helvetici, 76(l) :l-28, 2001. Zbl1036.53031MR1819659
  44. [Sch01c] J.-M. SCHLENKER, Hyperbolic manifolds with polyhedral boundary. math. GT/0111136, available at, 2001. 
  45. [Sch02a] J.-M. SCHLENKER, Hyperbolic manifolds with convex boundary. preprint, math. DG/0205305, available at, 2002. MR2208419
  46. [Sch02b] J.-M. SCHLENKER, Hyperideal polyhedra in hyperbolic manifolds. Preprint math. GT/0212355., 2002. 
  47. [Sch03] J.-M. SCHLENKER, Hyperbolic manifolds with constant curvature boundaries. In preparation, 2003. 
  48. [Sul79] D. SULLIVAN, The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ Math., (50):171-202, 1979. Zbl0439.30034MR556586
  49. [Thu97] W.-P. THURSTON, Three-dimensional geometry and topology. Recent version of the 1980 notes., 1997. Zbl0873.57001
  50. [Tro91] M. TROYANOVPrescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc, 324(2):793-821, 1991. Zbl0724.53023MR1005085
  51. [Wei60] A. WEIL, On discrete subgroups of Lie groups. Annals of Math., 72(l):369-384, 1960. Zbl0131.26602MR137792
  52. [Wei02] H. WEISS, Local rigidity of 3-dimensional cone-manifolds. PhD thesis, available at http://www.mathematik.uni-muenchen. de/personen/weiss.html, 2002. 

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.