Geometriae Dedicata

, Volume 126, Issue 1, pp 187–254 | Cite as

Minimal surfaces and particles in 3-manifolds

Original Paper


We consider 3-dimensional anti-de Sitter manifolds with conical singularities along time-like lines, which is what in the physics literature is known as manifolds with particles. We show that the space of such cone-manifolds is parametrized by the cotangent bundle of Teichmüller space, and that moreover such cone-manifolds have a canonical foliation by space-like surfaces. We extend these results to de Sitter and Minkowski cone-manifolds, as well as to some related “quasifuchsian” hyperbolic manifolds with conical singularities along infinite lines, in this later case under the condition that they contain a minimal surface with principal curvatures less than 1. In the hyperbolic case the space of such cone-manifolds turns out to be parametrized by an open subset in the cotangent bundle of Teichmüller space. For all settings, the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichmüller space is the same as the 3-dimensional gravity one. The proofs use minimal (or maximal, or CMC) surfaces, along with some results of Mess on AdS manifolds, which are recovered here in a different way, using differential-geometric methods and a result of Labourie on some mappings between hyperbolic surfaces, that allows an extension to cone-manifolds.


Anti-de sitter Cone-manifolds Quasi-fuchsian Hyperbolic Minimal surfaces 

Mathematics Subject Classifications (2000)

Primary 53C42 Secondary 53C50 Secondary 53C80 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahlfors, L.V.: Lectures on Quasiconformal Mappings. D. Van Nostrand Co., Inc., Toronto, Ont (1966). Manuscript prepared with the assistance of Clifford J. Earle, Jr. Van Nostrand Mathematical Studies, No. 10Google Scholar
  2. 2.
    Benedetti, R., Bonsante, F.: Canonical wick rotations in 3-dimensional gravity. math.DG/0508485. To appear, Mem. Amer. Math. Soc., (2005)Google Scholar
  3. 3.
    Barbot T., Béguin F. and Zeghib A. (2003). Feuilletages des espaces temps globalement hyperboliques par des hypersurfaces à courbure moyenne constante. C. R. Math. Acad. Sci. Paris 336(3): 245–250 MATHMathSciNetGoogle Scholar
  4. 4.
    Bérard P., do Carmo M. and Santos W. (1997). The index of constant mean curvature surfaces in hyperbolic 3-space. Math. Z. 224(2): 313–326 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Besse, A.: Einstein Manifolds. Springer (1987)Google Scholar
  6. 6.
    Bromberg K. (2004). Rigidity of geometrically finite hyperbolic cone-manifolds. Geom. Dedicata 105: 143–170 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Barbot, T., Zeghib, A.: Group actions on Lorentz spaces, mathematical aspects: a survey. In: The Einstein Equations and the Large Scale Behavior of Gravitational Fields, pp. 401–439. Birkhäuser, Basel (2004)Google Scholar
  8. 8.
    Epstein C.L. (1986). The hyperbolic Gauss map and quasiconformal reflections. J. Reine Angew. Math. 372: 96–135 MATHMathSciNetGoogle Scholar
  9. 9.
    Fock, V.: Talk given at “quantum hyperbolic geometry” workshop, AEI-Potsdam (2004)Google Scholar
  10. 10.
    Gray, A.: Tubes. Addison-Wesley (1990)Google Scholar
  11. 11.
    Gromov, M.: Hyperbolic manifolds (according to Thurston and Jø rgensen). In: Bourbaki Seminar, vol. 1979/80, volume 842 of Lecture Notes in Math., pp. 40–53. Springer, Berlin (1981)Google Scholar
  12. 12.
    Hodgson C.D. and Kerckhoff S.P. (1998). Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. J. Differential Geom. 48: 1–60 MATHMathSciNetGoogle Scholar
  13. 13.
    Hodge, T.W.S.: Hyperkähler geometry and Teichmüller space. PhD thesis, Imperial College (2005)Google Scholar
  14. 14.
    Hopf H. (1951). Über Flächen mit einer Relation zwischen den Hauptkrümmungen. Math. Nachr. 4: 232–249 MATHMathSciNetGoogle Scholar
  15. 15.
    Kapovich, M.: Hyperbolic Manifolds and Discrete Groups, vol. 183 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA (2001)Google Scholar
  16. 16.
    Krasnov K. (2000). Holography and Riemann surfaces. Adv. Theor. Math. Phys. 4(4): 929–979 MATHMathSciNetGoogle Scholar
  17. 17.
    Krasnov K. (2002). Analytic continuation for asymptotically AdS 3D gravity. Classical Quant Gravity 19(9): 2399–2424 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Labourie F. (1991). Problème de Minkowski et surfaces à courbure constante dans les variétés hyperboliques. Bull. Soc. Math. France 119(3): 307–325 MATHMathSciNetGoogle Scholar
  19. 19.
    Labourie F. (1992). Surfaces convexes dans l’espace hyperbolique et CP1-structures. J. London Math. Soc. II. Ser. 45: 549–565 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Mess, G.: Lorentz spacetimes of constant curvature. To appear, Geom. Dedicata. Preprint I.H.E.S./M/90/28, 1990.Google Scholar
  21. 21.
    Moncrief V. (1989). Reduction of the Einstein equations in 2 + 1 dimensions to a Hamiltonian system over Teichmüller space. J. Math. Phys. 30(12): 2907–2914 MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    O’Neill, B.: Semi-Riemannian Geometry. Academic Press (1983)Google Scholar
  23. 23.
    Otal J.-P. (1996). Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3. Astérisque 235: x–159 Google Scholar
  24. 24.
    Rivin I. and Hodgson C.D. (1993). A characterization of compact convex polyhedra in hyperbolic 3-space. Invent. Math. 111: 77–111 MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Rivin, I.: Thesis. PhD thesis, Princeton University (1986)Google Scholar
  26. 26.
    Rubinstein, J.H.: Minimal surfaces in geometric 3-manifolds. In: Global Theory of Minimal Surfaces, volume 2 of Clay Math. Proc., pp. 725–746. Amer. Math. Soc., Providence, RI (2005)Google Scholar
  27. 27.
    Schlenker J.-M. (1998). Métriques sur les polyèdres hyperboliques convexes. J. Differ. Geom. 48(2): 323–405 MATHMathSciNetGoogle Scholar
  28. 28.
    Schlenker J.-M. (2002). Hypersurfaces in H n and the space of its horospheres. Geom. Funct. Anal. 12(2): 395–435 MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Spivak, M.: A Comprehensive Introduction to Geometry, vol. I–V. Publish or Perish (1970–1975).Google Scholar
  30. 30.
    Taubes C.H. (2004). Minimal surfaces in germs of hyperbolic 3-manifolds. Geom. Topol. Monogr. 7: 69 CrossRefMathSciNetGoogle Scholar
  31. 31.
    Thurston, W.P.: Three-dimensional geometry and topology. Recent version available on (1980)Google Scholar
  32. 32.
    Troyanov M. (1991). Prescribing curvature on compact surfaces with conical singularities. Trans. Am. Math. Soc. 324(2): 793–821 MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Takhtajan L.A. and Teo L.P. (2003). Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography. Comm. Math. Phys. 239(1–2): 183–240 MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Uhlenbeck, K.K.: Closed minimal surfaces in hyperbolic 3-manifolds. In: Seminar on Minimal Submanifolds, volume 103 of Ann. of Math. Stud., pp. 147–168. Princeton Univ. Press, Princeton, NJ (1983)Google Scholar
  35. 35.
    Witten E. (1989). Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121(3): 351–399MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK
  2. 2.Institut de Mathématiques, UMR CNRS 5219Université Paul SabatierToulouse Cedex 9France

Personalised recommendations