Skip to main content
SpringerLink
Log in

Minimal surfaces and particles in 3-manifolds

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. 10

  2. Benedetti, R., Bonsante, F.: Canonical wick rotations in 3-dimensional gravity. math.DG/0508485. To appear, Mem. Amer. Math. Soc., (2005)

  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

    MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  5. Besse, A.: Einstein Manifolds. Springer (1987)

  6. Bromberg K. (2004). Rigidity of geometrically finite hyperbolic cone-manifolds. Geom. Dedicata 105: 143–170

    Article  MATH  MathSciNet  Google Scholar 

  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)

  8. Epstein C.L. (1986). The hyperbolic Gauss map and quasiconformal reflections. J. Reine Angew. Math. 372: 96–135

    MATH  MathSciNet  Google Scholar 

  9. Fock, V.: Talk given at “quantum hyperbolic geometry” workshop, AEI-Potsdam (2004)

  10. Gray, A.: Tubes. Addison-Wesley (1990)

  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)

  12. Hodgson C.D. and Kerckhoff S.P. (1998). Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. J. Differential Geom. 48: 1–60

    MATH  MathSciNet  Google Scholar 

  13. Hodge, T.W.S.: Hyperkähler geometry and Teichmüller space. PhD thesis, Imperial College (2005)

  14. Hopf H. (1951). Über Flächen mit einer Relation zwischen den Hauptkrümmungen. Math. Nachr. 4: 232–249

    MATH  MathSciNet  Google Scholar 

  15. Kapovich, M.: Hyperbolic Manifolds and Discrete Groups, vol. 183 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA (2001)

  16. Krasnov K. (2000). Holography and Riemann surfaces. Adv. Theor. Math. Phys. 4(4): 929–979

    MATH  MathSciNet  Google Scholar 

  17. Krasnov K. (2002). Analytic continuation for asymptotically AdS 3D gravity. Classical Quant Gravity 19(9): 2399–2424

    Article  MATH  MathSciNet  Google Scholar 

  18. Labourie F. (1991). Problème de Minkowski et surfaces à courbure constante dans les variétés hyperboliques. Bull. Soc. Math. France 119(3): 307–325

    MATH  MathSciNet  Google Scholar 

  19. Labourie F. (1992). Surfaces convexes dans l’espace hyperbolique et CP1-structures. J. London Math. Soc. II. Ser. 45: 549–565

    Article  MATH  MathSciNet  Google Scholar 

  20. Mess, G.: Lorentz spacetimes of constant curvature. To appear, Geom. Dedicata. Preprint I.H.E.S./M/90/28, 1990.

  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

    Article  MATH  MathSciNet  Google Scholar 

  22. O’Neill, B.: Semi-Riemannian Geometry. Academic Press (1983)

  23. Otal J.-P. (1996). Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3. Astérisque 235: x–159

    Google Scholar 

  24. Rivin I. and Hodgson C.D. (1993). A characterization of compact convex polyhedra in hyperbolic 3-space. Invent. Math. 111: 77–111

    Article  MATH  MathSciNet  Google Scholar 

  25. Rivin, I.: Thesis. PhD thesis, Princeton University (1986)

  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)

  27. Schlenker J.-M. (1998). Métriques sur les polyèdres hyperboliques convexes. J. Differ. Geom. 48(2): 323–405

    MATH  MathSciNet  Google Scholar 

  28. Schlenker J.-M. (2002). Hypersurfaces in H n and the space of its horospheres. Geom. Funct. Anal. 12(2): 395–435

    Article  MATH  MathSciNet  Google Scholar 

  29. Spivak, M.: A Comprehensive Introduction to Geometry, vol. I–V. Publish or Perish (1970–1975).

  30. Taubes C.H. (2004). Minimal surfaces in germs of hyperbolic 3-manifolds. Geom. Topol. Monogr. 7: 69

    Article  MathSciNet  Google Scholar 

  31. Thurston, W.P.: Three-dimensional geometry and topology. Recent version available on http://www.msri.org/publications/books/gt3m/ (1980)

  32. Troyanov M. (1991). Prescribing curvature on compact surfaces with conical singularities. Trans. Am. Math. Soc. 324(2): 793–821

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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)

  35. Witten E. (1989). Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121(3): 351–399

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK

    Kirill Krasnov

  2. Institut de Mathématiques, UMR CNRS 5219, Université Paul Sabatier, 31062, Toulouse Cedex 9, France

    Jean-Marc Schlenker

Authors
  1. Kirill Krasnov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Marc Schlenker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Marc Schlenker.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Krasnov, K., Schlenker, JM. Minimal surfaces and particles in 3-manifolds. Geom Dedicata 126, 187–254 (2007). https://doi.org/10.1007/s10711-007-9132-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-007-9132-1

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation