Geometriae Dedicata

, Volume 154, Issue 1, pp 47–80 | Cite as

Moduli of flat conformal structures of hyperbolic type

  • Graham Smith
Original Paper


To each flat conformal structure (FCS) of hyperbolic type in the sense of Kulkarni-Pinkall, we associate, for all \({\theta\in[(n-1)\pi/2,n\pi/2[}\) and for all r > tan(θ/n) a unique immersed hypersurface \({\Sigma_{r,\theta}=(M,i_{r,\theta})}\) in \({\mathbb{H}^{n+1}}\) of constant θ-special Lagrangian curvature equal to r. We show that these hypersurfaces smoothly approximate the boundary of the canonical hyperbolic end associated to the FCS by Kulkarni and Pinkall and thus obtain results concerning the continuous dependance of the hyperbolic end and of the Kulkarni-Pinkall metric on the flat conformal structure.


Möbius manifolds Flat conformal structures Special Lagrangian Immersions Foliations 

Mathematics Subject Classification (2000)

53A30 (35J60, 53C21, 53C42, 58J05) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andersson L., Barbot T., Béguin F., Zeghib A.: Cosmological time versus CMC time in spacetimes of constant curvatureGoogle Scholar
  2. 2.
    Aubin T.: Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Die Grundlehren der Mathematischen Wissenschaften, 252. Springer, New York (1982)Google Scholar
  3. 3.
    Caffarelli L., Nirenberg L., Spruck J.: The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Comm. Pure Appl. Math. 37(3), 369–402 (1984)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Chen S., Greenberg L.: Hyperbolic Spaces, Contribution to Analysis, pp. 49–87. Academic Press, New York (1974)Google Scholar
  5. 5.
    Epstein, D.B.A., Marden, A.: Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, In: Fundamentals of Hyperbolic Geometry: Selected Expositions, London Mathematical Society Lecture Note Series, vol. 328. Cambridge University Press, Cambridge (2006)Google Scholar
  6. 6.
    Fried D.: Closed similarity manifolds. Comment. Math. Helvet. 55, 576–582 (1980)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Guan B., Spruck J.: The existence of hypersurfaces of constant Gauss curvature with prescribed boundary. J. Diff. Geom. 62(2), 259–287 (2002)MathSciNetMATHGoogle Scholar
  8. 8.
    Harvey R., Lawson H.B. Jr.: Calibrated geometries. Acta. Math. 148, 47–157 (1982)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kamishima T.: Conformally flat manifolds whose development maps are not surjective. Trans. Am. Math. Soc. 294(2), 607–623 (1986)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kamishima, Y., Tan, S.: Deformation spaces on geometric structures, In: Aspects of Low-Dimensional Manifolds. Adv. Stud. Pure Math. 20. Kinokuniya, Tokyo (1992)Google Scholar
  11. 11.
    Kapovich, M.: Deformation spaces of flat conformal structures. In: Proceedings of the Second Soviet-Japan Joint Symposium of Topology (Khabarovsk, 1989), Questions Answers Gen. Topology 8(1), 253–264 (1990)Google Scholar
  12. 12.
    Krasnov, K., Schlenker, J.M.: On the renormalized volume of hyperbolic 3-manifolds, math.DG/0607081Google Scholar
  13. 13.
    Kulkarni R.S., Pinkall U.: A canonical metric for Möbius structures and its applications. Math. Zeitschrift 216(1), 89–129 (1994)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Labourie F.: Un lemme de Morse pour les surfaces convexes. Invent. Math. 141, 239–297 (2000)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Labourie F.: Problème de Minkowski et surfaces à courbure constante dans les variétés hyperboliques. Bull. Soc. Math. Fr. 119, 307–325 (1991)MathSciNetMATHGoogle Scholar
  16. 16.
    Mazzeo, R., Pacard, F.: Constant curvature foliations in asymptotically hyperbolic spacesGoogle Scholar
  17. 17.
    Smith, G.: Special Legendrian Structures and Weingarten Problems, Preprint, Orsay (2005)Google Scholar
  18. 18.
    Smith, G.: The non-linear Dirichlet problem in Hadamard manifolds, arXiv:0908.3590Google Scholar
  19. 19.
    Smith, G.: A Brief Note on Foliations of Constant Gaussian Curvature, arXiv:0802.2202Google Scholar
  20. 20.
    Thurston W.: Three-Dimensional Geometry and Topology, Princeton Mathematical Series, 35. Princeton University Press, Princeton (1997)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Centre de Recerca Matemàtica, Facultat de Ciències, Edifici CUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain

Personalised recommendations