Geometriae Dedicata

, Volume 134, Issue 1, pp 177–196 | Cite as

Polyhedral hyperbolic metrics on surfaces

Original Paper

Abstract

Let S be a topologically finite surface, and g be a hyperbolic metric on S with a finite number of conical singularities of positive singular curvature, cusps and complete ends of infinite area. We prove that there exists a convex polyhedral surface P in hyperbolic space \({\mathbb{H}^{3}}\) and a group G of isometries of \({\mathbb{H}^{3}}\) such that the induced metric on the quotient P/G is isometric to g. Moreover, the pair (P, G) is unique among a particular class of convex polyhedra.

Keywords

Hyperbolic generalized polyhedra Equivariant polyhedral realization Complete hyperbolic metrics Alexandrov Theorem Hyperbolic–de Sitter space 

Mathematics Subject Classification (2000)

57M50 53C24 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abikoff, W.: The Real Analytic Theory of Teichmüller space, Volume 820 of Lecture Notes in Mathematics. Springer, Berlin (1980)Google Scholar
  2. 2.
    Alexandrov, A.D.: Existence of a convex polyhedron and of a convex surface with a given metric. Rec. Math. [Mat. Sbornik] N.S. 11(53), 15–65 (1942) (Russian)MathSciNetGoogle Scholar
  3. 3.
    Alexandrov, A.D.: Convex Polyhedra. Springer Monographs in Mathematics. Springer-Verlag, Berlin (2005)Google Scholar
  4. 4.
    Andreev, E.M.: Convex polyhedra in Lobačevskiĭ spaces. Mat. Sb. (N.S.) 81(123), 445–478 (1970)MathSciNetGoogle Scholar
  5. 5.
    Andreev, E.M.: Convex polyhedra of finite volume in Lobačevskiĭ space. Mat. Sb. (N.S.) 83(125), 256–260 (1970)MathSciNetGoogle Scholar
  6. 6.
    Bao, X., Bonahon, F.: Hyperideal polyhedra in hyperbolic 3-space. Bull. Soc. Math. France 130(3), 457–491 (2002)MathSciNetMATHGoogle Scholar
  7. 7.
    Bobenko, A.I.: Ivan Izmestiev Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes. Ann. Inst. Fourier (Grenoble) 58(2), 447–505 (2008)MATHMathSciNetGoogle Scholar
  8. 8.
    Buser, P.: Geometry and Spectra of Compact Riemann Surfaces, Volume 106 of Progress in Mathematics. Birkhäuser Boston Inc., Boston (1992)Google Scholar
  9. 9.
    Crapo, H., Whiteley, W.: Statics of frameworks and motions of panel structures, a projective geometric introduction. Struct. Topology 6, 43–82 (1982) (With a French translation)Google Scholar
  10. 10.
    Darboux, G.: Leçons sur la théorie générale des surfaces. III, IV. Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. Éditions Jacques Gabay, Sceaux, 1993. Lignes géodésiques et courbure géodésique. Paramètres différentiels. Déformation des surfaces. [Geodesic lines and geodesic curvature. Differential parameters. Deformation of surfaces], Déformation infiniment petite et représentation sphérique. [Infinitely small deformation and spherical representation], Reprint of the 1894 original (III) and the 1896 original (IV), Cours de Géométrie de la Faculté des Sciences. [Course on Geometry of the Faculty of Science]Google Scholar
  11. 11.
    Fillastre, F.: Fuchsian polyhedra in Lorentzian space-forms. math.DG/0702532 (2007)Google Scholar
  12. 12.
    Fillastre, F.: Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces. Ann. Inst. Fourier (Grenoble) 57(1), 163–195 (2007)MathSciNetMATHGoogle Scholar
  13. 13.
    Fillastre, F., Izmestiev, I.: Hyperbolic cusps with convex polyhedral boundary. arXiv:0708.2666v1 (2007)Google Scholar
  14. 14.
    Fillastre, F., Izmestiev, I.: Parabolic convex polyhedra in de Sitter space. in preparation (2007)Google Scholar
  15. 15.
    Gluck, H.: Almost all simply connected closed surfaces are rigid. In: Geometric Topology (Proc. Conf., Park City, Utah, 1974), Lecture Notes in Math., Vol. 438, pp. 225–239. Springer, Berlin (1975)Google Scholar
  16. 16.
    Hodgson, C.D.: Deduction of Andreev’s theorem from Rivin’s characterization of convex hyperbolic polyhedra. In: Topology ’90 (Columbus, OH, 1990), Volume 1 of Ohio State Univ. Math. Res. Inst. Publ., pp. 185–193. de Gruyter, Berlin (1992)Google Scholar
  17. 17.
    Hulin, D., Troyanov, M.: Sur la courbure des surfaces de Riemann ouvertes. C. R. Acad. Sci. Paris Sér. I Math. 310(4), 203–206 (1990)MathSciNetMATHGoogle Scholar
  18. 18.
    Hulin, D., Troyanov, M.: Prescribing curvature on open surfaces. Math. Ann. 293(2), 277–315 (1992)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Izmestiev, I.: A variational proof of Alexandrov’s convex cap theorem. arXiv.org:math/0703169 (2007)Google Scholar
  20. 20.
    Katok, S.: Fuchsian Groups. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1992)Google Scholar
  21. 21.
    Naber, G.L.: The Geometry of Minkowski Spacetime. Dover Publications Inc., Mineola (2003). An introduction to the mathematics of the special theory of relativity, Reprint of the 1992 editionGoogle Scholar
  22. 22.
    Nag, S.: The Complex Analytic Theory of Teichmüller spaces. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1988). A Wiley-Interscience PublicationGoogle Scholar
  23. 23.
    O’Neill, B.: Semi-Riemannian Geometry, Volume 103 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1983). (With applications to relativity)Google Scholar
  24. 24.
    Rivin, I.: On geometry of convex polyhedra in hyperbolic 3-space. PhD thesis, Princeton University, June 1986Google Scholar
  25. 25.
    Rivin, I.: Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. Math. (2) 139(3), 553–580 (1994)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Rivin, I.: Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space. In: Analysis, Algebra, and Computers in Mathematical Research (Lule  1992), Volume 156 of Lecture Notes in Pure and Appl. Math., pp. 275–291. Dekker, New York (1994)Google Scholar
  27. 27.
    Rivin, I.: A characterization of ideal polyhedra in hyperbolic 3-space. Ann. of Math. (2) 143(1), 51–70 (1996)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Rivin, I., Hodgson, C.D.: A characterization of compact convex polyhedra in hyperbolic 3-space. Invent. Math. 111(1), 77–111 (1993)CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Roeder, R.K.W., Hubbard, J.H., Dunbar, W.D.: Andreev’s theorem on hyperbolic polyhedra. Ann. Inst. Fourier (Grenoble) 57(3), 825–882 (2007)MathSciNetMATHGoogle Scholar
  30. 30.
    Rousset, M.: Sur la rigidité de polyèdres hyperboliques en dimension 3: cas de volume fini, cas hyperidéal, cas fuchsien. Bull. Soc. Math. France 132(2), 233–261 (2004)MathSciNetMATHGoogle Scholar
  31. 31.
    Sabitov, I.Kh.: Local theory of bendings of surfaces [MR1039820 (91c:53004)]. In: Geometry, III, Volume 48 of Encyclopaedia Math. Sci., pp. 179–256. Springer, Berlin (1992)Google Scholar
  32. 32.
    Sauer, R.: Infinitesimale Verbiegungen zueinander projektiver Flächen. Math. Ann. 111(1), 71–82 (1935)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Schlenker, J.-M.: Métriques sur les polyèdres hyperboliques convexes. J. Diff. Geom. 48(2), 323–405 (1998)MathSciNetMATHGoogle Scholar
  34. 34.
    Schlenker, J.-M.: Convex polyhedra in Lorentzian space-forms. Asian J. Math. 5(2), 327–363 (2001)MathSciNetMATHGoogle Scholar
  35. 35.
    Schlenker, J.-M.: Hyperideal polyhedra in hyperbolic manifolds. arXiv:math.GT/0212355 (2003)Google Scholar
  36. 36.
    Schlenker, J.-M.: Hyperbolic manifolds with polyhedral boundary. http://www.picard.ups-tlse.fr/schlenker/texts/ideal.pdf (2004)
  37. 37.
    Schlenker, J.-M.: Hyperbolic manifolds with convex boundary. Invent. Math. 163(1), 109–169 (2006)CrossRefMathSciNetMATHGoogle Scholar
  38. 38.
    Schlenker, J.-M.: Small deformations of polygons and polyhedra. Trans. Am. Math. Soc. 359(5), 2155–2189 (2007)CrossRefMathSciNetMATHGoogle Scholar
  39. 39.
    Thurston, W.P.: The Geometry and Topology of Three-manifolds. Recent Version of the 1980 Notes. http://www.msri.org/gt3m (1997)
  40. 40.
    Zieschang, H., Vogt, E., Coldewey, H.: Surfaces and Planar Discontinuous Groups, Volume 835 of Lecture Notes in Mathematics. Springer, Berlin (1980). (Translated from the German by John Stillwell)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Fribourg, PérollesFribourgSwitzerland

Personalised recommendations