Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces

Fillastre, François

Annales de l'Institut Fourier, Tome 57 (2007) no. 1, pp. 163-195.

Un polyèdre fuchsien de l’espace hyperbolique est une surface polyédrale invariante sous l’action d’un groupe fuchsien d’isométries (c.a.d. un groupe d’isométries qui laissent globalement invariante une surface totalement géodésique et sur laquelle il agit de manière cocompacte). La métrique induite sur un polyèdre fuchsien convexe est isométrique à une métrique hyperbolique avec des singularités coniques de courbure singulière positive sur une surface compacte de genre plus grand que un. On démontre que ces métriques sont en fait réalisées par un unique polyèdre fuchsien convexe (modulo les isométries globales). Ce résultat étend un théorème célèbre de A.D. Alexandrov.

A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric to a hyperbolic metric with conical singularities of positive singular curvature on a compact surface of genus greater than one. We prove that these metrics are actually realised by exactly one convex Fuchsian polyhedron (up to global isometries). This extends a famous theorem of A.D. Alexandrov.

Classification : 53C45, 52A55, 52B70, 53C24
Keywords: Fuchsian, convex, polyhedron, hyperbolic, conical singularities, infinitesimal rigidity, Pogorelov map, Alexandrov
Mot clés : Fuchsien, convexe, polyèdre, hyperbolique, singularités coniques, rigidité infinitésimale, application de Pogorelov, Alexandrov

[1] Alexandrov, A. D. Convex polyhedra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005 (Translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky, With comments and bibliography by V. A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov) | MR | Zbl

[2] Berger, Marcel Géométrie. Vol. 5, CEDIC, Paris, 1977 (La sphère pour elle-même, géométrie hyperbolique, l’espace des sphères. [The sphere itself, hyperbolic geometry, the space of spheres]) | MR | Zbl

[3] Busemann, Herbert Convex surfaces, Interscience Tracts in Pure and Applied Mathematics, no. 6, Interscience Publishers, Inc., New York, 1958 | MR | Zbl

[4] Buser, Peter Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, 106, Birkhäuser Boston Inc., Boston, MA, 1992 | MR | Zbl

[5] Fillastre, François Polyhedral realisation of metrics with conical singularities on compact surfaces in Lorentzian space-forms (In preparation)

[6] Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques Riemannian geometry, Universitext, Springer-Verlag, Berlin, 1990 | MR | Zbl

[7] Gromov, Mikhael Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9, Springer-Verlag, Berlin, 1986 | MR | Zbl

[8] Indermitte, C.; Liebling, Th. M.; Troyanov, M.; Clémençon, H. Voronoi diagrams on piecewise flat surfaces and an application to biological growth, Theoret. Comput. Sci., Volume 263 (2001) no. 1-2, pp. 263-274 Combinatorics and computer science (Palaiseau, 1997) | DOI | MR | Zbl

[9] Labourie, François Métriques prescrites sur le bord des variétés hyperboliques de dimension $3$, J. Differential Geom., Volume 35 (1992) no. 3, pp. 609-626 | MR | Zbl

[10] Labourie, François; Schlenker, Jean-Marc Surfaces convexes fuchsiennes dans les espaces lorentziens à courbure constante, Math. Ann., Volume 316 (2000) no. 3, pp. 465-483 | DOI | MR | Zbl

[11] McOwen, Robert C. Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc., Volume 103 (1988) no. 1, pp. 222-224 | DOI | MR | Zbl

[12] Nag, Subhashis The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons Inc., New York, 1988 (A Wiley-Interscience Publication) | MR | Zbl

[13] O’Neill, Barrett Semi-Riemannian geometry, Pure and Applied Mathematics, 103, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983 (With applications to relativity) | MR | Zbl

[14] Pogorelov, A. V. Extrinsic geometry of convex surfaces, American Mathematical Society, Providence, R.I., 1973 (Translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, Vol. 35) | MR | Zbl

[15] Rivin, Igor Extra-large metrics

[16] Rivin, Igor On geometry of convex polyhedra in hyperbolic 3-space, Princeton University (June 1986) (Ph. D. Thesis)

[17] Rivin, Igor; Hodgson, Craig D. A characterization of compact convex polyhedra in hyperbolic $3$-space, Invent. Math., Volume 111 (1993) no. 1, pp. 77-111 | DOI | MR | Zbl

[18] Rousset, Mathias Sur la rigidité de polyèdres hyperboliques en dimension 3: cas de volume fini, cas hyperidéal cas fuchsien, Bull. Soc. Math. France, Volume 132 (2004) no. 2, pp. 233-261 | Numdam | MR | Zbl

[19] Sabitov, I. Kh. Around the proof of the Legendre-Cauchy lemma on convex polygons, Sibirsk. Mat. Zh., Volume 45 (2004) no. 4, pp. 892-919 | MR | Zbl

[20] Schlenker, Jean-Marc Hyperbolic manifolds with polyhedral boundary (arXiv:math.GT/0111136)

[21] Schlenker, Jean-Marc Hyperbolic manifolds with convex boundary, Invent. Math., Volume 163 (2006) no. 1, pp. 109-169 | DOI | MR | Zbl

[22] Spivak, Michael A comprehensive introduction to differential geometry. Vol. V, Publish or Perish Inc., Wilmington, Del., 1979 | MR | Zbl

[23] Thurston, William P. Shapes of polyhedra and triangulations of the sphere, The Epstein birthday schrift (Geom. Topol. Monogr.), Volume 1, Geom. Topol. Publ., Coventry, 1998, p. 511-549 (electronic) | MR | Zbl

[24] Troyanov, Marc Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc., Volume 324 (1991) no. 2, pp. 793-821 | DOI | MR | Zbl

[25] Zieschang, Heiner; Vogt, Elmar; Coldewey, Hans-Dieter Surfaces and planar discontinuous groups, Lecture Notes in Mathematics, 835, Springer, Berlin, 1980 (Translated from the German by John Stillwell) | MR | Zbl