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.
@article{AIF_2007__57_1_163_0,
author = {Fillastre, Fran\c cois},
title = {Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces},
journal = {Annales de l'Institut Fourier},
volume = {57},
year = {2007},
pages = {163-195},
doi = {10.5802/aif.2255},
zbl = {1123.53033},
mrnumber = {2313089},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2007__57_1_163_0}
}
Fillastre, François. Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces. Annales de l'Institut Fourier, Tome 57 (2007) pp. 163-195. doi : 10.5802/aif.2255. http://gdmltest.u-ga.fr/item/AIF_2007__57_1_163_0/
[1] Convex polyhedra, Springer-Verlag, Berlin, Springer Monographs in Mathematics (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 2127379 | Zbl 1067.52011
[2] 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 536874 | Zbl 0423.51003
[3] Convex surfaces, Interscience Publishers, Inc., New York, Interscience Tracts in Pure and Applied Mathematics, no. 6 (1958) | MR 105155 | Zbl 0196.55101
[4] Geometry and spectra of compact Riemann surfaces, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 106 (1992) | MR 1183224 | Zbl 0770.53001
[5] Polyhedral realisation of metrics with conical singularities on compact surfaces in Lorentzian space-forms (In preparation)
[6] Riemannian geometry, Springer-Verlag, Berlin, Universitext (1990) | MR 1083149 | Zbl 0716.53001
[7] Partial differential relations, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 9 (1986) | MR 864505 | Zbl 0651.53001
[8] Voronoi diagrams on piecewise flat surfaces and an application to biological growth, Theoret. Comput. Sci., Tome 263 (2001) no. 1-2, pp. 263-274 (Combinatorics and computer science (Palaiseau, 1997)) | Article | MR 1846934 | Zbl 0974.68222
[9] Métriques prescrites sur le bord des variétés hyperboliques de dimension , J. Differential Geom., Tome 35 (1992) no. 3, pp. 609-626 | MR 1163450 | Zbl 0768.53017
[10] Surfaces convexes fuchsiennes dans les espaces lorentziens à courbure constante, Math. Ann., Tome 316 (2000) no. 3, pp. 465-483 | Article | MR 1752780 | Zbl 0968.53047
[11] Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc., Tome 103 (1988) no. 1, pp. 222-224 | Article | MR 938672 | Zbl 0657.30033
[12] The complex analytic theory of Teichmüller spaces, John Wiley & Sons Inc., New York, Canadian Mathematical Society Series of Monographs and Advanced Texts (1988) (A Wiley-Interscience Publication) | MR 927291 | Zbl 0667.30040
[13] Semi-Riemannian geometry, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, Pure and Applied Mathematics, Tome 103 (1983) (With applications to relativity) | MR 719023 | Zbl 0531.53051
[14] 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 346714 | Zbl 0311.53067
[15] Extra-large metrics
[16] On geometry of convex polyhedra in hyperbolic 3-space, Princeton University (June 1986) (Ph. D. Thesis)
[17] A characterization of compact convex polyhedra in hyperbolic -space, Invent. Math., Tome 111 (1993) no. 1, pp. 77-111 | Article | MR 1193599 | Zbl 0784.52013
[18] Sur la rigidité de polyèdres hyperboliques en dimension 3: cas de volume fini, cas hyperidéal cas fuchsien, Bull. Soc. Math. France, Tome 132 (2004) no. 2, pp. 233-261 | Numdam | MR 2075567 | Zbl 1061.52007
[19] Around the proof of the Legendre-Cauchy lemma on convex polygons, Sibirsk. Mat. Zh., Tome 45 (2004) no. 4, pp. 892-919 | MR 2091654 | Zbl 1051.52015
[20] Hyperbolic manifolds with polyhedral boundary (arXiv:math.GT/0111136)
[21] Hyperbolic manifolds with convex boundary, Invent. Math., Tome 163 (2006) no. 1, pp. 109-169 | Article | MR 2208419 | Zbl 05000067
[22] A comprehensive introduction to differential geometry. Vol. V, Publish or Perish Inc., Wilmington, Del. (1979) | MR 394453 | Zbl 0439.53005
[23] Shapes of polyhedra and triangulations of the sphere, The Epstein birthday schrift, Geom. Topol. Publ., Coventry (Geom. Topol. Monogr.) Tome 1 (1998), p. 511-549 (electronic) | MR 1668340 | Zbl 0931.57010
[24] Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc., Tome 324 (1991) no. 2, pp. 793-821 | Article | MR 1005085 | Zbl 0724.53023
[25] Surfaces and planar discontinuous groups, Springer, Berlin, Lecture Notes in Mathematics, Tome 835 (1980) (Translated from the German by John Stillwell) | MR 606743 | Zbl 0438.57001