Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces

François Fillastre[1]

  • [1] Université de Neuchâtel Institut de Mathématiques rue Emile-Argand 11, cp 158 2009 Neuchâtel (Switzerland) et Université Paul Sabatier Laboratoire Emile Picard 118 route de Narbonne 31062 Toulouse Cedex 4 (France)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 1, page 163-195
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Fillastre, François. "Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces." Annales de l’institut Fourier 57.1 (2007): 163-195. <http://eudml.org/doc/10217>.

@article{Fillastre2007,
abstract = {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.},
affiliation = {Université de Neuchâtel Institut de Mathématiques rue Emile-Argand 11, cp 158 2009 Neuchâtel (Switzerland) et Université Paul Sabatier Laboratoire Emile Picard 118 route de Narbonne 31062 Toulouse Cedex 4 (France)},
author = {Fillastre, François},
journal = {Annales de l’institut Fourier},
keywords = {Fuchsian; convex; polyhedron; hyperbolic; conical singularities; infinitesimal rigidity; Pogorelov map; Alexandrov; infinitesimal},
language = {eng},
number = {1},
pages = {163-195},
publisher = {Association des Annales de l’institut Fourier},
title = {Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces},
url = {http://eudml.org/doc/10217},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Fillastre, François
TI - Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 1
SP - 163
EP - 195
AB - 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.
LA - eng
KW - Fuchsian; convex; polyhedron; hyperbolic; conical singularities; infinitesimal rigidity; Pogorelov map; Alexandrov; infinitesimal
UR - http://eudml.org/doc/10217
ER -

References

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