# Réalisation de métriques sur les surfaces compactes

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 $>1$. 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.
We also prove that any constant curvature metric with conical singularities of negative singular curvature on a compact surface of genus $>1$ can be realised by a unique convex Fuchsian'' polyhedron in a Lorentzian space-form.
Finally we present some possible expansion of these results, and this leads to general statements about realisation of metrics on surfaces.
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https://tel.archives-ouvertes.fr/tel-00122383
Contributor : François Fillastre <>
Submitted on : Friday, January 12, 2007 - 12:40:57 PM
Last modification on : Friday, January 10, 2020 - 9:08:06 PM
Long-term archiving on: : Tuesday, September 21, 2010 - 11:59:51 AM

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• HAL Id : tel-00122383, version 2

### Citation

Francois Fillastre. Réalisation de métriques sur les surfaces compactes. Mathématiques [math]. Université de Neuchâtel; Université Paul Sabatier - Toulouse III, 2006. Français. ⟨tel-00122383v2⟩

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