Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces

François Fillastre (1, 2)

1 UNINE - Institut de Mathématiques
2 LEP - Laboratoire Émile Picard

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.

Mots clés: Fuchsian; Convex; Polyhedron; Hyperbolic; Conical singularity; Infinitesimal Rigidity; Pogorelov Map; Alexandrov

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]

François Fillastre. Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces. Annales de l'Institut Fourier, 2007, 57 (1), pp.163--195. ⟨hal-00068960v2⟩