Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces

Details Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces

2006-05-15

arxiv.org/abs/math/0605403v2

Ann. Inst. Fourier (Grenoble) 57 (03/2007) 163--195

Mathematics

Differential Geometry

Some little corrections from the preceding version. To appear in Les Annales de l'Institut Fourier

Scientific paper

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.

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