Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces
Differential Geometry
2007-05-23 v2 Metric Geometry
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.
Cite
@article{arxiv.math/0605403,
title = {Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces},
author = {François Fillastre},
journal= {arXiv preprint arXiv:math/0605403},
year = {2007}
}
Comments
Some little corrections from the preceding version. To appear in Les Annales de l'Institut Fourier