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Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces

François Fillastre
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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.

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François Fillastre. 2006-08-31. Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces. https://arxiv.org/abs/math/0605403

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