Geometriae Dedicata

, Volume 134, Issue 1, pp 177–196 | Cite as

Polyhedral hyperbolic metrics on surfaces

Original Paper


Let S be a topologically finite surface, and g be a hyperbolic metric on S with a finite number of conical singularities of positive singular curvature, cusps and complete ends of infinite area. We prove that there exists a convex polyhedral surface P in hyperbolic space \({\mathbb{H}^{3}}\) and a group G of isometries of \({\mathbb{H}^{3}}\) such that the induced metric on the quotient P/G is isometric to g. Moreover, the pair (P, G) is unique among a particular class of convex polyhedra.


Hyperbolic generalized polyhedra Equivariant polyhedral realization Complete hyperbolic metrics Alexandrov Theorem Hyperbolic–de Sitter space 

Mathematics Subject Classification (2000)

57M50 53C24 


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© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Fribourg, PérollesFribourgSwitzerland

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