Inventiones mathematicae

, Volume 111, Issue 1, pp 77-111

First online:

A characterization of compact convex polyhedra in hyperbolic 3-space

  • Craig D. HodgsonAffiliated withMathematics Department, University of Melbourne
  • , Igor RivinAffiliated withNEC Research Institute and Mathematics Department, Princeton University

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In this paper we study the extrinsic geometry of convex polyhedral surfaces in three-dimensional hyperbolic spaceH 3. We obtain a number of new uniqueness results, and also obtain a characterization of the shapes of convex polyhedra inH 3 in terms of a generalized Gauss map. This characterization greatly generalizes Andre'ev's Theorem.