Polyhedral hyperbolic metrics on surfaces

Abstract

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.