Polyhedral hyperbolic metrics on surfaces
Résumé
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 $\H^3$ and a group $G$ of isometries of $\H^3$ 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.
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