Hyperideal polyhedra in hyperbolic manifolds

Details Hyperideal polyhedra in hyperbolic manifolds Hyperideal polyhedra in hyperbolic manifolds

2002-12-27

arxiv.org/abs/math/0212355v2

Mathematics

Geometric Topology

Updated versions will be posted on http://picard.ups-tlse.fr/~schlenker v2: no major change but many corrections

Scientific paper

Let $(M, \dr M)$ be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that $\dr M$ looks locally like a hyperideal polyhedron, and we characterize the possible dihedral angles. We find as special cases the results of Bao and Bonahon on hyperideal polyhedra, and those of Rousset on fuchsian hyperideal polyhedra. Our results can also be stated in terms of circle configurations on $\dr M$, they provide an extension of the Koebe theorem on circle packings. The proof uses some elementary properties of the hyperbolic volume, in particular the Schl\"afli formula and the fact that the volume of (truncated) hyperideal simplices is a concave function of the dihedral angles.

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