Skinning maps

Details Skinning maps Skinning maps

2006-12-19

arxiv.org/abs/math/0612566v3

Mathematics

Geometric Topology

50 pages, 4 figures. v3. Major revision incorporating referees' comments. To appear in the Duke Mathematical Journal. v2. Cosm

Scientific paper

Let M be a hyperbolic 3-manifold with nonempty totally geodesic boundary. We prove that there are upper and lower bounds on the diameter of the skinning map of M that depend only on the volume of the hyperbolic structure with totally geodesic boundary, answering a question of Y. Minsky. This is proven via a filling theorem, which states that as one performs higher and higher Dehn fillings, the skinning maps converge uniformly on all of Teichmuller space. We also exhibit manifolds with totally geodesic boundaries whose skinning maps have diameter tending to infinity, as well as manifolds whose skinning maps have diameter tending to zero (the latter are due to K. Bromberg and the author). In the final section, we give a proof of Thurston's Bounded Image Theorem.

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