Skinning maps

Abstract

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.

50 pages, 4 figures. v3. Major revision incorporating referees' comments. To appear in the Duke Mathematical Journal. v2. Cosmetic changes, minor corrections, inclusion of theorem with K. Bromberg

References in corpus (9)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
• Tameness of hyperbolic 3-manifolds
• Hyperbolic Structures on 3-manifolds, III: Deformations of 3-manifolds with incompressible boundary
• Negatively Oriented Ideal Triangulations and a Proof of Thurston's Hyperbolic Dehn Filling Theorem
• The classification of Kleinian surface groups, II: The Ending Lamination Conjecture
• Geometric inflexibility and 3-manifolds that fiber over the circle
• Envelopes of Horospheres and Weingarten Surfaces in Hyperbolic 3-Space

Cited by in corpus (3)

• Local topology in deformation spaces of hyperbolic 3-manifolds
• The Double Limit Theorem and its Legacy
• Thurston's Bounded image theorem