4/12/2007

Richard Kent
IV

Skinning Maps

The skinning map is a holomorphic self map of the Teichmuller space that arises naturally in Thurston's proof of Geometrization for Haken manifolds. Thurston's Bounded Image Theorem says that the skinning map of a hyperbolic manifold with totally geodesic boundary has bounded image. Yair Minsky has asked if bounds on the diameter may be obtained given topological information about the manifold.

I'll discuss "sharp" upper and lower bounds that only depend on the volume of the metric with totally geodesic boundary. These follow from: a filling theorem, which says that skinning maps converge uniformly as higher Dehn fillings are performed; the Bounded Image Theorem, together with a finiteness theorem of Jorgensen; and a theorem (joint with D. Dumas) that skinning maps are never constant. The theorem is sharp in the sense that the upper and lower bounds tend to infinity and zero, respectively, as the volume grows.