  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.
