Thick-skinned 3-manifolds

TL;DR

This paper establishes a relationship between the collar depth of the boundary of a hyperbolic 3-manifold and the diameter of its skinning map, showing exponential decay as the collar deepens.

Contribution

It provides a new exponential bound on the skinning map diameter based on boundary collar depth in hyperbolic 3-manifolds.

Findings

Diameter of skinning map decreases exponentially with boundary collar depth

Bound depends only on boundary genus and injectivity radius

Results apply to compact hyperbolic 3-manifolds with geodesic boundary

Abstract

We show that if the totally geodesic boundary of a compact hyperbolic 3-manifold M has a large collar of depth d, then the diameter of the skinning map of M is no more than A exp(-d) for some A depending only on the genus and injectivity radius of the boundary of M.