A family of non-injective skinning maps with critical points

TL;DR

This paper constructs examples of non-injective skinning maps with critical points for certain 3-manifolds, revealing complex behaviors in the associated Teichmüller spaces and extending known phenomena to higher genus surfaces.

Contribution

It demonstrates the existence of non-injective skinning maps with critical points for a family of 3-manifolds and their finite covers, expanding understanding of skinning map dynamics.

Findings

Skinning map is non-monotonic and has critical points.

Constructs examples with non-immersion skinning maps.

Extends phenomena to higher genus surfaces with punctures.

Abstract

Certain classes of 3-manifolds, following Thurston, give rise to a 'skinning map', a self-map of the Teichm\"{u}ller space of the boundary. This paper examines the skinning map of a 3-manifold M, a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry of M to conclude that the skinning map associated to M sends a specified path to itself, and use estimates on extremal length functions to show non-monotonicity and the existence of a critical point. A family of finite covers of M produces examples of non-immersion skinning maps on the Teichm\"{u}ller spaces of surfaces in each even genus, and with either 4 or 6 punctures.