A family of non-injective skinning maps with critical points (Q2787960)

The author considers Thurston's skinning maps associated to compact 3-manifolds. The skinning map \(\sigma_M\) is defined for a 3-manifold \(M\) with some conditions as a mapping from the Teichmüller space \(\mathcal{T}(\Sigma)\) to \(\mathcal{T}(\overline{\Sigma})\), where \(\Sigma\) denotes the boundary of \(M\) and \(\overline{\Sigma}\) denotes the orientation-reversed one.NEWLINENEWLINEThe main result of the paper under review is that there exists a skinning map which is non-injective and has a critical point for a certain 3-manifold. Precisely, there exists a pared 3-manifold \(M=(H_2,P)\) admitting such a skinning map, where \(H_2\) is a genus-2 handlebody and the paring locus \(P\) consists of two rank-1 cusps. This is proved by showing that the anti-holomorphic map \(\overline{\sigma_M}\) sends a certain real 1-dimensional submanifold \(\mathcal{R}\) of \(\mathcal{T}(\Sigma)\) to itself. Then the composition of an extremal length function with \(\overline{\sigma_M}\) is non-monotonic and continuously differentiable on \(\mathcal{R}\), while the extremal length function is a diffeomorphism on \(\mathcal{R}\). As a corollary, the author gives a family of pared 3-manifolds \(\{(H_n,P_n)\}_{n=2}^{\infty}\) admitting geometrically finite hyperbolic structures such that each skinning map has a critical point, where each boundary has even genus and 4 or 6 punctures, and each paring locus consists of two or three rank-1 cusps.