Nonstandard uniformized conformal structures on hyperbolic manifolds
Let \(M^ n\) denote a hyperbolic manifold of finite volume (or closed) and \({\mathcal C}(M^ n)\) the space of uniformized conformal structures on \(M^ n\). For \(n=2\) it is well known that this space is connected. The author discusses the problem whether \({\mathcal C}(M^ 3)\) is connected. He constructs a closed hyperbolic manifold \(M=H^ 3/\Gamma\), \(\Gamma\) a discrete subgroup of the isometry group of the hyperbolic space \(H^ 3\), and proves: Theorem. There exists a uniformized conformal structure \(c\in {\mathcal C}(M)\) with the following properties: a) c is a quasiconformal image \(f(c_ 0)\) of the canonical conformal structure \(c_ 0\in {\mathcal C}(M)\) induced by the hyperbolic metric. - b) The holonomy group \(G_*=d^*_ c(\pi_ 1(M))\subset Moeb_ 3\) has two invariant components \(\Omega_ 0\) and \(\Omega\) in its discontinuity set, one of which is the quasiconformal ball \(\Omega_ 0=\tilde f(B^ 3)\). - c) The holonomy group \(G_*\) has no parabolic elements. - d) The structure c is not a limit of any sequence of uniformized conformal structures \((c_ i)\subset {\mathcal C}(M)\) obtained from the canonical structure \(c_ 0\) through bending or stamping deformations. The construction of \(\Gamma\) and c is described in detail but it is too complicated to review it here.
scientific article; zbMATH DE number 4208978