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Pleated surfaces in hyperbolic manifolds, and Teichmüller spaces. (English. Russian original) Zbl 0851.53003
Russ. Acad. Sci., Dokl., Math. 50, No. 1, 1-4 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 337, No. 1, 7-9 (1994).
The authors introduce a new class of so-called “hinged” deformations of a conformal (in particular, hyperbolic) \(n\)-manifold \(M\), that give smooth curves in the Teichmüller space \(C(M)\) (the space of homotopy classes of uniformized conformal structures on \(M\)). This is, equivalently, by smooth families \(\{f_\alpha\}_{\alpha \in I}\) of quasiconformal automorphisms \(f_\alpha : \overline{H^{n+1}} \to \overline{H^{n+1}}\) of the closure \(\overline {H^{n+1}} = H^{n+1} \cup S^n\) of hyperbolic space \(H^{n+1}\) compatible with the isometric action of the holonomy group \(G \cong \pi_1(M)\) in the \((n+1)\)-dimensional space \(H^{n+1}\). Such “hinged” deformations of the manifold \(M\), distinct from all previously known types of deformations (bendings, stampings, and stampings with torsion of the hyperbolic manifold \(M\) (\(\text{vol } M < \infty\)), along totally geodesic submanifolds). The authors provide an answer to the well-known question of bending a hyperbolic manifold \(M\) along pleated geodesic hypersurfaces. They obtain real parameters for such deformations.
53A30 Conformal differential geometry (MSC2010)
53C20 Global Riemannian geometry, including pinching