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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.
##### MSC:
 53A30 Conformal differential geometry (MSC2010) 53C20 Global Riemannian geometry, including pinching