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Bending in the space of quasi-Fuchsian structures. (English) Zbl 0718.57010

The author presents a generalization of the idea of bending a hyperbolic n-manifold M along an embedded totally geodesic hypersurface [see W. Thurston, The geometry and topology of 3-manifolds, Princeton Duplicated Notes, 1978-1979; B. Apanasov, Proc. 1978 Stony Brook Conf., Ann. Math. Stud. 97, 21-31 (1981; Zbl 0464.30037); the author, Math. Proc. Camb. Philos. Soc. 98, 247-261 (1985; Zbl 0577.53041)]. This generalization is possible only for the case \(n=2\) (i.e. M is a surface) and deals with a deformation of M along any geodesic lamination on the surface M. In the case of a lamination, the author does not obtain an explicit bending embedding of the hyperbolic plane \({\mathbb{H}}^ 2\) (universal covering of M) into \({\mathbb{H}}^ 3\) but defines a deformation of M by an approximation construction of its infinitesimal deformation, i.e. its cocycles.

MSC:

57R40 Embeddings in differential topology
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
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References:

[1] Kourouniotis, Bending punctured tori.
[2] Epstein, Analytical and geometric aspects of hyperbolic space 111 pp 113– (1987)
[3] Thurston, The geometry and topology of 3-manifolds (1978)
[4] DOI: 10.1017/S030500410006343X · Zbl 0577.53041 · doi:10.1017/S030500410006343X
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