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Laminations et hypersurfaces géodésiques des variétés hyperboliques. (Laminations and geodesic hypersurfaces of hyperbolic varieties). (French) Zbl 0738.53019

Geodesic laminations and geodesic hypersurfaces in hyperbolic spaces of dimension \(>2\) are studied. Some elementary results and facts of hyperbolic geometry stated in [W. Thurston, Geometry and topology of 3-manifolds (Princeton 1978)] are used. A lamination on a manifold is a foliation defined on some closed domain. Interesting results are obtained including three main theorems one of which states the following. Let \(S^ n\) be a geodesic hypersurface in a hyperbolic space \(V^{n+1}\). Then there exist \(x\in S^ n\) and a constant \(C_ n\) which depends only on the number \(n\) such that for any \(r\geq 0\) \[ \hbox{Vol}_ n(B_ S(x,r))\leq C_ n\hbox{Vol}_{n+1}(B_ V(x,r)) \] where \(B_ S\) and \(B_ V\) are balls in \(S\) and \(V\) respectively.
Reviewer: Y.Mutô (Yokohama)

MSC:

53C12 Foliations (differential geometric aspects)
53C20 Global Riemannian geometry, including pinching
53C40 Global submanifolds
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References:

[1] M. GROMOV , Structures métriques pour les variétés riemanniennes . Notes de cours rédigées par J. LAFONTAINE et P. PANSU, C.E.D.I.C., Fernand Nathan, Paris, 1981 . MR 85e:53051 | Zbl 0509.53034 · Zbl 0509.53034
[2] W. THURSTON , Geometry and Topology of 3-Manifolds , Princeton, 1978 . · Zbl 0399.73039
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