TAILIEUCHUNG - Đề tài " Laminations mesur´ees de plissage des vari´et´es hyperboliques de dimension 3 "

R´sum´ anglais e e For a hyperbolic metric on a 3-dimensional manifold, the boundary of its convex core is a surface which is almost everywhere totally geodesic, but which is bent along a family of disjoint geodesics. The locus and intensity of this bending is described by a measured geodesic lamination, which is a topological object. We consider two problems: the topological characterization of those measured geodesic laminations which can occur as bending measured laminations of hyperbolic metrics; and the uniqueness problem which asks whether a hyperbolic metric is uniquely determined by its bending measured lamination. . | Annals of Mathematics Laminations mesur ees de plissage des vari et es hyperboliques de dimension 3 Par Francis Bonahon et Jean-Pierre Otal Annals of Mathematics 160 2004 1013 1055 Laminations mesurees de plissage des varietes hyperboliques de dimension 3 Par Francis Bonahon et Jean-Pierre Otal Resume anglais For a hyperbolic metric on a 3-dimensional manifold the boundary of its convex core is a surface which is almost everywhere totally geodesic but which is bent along a family of disjoint geodesics. The locus and intensity of this bending is described by a measured geodesic lamination which is a topological object. We consider two problems the topological characterization of those measured geodesic laminations which can occur as bending measured laminations of hyperbolic metrics and the uniqueness problem which asks whether a hyperbolic metric is uniquely determined by its bending measured lamination. Table des matieres 1. Definitions et conditions necessaires 2. Le lemme de fermeture 3. Les longueurs des laminations mesurees de plissage sont bornees 4. Convergence algebrique des metriques 5. Courbes de petites longueurs 6. Convergence des bords des creurs convexes 7. Fin de la demonstration du lemme de fermeture 8. Demonstration des theoremes 2 et 3 9. Demonstration du theoreme 1 Soit M une variete compacte de dimension 3 à bord dont l interieur M admet une metrique hyperbolique. Si au moins l une des composantes de dM est de caracteristique d Euler strictement negative le theoreme d hyperbolisa-tion de Thurston Th2 et le theoreme d uniformisation double d Ahlfors-Bers Ces travaux ont été subventionnes par les bourses DMS-9504282 et DMS-9803445 de la . et par 1 Unite Mixte de Recherche 5669 du . 1014 FRANCIS BONAHON AND JEAN-PIERRE OTAL Be montrent que M admet beaucoup de metriques hyperboliques geometrique-ment finies voir le 1 pour definitions et references . L un des invariants d une metrique hyperbolique sur M est la lamination mesuree de .

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