TAILIEUCHUNG - Đề tài " Geometrization of 3dimensional orbifolds "

This paper is devoted to the proof of the orbifold theorem: If O is a compact connected orientable irreducible and topologically atoroidal 3-orbifold with nonempty ramification locus, then O is geometric (. has a metric of constant curvature or is Seifert fibred). As a corollary, any smooth orientationpreserving nonfree finite group action on S 3 is conjugate to an orthogonal action. Contents 1. Introduction 2. 3-dimensional orbifolds . Basic definitions . Spherical and toric decompositions . Finite group actions on spheres with fixed points . Proof of the orbifold theorem from the main theorem 3. . | Annals of Mathematics Geometrization of 3dimensional orbifolds By Michel Boileau Bernhard Leeb and Joan Porti Annals of Mathematics 162 2005 195 290 Geometrization of 3-dimensional orbifolds By Michel Boileau Bernhard Leeb and Joan PoRTi Abstract This paper is devoted to the proof of the orbifold theorem If O is a compact connected orientable irreducible and topologically atoroidal 3-orbifold with nonempty ramification locus then O is geometric . has a metric of constant curvature or is Seifert fibred . As a corollary any smooth orientationpreserving nonfree finite group action on S3 4 5 is conjugate to an orthogonal action. Contents 1. Introduction 2. 3-dimensional orbifolds . Basic definitions . Spherical and toric decompositions . Finite group actions on spheres with fixed points . Proof of the orbifold theorem from the main theorem 3. 3-dimensional cone manifolds . Basic definitions . Exponential map cut locus cone injectivity radius . Spherical cone surfaces with cone angles n . Compactness for spaces of thick cone manifolds 4. Noncompact Euclidean cone 3-manifolds 5. The local geometry of cone 3-manifolds with lower diameter bound . Umbilic tubes . Statement of the main geometric results . A local Margulis lemma for imcomplete manifolds . Near singular vertices and short closed singular geodesics . Near embedded umbilic surfaces . Finding umbilic turnovers . Proof of Theorem Analysis of the thin part . Totally geodesic boundary 196 MICHEL BOILEAU BERNHARD LEEB AND JOAN PORTI 6. Proof of the main theorem . Reduction to the case when the smooth part is hyperbolic . Deformations of hyperbolic cone structures . Degeneration of hyperbolic cone structures 7. Topological stability of geometric limits . The case of cone angles a n . The case when cone angles approach the orbifold angles . Putting a CAT 1 -structure on the smooth part of a cone manifold 8. Spherical uniformization . .

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