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Local collapsing, orbifolds, and geometrization. (English) Zbl 1315.53034

Astérisque 365. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-795-7/pbk). 177 p. (2014).
The volume contains two papers: the first paper “Locally collapsed three-manifolds” discusses local collapsing in Riemannian geometry. The authors prove that a three-dimensional compact Riemannian manifold which is locally collapsed with respect to a lower curvature bound is a graph manifold. The main result is clearly stated and the description of some of the issues involved in the proof are also included. This theorem was stated by Perelman without proof and was used to show that certain collapsed manifolds arising in this proof of the geometrization conjecture are graph manifolds. One of the goals of this paper is to provide a proof of Perelman’s collapsing theorem which is streamlined, self-contained and accessible. A history of the problem is also presented. In addition, the authors provide three appendices on choosing ball covers, cloudy submanifolds and an isotopy lemma.
The second paper “Geometrization of three-dimensional manifolds via Ricci flow” discusses the geometrization of orbifolds. A three-dimensional closed orientable orbifold, which has no bad suborbifolds is known to have a geometric decomposition from the work of Perelman in the manifold case, along with earlier work of Boileau-Leeb-Porti, Boilleau-Maillot-Porti, Boilleau-Porti, Cooper-Hodgyson-Kerchoff, and Thurston. The authors propose a new, logically independent, unified proof of the geometrization of orbifolds, using Ricci flow. Along the paper they develop some tools for the geometry of orbifolds that may be of independent interest. The appendix “Weak and strong graph orbifolds” contains topological remarks about graph orbifolds with complete proofs.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C20 Global Riemannian geometry, including pinching
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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