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Geometrization of three manifolds and Perelman’s proof. (English) Zbl 1170.57016

This is a survey on Thurston’s geometrization conjecture of 3-manifolds and the proof of G. Perelman using the Ricci flow.
In the first two sections of the paper the author presents the main results concerning the topological decomposition of a 3-manifold, and gives various formulations of the Geometrization Conjecture. Then, classical examples of 3-manifolds are described, and results known before Perelman and consequences of Perelman’s proof are listed.
The third section deals with the Ricci flow, and the main contributions of R. Hamilton concerning the Ricci flow on closed manifolds admitting a Riemannian metric with non-negative Ricci curvature. The Hamilton-Ivey pinching of the curvature is also explained.
The fourth section concerns the study of the singularities of the Ricci flow. The parabolic rescaling of the flow at a singularity is first explained. Then the \(\kappa\)-noncollapsed theorem of Perelman to control the injectivity radius is described. The section ends with the notion of canonical neighborhoods and their gluing.
The fifth section of the survey deals with the existence of the Ricci flow with cut-off, and the sixth section with elliptization: the proof of the finite time extinction of the Ricci flow with cut-off on a manifold with finite fundamental group is outlined, following T. H. Colding and W. P. Minicozzi [Geom. Topol. 12, No. 5, 2537–2586 (2008; Zbl 1161.53352)].
The last section of the survey is devoted to the long-time behavior of the Ricci flow. After the exposition of the thick-thin decomposition, the geometrization of aspherical manifolds is explained, following recent work of the author of the survey and others [L. Bessières, G. Besson, M. Boileau, S. Maillot and J. Porti, “Weak collapsing and geometrisation of aspherical 3-manifolds”, preprint].
The survey is illustrated by numerous examples and figures which greatly contribute to the clarity of the exposition.

MSC:

57M50 General geometric structures on low-dimensional manifolds
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)

Citations:

Zbl 1161.53352
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