×

zbMATH — the first resource for mathematics

Proof of the Poincaré conjecture by deforming the metric by the Ricci curvature (after G. Perel’man). (Preuve de la conjecture de Poincaré en déformant la métrique par la courbure de Ricci [d’aprés G. Perel’man].) (French) Zbl 1181.53055
Séminaire Bourbaki. Volume 2004/2005. Exposés 938–951. Paris: Société Mathématique de France (ISBN 978-2-85629-224-2/pbk). Astérisque 307, 309-347, Exp. No. 947 (2006).
Summary: We present the proof of the Poincaré conjecture, on closed simply connected three-manifolds, proposed by G. Perel’man. It relies on the study of Riemannian metrics evoluting under the Ricci flow and on previous works by R. Hamilton. After an introduction to the analytical and geometrical techniques developed by R. Hamilton, we try to describe the technique of metric surgery used by G. Perel’man to go through the singular times for which the scalar curvature goes to infinity on certain parts of the manifold. The proof of the Poincaré conjecture then relies on the proof of the finite extinction time of the flow with surgeries, under certain assumptions, for which we present a version due to T. Colding and W. Minicozzi, J. Am. Math. Soc. 18, No. 3, 561–569 (2005; Zbl 1083.53058)].
For the entire collection see [Zbl 1105.00003].

MSC:
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
PDF BibTeX XML Cite