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Zbl 1130.53003
Perelman, Grisha
Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. (English)
[J] arXiv e-print service, Cornell University Library, Paper No. 0307245, 7 p., electronic only (2003)

In this short paper, posted in july 2003, G. Perelman shows that for any closed oriented three-manifold $M$, whose prime decomposition contains no aspherical factors, the Ricci flow with surgery stops in finite time. To stop means that the scalar curvature becomes large everywhere on $M$ so that it is covered by canonical neighbourhoods and hence the topology is completely known; it is also said that the flow becomes extinct in finite time. It is an important step in the proof of the Poincaré conjecture; indeed it is a short cut which avoids using the long time behaviour of the Ricci flow described in [{\it G. Perelman}, ``Ricci flow with surgery on three-manifolds", arXiv: math.DG/0303109 (2003; Zbl 1130.53002)] sections 6 to 8. The idea is to fill some suitably chosen loop by a minimal disc and let both the metric evolve by the Ricci flow and the loop by the curve shortening flow. This idea is similar to one used by R. Hamilton in [{\it R. Hamilton}, Commun. Anal. Geom. 7, No. 4, 695--729 (1999; Zbl 0939.53024)] and a very detailed proof is given in [{\it J. Morgan} and {\it G. Tian}, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3 (2007; Zbl pre05188193)]. An alternative approach using harmonic maps is given in [{\it T. Colding} and {\it W. Minicozzi}, J. Am. Math. Soc. 18, No. 3, 561--569 (2005; Zbl 1083.53058)] and with more details in [{\it T. Colding} and {\it W. Minicozzi}, ``Width and finite extinction time of Ricci flow", arXiv:0707.0108].
[Gérard Besson (Grenoble)]

MSC 2000:: *53-02 Research monographs (differential geometry)
53C44 Geometric evolution equations (mean curvature flow)
53C21 Methods of Riemannian geometry (global)
57M40 Characterizations of Euclidean 3-space and 3-sphere
57R60 Homotopy spheres, Poincare conjecture

Keywords: Ricci flow with surgery; scalar curvature; Poincaré conjecture; harmonic maps

Citations: Zbl 1130.53002; Zbl 0939.53024; Zbl 1083.53058

Cited in: Zbl 1165.53363 Zbl 1157.53035 Zbl 1154.30015 Zbl 1151.53061 Zbl 1148.53050 Zbl 1135.53026 Zbl 1157.53034 Zbl 1135.00004 Zbl 1155.00001 Zbl 1137.00319 Zbl 1108.53002

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