Finite extinction time for the solutions to the Ricci flow on certain three-manifolds.

*(English)*Zbl 1130.53003In 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 [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 [R. Hamilton, Commun. Anal. Geom. 7, No. 4, 695–729 (1999; Zbl 0939.53024)] and a very detailed proof is given in [J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3 (2007; Zbl 1179.57045)]. An alternative approach using harmonic maps is given in [T. Colding and W. Minicozzi, J. Am. Math. Soc. 18, No. 3, 561–569 (2005; Zbl 1083.53058)] and with more details in [T. Colding and W. Minicozzi, “Width and finite extinction time of Ricci flow”, arXiv:0707.0108].

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 [R. Hamilton, Commun. Anal. Geom. 7, No. 4, 695–729 (1999; Zbl 0939.53024)] and a very detailed proof is given in [J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3 (2007; Zbl 1179.57045)]. An alternative approach using harmonic maps is given in [T. Colding and W. Minicozzi, J. Am. Math. Soc. 18, No. 3, 561–569 (2005; Zbl 1083.53058)] and with more details in [T. Colding and W. Minicozzi, “Width and finite extinction time of Ricci flow”, arXiv:0707.0108].

Reviewer: Gérard Besson (Grenoble)

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

57M40 | Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010) |

57R60 | Homotopy spheres, Poincaré conjecture |