Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman.

*(English)*Zbl 1083.53058The paper affirmatively answers a question of Grisha Perelman (April 2003): Starting with an arbitrary metric on \(S^3\), does the Ricci flow become extinct in finite time? Part of Perelman’s interest in this question comes from the following: if one were interested in geometrization of a homotopy 3-sphere (or, more generally, a 3-manifold without aspherical submanifolds) and knew that the Ricci flow became extinct in finite time, then one would not need to analyze what happens to the flow as time goes to infinity (in particular, one would not need collapsing arguments). The actual answer is given as a corollary from an upper bound for the derivative of the width \(W(g(t))\) established by the authors for the Ricci flow \(g(t)\) on a closed orientable prime non-aspherical 3-manifold (Theorem 1.1), Corollary 1.2: Let \(M^3\) be a closed orientable 3-manifold whose prime decomposition has only non-aspherical factors and is equipped with a Riemannian metric \(g=g(0)\). Under the Ricci flow with surgery, \(g(t)\) must become extinct in finite time.

The authors note that Perelman has answered his original question in his posted preprint ”Finite extinction time for the solution to the Ricci flow on certain three-manifolds”, DG/0307245. Though they think that their slightly different approach may be of interest, in particular because it avoids using the curve shortening flow that Perelman simultaneously with the Ricci flow needed to invoke.

The authors note that Perelman has answered his original question in his posted preprint ”Finite extinction time for the solution to the Ricci flow on certain three-manifolds”, DG/0307245. Though they think that their slightly different approach may be of interest, in particular because it avoids using the curve shortening flow that Perelman simultaneously with the Ricci flow needed to invoke.

Reviewer: Boris N. Apanasov (Norman)

##### MSC:

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

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

57M50 | General geometric structures on low-dimensional manifolds |

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\textit{T. H. Colding} and \textit{W. P. Minicozzi II}, J. Am. Math. Soc. 18, No. 3, 561--569 (2005; Zbl 1083.53058)

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##### References:

[1] | D. Christodoulou and S.-T. Yau, Some remarks on the quasi-local mass, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 9 – 14. · doi:10.1090/conm/071/954405 · doi.org |

[2] | Tobias H. Colding and Camillo De Lellis, The min-max construction of minimal surfaces, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., vol. 8, Int. Press, Somerville, MA, 2003, pp. 75 – 107. · Zbl 1051.53052 · doi:10.4310/SDG.2003.v8.n1.a3 · doi.org |

[3] | Tobias H. Colding and William P. Minicozzi II, Minimal surfaces, Courant Lecture Notes in Mathematics, vol. 4, New York University, Courant Institute of Mathematical Sciences, New York, 1999. · Zbl 1175.53008 |

[4] | Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7 – 136. · Zbl 0867.53030 |

[5] | A. Hatcher, Notes on basic \(3\)-manifold topology, www.math.cornell.edu/hatcher/3M/ 3Mdownloads.html. |

[6] | Jürgen Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. A Wiley-Interscience Publication. · Zbl 0729.49001 |

[7] | William Meeks III, Leon Simon, and Shing Tung Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621 – 659. · Zbl 0521.53007 · doi:10.2307/2007026 · doi.org |

[8] | Mario J. Micallef and John Douglas Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), no. 1, 199 – 227. · Zbl 0661.53027 · doi:10.2307/1971420 · doi.org |

[9] | G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, math.DG/0307245. · Zbl 1130.53003 |

[10] | G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math.DG/0211159. |

[11] | G. Perelman, Ricci flow with surgery on three-manifolds, math.DG/0303109. · Zbl 1130.53002 |

[12] | R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; Preface translated from the Chinese by Kaising Tso. · Zbl 0830.53001 |

[13] | R. Schoen and S. T. Yau, Lectures on harmonic maps, Conference Proceedings and Lecture Notes in Geometry and Topology, II, International Press, Cambridge, MA, 1997. · Zbl 0886.53004 |

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