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Poincaré conjecture and Ricci flow An outline of the work of R Hamilton and G Perelman: Part II L Bessières Grenoble France


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Niveau: Supérieur, Doctorat, Bac+8
Poincaré conjecture and Ricci flow An outline of the work of R. Hamilton and G. Perelman: Part II L. Bessières (Grenoble, France) In the first section of this text (EMS-Newsletter 59, pp. 11– 15), we explained how to study singularities of the Ricci flow with sequences of parabolic rescaling. We showed that these sequences have limits with very strong geometric constraints and that these limits are called ?-solutions. We begin the final part of the article with their classification. ?-solutions Recall that a 3-dimensional ?-solution is a solution to the Ricci flow (M3,g(t)), defined for at least t ? (?∞,0], such that for any t ≤ 0, g(t) is a complete Riemannian metric with positive non-flat bounded sectional curvature. Moreover, it is ?-non-collapsed at any scale. For example, S3 and S2?R with their standard flows are ?-solutions while S2?S1 is not (when t ??∞, the S1 factor is very small compared to the sphere S 2). Perelman makes the full classification of 3-dimensional ?-solutions in chapter 11 of [PeI] and 1.5 in [PeII]. Theorem 1. A 3-dimensional ?-solution is isometric to: (A) S2 ?R with cylindrical flow, (B) B3 or RP3 ?B3 with Ricci flow of strictly positive sec- tional curvature, (C) S3/?, where ? is a finite subgroup of

  • connected sum

  • has strictly

  • sum decomposi- tion

  • scalar curvature

  • finite time

  • dimensional ?-solution

  • double ?-horns

  • has

  • connected

  • positive sectional



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