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Geometrization of three manifolds and Perelman’s proof

  • Joan Porti
Article

Abstract

This is a survey about Thurston’s geometrization conjecture of three manifolds and Perelman’s proof with the Ricci flow. In particular we review the essential contribution of Hamilton as well as some results in topology relevants for the proof.

Keywords

three manifolds geometrization Hamilton-Ricci flow 

Mathematics Subject Classifications

53C44 57M50 

Geometrizaci ón de variedades tridimensionales y la demostraci ón de Perelman

Resumen

Ésta es una exposición sobre la conjetura de geometrización de Thurston para variedades tridimensionales, así como de la demostración de Perelman mediante el flujo de Ricci. En particular se revisan la contribución esencial de Hamilton y algunos resultados de topología relevantes para la demostración.

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References

  1. [1]
    Bessières, L., Besson, G., Boileau, M., Maillot, S. M. and Porti, J., (2007). Weak collapsing and geometrisation of aspherical 3-manifolds, (preprint).Google Scholar
  2. [2]
    Bianchi, L., (1897). Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti,Mem. Soc. Ital. Scienze,11, 3, 267–352.MathSciNetGoogle Scholar
  3. [3]
    Bing, R. H., (1954). Locally tame sets are tame,Ann. of Math. (2),59, 145–158.CrossRefMathSciNetGoogle Scholar
  4. [4]
    Boileau, M., Leeb, B. and Porti, J., (2005). Geometrization of 3-dimensional orbifolds,Ann. of Math. (2),162, 1, 195–290.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Cao, H.-D. and Zhu, X.-P., (2006). A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow,Asian J. Math.,10, 2, 165–492.MATHMathSciNetGoogle Scholar
  6. [6]
    Casson, A. and Jungreis, D., (1994). Convergence groups and Seifert fibered 3-manifolds,Invent. Math.,118, 3, 441–456.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Cheeger, J. and Ebin, D. G., (1975).Comparison theorems in Riemannian geometry, North-Holland Publishing Co., Amsterdam. North-Holland Mathematical Library,9.Google Scholar
  8. [8]
    Cheeger, J. and Gromov, M., (1986). Collapsing Riemannian manifolds while keeping their curvature bounded. I,J. Differential Geom.,23, 3, 309–346.MATHMathSciNetGoogle Scholar
  9. [9]
    Cheeger, J. and Gromov,M., (1990). Collapsing Riemannian manifolds while keeping their curvature bounded. II,J. Differential Geom.,32, 1, 269–298.MATHMathSciNetGoogle Scholar
  10. [10]
    Colding, T. H. and Minicozzi, II, W. P., (2005). Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman,J. Amer. Math. Soc.,18, 3, 561–569, (electronic).MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Colding, T. H. and Minicozzi, II, W. P., (2007). Width and finite extinction time of Ricci flow. (preprint).Google Scholar
  12. [12]
    DeTurck, D. M., (1983). Deforming metrics in the direction of their Ricci tensors,J. Differential Geom.,18, 1, 157–162.MATHMathSciNetGoogle Scholar
  13. [13]
    Gabai, D., (1992). Convergence groups are Fuchsian groups,Ann. of Math. (2),136, 3, 447–510.CrossRefMathSciNetGoogle Scholar
  14. [14]
    Gromov, M., (1982). Volume and bounded cohomology,Inst. Hautes Études Sci. Publ. Math.,56, 5–99.MATHMathSciNetGoogle Scholar
  15. [15]
    Gromov, M. and Lawson, H. B., Jr., (1980). The classification of simply connected manifolds of positive scalar curvature,Ann. of Math. (2),111, 3, 423–434.CrossRefMathSciNetGoogle Scholar
  16. [16]
    Haken, W., (1962). Über das Homöomorphieproblem der 3-Mannigfaltigkeiten. I,Math. Z.,80, 89–120.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Hamilton, R. S, (1982). Three-manifolds with positive Ricci curvature,J. Differential Geom.,17, 2, 255–306.MATHMathSciNetGoogle Scholar
  18. [18]
    Hamilton, R. S., (1986). Four-manifolds with positive curvature operator,J. Differential Geom.,24, 2, 153–179.MATHMathSciNetGoogle Scholar
  19. [19]
    Hamilton, R. S., (1995). The formation of singularities in the Ricci flow, inSurveys in differential geometry, Vol. II (Cambridge, MA, 1993), Int. Press, Cambridge, MA, 7–136.Google Scholar
  20. [20]
    Hamilton, R. S., (1999). Non-singular solutions of the Ricci flow on three-manifolds,Comm. Anal. Geom.,7, 4, 695–729.MATHMathSciNetGoogle Scholar
  21. [21]
    Ivey, T., (1993). Ricci solitons on compact three-manifolds,Differential Geom. Appl.,3, 4, 301–307.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Jost, J., (1991).Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons Ltd., Chichester., A Wiley-Interscience Publication.MATHGoogle Scholar
  23. [23]
    Kapovich, M., (2001).Hyperbolic manifolds and discrete groups, Progress in Mathematics,183, Birkhäuser Boston Inc., Boston, MA.MATHGoogle Scholar
  24. [24]
    Kleiner, B. and Lott, J.Notes on Perelman’s papers, Preprint, math. DG/0605667.Google Scholar
  25. [25]
    Kneser, H., (1929). Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten.,Jahresbericht D. M. V.,38, 248–260.MATHGoogle Scholar
  26. [26]
    McMullen, C., (1990). Iteration on Teichmüller space,Invent. Math.,99, 2, 425–454.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    McMullen, C. T., (2005). Minkowski’s conjecture, well-rounded lattices and topological dimension,J. Amer. Math. Soc.,18, 3, 711–734, (electronic).MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    Meeks III,W. H. and Yau, S. T., (1981). The equivariant Dehn’s lemma and loop theorem,Comment. Math. Helv.,56, 2, 225–239.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    Mess, G., The Seifert conjecture and groups which are coarse quasiisometric to planes, (preprint).Google Scholar
  30. [30]
    Milnor, J., (2003). Towards the Poincaré conjecture and the classification of 3-manifolds,Notices Amer. Math. Soc.,50, 10, 1226–1233.MATHMathSciNetGoogle Scholar
  31. [31]
    Moise, E. E., (1952). Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung,Ann. of Math, (2),56, 96–114.CrossRefMathSciNetGoogle Scholar
  32. [32]
    Montesinos-Amilibia, J. M., (1984). Punts de vista sobre el problema de Poincaré, inThe development of mathematics in the nineteenth century, Arx. Sec. Cièn.,LXXV, Inst. Estudis Cat., Barcelona, 23–39.Google Scholar
  33. [33]
    Morgan, J. and Tian, G., Ricci Flow and the Poincare Conjecture, math. DG/0607607, (preprint).Google Scholar
  34. [34]
    Morgan, J. W. and Shalen, P. B., (1984). Valuations, trees, and degenerations of hyperbolic structures. I,Ann. of Math. (2),120, 3, 401–476.CrossRefMathSciNetGoogle Scholar
  35. [35]
    Mostow, G. D., (1968). Quasi-conformal mappings inn-space and the rigidity of hyperbolic space forms,Inst. Hautes Études Sci. Publ. Math.,34, 53–104.MATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    Müller, R., (2006).Differential Harnack inequalities and the Ricci flow, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich.MATHCrossRefGoogle Scholar
  37. [37]
    Munkres, J., (1960). Obstructions to the smoothing of piecewise-differentiable homeomorphisms,Ann. of Math. (2),72, 521–554.CrossRefMathSciNetGoogle Scholar
  38. [38]
    Myers, R., (1982). Simple knots in compact, orientable 3-manifolds,Trans. Amer. Math. Soc.,273, 1, 75–91.MATHMathSciNetGoogle Scholar
  39. [39]
    Otal, J.-P., (1996). Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3,Astérisque, 235, x+159.Google Scholar
  40. [40]
    Otal, J.-P., (1998). Thurston’s hyperbolization of Haken manifolds, inSurveys in differential geometry,III (Cambridge, MA, 1996), Int. Press, Boston, MA, 77–194.Google Scholar
  41. [41]
    Perelman, G., The entropy formula for the Ricci flow and its geometric applications, math. DG/0211159, (preprint).Google Scholar
  42. [42]
    Perelman, G., Ricci flow with surgery on three-manifolds, math. DG/0303109, (preprint).Google Scholar
  43. [43]
    Perelman, G., Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, math. DG/0307245, (preprint).Google Scholar
  44. [44]
    Poincaré, H., (1904). Cinquième complément à l’analysis situs,Palermo Rend.,18, 45–110.MATHCrossRefGoogle Scholar
  45. [45]
    Riley, R., (1975). A quadratic parabolic group,Math. Proc. Cambridge Philos. Soc.,77, 281–288.MATHCrossRefMathSciNetGoogle Scholar
  46. [46]
    Schoen, R. and Yau, S. T., (1979). Existence of incompressible minimal surfaces and the topology of threedimensional manifolds with nonnegative scalar curvature,Ann. of Math. (2),110, 1, 127–142.CrossRefMathSciNetGoogle Scholar
  47. [47]
    Scott, P., (1983). The geometries of 3-manifolds,Bull. London Math. Soc.,15, 5, 401–487.MATHCrossRefMathSciNetGoogle Scholar
  48. [48]
    Seifert, H., (1933). Topologie dreidimensionaler gefaserter Räume.,Acta Math.,60, 147–238.CrossRefMathSciNetGoogle Scholar
  49. [49]
    Shioya, T. and Yamaguchi, T., (2005). Volume collapsed three-manifolds with a lower curvature bound,Math. Ann.,333, 1, 131–155.MATHCrossRefMathSciNetGoogle Scholar
  50. [50]
    Soma, T., (1981). The Gromov invariant of links,Invent. Math.,64, 3, 445–454.MATHCrossRefMathSciNetGoogle Scholar
  51. [51]
    Thurston, W., (1979).The geometry and topology of 3-manifolds, Princeton Math. Dept., Princeton NJ.Google Scholar
  52. [52]
    Thurston, W. P., (1982). Three-dimensional manifolds, Kleinian groups and hyperbolic geometry,Bull. Amer. Math. Soc. (N.S.),6, 3, 357–381.MATHCrossRefMathSciNetGoogle Scholar
  53. [53]
    Thurston, W. P., (1986). Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds,Ann. of Math. (2),124, 2, 203–246.CrossRefMathSciNetGoogle Scholar
  54. [54]
    Thurston, W. P., (1986), Hyperbolic structures on 3-manifolds. II: Surface groups and 3-manifolds which fiber over the circle. (preprint).Google Scholar
  55. [55]
    Thurston, W. P., (1986), Hyperbolic structures on 3-manifolds. III: Deformations of 3-manifolds with incompressible boundary. (preprint).Google Scholar
  56. [56]
    Thurston, W. P., (1988). On the geometry and dynamics of diffeomorphisms of surfaces,Bull. Amer. Math. Soc. (N.S.),19, 2, 417–431.MATHCrossRefMathSciNetGoogle Scholar
  57. [57]
    Tukia, P., (1988). Homeomorphic conjugates of Fuchsian groups,J. Reine Angew. Math.,391, 1–54.MATHMathSciNetGoogle Scholar
  58. [58]
    Waldhausen, F., (1967). Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II,Invent. Math.,3, (1967), 308–333; ibid., Waldhausen, F., (1967). Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II,4,Invent. Math.,3, (1967), 87–117.CrossRefMathSciNetGoogle Scholar
  59. [59]
    Waldhausen, F., (1968). On irreducible 3-manifolds which are sufficiently large,Ann. of Math. (2),87, 56–88.CrossRefMathSciNetGoogle Scholar
  60. [60]
    Whitehead, J., (1935). A certain open manifold whose group is unity.,Q. J. Math., Oxf. Ser.,6, 268–279.CrossRefGoogle Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaCerdanyola del VallèsSpain

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