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Recent progress on the Poincaré conjecture and the classification of 3-manifolds
Author:
John W. Morgan
Journal:
Bull. Amer. Math. Soc. 42 (2005), 57-78
MSC (2000):
Primary 57M50, 57M27, 58J35
Posted:
October 29, 2004
MathSciNet review:
2115067
Full-text PDF
References |
Similar Articles |
Additional Information
- 1.
Yu.
Burago, M.
Gromov, and G.
Perel′man, A. D. Aleksandrov spaces with curvatures bounded
below, Uspekhi Mat. Nauk 47 (1992), no. 2(284),
3–51, 222 (Russian, with Russian summary); English transl., Russian
Math. Surveys 47 (1992), no. 2, 1–58. MR 1185284
(93m:53035), http://dx.doi.org/10.1070/RM1992v047n02ABEH000877
- 2.
Jeff
Cheeger and Mikhael
Gromov, Collapsing Riemannian manifolds while keeping their
curvature bounded. I, J. Differential Geom. 23
(1986), no. 3, 309–346. MR 852159
(87k:53087)
Jeff
Cheeger and Mikhael
Gromov, Collapsing Riemannian manifolds while keeping their
curvature bounded. II, J. Differential Geom. 32
(1990), no. 1, 269–298. MR 1064875
(92a:53066)
- 3.
T. Colding and W. Minicozzi, ``Estimates for the extinction time for the Ricci flow on certain
-manifolds and a question of Perelman," arXiv.math.DG/0308090, August 25, 2003.
- 4.
Daryl
Cooper, Craig
D. Hodgson, and Steven
P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds,
MSJ Memoirs, vol. 5, Mathematical Society of Japan, Tokyo, 2000. With
a postface by Sadayoshi Kojima. MR 1778789
(2002c:57027)
- 5.
Richard
S. Hamilton, Three-manifolds with positive Ricci curvature, J.
Differential Geom. 17 (1982), no. 2, 255–306.
MR 664497
(84a:53050)
- 6.
Richard
S. Hamilton, The Harnack estimate for the Ricci flow, J.
Differential Geom. 37 (1993), no. 1, 225–243.
MR
1198607 (93k:58052)
- 7.
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. MR 1375255
(97e:53075)
- 8.
Richard
S. Hamilton, Four-manifolds with positive isotropic curvature,
Comm. Anal. Geom. 5 (1997), no. 1, 1–92. MR 1456308
(99e:53049)
- 9.
Richard
S. Hamilton, Non-singular solutions of the Ricci flow on
three-manifolds, Comm. Anal. Geom. 7 (1999),
no. 4, 695–729. MR 1714939
(2000g:53034)
- 10.
John
Hempel, 3-Manifolds, Princeton University Press, Princeton, N.
J., 1976. Ann. of Math. Studies, No. 86. MR 0415619
(54 #3702)
- 11.
H. Kneser, ``Geschlossene Flächen in drei-dimesnionalen Mannigfaltigkeiten," Jahresber. Deutsch. Math. Verbein 38 (1929), 248-260.
- 12.
John
Milnor, Groups which act on 𝑆ⁿ without fixed
points, Amer. J. Math. 79 (1957), 623–630. MR 0090056
(19,761d)
- 13.
John
Milnor, Towards the Poincaré conjecture and the
classification of 3-manifolds, Notices Amer. Math. Soc.
50 (2003), no. 10, 1226–1233. MR 2009455
(2004h:57022)
- 14.
G.
Perelman, Spaces with curvature bounded below, 2
(Zürich, 1994) Birkhäuser, Basel, 1995, pp. 517–525.
MR
1403952 (97g:53055)
- 15.
G. Perelman, ``The entropy formula for the Ricci flow and its geometric applications," arXiv.math.DG/0211159, November 11, 2002.
- 16.
G. Perelman, ``Ricci flow with surgery on three-manifolds," arXiv.math.DG/0303109, March 10, 2003.
- 17.
G. Perelman, ``Finite extinction time for the solutions to the Ricci flow on certain three-manifolds," arXiv.math.DG/0307245, July 17, 2003.
- 18.
Henri
Poincaré, Œuvres. Tome VI, Les Grands Classiques
Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions
Jacques Gabay, Sceaux, 1996 (French). Géométrie. Analysis
situs (topologie). [Geometry. Analysis situs (topology)]; Reprint of the
1953 edition. MR
1401792 (98m:01041)
- 19.
Peter
Scott, The geometries of 3-manifolds, Bull. London Math. Soc.
15 (1983), no. 5, 401–487. MR 705527
(84m:57009), http://dx.doi.org/10.1112/blms/15.5.401
- 20.
H. Seifert, ``Topologie dreidimensionaler gefaserter Räume," Acta Math. 60 (1933), 147-288.
- 21.
Wan-Xiong
Shi, Deforming the metric on complete Riemannian manifolds, J.
Differential Geom. 30 (1989), no. 1, 223–301.
MR
1001277 (90i:58202)
- 22.
Takashi
Shioya and Takao
Yamaguchi, Collapsing three-manifolds under a lower curvature
bound, J. Differential Geom. 56 (2000), no. 1,
1–66. MR
1863020 (2002k:53074)
- 23.
T. Shioya and T. Yamaguchi, ``Volume collapsed three-manifolds with a lower curvature bound," arXiv.math.DG/0304472, April 15, 2003.
- 24.
William
P. Thurston, Three-dimensional geometry and topology. Vol. 1,
Princeton Mathematical Series, vol. 35, Princeton University Press,
Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975
(97m:57016)
- 1.
- Y. Burago, M. Gromov and G. Perelman, ``A. D. Aleksandrov spaces with curvature bounded below," Russian Math. Surveys 47 (1992), 1-58. MR 1185284 (93m:53035)
- 2.
- J. Cheegar and M. Gromov, ``Collapsing Riemannian manifolds while keeping their curvatures bounded, I, II." J. Differential Geom. 23 (1986), 309-346; 32 (1990), 269-298. MR 0852159 (87k:53087); MR 1064875 (92a:53066)
- 3.
- T. Colding and W. Minicozzi, ``Estimates for the extinction time for the Ricci flow on certain
-manifolds and a question of Perelman," arXiv.math.DG/0308090, August 25, 2003.
- 4.
- D. Cooper, C. Hodgson, S. Kerchoff,
-dimensional orbifolds and cone-manifolds, Math. Soc. Japan Memoirs, Vol. 5, Tokyo, 2000. MR 1778789 (2002c:57027)
- 5.
- R. Hamilton, ``Three-manifolds with positive Ricci curvature," J. Differential Geom. 17 (1982), 255-306. MR 0664497 (84a:53050)
- 6.
- R. Hamilton, ``The Harnack estimate for the Ricci flow," J. Differential. Geom. 37 (1993), 225-243. MR 1198607 (93k:58052)
- 7.
- R. Hamilton, ``Formation of singularities in the Ricci flow," Surveys in Differential Geometry 2, 7-136, International Press, 1995. MR 1375255 (97e:53075)
- 8.
- R. Hamilton, ``Four-manifolds with positive isotropic curvature," Comm. Anal. Geom. 5 (1997), 1-92. MR 1456308 (99e:53049)
- 9.
- R. Hamilton, ``Non-singular solutions of the Ricci flow on three-manifolds," Comm. Analysis and Geometry 7 (1999), 695-729. MR 1714939 (2000g:53034)
- 10.
- J. Hempel,
-manifolds, Ann. of Math. Studies, Vol. 86, Princeton University Press, 1976. MR 0415619 (54:3702)
- 11.
- H. Kneser, ``Geschlossene Flächen in drei-dimesnionalen Mannigfaltigkeiten," Jahresber. Deutsch. Math. Verbein 38 (1929), 248-260.
- 12.
- J. Milnor, ``Groups which act on
without fixed points," Amer. Journal of Math. 79 (1957), 623-630. MR 0090056 (19:761d)
- 13.
- J. Milnor, ``Towards the Poincaré Conjecture and the classification of
-manifolds," Notices AMS 50 (2003), 1226-1233. MR 2009455 (2004h:57022)
- 14.
- G. Perelman, ``Spaces with curvature bounded below," Proceedings of ICM 1994, 517-525, Birkhäuser, 1995. MR 1403952 (97g:53055)
- 15.
- G. Perelman, ``The entropy formula for the Ricci flow and its geometric applications," arXiv.math.DG/0211159, November 11, 2002.
- 16.
- G. Perelman, ``Ricci flow with surgery on three-manifolds," arXiv.math.DG/0303109, March 10, 2003.
- 17.
- G. Perelman, ``Finite extinction time for the solutions to the Ricci flow on certain three-manifolds," arXiv.math.DG/0307245, July 17, 2003.
- 18.
- H. Poincaré, ``Cinquième complément à l'analysis situs," Rend. Circ. Mat. Palermo 18 (1904), 45-110. (See Oeuvres, Tome VI, Paris, 1953, p. 498.) MR 1401792 (98m:01041)
- 19.
- P. Scott, ``The geometries of
-manifolds," Bull. London Math. Soc. 15 (1983), 401-487. MR 0705527 (84m:57009)
- 20.
- H. Seifert, ``Topologie dreidimensionaler gefaserter Räume," Acta Math. 60 (1933), 147-288.
- 21.
- W-X. Shi, ``Deforming the metric on complete riemannian manifolds,'' J. Differential Geom. 30 (1989), 223-301. MR 1001277 (90i:58202)
- 22.
- T. Shioya and T. Yamaguchi, ``Collapsing three-manifolds under a lower curvature bound," J. Differential Geom. 56 (2000), 1-66. MR 1863020 (2002k:53074)
- 23.
- T. Shioya and T. Yamaguchi, ``Volume collapsed three-manifolds with a lower curvature bound," arXiv.math.DG/0304472, April 15, 2003.
- 24.
- W. Thurston, Three-dimensional Geometry and Topology, Vol. 1 (Silvio Levy, ed.) Princeton Math. Ser., vol. 35, Princeton Univ. Press, 1997. MR 1435975 (97m:57016)
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Additional Information
John W. Morgan
Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027
Email:
jm@math.columbia.edu
DOI:
http://dx.doi.org/10.1090/S0273-0979-04-01045-6
PII:
S 0273-0979(04)01045-6
Received by editor(s):
June 11, 2004
Received by editor(s) in revised form:
September 1, 2004
Posted:
October 29, 2004
Additional Notes:
Written version of a talk presented on January 9, 2004, in the “Current Events in Mathematics" session at the AMS national meeting in Phoenix, AZ
Article copyright:
© Copyright 2004 American Mathematical Society
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