Estimates for the extinction time for the Ricci flow on certain $3$-manifolds and a question of Perelman
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- by Tobias H. Colding and William P. Minicozzi II
- J. Amer. Math. Soc. 18 (2005), 561-569
- DOI: https://doi.org/10.1090/S0894-0347-05-00486-8
- Published electronically: April 13, 2005
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Abstract:
We show that the Ricci flow becomes extinct in finite time on any Riemannian $3$-manifold without aspherical summands.References
- 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. MR 954405, DOI 10.1090/conm/071/954405
- 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. MR 2039986, DOI 10.4310/SDG.2003.v8.n1.a3
- 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. MR 1683966
- 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 Hr A. Hatcher, Notes on basic $3$–manifold topology, www.math.cornell.edu/hatcher/3M/ 3Mdownloads.html.
- Jürgen Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. A Wiley-Interscience Publication. MR 1100926
- 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. MR 678484, DOI 10.2307/2007026
- 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. MR 924677, DOI 10.2307/1971420 Pe1 G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three–manifolds, math.DG/0307245. Pe2 G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math.DG/0211159. Pe3 G. Perelman, Ricci flow with surgery on three–manifolds, math.DG/0303109.
- 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; With a preface translated from the Chinese by Kaising Tso. MR 1333601
- 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. MR 1474501
Bibliographic Information
- Tobias H. Colding
- Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012
- MR Author ID: 335440
- Email: colding@cims.nyu.edu
- William P. Minicozzi II
- Affiliation: Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218
- MR Author ID: 358534
- Email: minicozz@math.jhu.edu
- Received by editor(s): October 6, 2003
- Published electronically: April 13, 2005
- Additional Notes: The authors were partially supported by NSF Grants DMS 0104453 and DMS 0104187
- © Copyright 2005 American Mathematical Society
- Journal: J. Amer. Math. Soc. 18 (2005), 561-569
- MSC (2000): Primary 53C44; Secondary 53C42, 57M50
- DOI: https://doi.org/10.1090/S0894-0347-05-00486-8
- MathSciNet review: 2138137