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

Authors:
Tobias H. Colding and William P. Minicozzi II

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

Published electronically:
April 13, 2005

MathSciNet review:
2138137

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the Ricci flow becomes extinct in finite time on any Riemannian -manifold without aspherical summands.

**1.**D. Christodoulou and S.T. Yau,*Some remarks on the quasi-local mass*, Mathematics and general relativity (Santa Cruz, CA, 1986), 9-14, Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988. MR**0954405 (89k:83050)****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. MR**2039986**, https://doi.org/10.4310/SDG.2003.v8.n1.a3**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. MR**1683966****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. MR**1375255****5.**A. Hatcher,*Notes on basic -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. MR**1100926****7.**W. Meeks III, L. Simon, and S.T. Yau,*Embedded minimal surfaces, exotic spheres and manifolds with positive Ricci curvature*, Ann. of Math. (2)**116**(1982) 621-659. MR**0678484 (84f:53053)****8.**M.J. Micallef and J.D. Moore,*Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes*, Ann. of Math. (2)**127**(1988) 199-227. MR**0924677 (89e:53088)****9.**G. Perelman,*Finite extinction time for the solutions to the Ricci flow on certain three-manifolds*, math.DG/0307245.**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.**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. MR**1333601****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. MR**1474501**

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2000):
53C44,
53C42,
57M50

Retrieve articles in all journals with MSC (2000): 53C44, 53C42, 57M50

Additional Information

**Tobias H. Colding**

Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012

Email:
colding@cims.nyu.edu

**William P. Minicozzi II**

Affiliation:
Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218

Email:
minicozz@math.jhu.edu

DOI:
https://doi.org/10.1090/S0894-0347-05-00486-8

Keywords:
Ricci flow,
finite extinction,
$3$-manifolds,
min--max surfaces

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

Article copyright:
© Copyright 2005
American Mathematical Society