Nonsingular Ricci flow on a noncompact manifold in dimension three

Li Ma; Anqiang Zhu

doi:10.1016/j.crma.2008.12.002

Topology/Geometry

Nonsingular Ricci flow on a noncompact manifold in dimension three
[Flot de Ricci non singulier sur une variété tridimensionnelle non compacte]

Présenté par : Étienne Ghys

Li Ma ¹ ; Anqiang Zhu ¹

¹ Department of Mathematical Sciences, Tsinghua University, Peking 100084, PR China

Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 185-190.

Résumé
Abstract

Nous considérons le flot de Ricci $\frac{\partial}{\partial t} g = - 2 Ric$ sur la variété tridimensionnelle complète de courbure non négatif, c'est-à-dire $Rm ⩾ 0$ et $| Rm (p) | \to 0$ si $d (o, p) \to \infty$ . Nous démontrons que le flot de Ricci sur une telle variété est non singular pour tout temps fini.

We consider the Ricci flow $\frac{\partial}{\partial t} g = - 2 Ric$ on the 3-dimensional complete noncompact manifold $(M, g (0))$ with nonnegative curvature operator, i.e., $Rm ⩾ 0$ , and $| Rm (p) | \to 0$ , as $d (o, p) \to \infty$ . We prove that the Ricci flow on such a manifold is nonsingular in any finite time.

Reçu le : 2008-07-07
Accepté le : 2008-11-03
Publié le : 2009-02-01

DOI : 10.1016/j.crma.2008.12.002

Affiliations des auteurs :

Li Ma ¹ ; Anqiang Zhu ¹

¹ Department of Mathematical Sciences, Tsinghua University, Peking 100084, PR China

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     author = {Li Ma and Anqiang Zhu},
     title = {Nonsingular {Ricci} flow on a noncompact manifold in dimension three},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {185--190},
     publisher = {Elsevier},
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     doi = {10.1016/j.crma.2008.12.002},
     language = {en},
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Li Ma; Anqiang Zhu. Nonsingular Ricci flow on a noncompact manifold in dimension three. Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 185-190. doi : 10.1016/j.crma.2008.12.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.12.002/

Bibliographie
Cité par

[1] A. Chau; L.F. Tam; C. Yu Pseudolocality for the Ricci flow and applications | arXiv

[2] J. Cheeger; D. Ebin Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam, 1975

[3] B. Chow; D. Knopf The Ricci Flow: An Introduction, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, RI, 2004

[4] X. Dai; L. Ma Mass under the Ricci flow, Comm. Math. Phys., Volume 274 (2007) no. 1, pp. 65-80

[5] D. Gromoll; W. Meyer On complete open manifolds of positive curvature, Ann. of Math., Volume 90 (1969) no. 1, pp. 75-90

[6] R.S. Hamilton The formation of singularities in the Ricci flow, Cambridge, MA, 1995, International Press, Cambridge, MA (1995), pp. 7-136

[7] R.S. Hamilton Three-manifolds with positive Ricci curvature, J. Differential Geometry, Volume 17 (1982) no. 2, pp. 255-306

[8] R.S. Hamilton A compactness property for solutions of the Ricci flow, Amer. J. Math., Volume 117 (1995) no. 3, pp. 545-572

[9] B. Kleiner; J. Lott Notes on Perelman's paper | arXiv

[10] J. Morgan; G. Tian Ricci Flow and the Poincaré conjecture | arXiv

[11] T.A. Oliynyk; E. Woolgar Asymptotically flat Ricci flows, 2008 | arXiv

[12] G. Perelman The entropy for Ricci flow and its geometry applications, 2002 | arXiv

[13] G. Perelman Ricci flow with surgery on three-manifolds, 2003 | arXiv

[14] G. Perelman Finite time extinction time for the solutions to the Ricci flow on certain three-manifold, 2003 | arXiv

[15] W.X. Shi Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geometry, Volume 30 (1989), pp. 303-394

[16] R. Ye On the l function and the reduced volume of Perelman, 2004 http://www.math.ucsb.edu/yer/reduced.pdf (Available at)

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