Nonsingular Ricci flow on a noncompact manifold in dimension three

Ma, Li; Zhu, Anqiang

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

Ma, Li ¹ ; Zhu, Anqiang ¹

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

Comptes Rendus. Mathématique, Tome 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 :

Ma, Li ¹ ; Zhu, Anqiang ¹

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

@article{CRMATH_2009__347_3-4_185_0,
     author = {Ma, Li and Zhu, Anqiang},
     title = {Nonsingular {Ricci} flow on a noncompact manifold in dimension three},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {185--190},
     publisher = {Elsevier},
     volume = {347},
     number = {3-4},
     year = {2009},
     doi = {10.1016/j.crma.2008.12.002},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2008.12.002/}
}

TY  - JOUR
AU  - Ma, Li
AU  - Zhu, Anqiang
TI  - Nonsingular Ricci flow on a noncompact manifold in dimension three
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 185
EP  - 190
VL  - 347
IS  - 3-4
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2008.12.002/
DO  - 10.1016/j.crma.2008.12.002
LA  - en
ID  - CRMATH_2009__347_3-4_185_0
ER  -

%0 Journal Article
%A Ma, Li
%A Zhu, Anqiang
%T Nonsingular Ricci flow on a noncompact manifold in dimension three
%J Comptes Rendus. Mathématique
%D 2009
%P 185-190
%V 347
%N 3-4
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2008.12.002/
%R 10.1016/j.crma.2008.12.002
%G en
%F CRMATH_2009__347_3-4_185_0

Ma, Li; Zhu, Anqiang. Nonsingular Ricci flow on a noncompact manifold in dimension three. Comptes Rendus. Mathématique, Tome 347 (2009) no. 3-4, pp. 185-190. doi : 10.1016/j.crma.2008.12.002. http://archive.numdam.org/articles/10.1016/j.crma.2008.12.002/

Bibliographie
Cité par

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :