[Flot de Ricci de variétés de dimension 3 non-effondrées : Deux applications]
Dans cette Note, on donne deux applications simples de résultats dûs à Miles Simon sur le flot de Ricci des variétés de dimension 3 non-effondrées. On montre dʼabord un nouveau théorème de finitude à difféomorphisme près pour les variétés de dimension 3 à courbure de Ricci minorée, diamètre majoré et volume minoré. Ensuite, on donne une nouvelle preuve dʼun résultat dû à Cheeger et Colding. Si une suite de variétés compactes de dimension 3 à courbure de Ricci minorée converge au sens de Gromov–Hausdorff vers une une variété compacte de dimension 3, alors tout les éléments de la suite sont difféomorphes à la variété limite à partir dʼun certain rang.
In this short Note, we give two simple applications of results of Miles Simon about the Ricci flow of non-collapsed 3-manifolds. First, we prove a new diffeomorphism finiteness result for 3-manifolds with Ricci curvature bounded from below, volume bounded from below and diameter bounded from above. Second, we give an alternate proof of a theorem of Cheeger and Colding. Namely, we prove that if a sequence of compact 3-manifolds with Ricci curvature bounded from below Gromov–Hausdorff converges to a compact 3-manifold M, then all the ʼs are diffeomorphic to M for i large enough.
Accepté le :
Publié le :
@article{CRMATH_2011__349_9-10_567_0,
author = {Richard, Thomas},
title = {Ricci flow of non-collapsed 3-manifolds: {Two} applications},
journal = {Comptes Rendus. Math\'ematique},
pages = {567--569},
publisher = {Elsevier},
volume = {349},
number = {9-10},
year = {2011},
doi = {10.1016/j.crma.2011.03.009},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.03.009/}
}
TY - JOUR AU - Richard, Thomas TI - Ricci flow of non-collapsed 3-manifolds: Two applications JO - Comptes Rendus. Mathématique PY - 2011 SP - 567 EP - 569 VL - 349 IS - 9-10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.03.009/ DO - 10.1016/j.crma.2011.03.009 LA - en ID - CRMATH_2011__349_9-10_567_0 ER -
%0 Journal Article %A Richard, Thomas %T Ricci flow of non-collapsed 3-manifolds: Two applications %J Comptes Rendus. Mathématique %D 2011 %P 567-569 %V 349 %N 9-10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.03.009/ %R 10.1016/j.crma.2011.03.009 %G en %F CRMATH_2011__349_9-10_567_0
Richard, Thomas. Ricci flow of non-collapsed 3-manifolds: Two applications. Comptes Rendus. Mathématique, Tome 349 (2011) no. 9-10, pp. 567-569. doi : 10.1016/j.crma.2011.03.009. http://www.numdam.org/articles/10.1016/j.crma.2011.03.009/
[1] Diffeomorphism finiteness for manifolds with Ricci curvature and -norm of curvature bounded, Geom. Funct. Anal., Volume 1 (1991) no. 3, pp. 231-252
[2] -compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom., Volume 35 (1992) no. 2, pp. 265-281
[3] A Panoramic View of Riemannian Geometry, Springer-Verlag, Berlin, 2003
[4] Finiteness theorems for Riemannian manifolds, Amer. J. Math., Volume 92 (1970), pp. 61-74
[5] On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom., Volume 46 (1997) no. 3, pp. 406-480
[6] Hamiltonʼs Ricci Flow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI, 2006
[7] Ricci curvature and volume convergence, Ann. of Math. (2), Volume 145 (1997) no. 3, pp. 477-501
[8] Three-manifolds with positive Ricci curvature, J. Differential Geom., Volume 17 (1982) no. 2, pp. 255-306
[9] A compactness property for solutions of the Ricci flow, Amer. J. Math., Volume 117 (1995) no. 3, pp. 545-572
[10] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, ArXiv Mathematics e-prints, Nov. 2002.
[11] M. Simon, Ricci flow of non-collapsed 3-manifolds whose Ricci curvature is bounded from below, ArXiv e-prints, Mar. 2009.
[12] On the homotopy type of positively-pinched manifolds, Arch. Math. (Basel), Volume 18 (1967), pp. 523-524
Cité par Sources :
