[Une inégalité de Sobolev uniforme pour le flot de Ricci avec chirurgie et applications]
Nous prouvons une inégalité de Sobolev uniforme pour le flot de Ricci, indépendante du nombre de chirurgies. Comme application, nous établissons, avec moins d'hypothèses, un résultat de non-explosion plus fort que celui de Perelman sur la non-explosion de κ avec chirurgie. La preuve est plus courte et semble plus accessible. Le résultat améliore également des résultats antérieurs où l'inégalité de Sobolev dépendait du nombre de chirurgies.
We prove a uniform Sobolev inequality for Ricci flow, which is independent of the number of surgeries. As an application, under less assumptions, a noncollapsing result stronger than Perelman's κ noncollapsing with surgery is derived. The proof is much shorter and seems more accessible. The result also improves some earlier ones where the Sobolev inequality depended on the number of surgeries.
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@article{CRMATH_2008__346_9-10_549_0,
author = {Zhang, Qi S.},
title = {A uniform {Sobolev} inequality for {Ricci} flow with surgeries and applications},
journal = {Comptes Rendus. Math\'ematique},
pages = {549--552},
publisher = {Elsevier},
volume = {346},
number = {9-10},
year = {2008},
doi = {10.1016/j.crma.2008.03.016},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.03.016/}
}
TY - JOUR AU - Zhang, Qi S. TI - A uniform Sobolev inequality for Ricci flow with surgeries and applications JO - Comptes Rendus. Mathématique PY - 2008 SP - 549 EP - 552 VL - 346 IS - 9-10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.03.016/ DO - 10.1016/j.crma.2008.03.016 LA - en ID - CRMATH_2008__346_9-10_549_0 ER -
%0 Journal Article %A Zhang, Qi S. %T A uniform Sobolev inequality for Ricci flow with surgeries and applications %J Comptes Rendus. Mathématique %D 2008 %P 549-552 %V 346 %N 9-10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.03.016/ %R 10.1016/j.crma.2008.03.016 %G en %F CRMATH_2008__346_9-10_549_0
Zhang, Qi S. A uniform Sobolev inequality for Ricci flow with surgeries and applications. Comptes Rendus. Mathématique, Tome 346 (2008) no. 9-10, pp. 549-552. doi : 10.1016/j.crma.2008.03.016. http://www.numdam.org/articles/10.1016/j.crma.2008.03.016/
[1] Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry, Volume 11 (1976) no. 4, pp. 573-598 (in French)
[2] Notes on Perelman's papers http://arXiv.org/math.DG/0605667v1 (May 25, 2006)
[3] A complete proof of Poincare and geometrization conjectures-application of the Hamilton–Perelman theory of the Ricci flow, Asian J. Math., Volume 10 (June 2006) no. 2, pp. 165-492
[4] Optimal Sobolev inequalities on complete Riemannian manifolds with Ricci curvature bounded below and positive injectivity radius, Amer. J. Math., Volume 118 (1996) no. 2, pp. 291-300
[5] Meilleures constantes dans le théoréme d'inclusion de Sobolev, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 13 (1996) no. 1, pp. 57-93 (in French)
[6] The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J., Volume 79 (1995) no. 1, pp. 235-279
[7] Ricci flow and the Poincaré conjecture, 25 July, 2006 http://arXiv.org/math.DG/0607607v1
[8] The Entropy formula for the Ricci flow and its geometric applications, 11 Nov. 2002 http://arXiv.org/math.DG/0211159v1
[9] Ricci flow with surgery on three manifolds http://arXiv.org/math.DG/0303109
[10] Addendum to: A uniform Sobolev inequality under Ricci flow, Int. Math. Res. Notices, Volume 138 (2007), pp. 1-12
[11] Strong non-collapsing and uniform Sobolev inequalities for Ricci flow with surgeries (submitted for publication) | arXiv
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