Comptes Rendus
no. 9-10
Differential Geometry
A uniform Sobolev inequality for Ricci flow with surgeries and applications
[Une inégalité de Sobolev uniforme pour le flot de Ricci avec chirurgie et applications]

Présenté par : Thierry Aubin

Qi S. Zhang1
1 Department of Mathematics, University of California, Riverside, CA 92521, USA
Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 549-552.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.03.016
Qi S. Zhang 1

1 Department of Mathematics, University of California, Riverside, CA 92521, USA 
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Qi S. Zhang. A uniform Sobolev inequality for Ricci flow with surgeries and applications. Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 549-552. doi : 10.1016/j.crma.2008.03.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.03.016/

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[3] H.-D. Cao; X.-P. Zhu 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] E. Hebey 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] E. Hebey; M. Vaugon 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] E. Hebey; M. Vaugon 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] J.W. Morgan; G. Tian Ricci flow and the Poincaré conjecture, 25 July, 2006 http://arXiv.org/math.DG/0607607v1

[8] G. Perelman The Entropy formula for the Ricci flow and its geometric applications, 11 Nov. 2002 http://arXiv.org/math.DG/0211159v1

[9] G. Perelman Ricci flow with surgery on three manifolds http://arXiv.org/math.DG/0303109

[10] Q.S. Zhang Addendum to: A uniform Sobolev inequality under Ricci flow, Int. Math. Res. Notices, Volume 138 (2007), pp. 1-12

[11] Q.S. Zhang Strong non-collapsing and uniform Sobolev inequalities for Ricci flow with surgeries (submitted for publication) | arXiv

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