Comptes Rendus
Differential Geometry
A uniform Sobolev inequality for Ricci flow with surgeries and applications
Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 549-552.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.03.016

Qi S. Zhang 1

1 Department of Mathematics, University of California, Riverside, CA 92521, USA
@article{CRMATH_2008__346_9-10_549_0,
     author = {Qi S. Zhang},
     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},
}
TY  - JOUR
AU  - Qi S. Zhang
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
DO  - 10.1016/j.crma.2008.03.016
LA  - en
ID  - CRMATH_2008__346_9-10_549_0
ER  - 
%0 Journal Article
%A Qi S. Zhang
%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
%R 10.1016/j.crma.2008.03.016
%G en
%F CRMATH_2008__346_9-10_549_0
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/

[1] T. Aubin Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry, Volume 11 (1976) no. 4, pp. 573-598 (in French)

[2] K. Bruce; J. Lott Notes on Perelman's papers http://arXiv.org/math.DG/0605667v1 (May 25, 2006)

[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

Cited by Sources:

Comments - Politique