Ricci flow of non-collapsed 3-manifolds: Two applications

Thomas Richard

doi:10.1016/j.crma.2011.03.009

Differential Geometry

Ricci flow of non-collapsed 3-manifolds: Two applications
[Flot de Ricci de variétés de dimension 3 non-effondrées : Deux applications]

Présenté par : Jean-Pierre Demailly

Thomas Richard ¹

¹ Institut Fourier, 100, rue des Maths, 38402 St Martin dʼHères, France

Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 567-569.

Résumé
Abstract

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 $M_{i}$ of compact 3-manifolds with Ricci curvature bounded from below Gromov–Hausdorff converges to a compact 3-manifold M, then all the $M_{i}$ ʼs are diffeomorphic to M for i large enough.

Reçu le : 2010-11-04
Accepté le : 2011-03-10
Publié le : 2011-04-01

DOI : 10.1016/j.crma.2011.03.009

Affiliations des auteurs :

Thomas Richard ¹

¹ Institut Fourier, 100, rue des Maths, 38402 St Martin dʼHères, France

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Thomas Richard. Ricci flow of non-collapsed 3-manifolds: Two applications. Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 567-569. doi : 10.1016/j.crma.2011.03.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.03.009/

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