Dans cet article, on passe en revue certains résultats dus à Miles Simon sur le flot de Ricci de certains espaces métriques de dimension 3 exposés dans [28] et [26].
On commence par voir le lien entre théorèmes de rigidité et convergence des variétés sur un exemple dû à Berger et Durumeric. On remarque ensuite que pour obtenir de tels théorèmes de rigidité en utilisant le flot de Ricci, il faut être capable de construire le flot pour des espaces peu lisses.
Les deux dernières partie sont consacrées à une explication de la construction de tels flots (en suivant [28] et [26]) et à des applications géométriques de cette construction.
In this article, we review some results of Miles Simon about the Ricci flow of some 3-dimensional metric spaces. These results are from [26] and [28]. We first explain the link between rigidity theorems and convergence of manifolds on an example from Berger and Durumeric. Then, we notice that in order to obtain such rigidity theorems using Ricci flow, one needs to build a Ricci flow for potentially non-smooth spaces. The last two sections expose how to construct such flows (following [26] and [28]) and give some geometric applications of this construction.
Mot clés : courbure de Ricci minorée, flot de Ricci, convergence au sens de Gromov-Hausdorff, dimension 3
Mots clés : Ricci curvature bounded from below, Ricci flow, Gromov-Hausdorff convergence, dimension 3
@article{TSG_2009-2010__28__121_0, author = {Richard, Thomas}, title = {Suites de flots de {Ricci} en dimension 3 et applications}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {121--145}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {28}, year = {2009-2010}, doi = {10.5802/tsg.281}, language = {fr}, url = {http://www.numdam.org/articles/10.5802/tsg.281/} }
TY - JOUR AU - Richard, Thomas TI - Suites de flots de Ricci en dimension 3 et applications JO - Séminaire de théorie spectrale et géométrie PY - 2009-2010 SP - 121 EP - 145 VL - 28 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/tsg.281/ DO - 10.5802/tsg.281 LA - fr ID - TSG_2009-2010__28__121_0 ER -
%0 Journal Article %A Richard, Thomas %T Suites de flots de Ricci en dimension 3 et applications %J Séminaire de théorie spectrale et géométrie %D 2009-2010 %P 121-145 %V 28 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/tsg.281/ %R 10.5802/tsg.281 %G fr %F TSG_2009-2010__28__121_0
Richard, Thomas. Suites de flots de Ricci en dimension 3 et applications. Séminaire de théorie spectrale et géométrie, Tome 28 (2009-2010), pp. 121-145. doi : 10.5802/tsg.281. http://www.numdam.org/articles/10.5802/tsg.281/
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