Suites de flots de Ricci en dimension 3 et applications

Richard, Thomas

Séminaire de théorie spectrale et géométrie, Tome 28 (2009-2010), pp. 121-145.

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.

Classification : 53C44, 53C20, 53C23
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

[1] Abresch, Uwe; Meyer, Wolfgang T. A sphere theorem with a pinching constant below $\frac{1}{4}$, J. Differential Geom., Volume 44 (1996) no. 2, pp. 214-261 | MR | Zbl

[2] Anderson, Michael T.; Cheeger, Jeff Diffeomorphism finiteness for manifolds with Ricci curvature and $L^{n/2}$-norm of curvature bounded, Geom. Funct. Anal., Volume 1 (1991) no. 3, pp. 231-252 | MR | Zbl

[3] Anderson, Michael T.; Cheeger, Jeff $C^{\alpha }$-compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom., Volume 35 (1992) no. 2, pp. 265-281 | MR | Zbl

[4] Berger, M. Les variétés Riemanniennes $(1/4)$-pincées, Ann. Scuola Norm. Sup. Pisa (3), Volume 14 (1960), pp. 161-170 | Numdam | MR | Zbl

[5] Berger, Marcel Sur les variétés riemanniennes pincées juste au-dessous de $1/4$, Ann. Inst. Fourier (Grenoble), Volume 33 (1983) no. 2, p. 135-150 (loose errata) | Numdam | MR | Zbl

[6] Berger, Marcel A panoramic view of Riemannian geometry, Springer-Verlag, Berlin, 2003 | MR | Zbl

[7] Burago, Dmitri; Burago, Yuri; Ivanov, Sergei A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001 | MR

[8] Cheeger, Jeff Finiteness theorems for Riemannian manifolds, Amer. J. Math., Volume 92 (1970), pp. 61-74 | MR | Zbl

[9] Cheeger, Jeff; Colding, Tobias H. On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom., Volume 46 (1997) no. 3, pp. 406-480 | MR | Zbl

[10] Chen, Bing-Long; Zhu, Xi-Ping Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geom., Volume 74 (2006) no. 1, pp. 119-154 | MR | Zbl

[11] Chow, Bennett; Knopf, Dan The Ricci flow : an introduction, Mathematical Surveys and Monographs, 110, American Mathematical Society, Providence, RI, 2004 | MR

[12] Chow, Bennett; Lu, Peng; Ni, Lei Hamilton’s Ricci flow, Graduate Studies in Mathematics, 77, American Mathematical Society, Providence, RI, 2006 | MR

[13] Colding, Tobias H. Ricci curvature and volume convergence, Ann. of Math. (2), Volume 145 (1997) no. 3, pp. 477-501 | MR | Zbl

[14] Durumeric, O. A generalization of Berger’s theorem on almost $\frac{1}{4}$-pinched manifolds. II, J. Differential Geom., Volume 26 (1987) no. 1, pp. 101-139 | MR | Zbl

[15] Greene, R. E.; Wu, H. Lipschitz convergence of Riemannian manifolds, Pacific J. Math., Volume 131 (1988) no. 1, pp. 119-141 | MR | Zbl

[16] Gromov, Misha Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, 152, Birkhäuser Boston Inc., Boston, MA, 1999 Based on the 1981 French original [ MR0682063 (85e :53051)], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates | MR | Zbl

[17] Hamilton, Richard S. Three-manifolds with positive Ricci curvature, J. Differential Geom., Volume 17 (1982) no. 2, pp. 255-306 | MR | Zbl

[18] Hamilton, Richard S. A compactness property for solutions of the Ricci flow, Amer. J. Math., Volume 117 (1995) no. 3, pp. 545-572 | MR | Zbl

[19] Klingenberg, Wilhelm Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung, Comment. Math. Helv., Volume 35 (1961), pp. 47-54 | MR | Zbl

[20] Perelman, G. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (ArXiv Mathematics e-prints arXiv :math/0307245, 2003) | Zbl

[21] Perelman, G. Ricci flow with surgery on three-manifolds (ArXiv Mathematics e-prints arXiv :math/0303109, 2003) | Zbl

[22] Perelman, G. The entropy formula for the Ricci flow and its geometric applications (ArXiv Mathematics e-prints arXiv :math/0211159, 2002) | Zbl

[23] Perelman, G. Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers, Comparison geometry (Berkeley, CA, 1993–94) (Math. Sci. Res. Inst. Publ.), Volume 30, Cambridge Univ. Press, Cambridge, 1997, pp. 157-163 | MR | Zbl

[24] Shi, Wan-Xiong Complete noncompact three-manifolds with nonnegative Ricci curvature, J. Differential Geom., Volume 29 (1989) no. 2, pp. 353-360 | MR | Zbl

[25] Shi, Wan-Xiong Deforming the metric on complete Riemannian manifolds, J. Differential Geom., Volume 30 (1989) no. 1, pp. 223-301 | MR | Zbl

[26] Simon, M. Ricci flow of non-collapsed 3-manifolds whose Ricci curvature is bounded from below (ArXiv e-prints arXiv :math/0903.2142, 2009) | MR

[27] Simon, Miles Deformation of $C^0$ Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom., Volume 10 (2002) no. 5, pp. 1033-1074 | MR | Zbl

[28] Simon, Miles Ricci flow of almost non-negatively curved three manifolds, J. Reine Angew. Math., Volume 630 (2009), pp. 177-217 | MR | Zbl

[29] Weinstein, Alan On the homotopy type of positively-pinched manifolds, Arch. Math. (Basel), Volume 18 (1967), pp. 523-524 | MR | Zbl