Suites de flots de Ricci en dimension 3 et applications

Thomas Richard[1]

  • [1] Université Grenoble 1 Institut Fourier 100 rue des Maths BP 74 8402 St Martin d’Hères cedex (France)

Séminaire de théorie spectrale et géométrie (2009-2010)

  • Volume: 28, page 121-145
  • ISSN: 1624-5458

Abstract

top
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.

How to cite

top

Richard, Thomas. "Suites de flots de Ricci en dimension 3 et applications." Séminaire de théorie spectrale et géométrie 28 (2009-2010): 121-145. <http://eudml.org/doc/116460>.

@article{Richard2009-2010,
abstract = {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.},
affiliation = {Université Grenoble 1 Institut Fourier 100 rue des Maths BP 74 8402 St Martin d’Hères cedex (France)},
author = {Richard, Thomas},
journal = {Séminaire de théorie spectrale et géométrie},
keywords = {Ricci curvature bounded from below; Ricci flow; Gromov-Hausdorff convergence; dimension 3},
language = {fre},
pages = {121-145},
publisher = {Institut Fourier},
title = {Suites de flots de Ricci en dimension 3 et applications},
url = {http://eudml.org/doc/116460},
volume = {28},
year = {2009-2010},
}

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
PB - Institut Fourier
VL - 28
SP - 121
EP - 145
AB - 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.
LA - fre
KW - Ricci curvature bounded from below; Ricci flow; Gromov-Hausdorff convergence; dimension 3
UR - http://eudml.org/doc/116460
ER -

References

top
  1. Uwe Abresch, Wolfgang T. Meyer, A sphere theorem with a pinching constant below 1 4 , J. Differential Geom. 44 (1996), 214-261 Zbl0873.53024MR1425576
  2. Michael T. Anderson, Jeff Cheeger, Diffeomorphism finiteness for manifolds with Ricci curvature and L n / 2 -norm of curvature bounded, Geom. Funct. Anal. 1 (1991), 231-252 Zbl0764.53026MR1118730
  3. Michael T. Anderson, Jeff Cheeger, C α -compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom. 35 (1992), 265-281 Zbl0774.53021MR1158336
  4. M. Berger, Les variétés Riemanniennes ( 1 / 4 ) -pincées, Ann. Scuola Norm. Sup. Pisa (3) 14 (1960), 161-170 Zbl0096.15502MR140054
  5. Marcel Berger, Sur les variétés riemanniennes pincées juste au-dessous de 1 / 4 , Ann. Inst. Fourier (Grenoble) 33 (1983), 135-150 (loose errata) Zbl0497.53044MR699491
  6. Marcel Berger, A panoramic view of Riemannian geometry, (2003), Springer-Verlag, Berlin Zbl1038.53002MR2002701
  7. Dmitri Burago, Yuri Burago, Sergei Ivanov, A course in metric geometry, 33 (2001), American Mathematical Society, Providence, RI Zbl1232.53037MR1835418
  8. Jeff Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61-74 Zbl0194.52902MR263092
  9. Jeff Cheeger, Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), 406-480 Zbl0902.53034MR1484888
  10. Bing-Long Chen, Xi-Ping Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geom. 74 (2006), 119-154 Zbl1104.53032MR2260930
  11. Bennett Chow, Dan Knopf, The Ricci flow : an introduction, 110 (2004), American Mathematical Society, Providence, RI Zbl1086.53085MR2061425
  12. Bennett Chow, Peng Lu, Lei Ni, Hamilton’s Ricci flow, 77 (2006), American Mathematical Society, Providence, RI MR2274812
  13. Tobias H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 (1997), 477-501 Zbl0879.53030MR1454700
  14. O. Durumeric, A generalization of Berger’s theorem on almost 1 4 -pinched manifolds. II, J. Differential Geom. 26 (1987), 101-139 Zbl0599.53039MR892033
  15. R. E. Greene, H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988), 119-141 Zbl0646.53038MR917868
  16. Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, 152 (1999), Birkhäuser Boston Inc., Boston, MA Zbl0953.53002MR1699320
  17. Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255-306 Zbl0504.53034MR664497
  18. Richard S. Hamilton, A compactness property for solutions of the Ricci flow, Amer. J. Math. 117 (1995), 545-572 Zbl0840.53029MR1333936
  19. Wilhelm Klingenberg, Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung, Comment. Math. Helv. 35 (1961), 47-54 Zbl0133.15005MR139120
  20. G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds Zbl1130.53003
  21. G. Perelman, Ricci flow with surgery on three-manifolds Zbl1130.53002
  22. G. Perelman, The entropy formula for the Ricci flow and its geometric applications Zbl1130.53001
  23. G. Perelman, Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers, Comparison geometry (Berkeley, CA, 1993–94) 30 (1997), 157-163, Cambridge Univ. Press, Cambridge Zbl0890.53038MR1452872
  24. Wan-Xiong Shi, Complete noncompact three-manifolds with nonnegative Ricci curvature, J. Differential Geom. 29 (1989), 353-360 Zbl0668.53026MR982179
  25. Wan-Xiong Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), 223-301 Zbl0676.53044MR1001277
  26. M. Simon, Ricci flow of non-collapsed 3-manifolds whose Ricci curvature is bounded from below Zbl1239.53085MR2526789
  27. Miles Simon, Deformation of C 0 Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom. 10 (2002), 1033-1074 Zbl1034.58008MR1957662
  28. Miles Simon, Ricci flow of almost non-negatively curved three manifolds, J. Reine Angew. Math. 630 (2009), 177-217 Zbl1165.53046MR2526789
  29. Alan Weinstein, On the homotopy type of positively-pinched manifolds, Arch. Math. (Basel) 18 (1967), 523-524 Zbl0166.17601MR220311

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.