Curvature cones and the Ricci flow.
Séminaire de théorie spectrale et géométrie, Tome 31 (2012-2014), pp. 197-220.

This survey reviews some facts about nonnegativity conditions on the curvature tensor of a Riemannian manifold which are preserved by the action of the Ricci flow. The text focuses on two main points.

First we describe the known examples of preserved curvature conditions and how they have been used to derive geometric results, in particular sphere theorems.

We then describe some recent results which give restrictions on general preserved conditions.

The paper ends with some open questions on these matters.

DOI : 10.5802/tsg.300
Richard, Thomas 1

1 Room 659, Huxley Building Mathematics Department Imperial College London SW7 2AZ (UK)
@article{TSG_2012-2014__31__197_0,
     author = {Richard, Thomas},
     title = {Curvature cones and the {Ricci} flow.},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {197--220},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {31},
     year = {2012-2014},
     doi = {10.5802/tsg.300},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/tsg.300/}
}
TY  - JOUR
AU  - Richard, Thomas
TI  - Curvature cones and the Ricci flow.
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2012-2014
SP  - 197
EP  - 220
VL  - 31
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/tsg.300/
DO  - 10.5802/tsg.300
LA  - en
ID  - TSG_2012-2014__31__197_0
ER  - 
%0 Journal Article
%A Richard, Thomas
%T Curvature cones and the Ricci flow.
%J Séminaire de théorie spectrale et géométrie
%D 2012-2014
%P 197-220
%V 31
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/tsg.300/
%R 10.5802/tsg.300
%G en
%F TSG_2012-2014__31__197_0
Richard, Thomas. Curvature cones and the Ricci flow.. Séminaire de théorie spectrale et géométrie, Tome 31 (2012-2014), pp. 197-220. doi : 10.5802/tsg.300. http://www.numdam.org/articles/10.5802/tsg.300/

[1] Andrews, Ben; Hopper, Christopher The Ricci flow in Riemannian geometry, Lecture Notes in Mathematics, 2011, Springer, Heidelberg, 2011, pp. xviii+296 (A complete proof of the differentiable 1/4-pinching sphere theorem) | DOI | MR | Zbl

[2] Bessières, Laurent; Besson, Gérard; Maillot, Sylvain; Boileau, Michel; Porti, Joan Geometrisation of 3-manifolds, EMS Tracts in Mathematics, 13, European Mathematical Society (EMS), Zürich, 2010, pp. x+237 | DOI | MR | Zbl

[3] Böhm, Christoph; Wilking, Burkhard Manifolds with positive curvature operators are space forms, Ann. of Math. (2), Volume 167 (2008) no. 3, pp. 1079-1097 | DOI | MR | Zbl

[4] Brendle, Simon Ricci flow and the sphere theorem, Graduate Studies in Mathematics, 111, American Mathematical Society, Providence, RI, 2010, pp. viii+176 | DOI | MR | Zbl

[5] Brendle, Simon; Schoen, Richard Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc., Volume 22 (2009) no. 1, pp. 287-307 | DOI | MR | Zbl

[6] Chen, Haiwen Pointwise 1 4-pinched 4-manifolds, Ann. Global Anal. Geom., Volume 9 (1991) no. 2, pp. 161-176 | DOI | MR | Zbl

[7] Chow, Bennett; Knopf, Dan The Ricci flow: an introduction, Mathematical Surveys and Monographs, 110, American Mathematical Society, Providence, RI, 2004, pp. xii+325 | DOI | MR | Zbl

[8] Chow, Bennett; Lu, Peng; Ni, Lei Hamilton’s Ricci flow, Graduate Studies in Mathematics, 77, American Mathematical Society, Providence, RI; Science Press, New York, 2006, pp. xxxvi+608 | DOI | MR | Zbl

[9] Gromov, M. Sign and geometric meaning of curvature, Rend. Sem. Mat. Fis. Milano, Volume 61 (1991), p. 9-123 (1994) | DOI | MR | Zbl

[10] Gururaja, H. A.; Maity, Soma; Seshadri, Harish On Wilking’s criterion for the Ricci flow, Math. Z., Volume 274 (2013) no. 1-2, pp. 471-481 | DOI | MR | Zbl

[11] Hamilton, Richard S. Three-manifolds with positive Ricci curvature, J. Differential Geom., Volume 17 (1982) no. 2, pp. 255-306 http://projecteuclid.org/euclid.jdg/1214436922 | MR | Zbl

[12] Hamilton, Richard S. Four-manifolds with positive curvature operator, J. Differential Geom., Volume 24 (1986) no. 2, pp. 153-179 http://projecteuclid.org/euclid.jdg/1214440433 | MR | Zbl

[13] Huisken, Gerhard Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom., Volume 21 (1985) no. 1, pp. 47-62 http://projecteuclid.org/euclid.jdg/1214439463 | MR | Zbl

[14] Kleiner, Bruce; Lott, John Notes on Perelman’s papers, Geom. Topol., Volume 12 (2008) no. 5, pp. 2587-2855 | DOI | MR | Zbl

[15] Margerin, Christophe Une caractérisation optimale de la structure différentielle standard de la sphère en terme de courbure pour (presque) toutes les dimensions. I. Les énoncés, C. R. Acad. Sci. Paris Sér. I Math., Volume 319 (1994) no. 6, pp. 605-607 | MR | Zbl

[16] Máximo, Davi Non-negative Ricci curvature on closed manifolds under Ricci flow, Proc. Amer. Math. Soc., Volume 139 (2011) no. 2, pp. 675-685 | DOI | MR | Zbl

[17] Micallef, Mario J.; Moore, John Douglas Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2), Volume 127 (1988) no. 1, pp. 199-227 | DOI | MR | Zbl

[18] Morgan, John; Tian, Gang Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs, 3, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007, pp. xlii+521 | MR | Zbl

[19] Nguyen, Huy T. Isotropic curvature and the Ricci flow, Int. Math. Res. Not. IMRN (2010) no. 3, pp. 536-558 | DOI | MR | Zbl

[20] Perelman, Grisha The entropy formula for the Ricci flow and its geometric applications (http://arxiv.org/abs/math/0211159) | Zbl

[21] Perelman, Grisha Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (http://arxiv.org/abs/math/0307245) | Zbl

[22] Perelman, Grisha Ricci flow with surgery on three-manifolds (http://arxiv.org/abs/math/0303109) | Zbl

[23] Richard, Thomas; Seshadri, Harish Noncoercive Ricci flow invariant curvature cones (http://arxiv.org/abs/1308.1190)

[24] Richard, Thomas; Seshadri, Harish Positive isotropic curvature and self-duality in dimension 4 (http://arxiv.org/abs/1311.5256)

[25] Topping, Peter Lectures on the Ricci flow, London Mathematical Society Lecture Note Series, 325, Cambridge University Press, Cambridge, 2006, pp. x+113 | DOI | MR | Zbl

[26] Wilking, Burkhard A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities, J. Reine Angew. Math., Volume 679 (2013), pp. 223-247 | MR

Cité par Sources :