Geodesic
Manifolds with 1/4-pinched curvature are space forms
Brendle, Simon1 ; Schoen, Richard1
1 Department of Mathematics, Stanford University, Stanford, California 94305
Journal of the American Mathematical Society, Tome 22 (2009) no. 1, pp. 287-307

Voir la notice de l'article provenant de la source American Mathematical Society

Let $(M,g_0)$ be a compact Riemannian manifold with pointwise $1/4$-pinched sectional curvatures. We show that the Ricci flow deforms $g_0$ to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Böhm and Wilking.
DOI : 10.1090/S0894-0347-08-00613-9

Brendle, Simon 1 ; Schoen, Richard 1

1 Department of Mathematics, Stanford University, Stanford, California 94305 
@article{10_1090_S0894_0347_08_00613_9,
     author = {Brendle, Simon and Schoen, Richard},
     title = {Manifolds with 1/4-pinched curvature are space forms},
     journal = {Journal of the American Mathematical Society},
     pages = {287--307},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2009},
     doi = {10.1090/S0894-0347-08-00613-9},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-08-00613-9/}
}
TY  - JOUR
AU  - Brendle, Simon
AU  - Schoen, Richard
TI  - Manifolds with 1/4-pinched curvature are space forms
JO  - Journal of the American Mathematical Society
PY  - 2009
SP  - 287
EP  - 307
VL  - 22
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-08-00613-9/
DO  - 10.1090/S0894-0347-08-00613-9
ID  - 10_1090_S0894_0347_08_00613_9
ER  - 
%0 Journal Article
%A Brendle, Simon
%A Schoen, Richard
%T Manifolds with 1/4-pinched curvature are space forms
%J Journal of the American Mathematical Society
%D 2009
%P 287-307
%V 22
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-08-00613-9/
%R 10.1090/S0894-0347-08-00613-9
%F 10_1090_S0894_0347_08_00613_9
Brendle, Simon; Schoen, Richard. Manifolds with 1/4-pinched curvature are space forms. Journal of the American Mathematical Society, Tome 22 (2009) no. 1, pp. 287-307. doi: 10.1090/S0894-0347-08-00613-9

[1] Berger, M. Les variétés Riemanniennes (1/4)-pincées Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 1960 161 170

[2] Brendle, Simon, Schoen, Richard M. Classification of manifolds with weakly 1/4-pinched curvatures Acta Math. 2008 1 13

[3] Chen, Bing-Long, Zhu, Xi-Ping Ricci flow with surgery on four-manifolds with positive isotropic curvature J. Differential Geom. 2006 177 264

[4] Chen, Haiwen Pointwise \frac14-pinched 4-manifolds Ann. Global Anal. Geom. 1991 161 176

[5] Fraser, Ailana M. Fundamental groups of manifolds with positive isotropic curvature Ann. of Math. (2) 2003 345 354

[6] Gromoll, Detlef Differenzierbare Strukturen und Metriken positiver Krümmung auf Sphären Math. Ann. 1966 353 371

[7] Grove, Karsten, Karcher, Hermann, Ruh, Ernst A. Group actions and curvature Invent. Math. 1974 31 48

[8] Grove, Karsten, Karcher, Hermann, Ruh, Ernst A. Jacobi fields and Finsler metrics on compact Lie groups with an application to differentiable pinching problems Math. Ann. 1974 7 21

[9] Hamilton, Richard S. Three-manifolds with positive Ricci curvature J. Differential Geometry 1982 255 306

[10] Hamilton, Richard S. Four-manifolds with positive curvature operator J. Differential Geom. 1986 153 179

[11] Hamilton, Richard S. The formation of singularities in the Ricci flow 1995 7 136

[12] Hamilton, Richard S. Four-manifolds with positive isotropic curvature Comm. Anal. Geom. 1997 1 92

[13] Huisken, Gerhard Ricci deformation of the metric on a Riemannian manifold J. Differential Geom. 1985 47 62

[14] Im Hof, Hans-Christoph, Ruh, Ernst A. An equivariant pinching theorem Comment. Math. Helv. 1975 389 401

[15] Karcher, Hermann A short proof of Berger’s curvature tensor estimates Proc. Amer. Math. Soc. 1970 642 644

[16] Klingenberg, Wilhelm Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung Comment. Math. Helv. 1961 47 54

[17] Margerin, Christophe Pointwise pinched manifolds are space forms 1986 307 328

[18] 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) 1988 199 227

[19] Micallef, Mario J., Wang, Mckenzie Y. Metrics with nonnegative isotropic curvature Duke Math. J. 1993 649 672

[20] Nishikawa, Seiki Deformation of Riemannian metrics and manifolds with bounded curvature ratios 1986 343 352

[21] Rauch, H. E. A contribution to differential geometry in the large Ann. of Math. (2) 1951 38 55

[22] Ruh, Ernst A. Krümmung und differenzierbare Struktur auf Sphären. II Math. Ann. 1973 113 129

[23] Ruh, Ernst A. Riemannian manifolds with bounded curvature ratios J. Differential Geometry 1982

[24] Sugimoto, M., Shiohama, K. On the differentiable pinching problem Math. Ann. 1971 1 16

Cité par Sources :