Differential Geometry
Complete gradient shrinking Ricci solitons have finite topological type
[Les solitons de Ricci contractants complets sont de type fini]
Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 653-656.

Dans cette Note nous montrons qu'une variété riemanienne complète est de type topologique fini – c'est-à-dire qu'elle est homéomorphe à une variété compacte à bord – si son tenseur de Bakry–Emery–Ricci est bornée inférieurement par une constante positive et vérifie l'une des conditions suivantes :

(i) la courbure de Ricci est bornée supérieurement.

(ii) la courbure de Ricci est bornée inférieurement et le rayon d'injectivité est positif.

De plus, un soliton de Ricci contractant complet est de type topologique fini si sa coubure scalaire est bornée.

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry–Émery Ricci tensor has a positive lower bound, and either of the following conditions:

(i) the Ricci curvature is bounded from above;

(ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero.

Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.03.021
Fang, Fu-quan 1 ; Man, Jian-wen 2 ; Zhang, Zhen-lei 2

1 Department of Mathematics, Capital Normal University, Beijing, PR China
2 Chern Institute of Mathematics, Weijin Road 94, Tianjin 300071, PR China
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Fang, Fu-quan; Man, Jian-wen; Zhang, Zhen-lei. Complete gradient shrinking Ricci solitons have finite topological type. Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 653-656. doi : 10.1016/j.crma.2008.03.021. http://www.numdam.org/articles/10.1016/j.crma.2008.03.021/

[1] Ambrose, W. A theorem of Myers, Duke Math. J., Volume 24 (1957), pp. 345-348

[2] Cao, H.D.; Hamilton, R.; Ilmanen, T. Gauss densities and stability for some Ricci solitons | arXiv

[3] Derdzinski, A. A Myers-type theorem and compact Ricci solitons, Proc. Amer. Math. Soc., Volume 134 (2006), pp. 3645-3648

[4] Fernández-López, M.; García-Río, E. A remark on compact Ricci solitons, Math. Ann., Volume 340 (2008), pp. 893-896

[5] Gromoll, D.; Meyer, W.T. Examples of complete manifolds with positive Ricci curvature, J. Differential Geom., Volume 21 (1985) no. 2, pp. 195-211

[6] Li, X.M. On extensions of Myers' theorem, Bull. London Math. Soc., Volume 27 (1995), pp. 392-396

[7] Lott, J. Some geometric properties of the Bakry–Émery–Ricci tensor, Comment. Math. Helv., Volume 78 (2003), pp. 865-883

[8] Milnor, J. A note on curvature and fundamental group, J. Differential Geometry, Volume 2 (1968), pp. 1-7

[9] Wei, G.F.; Wylie, W. Comparison geometry for the Bakry–Émery Ricci tensor | arXiv

[10] Wylie, W. Complete shrinking Ricci solitons have finite fundamental group, Proc. Amer. Math. Soc., Volume 136 (2008), pp. 1803-1806

[11] Zhang, Z.L. On the finiteness of the fundamental group of a compact shrinking Ricci soliton, Colloq. Math., Volume 107 (2007), pp. 297-299

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