[Les solitons de Ricci contractants complets sont de type fini]
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.
Accepté le :
Publié le :
@article{CRMATH_2008__346_11-12_653_0,
author = {Fu-quan Fang and Jian-wen Man and Zhen-lei Zhang},
title = {Complete gradient shrinking {Ricci} solitons have finite topological type},
journal = {Comptes Rendus. Math\'ematique},
pages = {653--656},
publisher = {Elsevier},
volume = {346},
number = {11-12},
year = {2008},
doi = {10.1016/j.crma.2008.03.021},
language = {en},
}
TY - JOUR AU - Fu-quan Fang AU - Jian-wen Man AU - Zhen-lei Zhang TI - Complete gradient shrinking Ricci solitons have finite topological type JO - Comptes Rendus. Mathématique PY - 2008 SP - 653 EP - 656 VL - 346 IS - 11-12 PB - Elsevier DO - 10.1016/j.crma.2008.03.021 LA - en ID - CRMATH_2008__346_11-12_653_0 ER -
Fu-quan Fang; Jian-wen Man; Zhen-lei Zhang. Complete gradient shrinking Ricci solitons have finite topological type. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 653-656. doi : 10.1016/j.crma.2008.03.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.03.021/
[1] A theorem of Myers, Duke Math. J., Volume 24 (1957), pp. 345-348
[2] Gauss densities and stability for some Ricci solitons | arXiv
[3] A Myers-type theorem and compact Ricci solitons, Proc. Amer. Math. Soc., Volume 134 (2006), pp. 3645-3648
[4] A remark on compact Ricci solitons, Math. Ann., Volume 340 (2008), pp. 893-896
[5] Examples of complete manifolds with positive Ricci curvature, J. Differential Geom., Volume 21 (1985) no. 2, pp. 195-211
[6] On extensions of Myers' theorem, Bull. London Math. Soc., Volume 27 (1995), pp. 392-396
[7] Some geometric properties of the Bakry–Émery–Ricci tensor, Comment. Math. Helv., Volume 78 (2003), pp. 865-883
[8] A note on curvature and fundamental group, J. Differential Geometry, Volume 2 (1968), pp. 1-7
[9] Comparison geometry for the Bakry–Émery Ricci tensor | arXiv
[10] Complete shrinking Ricci solitons have finite fundamental group, Proc. Amer. Math. Soc., Volume 136 (2008), pp. 1803-1806
[11] On the finiteness of the fundamental group of a compact shrinking Ricci soliton, Colloq. Math., Volume 107 (2007), pp. 297-299
Cité par Sources :
Commentaires - Politique
Lower bounds for the scalar curvatures of noncompact gradient Ricci solitons
Bennett Chow; Peng Lu; Bo Yang
C. R. Math (2011)
Compact blow-up limits of finite time singularities of Ricci flow are shrinking Ricci solitons
Zhei-lei Zhang
C. R. Math (2007)