[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.
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@article{CRMATH_2008__346_11-12_653_0,
author = {Fang, Fu-quan and Man, Jian-wen and Zhang, Zhen-lei},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.03.021/}
}
TY - JOUR AU - Fang, Fu-quan AU - Man, Jian-wen AU - Zhang, Zhen-lei 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 UR - http://www.numdam.org/articles/10.1016/j.crma.2008.03.021/ DO - 10.1016/j.crma.2008.03.021 LA - en ID - CRMATH_2008__346_11-12_653_0 ER -
%0 Journal Article %A Fang, Fu-quan %A Man, Jian-wen %A Zhang, Zhen-lei %T Complete gradient shrinking Ricci solitons have finite topological type %J Comptes Rendus. Mathématique %D 2008 %P 653-656 %V 346 %N 11-12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.03.021/ %R 10.1016/j.crma.2008.03.021 %G en %F CRMATH_2008__346_11-12_653_0
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/
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