Geometry
Remarks on compact shrinking Ricci solitons of dimension four
Comptes Rendus. Mathématique, Volume 351 (2013) no. 21-22, pp. 817-823.

In this paper, we study the topological restriction of gradient shrinking Ricci solitons (M,g) of dimension 4. Let s be the scalar curvature of the metric g. Then we have:

Msdvg=4ρvol(M),
where ρ>0 is the shrinking constant and vol(M) is the volume of (M,g). We also have two kinds of topology results. (1) If we assume that:
Ms224ρ2vol(M),
then
2χ(M)±3τ(M)0.
(2) If (M,g) is a natural oriented Kähler surface, then we have:
2χ(M)+3τ(M)=ρ2vol(M)2π2.
Actually, we shall show that the assumption in (1) above is equivalent to the fact that:
Mσ2(Rcs6g)0.
Here σ2(A):=σ2(g) is the 2nd symmetric function of the eigenvalues of the matrix A:=Rcs6g.

Nous étudions dans cette Note la restriction topologique des solitons de Ricci (M,g) contractant le gradient, de dimension 4. Soit s la courbure scalaire de la métrique g, alors on a :

Msdvg=4ρvol(M),
ρ>0 est la constante de contraction et vol(M) le volume de (M,g). Nous obtenons également deux types de résultats topologiques. (1) En supposant :
Ms224ρ2vol(M),
alors
2χ(M)±3τ(M)0.
(2) Si (M,g) est une surface de Kähler naturellement orientée, alors on a :
2χ(M)+3τ(M)=ρ2vol(M)2π2.
En fait, nous montrons que lʼhypothèse de (1) ci-dessus est équivalente à :
Mσ2(Rcs6g)0,
avec σ2(Rcs6g):=σ2(g) la seconde fonction symétrique des valeurs propres de la matrice Rcs6g.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.10.006
Ma, Li 1, 2

1 Department of Mathematics, Henan Normal University, Xinxiang 453007, China
2 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
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Ma, Li. Remarks on compact shrinking Ricci solitons of dimension four. Comptes Rendus. Mathématique, Volume 351 (2013) no. 21-22, pp. 817-823. doi : 10.1016/j.crma.2013.10.006. http://www.numdam.org/articles/10.1016/j.crma.2013.10.006/

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Cited by Sources:

The research is partially supported by the National Natural Science Foundation of China No. 11271111 and SRFDP 20090002110019. The version here is a revised version after a suggestion from Prof. C. LeBrun during Muenster conference in August of 2006.