Remarks on compact shrinking Ricci solitons of dimension four

Ma, Li

Comptes Rendus. Mathématique, Tome 351 (2013) no. 21-22, pp. 817-823.

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),

où 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 :

∫Ms2⩽24ρ2vol(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(Rc−s6g)⩾0,

avec σ2(Rc−s6g):=σ2(g) la seconde fonction symétrique des valeurs propres de la matrice Rc−s6g.

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:

∫Ms2⩽24ρ2vol(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(Rc−s6g)⩾0.

Here σ2(A):=σ2(g) is the 2nd symmetric function of the eigenvalues of the matrix A:=Rc−s6g.

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