Remarks on compact shrinking Ricci solitons of dimension four

Li Ma

doi:10.1016/j.crma.2013.10.006

Geometry

Remarks on compact shrinking Ricci solitons of dimension four
[Remarques sur les solitons de Ricci contractant, compacts, de dimension quatre]

Présenté par : the Editorial Board

Li Ma ^1,²

¹ Department of Mathematics, Henan Normal University, Xinxiang 453007, China
² Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

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

Résumé
Abstract

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 :

\int_{M} s d v_{g} = 4 ρ vol (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 :

\int_{M} s^{2} ⩽ 24 ρ^{2} vol (M),

alors

2 χ (M) \pm 3 τ (M) ⩾ 0 .

(2) Si

(M, g)

est une surface de Kähler naturellement orientée, alors on a :

2 χ (M) + 3 τ (M) = \frac{ρ^{2} vol (M)}{2 π^{2}} .

En fait, nous montrons que lʼhypothèse de (1) ci-dessus est équivalente à :

\int_{M} σ_{2} (Rc - \frac{s}{6} g) ⩾ 0,

avec

σ_{2} (Rc - \frac{s}{6} g) : = σ_{2} (g)

la seconde fonction symétrique des valeurs propres de la matrice

Rc - \frac{s}{6} g

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:

\int_{M} s d v_{g} = 4 ρ vol (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:

\int_{M} s^{2} ⩽ 24 ρ^{2} vol (M),

then

2 χ (M) \pm 3 τ (M) ⩾ 0 .

(2) If

(M, g)

is a natural oriented Kähler surface, then we have:

2 χ (M) + 3 τ (M) = \frac{ρ^{2} vol (M)}{2 π^{2}} .

Actually, we shall show that the assumption in (1) above is equivalent to the fact that:

\int_{M} σ_{2} (Rc - \frac{s}{6} g) ⩾ 0 .

Here

σ_{2} (A) : = σ_{2} (g)

is the 2nd symmetric function of the eigenvalues of the matrix

A : = Rc - \frac{s}{6} g

Reçu le : 2012-12-17
Accepté le : 2013-10-09
Publié le : 2013-10-30

DOI : 10.1016/j.crma.2013.10.006

Affiliations des auteurs :

Li Ma ^{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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     title = {Remarks on compact shrinking {Ricci} solitons of dimension four},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {817--823},
     publisher = {Elsevier},
     volume = {351},
     number = {21-22},
     year = {2013},
     doi = {10.1016/j.crma.2013.10.006},
     language = {en},
}

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Li Ma. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.10.006/

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Cité par

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Cité par 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.

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