On the prescribed <i>Q</i>-curvature problem on $ {S}^{n}$

Hichem Chtioui; Afef Rigane

doi:10.1016/j.crma.2010.03.018

Partial Differential Equations/Differential Geometry

On the prescribed Q-curvature problem on

S^{n}

Presented by: Pierre-Louis Lions

Hichem Chtioui ¹; Afef Rigane ¹

¹ Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia

Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 635-638.

Abstract
Résumé

In this Note we prescribe a fourth order curvature – the Q-curvature on the standard n-sphere, $n ⩾ 5$ . Under the “flatness condition” of order β, $n - 4 ⩽ β < n$ near each critical point of the prescribed Q-curvature function, we prove new existence result through an Euler–Hopf type formula. Our argument gives a lower bound on the number of conformal metrics having the same Q-curvature.

Dans cette Note nous prescrivons une courbure du quatrième order-la Q-courbure sur la sphère standard de dimension $n ⩾ 5$ . Sous une « condition de platitude » d'ordre $β \in [n - 4, n [$ au voisinage de chaque point critique de la fonction Q-courbure prescrite, nous prouvons un nouveau résultat d'existence grâce à une formule de type Euler–Hopf. Notre argument donne une minoration du nombre des métriques ayant la même Q-courbure.

Received: 2010-03-01
Accepted: 2010-03-29
Published online: 2010-04-29

DOI: 10.1016/j.crma.2010.03.018

Author's affiliations:

Hichem Chtioui ¹; Afef Rigane ¹

¹ Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia

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     author = {Hichem Chtioui and Afef Rigane},
     title = {On the prescribed {\protect\emph{Q}-curvature} problem on $ {S}^{n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {635--638},
     publisher = {Elsevier},
     volume = {348},
     number = {11-12},
     year = {2010},
     doi = {10.1016/j.crma.2010.03.018},
     language = {en},
}

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Hichem Chtioui; Afef Rigane. On the prescribed Q-curvature problem on $ {S}^{n}$. Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 635-638. doi : 10.1016/j.crma.2010.03.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.018/

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