On the prescribed scalar curvature problem on $ {S}^{n}$: The degree zero case

Randa Ben Mahmoud; Hichem Chtioui; Afef Rigane

doi:10.1016/j.crma.2012.06.012

Partial Differential Equations/Differential Geometry

On the prescribed scalar curvature problem on

S^{n}

: The degree zero case
[Sur le problème de courbure scalaire prescrite sur

S^{n}

: Le cas de degré zéro]

Présenté par : Jean-Pierre Demailly

Randa Ben Mahmoud ¹ ; Hichem Chtioui ² ; Afef Rigane ¹

¹ Department of Mathematics, Faculty of Sciences of Sfax, Route of Soukra, Sfax, Tunisia
² Department of Mathematics, King Abdulaziz University, P.O. 80230, Jeddah, Saudi Arabia

Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 583-586.

Résumé
Abstract

Dans cette Note nous considérons le problème dʼexistence de métriques conformes avec courbure scalaire prescrite, sur la sphère standard $S^{n}$ , $n ⩾ 3$ . Nous donnons de nouveaux résultats dʼexistence et de multiplicité reposant sur un nouveau type de formule dʼEuler–Hopf. Nos arguments ont également lʼavantage dʼétendre des résultats bien connus de Y. Li (1995) [10].

In this Note, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere $S^{n}$ , $n ⩾ 3$ . We give new existence and multiplicity results based on a new Euler–Hopf formula type. Our argument also has the advantage of extending the well known results due to Y. Li (1995) [10].

Reçu le : 2012-06-02
Accepté le : 2012-06-27
Publié le : 2012-06-01

DOI : 10.1016/j.crma.2012.06.012

Affiliations des auteurs :

Randa Ben Mahmoud ¹ ; Hichem Chtioui ² ; Afef Rigane ¹

¹ Department of Mathematics, Faculty of Sciences of Sfax, Route of Soukra, Sfax, Tunisia
² Department of Mathematics, King Abdulaziz University, P.O. 80230, Jeddah, Saudi Arabia

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     author = {Randa Ben Mahmoud and Hichem Chtioui and Afef Rigane},
     title = {On the prescribed scalar curvature problem on $ {S}^{n}$: {The} degree zero case},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {583--586},
     publisher = {Elsevier},
     volume = {350},
     number = {11-12},
     year = {2012},
     doi = {10.1016/j.crma.2012.06.012},
     language = {en},
}

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Randa Ben Mahmoud; Hichem Chtioui; Afef Rigane. On the prescribed scalar curvature problem on $ {S}^{n}$: The degree zero case. Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 583-586. doi : 10.1016/j.crma.2012.06.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.06.012/

Bibliographie
Cité par

[1] A. Ambrosetti; M. Badiale Homoclinics: Poincaré–Melnikov type results via variational approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 15 (1998), pp. 233-252

[2] A. Ambrosetti; J. Garcia Azorero; A. Peral Perturbation of $- Δ u + u^{\frac{(N + 2)}{(N - 2)}} = 0$ , the scalar curvature problem in $R^{N}$ and related topics, J. Funct. Anal., Volume 165 (1999), pp. 117-149

[3] A. Bahri Critical Point at Infinity in Some Variational Problems, Pitman Res. Notes Math., vol. 182, Longman Sci. Tech., Harlow, 1989

[4] A. Bahri An invariant for Yamabe-type flows with applications to scalar curvature problems in high dimensions, Duke Math. J., Volume 81 (1996), pp. 323-466

[5] R. Ben Mahmoud; H. Chtioui Existence results for the prescribed scalar curvature on $S^{3}$ , Ann. Inst. Fourier, Volume 61 (2011), pp. 971-986

[6] R. Ben Mahmoud; H. Chtioui Prescribing the scalar curvature problem on higher-dimensional manifolds, Discrete Contin. Dyn. Syst. A, Volume 32 (2012) no. 5, pp. 1857-1879

[7] R. Ben Mahmoud, H. Chtioui, A. Rigane, On the prescribed scalar curvature on $S^{n}$ : The degree zero case, in preparation.

[8] A. Chang; P. Yang A perturbation result in prescribing scalar curvature on $S^{n}$ , Duke Math. J., Volume 64 (1991), pp. 27-69

[9] J. Kazdan; J. Warner Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature, Ann. Math., Volume 101 (1975), pp. 317-331

[10] Y.Y. Li Prescribing scalar curvature on $S^{n}$ and related topics, Part I, J. Diff. Eq., Volume 120 (1995), pp. 319-410

[11] Y.Y. Li Prescribing scalar curvature on $S^{n}$ and related topics, Part II: Existence and compactness, Comm. Pure Appl. Math., Volume 49 (1996), pp. 541-579

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