Existence results for the prescribed Scalar curvature on S 3
Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 971-986.

This paper is devoted to the existence of conformal metrics on S 3 with prescribed scalar curvature. We extend well known existence criteria due to Bahri-Coron.

Ce papier est consacré à l’existence des métriques conforme sur S 3 avec courbure scalaire prescrite. Nous étendons les critères d’existence bien connus de Bahri-Coron.

DOI: 10.5802/aif.2634
Classification: 58E05,  35J65,  35C21,  35B40
Keywords: Scalar curvature, critical points at infinity, topological method
Mahmoud, Randa Ben 1; Chtioui, Hichem 

1 Faculté des Sciences de Sfax Département de Mathématiques Route Soukra 3018 Sfax (Tunisie)
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Mahmoud, Randa Ben; Chtioui, Hichem. Existence results for the prescribed Scalar curvature on $S^{3}$. Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 971-986. doi : 10.5802/aif.2634. http://archive.numdam.org/articles/10.5802/aif.2634/

[1] Aubin, T. Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures et Appl., Volume 55 (1976), pp. 269-296 | MR | Zbl

[2] Aubin, T.; Bahri, A. Une hypothèse topologique pour le problème de la courbure scalaire prescrite. (French) [A topological hypothesis for the problem of prescribed scalar curvature], J. Math. Pures Appl., Volume 76 (1997) no. 10, pp. 843-850 | DOI | MR | Zbl

[3] Bahri, A. Critical point at infinity in some variational problems, Pitman Res. Notes Math, Ser, 182, Longman Sci. Tech., Harlow, 1989 | MR | Zbl

[4] Bahri, A. An invariant for yamabe-type flows with applications to scalar curvature problems in high dimensions, A celebration of J. F. Nash Jr., Duke Math. J., Volume 81 (1996), pp. 323-466 | DOI | MR | Zbl

[5] Bahri, A.; Coron, J. M. The scalar curvature problem on the standard three dimensional spheres, J. Funct. Anal., Volume 95 (1991), pp. 106-172 | DOI | MR | Zbl

[6] Bahri, A.; Rabinowitz, P. Periodic orbits of hamiltonian systems of three body type, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 8 (1991), pp. 561-649 | Numdam | MR | Zbl

[7] Ben Ayed, M.; Chen, Y.; Chtioui, H.; Hammami, M. On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J., Volume 84 (1996), pp. 633-677 | DOI | MR | Zbl

[8] Chang, S. A.; Gursky, M. J.; Yang, P. C. The scalar curvature equation on 2 and 3 spheres, Calc. Var., Volume 1 (1993), pp. 205-229 | DOI | MR | Zbl

[9] Chang, S. Y.; Yang, P. A perturbation result in prescribing scalar curvature on S n , Duke Math. J., Volume 64 (1991), pp. 27-69 Addendum 71 (1993), p. 333–335 | DOI | MR | Zbl

[10] Chen, C. C.; Lin, C. S. Prescribing scalar curvature on S n , Part I: Apriori estimates, J. differential geometry, Volume 57 (2001), pp. 67-171 | MR | Zbl

[11] Chtioui, H. Prescribing the Scalar Curvature Problem on Three and Four Manifolds, Advanced Nonlinear Studies, Volume 3 (2003), pp. 457-470 | MR | Zbl

[12] Conley, C. C. Isolated invariant sets and the Morse index, CBMS Reg. conf-Series in Math, 38, AMS, 1978 | MR | Zbl

[13] Escobar, J.; Schoen, R. Conformal metrics with prescribed scalar curvature, Inventiones Math., Volume 86 (1986), pp. 243-254 | DOI | MR | Zbl

[14] Floer, A. Cuplength Estimates on Lagrangian intersections, Comm. Pure and Applied Math, Volume XLII (1989) no. 4, pp. 335-356 | DOI | MR | Zbl

[15] Kazdan, J.; Warner, J. Existence and conformal deformations of metrics with prescribed Gaussian and scalar curvature, Annals of Math., Volume 101 (1975), pp. 317-331 | DOI | MR | Zbl

[16] Li, Y. Y. Prescribing scalar curvature on S 3 , S 4 and related problems, J. Functional Analysis, Volume 118 (1993), pp. 43-118 | DOI | MR | Zbl

[17] Li, Y. Y. Prescribing scalar curvature on S n and related topics, Part I, Journal of Differential Equations, Volume 120 (1995), pp. 319-410 | DOI | MR | Zbl

[18] Li, Y. Y. Prescribing scalar curvature on S n and related topics, Part II: existence and compactness, Comm. Pure Appl. Math., Volume 49 (1996), pp. 541-579 | DOI | MR | Zbl

[19] Lin, C. S. On Liouville theorem and apriori estimates for the scalar curvature equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci, Volume 4 (1998), pp. 107-130 | Numdam | MR | Zbl

[20] Lions, P. L. The concentration compactness principle in the calculus of variations. The limit case, Rev. Mat. Iberoamericana, Volume 1 (1985), p. I: 165-201, II: 45–121 | MR | Zbl

[21] Milnor, J. Lectures on the h-cobordism theorem, Princeton University Press, 1965 | MR | Zbl

[22] Schoen, R. Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., Volume 20 (1984), pp. 479-495 | MR | Zbl

[23] Schoen, R.; Zhang, D. Prescribed scalar curvature on the n-sphere, Calculus of Variations and Partial Differential Equations, Volume 4 (1996), pp. 1-25 | DOI | MR | Zbl

[24] Struwe, M. A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z., Volume 187 (1984), pp. 511-517 | DOI | MR | Zbl

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