Prescribing the scalar curvature problem on higher-dimensional manifolds

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May 2012, 32(5): 1857-1879. doi: 10.3934/dcds.2012.32.1857

Prescribing the scalar curvature problem on higher-dimensional manifolds

Randa Ben Mahmoud ^1, and Hichem Chtioui ^2,

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

Received November 2010 Revised July 2011 Published January 2012

Abstract
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In this paper we consider the problem of existence of conformal metrics with prescribed scalar curvature on n-dimensional Riemannian manifolds, $n \geq 5 $. Using precise estimates on the losses of compactness, we characterize the critical points at infinity of the associated variational problem and we prove existence results for curvatures satisfying an assumption of Bahri-Coron type.

Keywords: topological method., critical points at infinity, Variational method, Scalar curvature, loss of compactness.

Mathematics Subject Classification: 58E05, 35J65, 35C21, 35B4.

Citation: Randa Ben Mahmoud, Hichem Chtioui. Prescribing the scalar curvature problem on higher-dimensional manifolds. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1857-1879. doi: 10.3934/dcds.2012.32.1857

References:

[1]	T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,, J. Math. Pures Appl. (9), 55 (1976), 269.
[2]	T. Aubin and A. Bahri, Méthodes de topologie algebrique pour le problème de la courbure scalaire prescrite,, J. Math. Pures Appl. (9), 76 (1997), 525. doi: 10.1016/S0021-7824(97)89961-8.
[3]	T. Aubin and A. Bahri, Une hypothése topologique pour le problème de la courbure scalaire prescrite, (French) [A topological hypothesis for the problem of prescribed scalar curvature],, J. Math. Pures Appl. (9), 76 (1997), 843. doi: 10.1016/S0021-7824(97)89973-4.
[4]	A. Ambrosetti, J. Garcia Azorero and I. Peral, Perturbation of $-\Delta u + u^{\frac{(N+2)}{(N-2)}} = 0$, the scalar curvature problem in $\mathbbR^N$, and related topics,, Journal of Functional Analysis, 165 (1999), 117. doi: 10.1006/jfan.1999.3390.
[5]	A. Bahri, "Critical Point at Infinity in Some Variational Problems,", Pitman Res. Notes Math, 182 (1989).
[6]	A. Bahri, An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimensions,, A celebration of J. F. Nash, 81 (1996), 323. doi: 10.1215/S0012-7094-96-08116-8.
[7]	A. Bahri and H. Brezis, Équations elliptiques non linéaires sur des variétés avec exposant de Sobolev critique,, C. R. Acad. Sci. Paris Sér. I Math., 307 (1988), 537.
[8]	A. Bahri and J.-M. Coron, The scalar curvature problem on the standard three-dimensional spheres,, J. Funct. Anal., 95 (1991), 106. doi: 10.1016/0022-1236(91)90026-2.
[9]	A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of topology of the domain,, Comm. Pure Appl. Math., 41 (1988), 255.
[10]	M. Ben Ayed, Y. Chen, H. Chtioui and M. Hammami, On the prescribed scalar curvature problem on 4-manifolds,, Duke Math. J., 84 (1996), 633. doi: 10.1215/S0012-7094-96-08420-3.
[11]	R. Ben Mahmoud and H. Chtioui, Existence results for the prescribed scalar curvature on $\mathbbS^3$,, Annales de l'Institut Fourier, (2010).
[12]	S.-Y. Chang and P. Yang, A perturbation result in prescribing scalar curvature on $S^n$,, Duke Math. J., 64 (1991), 27. doi: 10.1215/S0012-7094-91-06402-1.
[13]	S.-Y. Chang, M. J. Gursky and P. C. Yang, The scalar curvature equation on 2- and 3-spheres,, Calc. Var. Partial Differential Equations, 1 (1993), 205.
[14]	C.-C. Chen and C.-S. Lin, Estimates of the conformal scalar curvature equation via the method of moving planes,, Comm. Pure Appl. Math., 50 (1997), 971. doi: 10.1002/(SICI)1097-0312(199710)50:10<971::AID-CPA2>3.0.CO;2-D.
[15]	C.-C. Chen and C.-S. Lin, Estimates of the conformal scalar curvature equation via the method of moving planes. II,, J. Differential Geom., 49 (1998), 115.
[16]	C.-C. Chen and C.-S. Lin, Prescribing scalar curvature on $S^n$. I: A priori estimates,, J. Differential Geometry, 57 (2001), 67.
[17]	H. Chtioui, Prescribing the scalar curvature problem on three and four manifolds,, Advanced Nonlinear Studies, 3 (2003), 457.
[18]	A. Hatcher, "Algebraic Topology,", Campbridge University Press, (2002).
[19]	J. Kazdan and F. W. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures,, Annals of Math. (2), 101 (1975), 317. doi: 10.2307/1970993.
[20]	J. Lee and T. Parker, The Yamabe problem,, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37.
[21]	Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I,, Journal of Differential Equations, 120 (1995), 319. doi: 10.1006/jdeq.1995.1115.
[22]	Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems. II: Existence and compactness,, Comm. Pure Appl. Math., 49 (1996), 541. doi: 10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A.
[23]	R. Schoen and D. Zhang, Prescribed scalar curvature on the n-sphere,, Calculus of Variations and Partial Differential Equations, 4 (1996), 1. doi: 10.1007/BF01322307.
[24]	R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature,, J. Differential Geom., 20 (1984), 479.
[25]	R. Schoen, Courses at Stanford University (1988) and New York University (1989),, unpublished., ().
[26]	M. Struwe, "Variational Methods. Applications to Nonlinear PDE and Hamilton Systems,", Springer-Verlag, (1990).
[27]	N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds,, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265.

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References:

[1]	T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,, J. Math. Pures Appl. (9), 55 (1976), 269.
[2]	T. Aubin and A. Bahri, Méthodes de topologie algebrique pour le problème de la courbure scalaire prescrite,, J. Math. Pures Appl. (9), 76 (1997), 525. doi: 10.1016/S0021-7824(97)89961-8.
[3]	T. Aubin and A. Bahri, Une hypothése topologique pour le problème de la courbure scalaire prescrite, (French) [A topological hypothesis for the problem of prescribed scalar curvature],, J. Math. Pures Appl. (9), 76 (1997), 843. doi: 10.1016/S0021-7824(97)89973-4.
[4]	A. Ambrosetti, J. Garcia Azorero and I. Peral, Perturbation of $-\Delta u + u^{\frac{(N+2)}{(N-2)}} = 0$, the scalar curvature problem in $\mathbbR^N$, and related topics,, Journal of Functional Analysis, 165 (1999), 117. doi: 10.1006/jfan.1999.3390.
[5]	A. Bahri, "Critical Point at Infinity in Some Variational Problems,", Pitman Res. Notes Math, 182 (1989).
[6]	A. Bahri, An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimensions,, A celebration of J. F. Nash, 81 (1996), 323. doi: 10.1215/S0012-7094-96-08116-8.
[7]	A. Bahri and H. Brezis, Équations elliptiques non linéaires sur des variétés avec exposant de Sobolev critique,, C. R. Acad. Sci. Paris Sér. I Math., 307 (1988), 537.
[8]	A. Bahri and J.-M. Coron, The scalar curvature problem on the standard three-dimensional spheres,, J. Funct. Anal., 95 (1991), 106. doi: 10.1016/0022-1236(91)90026-2.
[9]	A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of topology of the domain,, Comm. Pure Appl. Math., 41 (1988), 255.
[10]	M. Ben Ayed, Y. Chen, H. Chtioui and M. Hammami, On the prescribed scalar curvature problem on 4-manifolds,, Duke Math. J., 84 (1996), 633. doi: 10.1215/S0012-7094-96-08420-3.
[11]	R. Ben Mahmoud and H. Chtioui, Existence results for the prescribed scalar curvature on $\mathbbS^3$,, Annales de l'Institut Fourier, (2010).
[12]	S.-Y. Chang and P. Yang, A perturbation result in prescribing scalar curvature on $S^n$,, Duke Math. J., 64 (1991), 27. doi: 10.1215/S0012-7094-91-06402-1.
[13]	S.-Y. Chang, M. J. Gursky and P. C. Yang, The scalar curvature equation on 2- and 3-spheres,, Calc. Var. Partial Differential Equations, 1 (1993), 205.
[14]	C.-C. Chen and C.-S. Lin, Estimates of the conformal scalar curvature equation via the method of moving planes,, Comm. Pure Appl. Math., 50 (1997), 971. doi: 10.1002/(SICI)1097-0312(199710)50:10<971::AID-CPA2>3.0.CO;2-D.
[15]	C.-C. Chen and C.-S. Lin, Estimates of the conformal scalar curvature equation via the method of moving planes. II,, J. Differential Geom., 49 (1998), 115.
[16]	C.-C. Chen and C.-S. Lin, Prescribing scalar curvature on $S^n$. I: A priori estimates,, J. Differential Geometry, 57 (2001), 67.
[17]	H. Chtioui, Prescribing the scalar curvature problem on three and four manifolds,, Advanced Nonlinear Studies, 3 (2003), 457.
[18]	A. Hatcher, "Algebraic Topology,", Campbridge University Press, (2002).
[19]	J. Kazdan and F. W. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures,, Annals of Math. (2), 101 (1975), 317. doi: 10.2307/1970993.
[20]	J. Lee and T. Parker, The Yamabe problem,, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37.
[21]	Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I,, Journal of Differential Equations, 120 (1995), 319. doi: 10.1006/jdeq.1995.1115.
[22]	Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems. II: Existence and compactness,, Comm. Pure Appl. Math., 49 (1996), 541. doi: 10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A.
[23]	R. Schoen and D. Zhang, Prescribed scalar curvature on the n-sphere,, Calculus of Variations and Partial Differential Equations, 4 (1996), 1. doi: 10.1007/BF01322307.
[24]	R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature,, J. Differential Geom., 20 (1984), 479.
[25]	R. Schoen, Courses at Stanford University (1988) and New York University (1989),, unpublished., ().
[26]	M. Struwe, "Variational Methods. Applications to Nonlinear PDE and Hamilton Systems,", Springer-Verlag, (1990).
[27]	N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds,, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265.

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