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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

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, 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), 269-296.  Google Scholar

[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), 525-549. doi: 10.1016/S0021-7824(97)89961-8.  Google Scholar

[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), 843-850. doi: 10.1016/S0021-7824(97)89973-4.  Google Scholar

[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), 117-149. doi: 10.1006/jfan.1999.3390.  Google Scholar

[5]

A. Bahri, "Critical Point at Infinity in Some Variational Problems," Pitman Res. Notes Math, Ser., 182, Longman Sci. Tech., Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1989.  Google Scholar

[6]

A. Bahri, 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., 81 (1996), 323-466. doi: 10.1215/S0012-7094-96-08116-8.  Google Scholar

[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), 537-576.  Google Scholar

[8]

A. Bahri and J.-M. Coron, The scalar curvature problem on the standard three-dimensional spheres, J. Funct. Anal., 95 (1991), 106-172. doi: 10.1016/0022-1236(91)90026-2.  Google Scholar

[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), 255-294.  Google Scholar

[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), 633-677. doi: 10.1215/S0012-7094-96-08420-3.  Google Scholar

[11]

R. Ben Mahmoud and H. Chtioui, Existence results for the prescribed scalar curvature on $\mathbbS^3$, Annales de l'Institut Fourier, 2010. Google Scholar

[12]

S.-Y. Chang and P. Yang, A perturbation result in prescribing scalar curvature on $S^n$, Duke Math. J., 64 (1991), 27-69. doi: 10.1215/S0012-7094-91-06402-1.  Google Scholar

[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), 205-229.  Google Scholar

[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), 971-1017. doi: 10.1002/(SICI)1097-0312(199710)50:10<971::AID-CPA2>3.0.CO;2-D.  Google Scholar

[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), 115-178.  Google Scholar

[16]

C.-C. Chen and C.-S. Lin, Prescribing scalar curvature on $S^n$. I: A priori estimates, J. Differential Geometry, 57 (2001), 67-171.  Google Scholar

[17]

H. Chtioui, Prescribing the scalar curvature problem on three and four manifolds, Advanced Nonlinear Studies, 3 (2003), 457-470.  Google Scholar

[18]

A. Hatcher, "Algebraic Topology," Campbridge University Press, Cambridge, 2002.  Google Scholar

[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), 317-331. doi: 10.2307/1970993.  Google Scholar

[20]

J. Lee and T. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91.  Google Scholar

[21]

Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I, Journal of Differential Equations, 120 (1995), 319-410. doi: 10.1006/jdeq.1995.1115.  Google Scholar

[22]

Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems. II: Existence and compactness, Comm. Pure Appl. Math., 49 (1996), 541-597. doi: 10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A.  Google Scholar

[23]

R. Schoen and D. Zhang, Prescribed scalar curvature on the n-sphere, Calculus of Variations and Partial Differential Equations, 4 (1996), 1-25. doi: 10.1007/BF01322307.  Google Scholar

[24]

R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.  Google Scholar

[25]

R. Schoen, Courses at Stanford University (1988) and New York University (1989),, unpublished., ().   Google Scholar

[26]

M. Struwe, "Variational Methods. Applications to Nonlinear PDE and Hamilton Systems," Springer-Verlag, Berlin, 1990.  Google Scholar

[27]

N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265-274.  Google Scholar

show all references

References:
[1]

T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9), 55 (1976), 269-296.  Google Scholar

[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), 525-549. doi: 10.1016/S0021-7824(97)89961-8.  Google Scholar

[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), 843-850. doi: 10.1016/S0021-7824(97)89973-4.  Google Scholar

[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), 117-149. doi: 10.1006/jfan.1999.3390.  Google Scholar

[5]

A. Bahri, "Critical Point at Infinity in Some Variational Problems," Pitman Res. Notes Math, Ser., 182, Longman Sci. Tech., Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1989.  Google Scholar

[6]

A. Bahri, 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., 81 (1996), 323-466. doi: 10.1215/S0012-7094-96-08116-8.  Google Scholar

[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), 537-576.  Google Scholar

[8]

A. Bahri and J.-M. Coron, The scalar curvature problem on the standard three-dimensional spheres, J. Funct. Anal., 95 (1991), 106-172. doi: 10.1016/0022-1236(91)90026-2.  Google Scholar

[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), 255-294.  Google Scholar

[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), 633-677. doi: 10.1215/S0012-7094-96-08420-3.  Google Scholar

[11]

R. Ben Mahmoud and H. Chtioui, Existence results for the prescribed scalar curvature on $\mathbbS^3$, Annales de l'Institut Fourier, 2010. Google Scholar

[12]

S.-Y. Chang and P. Yang, A perturbation result in prescribing scalar curvature on $S^n$, Duke Math. J., 64 (1991), 27-69. doi: 10.1215/S0012-7094-91-06402-1.  Google Scholar

[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), 205-229.  Google Scholar

[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), 971-1017. doi: 10.1002/(SICI)1097-0312(199710)50:10<971::AID-CPA2>3.0.CO;2-D.  Google Scholar

[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), 115-178.  Google Scholar

[16]

C.-C. Chen and C.-S. Lin, Prescribing scalar curvature on $S^n$. I: A priori estimates, J. Differential Geometry, 57 (2001), 67-171.  Google Scholar

[17]

H. Chtioui, Prescribing the scalar curvature problem on three and four manifolds, Advanced Nonlinear Studies, 3 (2003), 457-470.  Google Scholar

[18]

A. Hatcher, "Algebraic Topology," Campbridge University Press, Cambridge, 2002.  Google Scholar

[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), 317-331. doi: 10.2307/1970993.  Google Scholar

[20]

J. Lee and T. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91.  Google Scholar

[21]

Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I, Journal of Differential Equations, 120 (1995), 319-410. doi: 10.1006/jdeq.1995.1115.  Google Scholar

[22]

Y. Y. Li, Prescribing scalar curvature on $S^n$ and related problems. II: Existence and compactness, Comm. Pure Appl. Math., 49 (1996), 541-597. doi: 10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A.  Google Scholar

[23]

R. Schoen and D. Zhang, Prescribed scalar curvature on the n-sphere, Calculus of Variations and Partial Differential Equations, 4 (1996), 1-25. doi: 10.1007/BF01322307.  Google Scholar

[24]

R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.  Google Scholar

[25]

R. Schoen, Courses at Stanford University (1988) and New York University (1989),, unpublished., ().   Google Scholar

[26]

M. Struwe, "Variational Methods. Applications to Nonlinear PDE and Hamilton Systems," Springer-Verlag, Berlin, 1990.  Google Scholar

[27]

N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265-274.  Google Scholar

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