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Prescribing the scalar curvature problem on higher-dimensional manifolds
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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), 27-69.
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), 205-229. |
[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. |
[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. |
[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. |
[17] |
H. Chtioui, Prescribing the scalar curvature problem on three and four manifolds, Advanced Nonlinear Studies, 3 (2003), 457-470. |
[18] |
A. Hatcher, "Algebraic Topology," Campbridge University Press, Cambridge, 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), 317-331.
doi: 10.2307/1970993. |
[20] |
J. Lee and T. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91. |
[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. |
[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. |
[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. |
[24] |
R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495. |
[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, Berlin, 1990. |
[27] |
N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265-274. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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), 27-69.
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), 205-229. |
[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. |
[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. |
[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. |
[17] |
H. Chtioui, Prescribing the scalar curvature problem on three and four manifolds, Advanced Nonlinear Studies, 3 (2003), 457-470. |
[18] |
A. Hatcher, "Algebraic Topology," Campbridge University Press, Cambridge, 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), 317-331.
doi: 10.2307/1970993. |
[20] |
J. Lee and T. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), 17 (1987), 37-91. |
[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. |
[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. |
[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. |
[24] |
R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495. |
[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, Berlin, 1990. |
[27] |
N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265-274. |
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