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Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry
The scalar curvature problem on four-dimensional manifolds
1. | Department of mathematics, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia |
2. | California State University Los Angeles, 5151 University Drive, Los Angeles, Department of mathematics, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia |
We study the problem of existence of conformal metrics with prescribed scalar curvatures on a closed Riemannian $ 4 $-manifold not conformally diffeomorphic to the standard sphere $ S^{4} $. Using the critical points at infinity theory of A.Bahri [
References:
[1] |
M. Ahmedou and H. Chtioui, Conformal metrics of prescribed scalar curvature on 4-manifolds: the degree zero case, Arabian Journal of Mathematics, 6 (memorial Issue in Honor of Professor Abbas Bahri) (2017), 127–136.
doi: 10.1007/s40065-017-0169-1. |
[2] |
T. Aubin,
Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures et Appl., 55 (1976), 269-296.
|
[3] |
T. Aubin and A. Bahri, Méthodes de topologie algébrique pour le problème de la courbure scalaire prescrite, [Methods of algebraic topology for the problem of prescribed scalar curvature], J. Math. Pures Appl., 76 (1997), 525–549.
doi: 10.1016/S0021-7824(97)89961-8. |
[4] |
A. Ambrosetti, J. Garcia Azorero and A. Peral,
Perturbation of $-\Delta u + u^{\frac{(N+2)}{(N-2)}} = 0$, the Scalar Curvature Problem in $\mathbb{R}^N$ and related topics, Journal of Functional Analysis, 165 (1999), 117-149.
doi: 10.1006/jfan.1999.3390. |
[5] |
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. |
[6] |
A. Bahri, Critical Point at Infinity in Some Variational Problems, Pitman Res. Notes Math. Ser 182 Longman Sci. Tech. Harlow 1989. |
[7] |
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 Appli. Math., 41 (1988), 255-294.
doi: 10.1002/cpa.3160410302. |
[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 P. Rabinowitz,
Periodic orbits of hamiltonian systems of three body type, Ann. Inst. H. Poincar, Anal. Non Linéire, 8 (1991), 561-649.
doi: 10.1016/S0294-1449(16)30252-9. |
[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 S3, Annales de l'institut Fourier, 61 (2011), 971-986.
doi: 10.5802/aif.2634. |
[12] |
Ben Mahmoud and H. Chtioui,
Prescribing the scalar curvature problem on higher-dimensional manifolds, Discrete and Continuous Dynamical Systems - Series A, 32 (2012), 1857-1879.
doi: 10.3934/dcds.2012.32.1857. |
[13] |
S. A. Chang, M. J. Gursky and P. C. Yang,
The scalar curvature equation on 2 and 3 spheres, Calc. Var., 1 (1993), 205-229.
doi: 10.1007/BF01191617. |
[14] |
S. A. Chang and P. C. Yang,
A perturbation result in prescribing scalar curvature on Sn, Duke Math. J., 64 (1991), 27-69.
doi: 10.1215/S0012-7094-91-06402-1. |
[15] |
C. C. Chen and C. S. Lin,
Estimates of the scalar curvature via the method of moving planes Ⅰ, 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. |
[16] |
C. C. Chen and C. S. Lin,
Estimates of the scalar curvature via the method of moving planes Ⅱ, J. Diff. Geom., 49 (1998), 115-178.
|
[17] |
C. C. Chen and C. S. Lin,
Prescribing the scalar curvature on Sn, I. A priori estimates, J. Diff. Geom., 57 (2001), 67-171.
|
[18] |
H. Chtioui,
Prescribing the scalar curvature problem on three and four manifolds, Advanced Nonlinear Studies, 3 (2003), 457-470.
doi: 10.1515/ans-2003-0404. |
[19] |
H. Chtioui, R. Ben Mahmoud and D. A. Abuzaid,
Conformal transformation of metrics on the n-sphere, Nonlinear Analysis: TMA, 82 (2013), 66-81.
doi: 10.1016/j.na.2013.01.003. |
[20] |
H. Chtioui and Afef Rigane,
On the prescribed Q-curvature problem on Sn, Journal of Functional Analysis, 261 (2011), 2999-3043.
doi: 10.1016/j.jfa.2011.07.017. |
[21] |
O. Druet,
Generalized scalar curvature type equations on compact Riemannian manifolds, Proc. Roy. Soc. Edinburgh sect. A, 130 (2000), 767-788.
doi: 10.1017/S0308210500000408. |
[22] |
Escobar-Schoen,
Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986), 243-254.
doi: 10.1007/BF01389071. |
[23] |
Min Ji,
Scalar curvature equation on Sn, Part Ⅰ: Topological conditions, J. Diff. Eq., 246 (2009), 749-787.
doi: 10.1016/j.jde.2008.04.011. |
[24] |
J. Kazdan and F. Warner,
Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math., 101 (1975), 317-331.
doi: 10.2307/1970993. |
[25] |
J. M. Lee and M. Parker,
The Yamabe problem, Bull. Am. Math. Soc., 17 (1987), 37-91.
doi: 10.1090/S0273-0979-1987-15514-5. |
[26] |
Y. Y. Li,
Prescribing scalar curvature on Sn and related topics, Part Ⅰ, Journal of Differential Equations, 120 (1995), 319-410.
doi: 10.1006/jdeq.1995.1115. |
[27] |
Y. Y. Li,
Prescribing scalar curvature on Sn and related topics, Part Ⅱ : existence and compactness, Comm. Pure Appl. Math., 49 (1996), 541-579.
doi: 10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A. |
[28] |
Y. Y. Li,
Prescribing scalar curvature on S3, S4 and related problems, J. Functional Analysis, 118 (1993), 43-118.
doi: 10.1006/jfan.1993.1138. |
[29] |
M. Stuwe,
A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[30] |
R. Schoen,
Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom., 20 (1984), 479-495.
|
[31] |
R. Schoen and S. T. Yau,
On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., 65 (1979), 45-76.
|
[32] |
R. Schoen and S. T. Yau,
Proof of the positive action conjecture in quantum relativity, Phys. Rev. Lett., 42 (1979), 547-548.
|
[33] |
R. Schoen and S. T. Yau,
Proof of the positive mass theorem Ⅱ, Comm. Phys., 79 (1981), 231-260.
|
[34] |
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. |
[35] |
K. Sharaf,
On the prescribed scalar curvature problem on Sn: Part1, asymptotic estimates and existence results, Differential Geometry and its Applications, 49 (2016), 423-446.
doi: 10.1016/j.difgeo.2016.09.007. |
[36] |
N. Trudinger,
Remarks concerning the conformal deformation of Riemannian structures on compact-manifolds, Ann. Sc. Norm. Super. Pisa, 3 (1968), 265-274.
|
show all references
References:
[1] |
M. Ahmedou and H. Chtioui, Conformal metrics of prescribed scalar curvature on 4-manifolds: the degree zero case, Arabian Journal of Mathematics, 6 (memorial Issue in Honor of Professor Abbas Bahri) (2017), 127–136.
doi: 10.1007/s40065-017-0169-1. |
[2] |
T. Aubin,
Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures et Appl., 55 (1976), 269-296.
|
[3] |
T. Aubin and A. Bahri, Méthodes de topologie algébrique pour le problème de la courbure scalaire prescrite, [Methods of algebraic topology for the problem of prescribed scalar curvature], J. Math. Pures Appl., 76 (1997), 525–549.
doi: 10.1016/S0021-7824(97)89961-8. |
[4] |
A. Ambrosetti, J. Garcia Azorero and A. Peral,
Perturbation of $-\Delta u + u^{\frac{(N+2)}{(N-2)}} = 0$, the Scalar Curvature Problem in $\mathbb{R}^N$ and related topics, Journal of Functional Analysis, 165 (1999), 117-149.
doi: 10.1006/jfan.1999.3390. |
[5] |
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. |
[6] |
A. Bahri, Critical Point at Infinity in Some Variational Problems, Pitman Res. Notes Math. Ser 182 Longman Sci. Tech. Harlow 1989. |
[7] |
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 Appli. Math., 41 (1988), 255-294.
doi: 10.1002/cpa.3160410302. |
[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 P. Rabinowitz,
Periodic orbits of hamiltonian systems of three body type, Ann. Inst. H. Poincar, Anal. Non Linéire, 8 (1991), 561-649.
doi: 10.1016/S0294-1449(16)30252-9. |
[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 S3, Annales de l'institut Fourier, 61 (2011), 971-986.
doi: 10.5802/aif.2634. |
[12] |
Ben Mahmoud and H. Chtioui,
Prescribing the scalar curvature problem on higher-dimensional manifolds, Discrete and Continuous Dynamical Systems - Series A, 32 (2012), 1857-1879.
doi: 10.3934/dcds.2012.32.1857. |
[13] |
S. A. Chang, M. J. Gursky and P. C. Yang,
The scalar curvature equation on 2 and 3 spheres, Calc. Var., 1 (1993), 205-229.
doi: 10.1007/BF01191617. |
[14] |
S. A. Chang and P. C. Yang,
A perturbation result in prescribing scalar curvature on Sn, Duke Math. J., 64 (1991), 27-69.
doi: 10.1215/S0012-7094-91-06402-1. |
[15] |
C. C. Chen and C. S. Lin,
Estimates of the scalar curvature via the method of moving planes Ⅰ, 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. |
[16] |
C. C. Chen and C. S. Lin,
Estimates of the scalar curvature via the method of moving planes Ⅱ, J. Diff. Geom., 49 (1998), 115-178.
|
[17] |
C. C. Chen and C. S. Lin,
Prescribing the scalar curvature on Sn, I. A priori estimates, J. Diff. Geom., 57 (2001), 67-171.
|
[18] |
H. Chtioui,
Prescribing the scalar curvature problem on three and four manifolds, Advanced Nonlinear Studies, 3 (2003), 457-470.
doi: 10.1515/ans-2003-0404. |
[19] |
H. Chtioui, R. Ben Mahmoud and D. A. Abuzaid,
Conformal transformation of metrics on the n-sphere, Nonlinear Analysis: TMA, 82 (2013), 66-81.
doi: 10.1016/j.na.2013.01.003. |
[20] |
H. Chtioui and Afef Rigane,
On the prescribed Q-curvature problem on Sn, Journal of Functional Analysis, 261 (2011), 2999-3043.
doi: 10.1016/j.jfa.2011.07.017. |
[21] |
O. Druet,
Generalized scalar curvature type equations on compact Riemannian manifolds, Proc. Roy. Soc. Edinburgh sect. A, 130 (2000), 767-788.
doi: 10.1017/S0308210500000408. |
[22] |
Escobar-Schoen,
Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986), 243-254.
doi: 10.1007/BF01389071. |
[23] |
Min Ji,
Scalar curvature equation on Sn, Part Ⅰ: Topological conditions, J. Diff. Eq., 246 (2009), 749-787.
doi: 10.1016/j.jde.2008.04.011. |
[24] |
J. Kazdan and F. Warner,
Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math., 101 (1975), 317-331.
doi: 10.2307/1970993. |
[25] |
J. M. Lee and M. Parker,
The Yamabe problem, Bull. Am. Math. Soc., 17 (1987), 37-91.
doi: 10.1090/S0273-0979-1987-15514-5. |
[26] |
Y. Y. Li,
Prescribing scalar curvature on Sn and related topics, Part Ⅰ, Journal of Differential Equations, 120 (1995), 319-410.
doi: 10.1006/jdeq.1995.1115. |
[27] |
Y. Y. Li,
Prescribing scalar curvature on Sn and related topics, Part Ⅱ : existence and compactness, Comm. Pure Appl. Math., 49 (1996), 541-579.
doi: 10.1002/(SICI)1097-0312(199606)49:6<541::AID-CPA1>3.0.CO;2-A. |
[28] |
Y. Y. Li,
Prescribing scalar curvature on S3, S4 and related problems, J. Functional Analysis, 118 (1993), 43-118.
doi: 10.1006/jfan.1993.1138. |
[29] |
M. Stuwe,
A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[30] |
R. Schoen,
Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom., 20 (1984), 479-495.
|
[31] |
R. Schoen and S. T. Yau,
On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., 65 (1979), 45-76.
|
[32] |
R. Schoen and S. T. Yau,
Proof of the positive action conjecture in quantum relativity, Phys. Rev. Lett., 42 (1979), 547-548.
|
[33] |
R. Schoen and S. T. Yau,
Proof of the positive mass theorem Ⅱ, Comm. Phys., 79 (1981), 231-260.
|
[34] |
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. |
[35] |
K. Sharaf,
On the prescribed scalar curvature problem on Sn: Part1, asymptotic estimates and existence results, Differential Geometry and its Applications, 49 (2016), 423-446.
doi: 10.1016/j.difgeo.2016.09.007. |
[36] |
N. Trudinger,
Remarks concerning the conformal deformation of Riemannian structures on compact-manifolds, Ann. Sc. Norm. Super. Pisa, 3 (1968), 265-274.
|
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