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February  2020, 19(2): 723-746. doi: 10.3934/cpaa.2020034

The scalar curvature problem on four-dimensional manifolds

Hichem Chtioui 1, , Hichem Hajaiej 2, and  Marwa Soula 2,

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

Received  October 2018 Revised  July 2019 Published  October 2019

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 [6] and the positive mass theorem of R.Schoen and S.T.Yau [32], we prove compactness and existence results under the assumption that the prescribed function is flat near its critical points. These are the first results on the prescribed scalar curvature problem where no upper-bound condition on the flatness order is assumed.

Keywords: Conformal metrics, scalar curvature, variational method, loss of compactness, critical points at infinity.
Mathematics Subject Classification: Primary: 35J60; Secondary: 58E05.
Citation: Hichem Chtioui, Hichem Hajaiej, Marwa Soula. The scalar curvature problem on four-dimensional manifolds. Communications on Pure and Applied Analysis, 2020, 19 (2) : 723-746. doi: 10.3934/cpaa.2020034
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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