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

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

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.

Citation: Hichem Chtioui, Hichem Hajaiej, Marwa Soula. The scalar curvature problem on four-dimensional manifolds. Communications on Pure & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

A. AmbrosettiJ. 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.  Google Scholar

[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.  Google Scholar

[6]

A. Bahri, Critical Point at Infinity in Some Variational Problems, Pitman Res. Notes Math. Ser 182 Longman Sci. Tech. Harlow 1989.  Google Scholar

[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.  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 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.  Google Scholar

[10]

M. Ben AyedY. ChenH. 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 S3, Annales de l'institut Fourier, 61 (2011), 971-986.  doi: 10.5802/aif.2634.  Google Scholar

[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.  Google Scholar

[13]

S. A. ChangM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[19]

H. ChtiouiR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

Escobar-Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986), 243-254.  doi: 10.1007/BF01389071.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.   Google Scholar

[32]

R. Schoen and S. T. Yau, Proof of the positive action conjecture in quantum relativity, Phys. Rev. Lett., 42 (1979), 547-548.   Google Scholar

[33]

R. Schoen and S. T. Yau, Proof of the positive mass theorem Ⅱ, Comm. Phys., 79 (1981), 231-260.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

A. AmbrosettiJ. 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.  Google Scholar

[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.  Google Scholar

[6]

A. Bahri, Critical Point at Infinity in Some Variational Problems, Pitman Res. Notes Math. Ser 182 Longman Sci. Tech. Harlow 1989.  Google Scholar

[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.  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 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.  Google Scholar

[10]

M. Ben AyedY. ChenH. 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 S3, Annales de l'institut Fourier, 61 (2011), 971-986.  doi: 10.5802/aif.2634.  Google Scholar

[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.  Google Scholar

[13]

S. A. ChangM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[19]

H. ChtiouiR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

Escobar-Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math., 86 (1986), 243-254.  doi: 10.1007/BF01389071.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.   Google Scholar

[32]

R. Schoen and S. T. Yau, Proof of the positive action conjecture in quantum relativity, Phys. Rev. Lett., 42 (1979), 547-548.   Google Scholar

[33]

R. Schoen and S. T. Yau, Proof of the positive mass theorem Ⅱ, Comm. Phys., 79 (1981), 231-260.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

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