February  2020, 19(2): 715-722. doi: 10.3934/cpaa.2020033

Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry

Department of Mathematics, Sogang University, Seoul 04107, Korea

Received  September 2018 Revised  July 2019 Published  October 2019

Fund Project: The author is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2019041021)

Using the flow method, we prove an existence result for the problem of prescribing the $ Q $-curvature on the even dimensional sphere $ S^n $. More precisely, we prove that there exists a conformal metric on $ S^n $ such that its $ Q $-curvature is $ f $, when $ f $ possesses certain symmetry.

Citation: Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure & Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033
References:
[1]

P. BairdA. Fardoun and R. Regbaoui, The evolution of the scalar curvature of a surface to a prescribed function, Ann. Sc. Norm. Super. Pisa Cl. Sci., 3 (2004), 17-38.   Google Scholar

[2]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math., 138 (1993), 213-242.  doi: 10.2307/2946638.  Google Scholar

[3]

S. Brendle, Convergence of the Q-curvature flow on S4, Adv. Math., 205 (2006), 1-32.  doi: 10.1016/j.aim.2005.07.002.  Google Scholar

[4]

S. Brendle, Global existence and convergence for a higher order flow in conformal geometry, Ann. of Math., 158 (2003), 323-343.  doi: 10.4007/annals.2003.158.323.  Google Scholar

[5]

S. Brendle, Prescribing a higher order conformal invariant on Sn, Comm. Anal. Geom., 11 (2003), 837-858.  doi: 10.4310/CAG.2003.v11.n5.a2.  Google Scholar

[6]

S.-Y. A. ChangM. J. Gursky and P. C. Yang, The scalar curvature equation on 2- and 3-spheres, Calc. Var. Partial Differential Equations, 1 (1993), 205-229.  doi: 10.1007/BF01191617.  Google Scholar

[7]

S.-Y. 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

[8]

S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on S2, J. Differential Geom., 27 (1988), 259-296.   Google Scholar

[9]

S.-Y. A. Chang and P. C. Yang, Extremal metrics of zeta functional determinants on 4-manifolds, Ann. of Math., 142 (1995), 171-212.  doi: 10.2307/2118613.  Google Scholar

[10]

S.-Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on S2, Acta Math., 159 (1987), 215-259.  doi: 10.1007/BF02392560.  Google Scholar

[11]

W. Chen and C. Li, Prescribing scalar curvature on Sn, Pacific J. Math., 199 (2001), 61-78.  doi: 10.2140/pjm.2001.199.61.  Google Scholar

[12]

X. Chen and X. Xu, Q-curvature flow on the standard sphere of even dimension, J. Funct. Anal., 261 (2011), 934-980.  doi: 10.1016/j.jfa.2011.04.005.  Google Scholar

[13]

X. Chen and X. Xu, The scalar curvature flow on Sn–-perturbation theorem revisited, Invent. Math., 187 (2012), 395-506.  doi: 10.1007/s00222-011-0335-6.  Google Scholar

[14]

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

[15]

C. Fefferman and C. R. Graham, Conformal invariants, in, The Mathematical Heritage of Élie Cartan (Lyon, 1984), Astérisque (1985), 95–116.  Google Scholar

[16]

C. Fefferman and C. R. Graham, Q-curvature and Poincaré metrics, Math. Res. Lett., 9 (2002), 139-151.  doi: 10.4310/MRL.2002.v9.n2.a2.  Google Scholar

[17]

Z. C. Han, Prescribing Gaussian curvature on S2, Duke Math. J., 61 (1990), 679-703.  doi: 10.1215/S0012-7094-90-06125-3.  Google Scholar

[18]

Z. C. Han and Y. Y. Li, On the local solvability of the Nirenberg problem on S2, Discrete Contin. Dyn. Syst., 28 (2010), 607-615.  doi: 10.3934/dcds.2010.28.607.  Google Scholar

[19]

P. T. Ho, Prescribed mean curvature equation on the unit ball in the presence of reflection or rotation symmetry, Proc. Roy. Soc. Edinburgh Sect. A, (2017), to appear. doi: 10.1017/prm.2018.40.  Google Scholar

[20]

P. T. Ho, Prescribed Q-curvature flow on Sn, J. Geom. Phys., 62 (2012), 1233-1261.  doi: 10.1016/j.geomphys.2011.11.015.  Google Scholar

[21]

P. T. Ho, Prescribed Webster scalar curvature on S2n+1 in the presence of reflection or rotation symmetry, Bull. Sci. Math., 140 (2016), 506-518.  doi: 10.1016/j.bulsci.2015.06.001.  Google Scholar

[22]

P. T. Ho, Q-curvature flow on Sn, Comm. Anal. Geom., 18 (2010), 791-820.  doi: 10.4310/CAG.2010.v18.n4.a5.  Google Scholar

[23]

P. T. Ho, Results of prescribing Q-curvature on Sn, Arch. Math. (Basel), 100 (2013), 85-93.  doi: 10.1007/s00013-012-0472-1.  Google Scholar

[24]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith, Translations of Mathematical Monographs, 23 (1968) American Mathematical Society, Providence, R.I.  Google Scholar

[25]

M. C. Leung and F. Zhou, Prescribed scalar curvature equation on Sn in the presence of reflection or rotation symmetry, Proc. Amer. Math. Soc., 142 (2014), 1607-1619.  doi: 10.1090/S0002-9939-2014-11993-9.  Google Scholar

[26]

L. Ma and B. Liu, Q-curvature flow with indefinite nonlinearity, C. R. Math. Acad. Sci. Paris, 348 (2010), 403-406.  doi: 10.1016/j.crma.2010.02.014.  Google Scholar

[27]

A. Malchiodi and M. Struwe, Q-curvature flow on S4, J. Differential Geom., 73 (2006), 1-44.   Google Scholar

[28]

J. Moser, On a nonlinear problem in differential geometry, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973,273–280.  Google Scholar

[29]

Q. A. Ngô and H. Zhang, Global existence and convergence of Q-curvature flow on manifolds of even dimension, preprint. Google Scholar

[30]

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

[31]

V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, (Russian), Trudy Mat. Inst. Steklov., 83 (1965), 3-163.   Google Scholar

[32]

M. Struwe, A flow approach to Nirenberg's problem, Duke Math. J., 128 (2005), 19-64.  doi: 10.1215/S0012-7094-04-12812-X.  Google Scholar

[33]

J. C. Wei and X. Xu, On conformal deformations of metrics on Sn, J. Funct. Anal., 157 (1998), 292-325.  doi: 10.1006/jfan.1998.3271.  Google Scholar

[34]

X. Xu and P. C. Yang, Remarks on prescribing Gauss curvature, Trans. Amer. Math. Soc., 336 (1993), 831-840.  doi: 10.2307/2154378.  Google Scholar

show all references

References:
[1]

P. BairdA. Fardoun and R. Regbaoui, The evolution of the scalar curvature of a surface to a prescribed function, Ann. Sc. Norm. Super. Pisa Cl. Sci., 3 (2004), 17-38.   Google Scholar

[2]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math., 138 (1993), 213-242.  doi: 10.2307/2946638.  Google Scholar

[3]

S. Brendle, Convergence of the Q-curvature flow on S4, Adv. Math., 205 (2006), 1-32.  doi: 10.1016/j.aim.2005.07.002.  Google Scholar

[4]

S. Brendle, Global existence and convergence for a higher order flow in conformal geometry, Ann. of Math., 158 (2003), 323-343.  doi: 10.4007/annals.2003.158.323.  Google Scholar

[5]

S. Brendle, Prescribing a higher order conformal invariant on Sn, Comm. Anal. Geom., 11 (2003), 837-858.  doi: 10.4310/CAG.2003.v11.n5.a2.  Google Scholar

[6]

S.-Y. A. ChangM. J. Gursky and P. C. Yang, The scalar curvature equation on 2- and 3-spheres, Calc. Var. Partial Differential Equations, 1 (1993), 205-229.  doi: 10.1007/BF01191617.  Google Scholar

[7]

S.-Y. 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

[8]

S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on S2, J. Differential Geom., 27 (1988), 259-296.   Google Scholar

[9]

S.-Y. A. Chang and P. C. Yang, Extremal metrics of zeta functional determinants on 4-manifolds, Ann. of Math., 142 (1995), 171-212.  doi: 10.2307/2118613.  Google Scholar

[10]

S.-Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on S2, Acta Math., 159 (1987), 215-259.  doi: 10.1007/BF02392560.  Google Scholar

[11]

W. Chen and C. Li, Prescribing scalar curvature on Sn, Pacific J. Math., 199 (2001), 61-78.  doi: 10.2140/pjm.2001.199.61.  Google Scholar

[12]

X. Chen and X. Xu, Q-curvature flow on the standard sphere of even dimension, J. Funct. Anal., 261 (2011), 934-980.  doi: 10.1016/j.jfa.2011.04.005.  Google Scholar

[13]

X. Chen and X. Xu, The scalar curvature flow on Sn–-perturbation theorem revisited, Invent. Math., 187 (2012), 395-506.  doi: 10.1007/s00222-011-0335-6.  Google Scholar

[14]

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

[15]

C. Fefferman and C. R. Graham, Conformal invariants, in, The Mathematical Heritage of Élie Cartan (Lyon, 1984), Astérisque (1985), 95–116.  Google Scholar

[16]

C. Fefferman and C. R. Graham, Q-curvature and Poincaré metrics, Math. Res. Lett., 9 (2002), 139-151.  doi: 10.4310/MRL.2002.v9.n2.a2.  Google Scholar

[17]

Z. C. Han, Prescribing Gaussian curvature on S2, Duke Math. J., 61 (1990), 679-703.  doi: 10.1215/S0012-7094-90-06125-3.  Google Scholar

[18]

Z. C. Han and Y. Y. Li, On the local solvability of the Nirenberg problem on S2, Discrete Contin. Dyn. Syst., 28 (2010), 607-615.  doi: 10.3934/dcds.2010.28.607.  Google Scholar

[19]

P. T. Ho, Prescribed mean curvature equation on the unit ball in the presence of reflection or rotation symmetry, Proc. Roy. Soc. Edinburgh Sect. A, (2017), to appear. doi: 10.1017/prm.2018.40.  Google Scholar

[20]

P. T. Ho, Prescribed Q-curvature flow on Sn, J. Geom. Phys., 62 (2012), 1233-1261.  doi: 10.1016/j.geomphys.2011.11.015.  Google Scholar

[21]

P. T. Ho, Prescribed Webster scalar curvature on S2n+1 in the presence of reflection or rotation symmetry, Bull. Sci. Math., 140 (2016), 506-518.  doi: 10.1016/j.bulsci.2015.06.001.  Google Scholar

[22]

P. T. Ho, Q-curvature flow on Sn, Comm. Anal. Geom., 18 (2010), 791-820.  doi: 10.4310/CAG.2010.v18.n4.a5.  Google Scholar

[23]

P. T. Ho, Results of prescribing Q-curvature on Sn, Arch. Math. (Basel), 100 (2013), 85-93.  doi: 10.1007/s00013-012-0472-1.  Google Scholar

[24]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith, Translations of Mathematical Monographs, 23 (1968) American Mathematical Society, Providence, R.I.  Google Scholar

[25]

M. C. Leung and F. Zhou, Prescribed scalar curvature equation on Sn in the presence of reflection or rotation symmetry, Proc. Amer. Math. Soc., 142 (2014), 1607-1619.  doi: 10.1090/S0002-9939-2014-11993-9.  Google Scholar

[26]

L. Ma and B. Liu, Q-curvature flow with indefinite nonlinearity, C. R. Math. Acad. Sci. Paris, 348 (2010), 403-406.  doi: 10.1016/j.crma.2010.02.014.  Google Scholar

[27]

A. Malchiodi and M. Struwe, Q-curvature flow on S4, J. Differential Geom., 73 (2006), 1-44.   Google Scholar

[28]

J. Moser, On a nonlinear problem in differential geometry, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973,273–280.  Google Scholar

[29]

Q. A. Ngô and H. Zhang, Global existence and convergence of Q-curvature flow on manifolds of even dimension, preprint. Google Scholar

[30]

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

[31]

V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, (Russian), Trudy Mat. Inst. Steklov., 83 (1965), 3-163.   Google Scholar

[32]

M. Struwe, A flow approach to Nirenberg's problem, Duke Math. J., 128 (2005), 19-64.  doi: 10.1215/S0012-7094-04-12812-X.  Google Scholar

[33]

J. C. Wei and X. Xu, On conformal deformations of metrics on Sn, J. Funct. Anal., 157 (1998), 292-325.  doi: 10.1006/jfan.1998.3271.  Google Scholar

[34]

X. Xu and P. C. Yang, Remarks on prescribing Gauss curvature, Trans. Amer. Math. Soc., 336 (1993), 831-840.  doi: 10.2307/2154378.  Google Scholar

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