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 and 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. 

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[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.

[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. 

[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.

[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.

[34]

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

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. 

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[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.

[29]

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

[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.

[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. 

[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.

[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.

[34]

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

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