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

Pak Tung Ho

doi:10.3934/cpaa.2020033

Article Contents

2020, Volume 19, Issue 2: 715-722. Doi: 10.3934/cpaa.2020033

This issue Previous Article (1+2)-dimensional Black-Scholes equations with mixed boundary conditions Next Article The scalar curvature problem on four-dimensional manifolds

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

Pak Tung Ho

Department of Mathematics, Sogang University, Seoul 04107, Korea

Received: September 2018

Revised: July 2019

Published: October 2019

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)

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 53C44, 35K55, 53A30; Secondary: 53C21.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	P. Baird, A. 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. Chang, M. 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.