September  2021, 20(9): 3215-3234. doi: 10.3934/cpaa.2021103

Fractional Yamabe solitons and fractional Nirenberg problem

1. 

Department of Mathematics, Sogang University, Seoul 04107, Korea, Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea

2. 

Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea

3. 

Department of Mathematics, Johns Hopkins University, Baltimore MD 21218, USA

* Corresponding author

Received  January 2021 Revised  May 2021 Published  September 2021 Early access  June 2021

Fund Project: Ho's research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2019R1F1A1041021) and (NRF-2020R1A6A1A03047877), and by Korea Institute for Advanced Study (KIAS) grant funded by the Korea government (MSIP)

In this paper, we first study the fractional Yamabe solitons, which are the self-similar solutions to fractional Yamabe flow.We prove some rigidity results and Liouville type results for such solitons.We thenconsider the fractional Nirenberg problem:the problem of prescribing fractional order curvature on the sphere.More precisely, we prove that there exists a conformal metric on the unit sphere such that itsfractional order curvature is $ f $, when $ f $ possesses certain reflection or rotation symmetry.

Citation: Pak Tung Ho, Rong Tang. Fractional Yamabe solitons and fractional Nirenberg problem. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3215-3234. doi: 10.3934/cpaa.2021103
References:
[1]

J. Case and S. Y. A. Chang, On fractional GJMS operators, Commun. Pure Appl. Math., 69 (2016), 1017-1061. 

[2]

H. Chan, Y. Sire and L. Sun, Convergence of the fractional Yamabe flow for a class of initial data, preprint, arXiv: 1809.05753v1.

[3]

S. Y. A. Chang and M. González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. 

[4]

S. Y. A. ChangM. J. Gursky and P. C. Yang, The scalar curvature equation on 2-and 3-spheres, Calc. Var. Partial Differ. Equ., 1 (1993), 205-229. 

[5]

S. Y. A. Chang and P. C. Yang, A perturbation result in prescribing scalar curvature on $S^n$, Duke Math. J., 64 (1991), 27-69.  doi: 10.1215/S0012-7094-91-06402-1.

[6]

S. Y. A. Chang and P. C. Yang, Conformal deformation of metrics on $S^2$, J. Differ. Geom., 27 (1988), 259-296. 

[7]

S. Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on $S^2$, Acta Math., 159 (1987), 215-259.  doi: 10.1007/BF02392560.

[8]

W. ChenC. Li and B. Ou, Classifications of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[9]

X. Chen, P. T. Ho and J. Xiong, A fractional conformal curvature flow on the unit sphere, preprint, arXiv: 1906.08434.

[10]

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

[11]

Y. H. ChenC. Liu and Y. Zheng, Existence results for the fractional Nirenberg problem, J. Funct. Anal., 270 (2016), 4043-4086.  doi: 10.1016/j.jfa.2016.03.013.

[12]

Y. H. Chen and Y. Zheng, Peak solutions for the fractional Nirenberg problem, Nonlinear Anal., 122 (2015), 100-124.  doi: 10.1016/j.na.2015.04.002.

[13]

H. Chtioui and W. Abdelhedi, On a fractional Nirenberg type problem on the $n$-dimensional sphere, Complex Var. Elliptic Equ., 62 (2017), 1015-1036.  doi: 10.1080/17476933.2016.1260557.

[14]

H. Chtioui and W. Abdelhedi, On a fractional Nirenberg problem on $n$-dimensional spheres: existence and multiplicity results, Bull. Sci. Math., 140 (2016), 617-628.  doi: 10.1016/j.bulsci.2015.04.007.

[15]

P. Daskalopoulos and N. Sesum, The classification of locally conformally flat Yamabe solitons, Adv. Math., 240 (2013), 346-369.  doi: 10.1016/j.aim.2013.03.011.

[16]

P. DaskalopoulosY. Sire and J. L. Vázquez, Weak and smooth solutions for a fractional Yamabe flow: the case of general compact and locally conformally flat manifolds, Commun. Partial Differ. Equ., 42 (2017), 1481-1496.  doi: 10.1080/03605302.2017.1377230.

[17]

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

[18] C. Fefferman and C. R. Graham, The Ambient Metric, Princeton Univ. Press, Princeton, NJ, 2012. 
[19]

M. González and J. Qing, Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE, 6 (2013), 1535–1576. doi: 10.2140/apde.2013.6.1535.

[20]

M. GonzálezR. Mazzeo and Y. Sire, Singular solutions of fractional order conformal Laplacians, J. Geom. Anal., 22 (2012), 845-863.  doi: 10.1007/s12220-011-9217-9.

[21]

C. R. GrahamR. JenneL. Mason and G. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. Lond. Math. Soc. (2), 46 (1992), 557-565.  doi: 10.1112/jlms/s2-46.3.557.

[22]

C. R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118.  doi: 10.1007/s00222-002-0268-1.

[23]

P. T. Ho, Soliton to the fractional Yamabe flow, Nonlinear Anal., 139 (2016), 211-217.  doi: 10.1016/j.na.2016.02.026.

[24]

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, 149 (2019), 781-789.  doi: 10.1017/prm.2018.40.

[25]

P. T. Ho, Prescribed Webster scalar curvature on $S^{2n+1}$ in the presence of reflection or rotation symmetry, Bull. Sci. Math., 140 (2016), 506-518.  doi: 10.1016/j.bulsci.2015.06.001.

[26]

P. T. Ho, Prescribing $Q$-curvature on $S^n$ in the presence of symmetry, Commun. Pure Appl. Anal., 19 (2020), 715-722.  doi: 10.3934/cpaa.2020033.

[27]

S. Y. Hsu, Some properties of the Yamabe soliton and the related nonlinear elliptic equation, Calc. Var. Partial Differ. Equ., 49 (2014), 307-321.  doi: 10.1007/s00526-012-0583-3.

[28]

T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc. (JEMS), 16 (2014) 1111–1171. doi: 10.4171/JEMS/456.

[29]

T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, Part II: existence of solutions, Int. Math. Res. Not. IMRN, (2015) 1555–1589. doi: 10.1093/imrn/rnt260.

[30]

T. Jin and J. Xiong, A fractional Yamabe flow and some applications, J. Reine Angew. Math., 696 (2014), 187-223.  doi: 10.1515/crelle-2012-0110.

[31]

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

[32]

Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc. (JEMS), 6 (2004), 153-180. 

[33]

C. Liu and Q. Ren, Infinitely many non-radial solutions for fractional Nirenberg problem, Calc. Var. Partial Differ. Equ., 56 (2017), 40 pp. doi: 10.1007/s00526-017-1141-9.

[34]

C. Liu and Q. Ren, Multi-bump solutions for fractional Nirenberg problem, Nonlinear Anal., 171 (2018), 177-207.  doi: 10.1016/j.na.2018.02.001.

[35]

Z. Liu, Concentration of solutions for the fractional Nirenberg problem, Commun. Pure Appl. Anal., 15 (2016), 563-576.  doi: 10.3934/cpaa.2016.15.563.

[36]

L. Ma and V. Miquel, Remarks on scalar curvature of Yamabe solitons, Ann. Global Anal. Geom., 42 (2012), 195-205.  doi: 10.1007/s10455-011-9308-7.

[37]

S. Maeta, Three-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor, Ann. Global Anal. Geom., 58 (2020), 227–237. doi: 10.1007/s10455-020-09722-9.

[38]

R. Mazzeo and R. B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., 75 (1987), 260-310.  doi: 10.1016/0022-1236(87)90097-8.

[39]

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

[40]

P. Pavlov and S. Samko, A description of spaces $L^\alpha_p(S_{n-1})$ in terms of spherical hypersingular integrals (Russian), Soviet Math. Dokl., 29 (1984), 549-553. 

[41]

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

[42]

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

[43]

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

show all references

References:
[1]

J. Case and S. Y. A. Chang, On fractional GJMS operators, Commun. Pure Appl. Math., 69 (2016), 1017-1061. 

[2]

H. Chan, Y. Sire and L. Sun, Convergence of the fractional Yamabe flow for a class of initial data, preprint, arXiv: 1809.05753v1.

[3]

S. Y. A. Chang and M. González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. 

[4]

S. Y. A. ChangM. J. Gursky and P. C. Yang, The scalar curvature equation on 2-and 3-spheres, Calc. Var. Partial Differ. Equ., 1 (1993), 205-229. 

[5]

S. Y. A. Chang and P. C. Yang, A perturbation result in prescribing scalar curvature on $S^n$, Duke Math. J., 64 (1991), 27-69.  doi: 10.1215/S0012-7094-91-06402-1.

[6]

S. Y. A. Chang and P. C. Yang, Conformal deformation of metrics on $S^2$, J. Differ. Geom., 27 (1988), 259-296. 

[7]

S. Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on $S^2$, Acta Math., 159 (1987), 215-259.  doi: 10.1007/BF02392560.

[8]

W. ChenC. Li and B. Ou, Classifications of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[9]

X. Chen, P. T. Ho and J. Xiong, A fractional conformal curvature flow on the unit sphere, preprint, arXiv: 1906.08434.

[10]

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

[11]

Y. H. ChenC. Liu and Y. Zheng, Existence results for the fractional Nirenberg problem, J. Funct. Anal., 270 (2016), 4043-4086.  doi: 10.1016/j.jfa.2016.03.013.

[12]

Y. H. Chen and Y. Zheng, Peak solutions for the fractional Nirenberg problem, Nonlinear Anal., 122 (2015), 100-124.  doi: 10.1016/j.na.2015.04.002.

[13]

H. Chtioui and W. Abdelhedi, On a fractional Nirenberg type problem on the $n$-dimensional sphere, Complex Var. Elliptic Equ., 62 (2017), 1015-1036.  doi: 10.1080/17476933.2016.1260557.

[14]

H. Chtioui and W. Abdelhedi, On a fractional Nirenberg problem on $n$-dimensional spheres: existence and multiplicity results, Bull. Sci. Math., 140 (2016), 617-628.  doi: 10.1016/j.bulsci.2015.04.007.

[15]

P. Daskalopoulos and N. Sesum, The classification of locally conformally flat Yamabe solitons, Adv. Math., 240 (2013), 346-369.  doi: 10.1016/j.aim.2013.03.011.

[16]

P. DaskalopoulosY. Sire and J. L. Vázquez, Weak and smooth solutions for a fractional Yamabe flow: the case of general compact and locally conformally flat manifolds, Commun. Partial Differ. Equ., 42 (2017), 1481-1496.  doi: 10.1080/03605302.2017.1377230.

[17]

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

[18] C. Fefferman and C. R. Graham, The Ambient Metric, Princeton Univ. Press, Princeton, NJ, 2012. 
[19]

M. González and J. Qing, Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE, 6 (2013), 1535–1576. doi: 10.2140/apde.2013.6.1535.

[20]

M. GonzálezR. Mazzeo and Y. Sire, Singular solutions of fractional order conformal Laplacians, J. Geom. Anal., 22 (2012), 845-863.  doi: 10.1007/s12220-011-9217-9.

[21]

C. R. GrahamR. JenneL. Mason and G. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. Lond. Math. Soc. (2), 46 (1992), 557-565.  doi: 10.1112/jlms/s2-46.3.557.

[22]

C. R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118.  doi: 10.1007/s00222-002-0268-1.

[23]

P. T. Ho, Soliton to the fractional Yamabe flow, Nonlinear Anal., 139 (2016), 211-217.  doi: 10.1016/j.na.2016.02.026.

[24]

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, 149 (2019), 781-789.  doi: 10.1017/prm.2018.40.

[25]

P. T. Ho, Prescribed Webster scalar curvature on $S^{2n+1}$ in the presence of reflection or rotation symmetry, Bull. Sci. Math., 140 (2016), 506-518.  doi: 10.1016/j.bulsci.2015.06.001.

[26]

P. T. Ho, Prescribing $Q$-curvature on $S^n$ in the presence of symmetry, Commun. Pure Appl. Anal., 19 (2020), 715-722.  doi: 10.3934/cpaa.2020033.

[27]

S. Y. Hsu, Some properties of the Yamabe soliton and the related nonlinear elliptic equation, Calc. Var. Partial Differ. Equ., 49 (2014), 307-321.  doi: 10.1007/s00526-012-0583-3.

[28]

T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc. (JEMS), 16 (2014) 1111–1171. doi: 10.4171/JEMS/456.

[29]

T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, Part II: existence of solutions, Int. Math. Res. Not. IMRN, (2015) 1555–1589. doi: 10.1093/imrn/rnt260.

[30]

T. Jin and J. Xiong, A fractional Yamabe flow and some applications, J. Reine Angew. Math., 696 (2014), 187-223.  doi: 10.1515/crelle-2012-0110.

[31]

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

[32]

Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc. (JEMS), 6 (2004), 153-180. 

[33]

C. Liu and Q. Ren, Infinitely many non-radial solutions for fractional Nirenberg problem, Calc. Var. Partial Differ. Equ., 56 (2017), 40 pp. doi: 10.1007/s00526-017-1141-9.

[34]

C. Liu and Q. Ren, Multi-bump solutions for fractional Nirenberg problem, Nonlinear Anal., 171 (2018), 177-207.  doi: 10.1016/j.na.2018.02.001.

[35]

Z. Liu, Concentration of solutions for the fractional Nirenberg problem, Commun. Pure Appl. Anal., 15 (2016), 563-576.  doi: 10.3934/cpaa.2016.15.563.

[36]

L. Ma and V. Miquel, Remarks on scalar curvature of Yamabe solitons, Ann. Global Anal. Geom., 42 (2012), 195-205.  doi: 10.1007/s10455-011-9308-7.

[37]

S. Maeta, Three-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor, Ann. Global Anal. Geom., 58 (2020), 227–237. doi: 10.1007/s10455-020-09722-9.

[38]

R. Mazzeo and R. B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal., 75 (1987), 260-310.  doi: 10.1016/0022-1236(87)90097-8.

[39]

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

[40]

P. Pavlov and S. Samko, A description of spaces $L^\alpha_p(S_{n-1})$ in terms of spherical hypersingular integrals (Russian), Soviet Math. Dokl., 29 (1984), 549-553. 

[41]

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

[42]

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

[43]

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

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