# American Institute of Mathematical Sciences

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. Chang, M. 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. Chen, C. 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. Chen, C. 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. Daskalopoulos, Y. 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. González, R. 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. Graham, R. Jenne, L. 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. Chang, M. 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. Chen, C. 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. Chen, C. 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. Daskalopoulos, Y. 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. González, R. 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. Graham, R. Jenne, L. 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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