A Dual Yamabe Flow and Related Integral Flows

Jingang Xiong

Chinese Annals of Mathematics, Series B ›› 2024, Vol. 45 ›› Issue (3) : 319 -348.

PDF
Chinese Annals of Mathematics, Series B ›› 2024, Vol. 45 ›› Issue (3) : 319 -348. DOI: 10.1007/s11401-024-0019-3
Article

A Dual Yamabe Flow and Related Integral Flows

Author information +
History +
PDF

Abstract

The author studies a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS for short) subcritical regime, he presents a precise blow-up profile exhibited by the flows. In the HLS critical regime, by introducing a dual Q curvature he demonstrates the concentration-compactness phenomenon. If, in addition, the integral kernel matches with the Green’s function of a conformally invariant elliptic operator, this critical flow can be considered as a dual Yamabe flow. Convergence is then established on the unit spheres, which is also valid on certain locally conformally flat manifolds.

Keywords

Hardy-Littlewood-Sobolev functional / Dual Q curvature / Integral flow

Cite this article

Download citation ▾
Jingang Xiong. A Dual Yamabe Flow and Related Integral Flows. Chinese Annals of Mathematics, Series B, 2024, 45(3): 319-348 DOI:10.1007/s11401-024-0019-3

登录浏览全文

4963

注册一个新账户 忘记密码

References

[1]

Aubin T. Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl., 1976, 55: 269-296

[2]

Bahri A, Coron J-M. The scalar-curvature problem on the standard three-dimensional sphere. J. Funct. Anal., 1991, 95(1): 106-172

[3]

Bonforte M, Endal J. Nonlocal nonlinear diffusion equations, Smoothing effects, Green functions, and functional inequalities. J. Funct. Anal., 2023, 284: 104

[4]

Branson T P. Differential operators canonically associated to a conformal structure. Math. Scand., 1985, 57(2): 295-345

[5]

Brendle S. Convergence of the Yamabe flow for arbitrary initial energy. J. Differential Geom., 2005, 69: 217-278

[6]

Brendle S. Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math., 2007, 170(3): 541-576

[7]

Chan H, Sire Y, Sun L. Convergence of the fractional Yamabe flow for a class of initial data. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2020, 21: 1703-1740

[8]

Chang S-Y, González M. Fractional Laplacian in conformal geometry. Adv. Math., 2011, 226: 1410-1432

[9]

Chang S-Y, Yang R. On a class of non-local operators in conformal geometry. Chinese Ann. Math. Ser. B, 2017, 38(1): 215-234

[10]

Chen W, Li C, Ou B. Classification of solutions for an integral equation. Comm. Pure Appl. Math., 2006, 59: 330-343

[11]

Chill R. On the Lojasiewicz-Simon gradient inequality. J. Funct. Anal., 2003, 201(2): 572-601

[12]

Chow B. The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Comm. Pure. Appl. Math., 1992, 45: 1003-1014

[13]

Daskalopoulos P, Sire Y, Vázquez J-L. Weak and smooth solutions for a fractional Yamabe flow: The case of general compact and locally conformally flat manifolds. Comm. Partial Differential Equations, 2017, 42(9): 1481-1496

[14]

Dou J, Zhu M. Reversed Hardy-Littewood-Sobolev inequality. Int. Math. Res. Not. IMRN, 2015, 2015: 9696-9726

[15]

Fefferman C, Graham C R. The Ambient Metric, Annals of Mathematics Studies, 178, 2012, Princeton, NJ: Princeton University Press

[16]

Giaquinta M, Giusti E. On the regularity of the minima of variational integrals. Acta Math., 1982, 148: 31-46

[17]

Gursky M, Malchiodi A. A strong maximum principle for the Paneitz operator and a non-local flow for the Q-curvature. J. Eur. Math. Soc. (JEMS), 2015, 17(9): 2137-2173

[18]

Graham C R, Jenne R, Mason L J, Sparling G A J. Conformally invariant powers of the Laplacian, I, Existence. J. London Math. Soc.(2), 1992, 46: 557-565

[19]

Graham C R, Zworski M. Scattering matrix in conformal geometry. Invent. Math., 2003, 152: 89-118

[20]

Han Y, Zhu M. Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications. J. Differential Equations, 2016, 260: 1-25

[21]

Hang F, Yang P. Q-curvature on a class of manifolds of dimension at least 5. Comm. Pure Appl. Math., 2016, 69(8): 1452-1491

[22]

Hebey E, Robert F. Compactness and global estimates for the geometric Paneitz equation in high dimensions. Electron. Res. Announc. AMS, 2004, 10: 135-141

[23]

Jin T, Li Y Y, Xiong J. The Nirenberg problem and its generalizations: A unified approach. Math. Ann., 2017, 369(1–2): 109-151

[24]

Jin T, Xiong J. A fractional Yamabe flow and some applications. J. Reine Angew. Math., 2014, 696: 187-223

[25]

Jin T, Xiong J. Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2021, 38: 1167-1216

[26]

Jin T, Xiong J, Yang X. Stability of the separable solutions for a nonlinear boundary diffusion problem. J. Math. Pures Appl.(9), 2024, 183: 1-43

[27]

Li A, Li Y Y. On some conformally invariant fully nonlinear equations, II, Liouville, Harnack and Yamabe. Acta Math., 2005, 195: 117-154

[28]

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

[29]

Li Y Y, Xiong J. Compactness of conformal metrics with constant Q-curvature, I. Adv. Math., 2019, 345: 116-160

[30]

Lieb E H. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. of Math. (2), 1983, 118: 349-374

[31]

Malchiodi A. On conformal metrics with constant Q-curvature. Anal. Theory Appl., 2019, 35: 117-143

[32]

Paneitz, S., A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, SIGMA Symmetry Integrability Geom. Methods Appl., 4, 2008.

[33]

Qing, J. and Raske, D., On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds, Int. Math. Res. Not., 2006, 20 PP.

[34]

Schoen R. Conformal deformation of a Riemannian metric to constant scalar curvature. J. Diff. Geom., 1984, 20: 479-495

[35]

Schwetlick H, Struwe M. Convergence of the Yamabe flow for “large” energies. J. Reine Angew. Math., 2003, 562: 59-100

[36]

Simon L. Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. of Math. (2), 1983, 118: 525-571

[37]

Struwe M. A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z., 1984, 187(4): 511-517

[38]

Trudinger N. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Annali Scuola Norm. Sup. Pisa, 1968, 22: 265-274

[39]

Yamabe H. On a deformation of Riemannian structures on compact manifolds. Osaka Math J., 1960, 12: 21-37

[40]

Ye R. Global existence and convergence of Yamabe flow. J. Differential Geom., 1994, 39: 35-50

[41]

Zhu M. Prescribing integral curvature equation. Differential Integral Equations, 2016, 29: 889-904

AI Summary AI Mindmap
PDF

149

Accesses

0

Citation

Detail

Sections
Recommended

AI思维导图

/