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A new proof of the classification of solutions to some linearized fractional Yamabe type equations

  • School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

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https://doi.org/10.3969/j.issn.0253-2778.2015.12.001
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  • Author Bio:

    FANG Yi, male, born in 1985, PhD candidate. Research field: geometric analysis. E-mail: yif@mail.ustc.edu.cn

  • Received Date: 07 April 2015
  • Accepted Date: 10 November 2015
  • Rev Recd Date: 10 November 2015
  • Publish Date: 30 December 2015
    Abstract

  • Abstract

    The classifications of all the solutions to the linearized Yamabe equations and fractional Yamabe type equations are crucial to the proof of the compactness of the scalar curvature problems and the fractional scalar curvature problems respectively. These classifications, though having been proved in an analytical way before, have been proved by adopting some new geometric approaches from the perspective of conformal geometry.

    Abstract

    The classifications of all the solutions to the linearized Yamabe equations and fractional Yamabe type equations are crucial to the proof of the compactness of the scalar curvature problems and the fractional scalar curvature problems respectively. These classifications, though having been proved in an analytical way before, have been proved by adopting some new geometric approaches from the perspective of conformal geometry.
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    • References(18)
  • [1]
    Aubin T. Problèms isopérimétriques et espaces de Sobolev[J]. J Differential Geom, 1976, 11(4): 573-598.
    [2]
    Talenti G. Best constant in Sobolev inequality[J]. Ann Mat Pura Appl, 1976, 110 (1): 353-372.
    [3]
    Caffarelli L A, Gidas B, Spruck J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[J]. Comm Pure Appl Math, 1989, 42 (3): 271-297.
    [4]
    Aubin T. Equations diérentielles non linéaires et problème de Yamabe concernant la courbure scalaire[J]. J Math Pures Appl, 1976, 55(3): 269-296.
    [5]
    Schoen R. Conformal deformation of a Riemannian metric to constant scalar curvature[J]. J Differential Geom, 1984, 20(2): 479-495.
    [6]
    Chen C C, Lin C S. Estimate of the conformal scalar curvature equation via the method of moving planes. Ⅱ[J]. J Differential Geom, 1998, 49(1): 115-178.
    [7]
    Marques F C. A priori estimates for the Yamabe problem in the non-locally conformally at case[J]. J Differential Geom, 2005, 71(2): 315-346.
    [8]
    Gonzlez Nogueras M M, Qing J. Fractional conformal Laplacians and fractional Yamabe problems[J]. Anal PDE, 2003, 6(7): 1 535-1 576.
    [9]
    Lieb E H, Loss M. Analysis[M]. Second edition. Providence, RI: Amer Math Soc, 2001.
    [10]
    Dvila J, del Pino M, Sire Y. Nondegeneracy of the bubble in the critical case for nonlocal equations[J]. Proc Amer Math Soc, 2013,141(11): 3 865-3 870.
    [11]
    Graham C R, Zworski M. Scattering matrix in conformal geometry[J]. Invent Math, 2003, 152(1): 89-118.
    [12]
    Mazzeo R R, Melrose R B. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[J]. J Funct Anal, 1987, 75 (2): 260-310.
    [13]
    Graham C R, Jenne R, Mason L J, et al. Conformally invariant powers of the Laplacian, I: Existence[J]. J London Math Soc, 1992, 46(3): 557-565.
    [14]
    Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian[J]. Comm Partial Differential Equations, 2007, 2(7-9):1 245-1 260.
    [15]
    Chang S A, del Mar Gonzlez M. Fractional Laplacian in conformal geometry[J]. Adv Math, 2011, 226(2): 1 410-1 432.
    [16]
    Branson T P. Sharp inequalities, the functional determinant, and the complementary series[J]. Trans Amer Math Soc, 1995, 347 (10): 3 671-3 742.
    [17]
    Lee J M, Parker T H. The Yamabe problem[J]. Bull Amer Math Soc (N.S.), 1987, 17(1): 37-91.
    [18]
    Chavel I. Eigenvalues in Riemannian Geometry[M]. Orlando, FL: Academic Press, 1984:35.
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    [1]
    Aubin T. Problèms isopérimétriques et espaces de Sobolev[J]. J Differential Geom, 1976, 11(4): 573-598.
    [2]
    Talenti G. Best constant in Sobolev inequality[J]. Ann Mat Pura Appl, 1976, 110 (1): 353-372.
    [3]
    Caffarelli L A, Gidas B, Spruck J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[J]. Comm Pure Appl Math, 1989, 42 (3): 271-297.
    [4]
    Aubin T. Equations diérentielles non linéaires et problème de Yamabe concernant la courbure scalaire[J]. J Math Pures Appl, 1976, 55(3): 269-296.
    [5]
    Schoen R. Conformal deformation of a Riemannian metric to constant scalar curvature[J]. J Differential Geom, 1984, 20(2): 479-495.
    [6]
    Chen C C, Lin C S. Estimate of the conformal scalar curvature equation via the method of moving planes. Ⅱ[J]. J Differential Geom, 1998, 49(1): 115-178.
    [7]
    Marques F C. A priori estimates for the Yamabe problem in the non-locally conformally at case[J]. J Differential Geom, 2005, 71(2): 315-346.
    [8]
    Gonzlez Nogueras M M, Qing J. Fractional conformal Laplacians and fractional Yamabe problems[J]. Anal PDE, 2003, 6(7): 1 535-1 576.
    [9]
    Lieb E H, Loss M. Analysis[M]. Second edition. Providence, RI: Amer Math Soc, 2001.
    [10]
    Dvila J, del Pino M, Sire Y. Nondegeneracy of the bubble in the critical case for nonlocal equations[J]. Proc Amer Math Soc, 2013,141(11): 3 865-3 870.
    [11]
    Graham C R, Zworski M. Scattering matrix in conformal geometry[J]. Invent Math, 2003, 152(1): 89-118.
    [12]
    Mazzeo R R, Melrose R B. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[J]. J Funct Anal, 1987, 75 (2): 260-310.
    [13]
    Graham C R, Jenne R, Mason L J, et al. Conformally invariant powers of the Laplacian, I: Existence[J]. J London Math Soc, 1992, 46(3): 557-565.
    [14]
    Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian[J]. Comm Partial Differential Equations, 2007, 2(7-9):1 245-1 260.
    [15]
    Chang S A, del Mar Gonzlez M. Fractional Laplacian in conformal geometry[J]. Adv Math, 2011, 226(2): 1 410-1 432.
    [16]
    Branson T P. Sharp inequalities, the functional determinant, and the complementary series[J]. Trans Amer Math Soc, 1995, 347 (10): 3 671-3 742.
    [17]
    Lee J M, Parker T H. The Yamabe problem[J]. Bull Amer Math Soc (N.S.), 1987, 17(1): 37-91.
    [18]
    Chavel I. Eigenvalues in Riemannian Geometry[M]. Orlando, FL: Academic Press, 1984:35.
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