TY - JOUR

T1 - Classification of singular radial solutions to the σk Yamabe equation on annular domains

AU - Chang, S. Y.Alice

AU - Han, Zheng Chao

AU - Yang, Paul

N1 - Funding Information: E-mail addresses: chang@math.princeton.edu (S.-Y. Alice Chang), zchan@math.rutgers.edu (Zheng-Chao Han), yang@math.princeton.edu (Paul Yang). 1S.-Y. Alice Chang was partially supported by NSF through Grant DMS0245266. 2Zheng-Chao Han was partially supported by NSF through Grant DMS-0103888. 3Paul Yang was partially supported by NSF through Grant DMS0245266.

PY - 2005/9/15

Y1 - 2005/9/15

N2 - The study of the kth elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so-called σk curvature, has produced many fruitful results in conformal geometry in recent years. In these studies in conformal geometry, the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic PDE. Important advances have been made in recent years in the understanding of the analytic behavior of solutions of the PDE. However, the singular behavior of these solutions, which is important in describing many important questions in conformal geometry, is little understood. This note classifies all possible radial solutions, in particular, the singular solutions of the σk Yamabe equation, which describes conformal metrics whose σk curvature equals a constant. Although the analysis involved is of elementary nature, these results should provide useful guidance in studying the behavior of singular solutions in the general situation.

AB - The study of the kth elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so-called σk curvature, has produced many fruitful results in conformal geometry in recent years. In these studies in conformal geometry, the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic PDE. Important advances have been made in recent years in the understanding of the analytic behavior of solutions of the PDE. However, the singular behavior of these solutions, which is important in describing many important questions in conformal geometry, is little understood. This note classifies all possible radial solutions, in particular, the singular solutions of the σk Yamabe equation, which describes conformal metrics whose σk curvature equals a constant. Although the analysis involved is of elementary nature, these results should provide useful guidance in studying the behavior of singular solutions in the general situation.

KW - Conformal metric

KW - Generalized Yamabe equation

KW - Schouten curvature

KW - Singular radial solution

KW - σ curvature

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U2 - https://doi.org/10.1016/j.jde.2005.05.005

DO - https://doi.org/10.1016/j.jde.2005.05.005

M3 - Article

VL - 216

SP - 482

EP - 501

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 2

ER -