2004/06/02 by S. -Y. Alice Chang, Sun Chang, Chang, S. -Y. Alice +4
Mathematics · Physics and Astronomy · #35B05 #35B40 #35J65 (Secondary) #58J05 (Primary) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Nonlinear Waves and Solitons #math.AP #math.DG #msc:35B05 #msc:35B40 #msc:35J65 #msc:58J05
paper · pdf · doi:10.48550/arxiv.math/0406028
12 figures
arxiv created 2004/06/02 · openalex publication_date 2004/06/02 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
The study of the k-th 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, 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.