Classification of singular radial solutions to the $\sigma_k$ Yamabe equation on annular domains

Abstract

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $\sigma_k$ 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 \emph{singular} solutions of the $\sigma_k$ Yamabe equation, which describes conformal metrics whose $\sigma_k$ 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.