Conformal Laplacian and Conical Singularities

Abstract

We study a behavior of the conformal Laplacian operator $\L_g$ on a manifold with \emph{tame conical singularities}: when each singularity is given as a cone over a product of the standard spheres. We study the spectral properties of the operator $\L_g$ on such manifolds. We describe the asymptotic of a general solution of the equation $\L_g u = Q u^{\alpha}$ with $1\leq \alpha\leq \frac{n+2}{n-2}$ near each singular point. In particular, we derive the asymptotic of the Yamabe metric near such singularity.