Singular Yamabe metrics on complements of closed sets

Abstract

Let $n\geq3$ and let $Λ\subset \mathbb S^n$ be closed, contain at least two points, and satisfy $\dim_{\mathrm H}Λ<(n-2)/2$. We prove that there exists a complete metric of scalar curvature $n(n-1)$ on $\mathbb S ^n\setminusΛ$, conformal to the round metric.