Singular Yamabe metrics on complements of closed sets

Let $n\ge 3$ and let $\mathrm{\Lambda }\subset {\mathbb{S}}^n$ be closed, contain at least two points, and satisfy ${\dim }_{\mathrm{H}}\mathrm{\Lambda }<(n-2)\mathrm{/}2$. We prove that there exists a complete metric of scalar curvature $n(n-1)$ on ${\mathbb{S}}^n\setminus \mathrm{\Lambda }$, conformal to the round metric.