Singular Yamabe problem for scalar flat metrics on the sphere

Let $\mathrm{\Omega }$ be a domain on the unit $n$-sphere ${\mathbb{S}}^n$ and $\mathring{g}$ the standard metric of ${\mathbb{S}}^n$, $n\ge 3$. We show that there exists a conformal metric $g$ with vanishing scalar curvature $R(g)=0$ such that $(\mathrm{\Omega },g)$ is complete if and only if the Bessel capacity ${\mathcal{C}}_{\alpha ,q}({\mathbb{S}}^n\setminus \mathrm{\Omega })=0$, where $\alpha =1+\frac{2}{n}$ and $q=\frac{n}{2}$. Our analysis utilizes some well known properties of capacity and Wolff potentials, as well as a version of the Hopf-Rinow theorem for the divergent curves.