Singular Yamabe problem for scalar flat metrics on the sphere

Let \(\Omega \) $\mathrm{\Omega }$ be a domain on the unit n-sphere \( {\mathbb {S}}^n\) ${\mathbb{S}}^n$ and \( \overset{{\,}_\circ }{g}\) $\stackrel{{}_{\circ }}{g}$ the standard metric of \({\mathbb {S}}^n\) ${\mathbb{S}}^n$ , \(n\ge 3\) $n\ge 3$ . We show that there exists a conformal metric g with vanishing scalar curvature \(R(g)=0\) $R(g)=0$ such that \((\Omega , g)\) $(\mathrm{\Omega },g)$ is complete if and only if the Bessel capacity \({\mathcal {C}}_{\alpha , q}({\mathbb {S}}^n\setminus \Omega )=0\) ${\mathcal{C}}_{\alpha ,q}({\mathbb{S}}^n\backslash \mathrm{\Omega })=0$ , where \(\alpha =1+\frac{2}{n}\) $\alpha =1+\frac{2}{n}$ and \(q=\frac{n}{2}\) $q=\frac{n}{2}$ . 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.