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Let $\Omega$ 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 thatthere exists a conformal metric $g$ with vanishing scalar curvature $R(g)=0$such that $(\Omega, g)$ is complete if and only if the Bessel capacity$\mathcal C_{\alpha, q}(\mathbb S^n\setminus \Omega)=0$, where$\alpha=1+\frac2n$ and $q=\frac n2$. Our analysis utilizes some well knownproperties of capacity and Wolff potentials, as well as a version of theHopf-Rinow theorem for the divergent curves.

Singular Yamabe problem for scalar flat metrics on the sphere