Singular Yamabe problem for scalar flat metrics on the sphere

Abstract

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