Complete conformal metrics of constant positive scalar curvature with prescribed isolated singularities

Let $(M^n,g)$ be a smooth closed connected Riemannian manifold of dimension $n\ge 3$ with positive Yamabe constant. For any prescribed finite set $\mathrm{\Lambda }\subset M$ containing at least two points, we construct a complete metric on $M\setminus \mathrm{\Lambda }$, conformal to $g$, with scalar curvature $n(n-1)$. At each singularity, the local geometric term in the reduced action comes from the first nonzero homogeneous metric term of degree at most $(n-2)\mathrm{/}2$ in finite conformal normal coordinates, or from the mass of the Green function metric when all such terms vanish. We completely solve Schoen’s conjecture about singular Yamabe problem.