Constant scalar curvature metrics with isolated singularities

Abstract

We extend the results and methods of \cite{MP} to prove the existence of constant positive scalar curvature metrics $g$ which are complete and conformal to the standard metric on $S^N \setminus Λ$, where $Λ$ is a disjoint union of submanifolds of dimensions between 0 and $(N-2)/2$. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by Schoen \cite{S}, but the proof we give here, based on the techniques of \cite{MP}, is more direct, and provides more information about their geometry. When $Λ$ is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in \cite{MPU1} and \cite{MPU2}