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
∖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}