  Complete, conformally flat metrics of constant positive scalar curvature on
the complement of $k$ points in the $n$-sphere, $k \ge 2$, $n \ge 3$, were
constructed by R\. Schoen [S2]. We consider the problem of determining the
moduli space of all such metrics. All such metrics are asymptotically periodic,
and we develop the linear analysis necessary to understand the nonlinear
problem. This includes a Fredholm theory and asymptotic regularity theory for
the Laplacian on asymptotically periodic manifolds, which is of independent
interest. The main result is that the moduli space is a locally real analytic
variety of dimension $k$. For a generic set of nearby conformal classes the
moduli space is shown to be a $k-$dimensional real analytic manifold. The
structure as a real analytic variety is obtained by writing the space as an
intersection of a Fredholm pair of infinite dimensional real analytic
manifolds.

Differential Geometry;Differential Geometry