MODULI SPACES OF SINGULAR YAMABE METRICS

Complete, conformally flat metrics of constant positive scalar curvatu re on the complement of k points in the n-sphere, k greater than or eq ual to 2, n greater than or equal to 3, were constructed by R. Schoen in 1988. We consider the problem of determining the moduli space of al l such metrics. All such metrics are asymptotically periodic, and we d evelop the linear analysis necessary to understand the nonlinear probl em. This includes a Fredholm theory and asymptotic regularity theory f or the Laplacian on asymptotically periodic manifolds, which is of ind ependent interest. The main result is that the moduli space is a local ly real analytic variety of dimension k. For a generic set of nearby c onformal classes the moduli space is shown to be a k-dimensional real analytic manifold. The structure as areal analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinit e dimensional real analytic manifolds.