COMPACTNESS RESULTS FOR COMPLETE METRICS OF CONSTANT POSITIVE SCALAR CURVATURE ON SUBDOMAINS OF S(N)

In this paper we prove that the set of metrics conformal to the standa rd metric on S(n)\{P1,...,P(k)}, where {p1,...,p(k)} is an arbitrary s et of k > 1 points, is compact in the C(m,alpha) topology, provided th e Dilational Pohozaev invariants are bounded away from zero. Such metr ics are necessarily complete on S(n)\{p1,...,p(k)} and asymptotic, at each singular point, to one of the periodic, spherically symmetric sol utions on the positive n-dimensional half cylinder, which are known as Delaunay metrics. The Pohozaev invariants are defined at each singula r point, and their dilational component is shown to be equal to a dime nsional constant times the Hamiltonian energy of the asymptotic Delaun ay metric. We also address the issue of convergence when the invariant s tend to zero for a proper subset of the singular points.