A compactness theorem for scalar-flat metrics on manifolds with boundary

Let (M ⁿ , g) be a compact Riemannian manifold with boundary ?M. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have ?M as a constant mean curvature hypersurface. We prove that this set is compact for dimensions n ?? 7 under the generic condition that the trace-free 2nd fundamental form of ?M is nonzero everywhere.