  We apply scattering theory on asymptotically hyperbolic manifolds to singular
Yamabe metrics, applying the results to the study of the conformal geometry of
compact manifolds with boundary. In particular, we define extrinsic versions of
the conformally invariant powers of the Laplacian, or GJMS operators, on the
boundary of any such manifold, along with associated extrinsic Q-curvatures. We
use the existence and uniqueness of a singular Yamabe metric to define also
nonlocal extrinsic fractional GJMS operators on the boundary, and draw other
global conclusions about the scattering operator, including a Gauss-Bonnet
theorem in dimension four.
