  For an embedded conformal hypersurface with boundary, we construct critical
order local invariants and their canonically associated differential operators.
These are obtained holographically in a construction that uses a singular
Yamabe problem and a corresponding minimal hypersurface with boundary. They
include an extrinsic Q-curvature for the boundary of the embedded conformal
manifold and, for its interior, the Q-curvature and accompanying boundary
transgression curvatures. This gives universal formulae for extrinsic analogs
of Branson Q-curvatures that simultaneously generalize the Willmore energy
density, including the boundary transgression terms required for conformal
invariance. It also gives extrinsic conformal Laplacian power type operators
associated with all these curvatures. The construction also gives formulae for
the divergent terms and anomalies in the volume and hyper-area asymptotics
determined by minimal hypersurfaces having boundary at the conformal infinity.
A main feature is the development of a universal, distribution-based, boundary
calculus for the treatment of these and related problems.
