  The invariant theory for conformal hypersurfaces is studied by treating these
as the conformal infinity of a conformally compact manifold: For a given
conformal hypersurface embedding, a distinguished ambient metric is found
(within its conformal class) by solving a singular version of the Yamabe
problem. Using existence results for asymptotic solutions to this problem, we
develop the details of how to proliferate conformal hypersurface invariants. In
addition we show how to compute the the solution's asymptotics. We also develop
a calculus of conformal hypersurface invariant differential operators and in
particular, describe how to compute extrinsically coupled analogues of
conformal Laplacian powers. Our methods also enable the study of integrated
conformal hypersurface invariants and their functional variations. As a main
application we develop new higher dimensional analogues of the Willmore energy
for embedded surfaces. This complements recent progress on the existence and
construction of such functionals.
