  The Willmore energy, alias bending energy or rigid string action, and its
variation-the Willmore invariant-are important surface conformal invariants
with applications ranging from cell membranes to the entanglement entropy in
quantum gravity. In work of Andersson, Chrusciel, and Friedrich, the same
invariant arises as the obstruction to smooth boundary asymptotics to the
Yamabe problem of finding a metric in a conformal class with constant scalar
curvature. We use conformal geometry tools to describe and compute the
asymptotics of the Yamabe problem on a conformally compact manifold and thus
produce higher order hypersurface conformal invariants that generalise the
Willmore invariant. We give a holographic formula for these as well as
variational principles for the lowest lying examples.
