  An important problem is to determine under which circumstances a metric on a
conformally compact manifold is conformal to a Poincar\'e--Einstein metric.
Such conformal rescalings are in general obstructed by conformal invariants of
the boundary hypersurface embedding, the first of which is the trace-free
second fundamental form and then, at the next order, the trace-free Fialkow
tensor. We show that these tensors are the lowest order examples in a sequence
of conformally invariant higher fundamental forms determined by the data of a
conformal hypersurface embedding. We give a construction of these canonical
extrinsic curvatures. Our main result is that the vanishing of these
fundamental forms is a necessary and sufficient condition for a conformally
compact metric to be conformally related to an asymptotically
Poincar\'e--Einstein metric. More generally, these higher fundamental forms are
basic to the study of conformal hypersurface invariants. Because Einstein
metrics necessarily have constant scalar curvature, our method employs
asymptotic solutions of the singular Yamabe problem to select an asymptotically
distinguished conformally compact metric. Our approach relies on conformal
tractor calculus as this is key for an extension of the general theory of
conformal hypersurface embeddings that we further develop here. In particular,
we give in full detail tractor analogs of the classical Gauss Formula and Gauss
Theorem for Riemannian hypersurface embeddings.
