Given a conformally compact manifold an important question is whether
the metric is conformally related to a conformally compact Einstein
metric (i.e. a Poincar\'e--Einstein metric). In general such a
conformal rescaling is 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, it
is the Fialkow tensor, or equivalently its trace-free part. We show
that the trace-free second fundamental form and the trace-free part
of the Fialkow tensor are the lowest order examples in a a sequence
of conformally invariant higher fundamental forms determined by the
data of a conformal hypersurface embedding, and we construct these
trace-free symmetric rank two tensors here. The vanishing of these
fundamental forms is a necessary and sufficient condition for a
conformally compact metric to be conformally related to an
asyptotically Poincare-Einstein metric.
This is joint work with Sam Blitz and Andrew Waldron