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Description

Using variational considerations, we establish that there exists a

new symmetric trace-free tensor conformal invariant of hypersurfaces

embeddings in even dimensional conformal manifolds. This conformal

invariant completes the family of conformal invariants known as

conformal fundamental forms.

The object has important links to global problems. In the context

of the even dimensional boundary-value Poincar\'e--Einstein problem,

the image of the Dirichlet--to--Neumann map is conformally

invariant. Recent investigations established that this

image is the pullback of a particular Riemannian invariant to the

odd-dimensional boundary. We show here that, in fact, that image

arises as the restriction of the new conformal invariant constructed

here. As a consequence of the proof, we are able to construct several

new global conformal invariants of the boundary. Finally, we use

our variational results to establish that compact Bach-flat manifolds

with umbilic boundary must admit a (formal to all orders)

Poincar\'e--Einstein metric in the conformal class of its interior.