Conformal hypersurface invariants and Bach-type Boundary Problems

Abstract

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.