  A conformally invariant generalization of the Willmore energy for compact
immersed submanifolds of even dimension in a Riemannian manifold is derived and
studied. The energy arises as the coefficient of the log term in the
renormalized area expansion of a minimal submanifold in a Poincare-Einstein
space with prescribed boundary at infinity. Its first variation is identified
as the obstruction to smoothness of the minimal submanifold. The energy is
explicitly identified for the case of submanifolds of dimension four.
Variational properties of this four-dimensional energy are studied in detail
when the background is a Euclidean space or a sphere, including identifications
of critical embeddings, questions of boundedness above and below for various
topologies, and second variation.
