  A four dimensional conformally invariant energy is studied. This energy
generalises the well known two-dimensional Willmore energy. Although not
positive definite, it includes minimal hypersurfaces as critical points. We
compute its first variation and by applying the Noether theorem to the
invariances, we derive some conservation laws which are satisfied by its
critical points and with good analytical dispositions. In particular, we show
that its critical points are smooth. We also investigate other possible four
dimensional generalisations of the Willmore energy, and give strong evidence
that critical points of such energies do not include minimal hypersurfaces.
