  In this thesis, 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 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.
