  The most general conformally invariant bending energy of a closed
four-dimensional surface, polynomial in the extrinsic curvature and its
derivatives, is constructed. This invariance manifests itself as a set of
constraints on the corresponding stress tensor. If the topology is fixed, there
are three independent polynomial invariants: two of these are the
straighforward quartic analogues of the quadratic Willmore energy for a
two-dimensional surface; one is intrinsic (the Weyl invariant), the other
extrinsic; the third invariant involves a sum of a quadratic in gradients of
the extrinsic curvature -- which is not itself invariant -- and a quartic in
the curvature. The four-dimensional energy quadratic in extrinsic curvature
plays a central role in this construction.
