Conformally invariant bending energy for hypersurfaces

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 twodimensional 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. February 2, 2008 PACS: 04.60.Ds, 87.16.Dg, 46.70.Hg, 02.40.Hw