A New Conformal Invariant for four-Dimensional Hypersurfaces

A new conformally invariant energy for four-dimensional hypersurfaces is devised. It renders possible the study of a large class of curvature energies, and we show that their critical points are smooth. As corollaries, we obtain the regularity of the critical points of the four-dimensional analogues of the Willmore energy, of the $Q$-curvature energy, but also that Bach-flat hypersurfaces are smooth, along with relevant estimates.