Analysis of Conformally Invariant Energies on four-Dimensional Hypersurfaces

We discuss a large class of conformally invariant curvature energies for immersed hypersurfaces of dimension 4. The class under study includes various examples that have appeared in the recent literature and which arise from different contexts. We show that under natural small-energy hypotheses, critical points satisfy improved energy estimates. Nearly all the PDEs which we consider are quasilinear and fourth-order in the mean curvature. We approach the problem a la T. Riviere by generating first from Noether's theorem divergence-free "potentials", and then by exhibiting an underlying analytically favourable algebraic structure relating them. Similar ideas are used to obtain energy estimates for the Weyl tensor of a Bach-flat immersion. Finally, we show how to use Noether's theorem and the Gauss-Bonnet theorem to construct on a hypersurface non-trivial divergence-free and Codazzi symmetric two-tensors, cubic in the second fundamental form.