Analysis of Critical Points of Conformally Invariant Curvature Energies in 4d

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. Rivi\`ere by generating first from Noether's theorem divergence-free "potentials", and then by exhibiting an underlying analytically favourable algebraic structure relating them. We also consider local Palais-Smale sequences and show they converge to a solution of a constrained Euler-Lagrange equation with Lagrange multiplier appearing in the form of a TT-tensor.