Structural Equations for Critical Points of Conformally Invariant Curvature Energies in 4d

This paper considers the Euler-Lagrange equations satisfied by the critical points of a large class of conformally invariant extrinsic energies for 4-manifolds immersed into Euclidean space (any codimension). Using invariances and Noether's theorem, we convert the Euler-Lagrange equation in a system of equations with analytically favourable structures. The present paper generalises to the four-dimensional setting ideas originally developed by Tristan Rivi\`ere in his study of the Willmore energy in two dimensions.