The Regularity of Critical Points to Scale-Invariant Curvature Energies in Dimension 4

We consider a class of scale-invariant curvature energies defined on immersed $4$-dimensional manifolds and prove that weak immersions that are critical points of such energies are analytic in any given local harmonic chart. Because of the criticality of this variational problem, the regularity result is obtained through the identification of conservation laws by applying Noether theorem. The resulting identities generate a lower order elliptic system of PDEs to which methods from integrability by compensation and interpolation theory are applied.