Analysis of Conformally Invariant Curvature Energies in Dimension Four

We prove that weak immersions of four dimensional manifolds into Euclidean space which are critical points to a large class of conformally invariant curvature energies are analytic in any given local harmonic chart. Because of the criticality of this variational problem, the regularity result is obtained through rewriting the critical Euler-Lagrange equation in divergence form via Noether’s theorem, and calling upon an abundant use of methods from integrability by compensation theory such as interpolation spaces estimates.

This is joint work with Tian Lan, Dorian Martino, and Tristan Rivière.