Regularity of Critical Points of Scale-Invariant Geometric Energies for Even-Dimensional Submanifolds of $\mathbb R^m$

We consider scale-invariant curvature energies for immersions of closed manifolds of even dimension $n=2h$ into $\mathbb R^m$, with principal term $\int_Σ \big|\nabla^{(h-1)} \vec{\mathrm{I\!I}}\big|_g^2\,d\text{vol}_g$ and arbitrary lower-order polynomial extrinsic invariants of the same scaling. F