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 ${\int }_{\mathrm{\Sigma }}\mid {\mathrm{\nabla }}^{(h-1)}\overrightarrow{\mathrm{I}\text{\hspace{0.17em}⁣}\mathrm{I}}{\mid }_g^{2}d{\text{vol}}_g$ and arbitrary lower-order polynomial extrinsic invariants of the same scaling. Following the four-dimensional approach developed in joint work with Bernard, Martino, and Rivière, we prove that every weak critical immersion in the natural Sobolev class $W^{h+1,2}$, whose induced metric and its inverse have $L^{\mathrm{\infty }}$ coefficients, is real-analytic in harmonic coordinates. The proof combines geometric conservation laws, additional structural identities, and elliptic estimates with critical Sobolev coefficients to obtain Morrey decay and bootstrap to full regularity.