Uniform regularity results for critical and subcritical surface energies

We establish regularity results for critical points to energies of immersed surfaces depending on the first and the second fundamental form exclusively. These results hold for a large class of intrinsic elliptic Lagrangians which are sub-critical or critical. They are derived using uniform ϵ -regularity estimates which do not degenerate as the Lagrangians approach the critical regime given by the Willmore integrand.