Uniform Regularity Results for Critical and Subcritical Surface Energies

TL;DR

This paper proves regularity results for immersed surfaces with energies depending on their fundamental forms, applicable to a broad class of elliptic Lagrangians near the critical Willmore energy, using uniform epsilon-regularity estimates.

Contribution

It introduces uniform epsilon-regularity estimates for critical points of surface energies, extending regularity results to sub-critical and critical regimes without degeneracy.

Findings

Regularity results for immersed surfaces with critical and subcritical energies.

Uniform epsilon-regularity estimates that remain stable near the critical regime.

Applicability to a large class of intrinsic elliptic Lagrangians.

Abstract

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 $\epsilon-$regularity estimates which do not degenerate as the Lagrangians approach the critical regime given by the Willmore integrand