Stationary surfaces with boundaries

This article investigates stationary surfaces with boundaries, which arise as the critical points of functionals dependent on curvature. Precisely, a generalized “bending energy” functional 𝒲 is considered which involves a Lagrangian that is symmetric in the principal curvatures. The first variation of 𝒲 is computed, and a stress tensor is extracted whose divergence quantifies deviation from 𝒲 -criticality. Boundary-value problems are then examined, and a characterization of free-boundary 𝒲 -surfaces with rotational symmetry is given for scaling-invariant 𝒲 -functionals. In case the functional is not scaling-invariant, certain boundary-to-interior consequences are discussed. Finally, some applications to the conformal Willmore energy and the p-Willmore energy of surfaces are presented.