Stationary Surfaces with Boundaries

TL;DR

This paper studies stationary surfaces with boundaries as critical points of curvature-dependent functionals, providing characterizations, boundary conditions, and applications to conformal and p-Willmore energies.

Contribution

It introduces a generalized bending energy functional involving principal curvatures, computes its first variation, and characterizes free-boundary solutions with rotational symmetry.

Findings

Derived the first variation and stress tensor for the generalized energy functional.

Characterized free-boundary stationary surfaces with rotational symmetry.

Discussed implications for conformal Willmore and p-Willmore energies.

Abstract

This article investigates stationary surfaces with boundaries, which arise as the critical points of functionals dependent on curvature. Precisely, a generalized "bending energy" functional $\mathcal{W}$ is considered which involves a Lagrangian that is symmetric in the principal curvatures. The first variation of $\mathcal{W}$ is computed, and a stress tensor is extracted whose divergence quantifies deviation from $\mathcal{W}$-criticality. Boundary-value problems are then examined, and a characterization of free-boundary $\mathcal{W}$-surfaces with rotational symmetry is given for scaling-invariant $\mathcal{W}$-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.