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arXiv · 1912.07103

Stationary Surfaces with Boundaries

Anthony Gruber·Magdalena Toda·Hung Tran
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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.

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Anthony Gruber, Magdalena Toda, Hung Tran. 2019-12-15. Stationary Surfaces with Boundaries. https://arxiv.org/abs/1912.07103

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