• DOI:10.1007/s10455-022-09850-4
• Corpus ID: 209377046

Stationary surfaces with boundaries

@article{Gruber2019StationarySW,
  title={Stationary surfaces with boundaries},
  author={Anthony Gruber and M D Toda and Hung Tran},
  journal={Annals of Global Analysis and Geometry},
  year={2019},
  volume={62},
  pages={305 - 328}
}

• Anthony Gruber, M D Toda, Hung Tran
• Published 15 December 2019
• Mathematics
• Annals of Global Analysis and Geometry

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document} is considered which involves a Lagrangian that is symmetric in the principal curvatures. The first variation of W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document} is computed, and a stress tensor is extracted whose divergence quantifies deviation from W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document}-criticality. Boundary-value problems are then examined, and a characterization of free-boundary W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document}-surfaces with rotational symmetry is given for scaling-invariant W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document}-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.