Curvature Varifolds with Orthogonal Boundary
Abstract
We consider the class $S^m_\perp(\Omega)$ of $m$dimensional surfaces in $\bar{\Omega} \subset {\mathbb R}^n$ which intersect $S = \partial \Omega$ orthogonally along the boundary. A piece of an affine $m$plane in $S^m_\perp(\Omega)$ is called an orthogonal slice. We prove estimates for the area by the $L^p$integral of the second fundamental form in three cases: first when $\Omega$ admits no orthogonal slices, second for $m = p = 2$ if all orthogonal slices are topological disks, and finally for all $\Omega$ if the surfaces are confined to a neighborhood of $S$. The orthogonality constraint has a weak formulation for curvature varifolds, we classify those varifolds of vanishing curvature. As an application, we prove for any $\Omega$ the existence of an orthogonal $2$varifold which minimizes the $L^2$ curvature in the integer rectifiable class.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 DOI:
 10.48550/arXiv.2203.08045
 arXiv:
 arXiv:2203.08045
 Bibcode:
 2022arXiv220308045K
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs