Curvature Varifolds with Orthogonal Boundary

Abstract

We consider the class $S_{\perp }^m(\hat{I}\text{c◯})$ of $m$-dimensional surfaces in $\bar{\hat{I}}\text{c◯}\subset {\mathbb{R}}^n$ which intersect $S=\partial \hat{I}\text{c◯}$ orthogonally along the boundary. A piece of an affine $m$-plane in $S_{\perp }^m(\hat{I}\text{c◯})$ 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 $\hat{I}\text{c◯}$ admits no orthogonal slices, second for $m=p=2$ if all orthogonal slices are topological disks, and finally for all $\hat{I}\text{c◯}$ 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 $\hat{I}\text{c◯}$ the existence of an orthogonal $2$-varifold which minimizes the $L^2$ curvature in the integer rectifiable class.