Curvature varifolds with orthogonal boundary

Abstract

We consider the class $S_{\perp }^m\left(\Omega \right)$ of $m$-dimensional surfaces in $\overline{\Omega }\subset R^n$ that intersect $S=\partial \Omega$ orthogonally along the boundary. A piece of an affine $m$-plane in $S_{\perp }^m\left(\Omega \right)$ 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 that minimizes the $L^2$ curvature in the integer rectifiable class.