On the regularity of area minimizing currents at boundaries with arbitrary multiplicity

Abstract

In this paper, we consider an area minimizing integral $m$-current $T$ within a submanifold $\Sigma$ of $\mathbb{R}^{m+n}$, taking a boundary $\Gamma$ with arbitrary multiplicity $Q \in \mathbb{N} \setminus \{0\}$, where $\Gamma$ and $\Sigma$ are $C^{3, \kappa}$. We prove a sharp generalization of Allard's boundary regularity theorem to a higher multiplicity setting. Precisely, we prove that the set of density $Q/2$ singular boundary points of $T$ is $\mathcal{H}^{m-3}$-rectifiable. As a consequence, we show that the entire boundary regular set, without any assumptions on the density, is open and dense in $\Gamma$ which is also dimensionally sharp. Moreover, we prove that if $p \in \Gamma$ admits an open neighborhood in $\Gamma$ consisting of density $Q/2$ points with a tangent cone supported in a half $m$-plane, then $p$ is regular. Furthermore, we show that if the convex barrier condition is satisfied-namely, if $\Gamma$ is a closed manifold that lies at the boundary of a uniformly convex set and $\Sigma = \mathbb{R}^{m+n}$-then the entire boundary singular set is $\mathcal{H}^{m-3}$-rectifiable. Additionally, we investigate certain assumptions on $\Gamma$ that enable us to provide further information about the singular boundary set.