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

In this paper, we consider an area minimizing integral $m$-current $T$ within a submanifold $\mathrm{\Sigma }$ of ${\mathbb{R}}^{m+n}$, taking a boundary $\mathrm{\Gamma }$ with arbitrary multiplicity $Q\in \mathbb{N}\setminus \{0\}$, where $\mathrm{\Gamma }$ and $\mathrm{\Sigma }$ 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 $Q\mathrm{/}2$ 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 $\mathrm{\Gamma }$ which is also dimensionally sharp. Moreover, we prove that if $p\in \mathrm{\Gamma }$ admits an open neighborhood in $\mathrm{\Gamma }$ consisting of density $Q\mathrm{/}2$ 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 $\mathrm{\Gamma }$ is a closed manifold that lies at the boundary of a uniformly convex set and $\mathrm{\Sigma }={\mathbb{R}}^{m+n}$-then the entire boundary singular set is ${\mathcal{H}}^{m-3}$-rectifiable. Additionally, we investigate certain assumptions on $\mathrm{\Gamma }$ that enable us to provide further information about the singular boundary set.