We consider integral area-minimizing $2$-dimensional currents $T$ in $U\subset \mathbb R^{ 2+n}$ with $\partial T = Q[\![\Gamma]\!]$, where $Q\in \mathbb N \setminus \{ 0\}$ and $\Gamma$ is sufficiently smooth. We prove that, if $q\in \Gamma$ is a point where the density of $T$ is strictly below $\frac{ Q+1}{ 2}$, then the current is regular at $q$. The regularity is understood in the following sense: there is a neighborhood of $q$ in which $T$ consists of a finite number of regular minimal submanifolds meeting transversally at $\Gamma$ (and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for $Q=1$. As a corollary, if $\Omega\subset \mathbb R^{ 2+n}$ is a bounded uniformly convex set and $\Gamma\subset \partial \Omega$ a smooth $1$-dimensional closed submanifold, then any area-minimizing current $T$ with $\partial T = Q [\![\Gamma]\!]$ is regular in a neighborhood of $\Gamma$.
Categories: math.AP