An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity

We consider integral area-minimizing 2-dimensional currents \(T\) in \(U\subset \mathbf {R}^{2+n}\) with \(\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]\), where \(Q\in \mathbf {N} \setminus \{0\}\) 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 \(\Omega \subset \mathbf {R}^{2+n}\) is a bounded uniformly convex set and \(\Gamma \subset \partial \Omega \) a smooth 1-dimensional closed submanifold, then any area-minimizing current \(T\) with \(\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]\) is regular in a neighborhood of \(\Gamma \).