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 \(\Gamma \).

我们考虑在\(U\subset \mathbf {R}^{2+n}\)中积分面积最小化二维电流\(T\)，其中\(\partial T = Q\left [\!\![ {\Gamma }\right ]\!\!]\)，其中\(Q\in \mathbf {N} \setminus \{0\}\)和\(\Gamma \)足够平滑。我们证明，如果\(q\in \Gamma \)是一个点，其中\(T\)的密度严格低于\(\frac{Q+1}{2}\)，则电流在\(q\)。正则性可以这样理解：存在一个\(q\)邻域，其中\(T\)由有限数量的正则最小子流形在\(\Gamma \)横向相遇（并用适当的整数重数）。鉴于众所周知的例子，我们的结果是最优的，并且它是 Allard 经典定理对\(Q=1\)的第一个非平凡推广。作为推论，如果\(\Omega \subset \mathbf {R}^{2+n}\)是有界一致凸集，并且\(\Gamma \subset \partial \Omega \) 是一个平滑的一维闭子流形，则任何具有\(\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]\) 的面积最小化电流\(T \) 在\(邻域内是正则的 \伽玛 \)。