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 of R2+n with partial derivative T = Q parallel to Gamma parallel to, where Q is an element of N\{0} and Gamma is sufficiently smooth. We prove that, if q is an element of Gamma is a point where the density of T is strictly below 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 of R2+n is a bounded uniformly convex set and Gamma subset of partial derivative Omega a smooth 1-dimensional closed submanifold, then any area-minimizing current T with partial derivative T = Q parallel to Gamma parallel to is regular in a neighborhood of Gamma.