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 T in U⊂ R²⁺ⁿ U ⊂ R 2 + n with ∂ T = Q [\!\![Γ ]\!\!] ∂ T = Q 〚 Γ 〛 , where Q∈ N \0\ Q ∈ N ∖ { 0 } and Γ Γ is sufficiently smooth. We prove that, if q∈ Γ q ∈ Γ is a point where the density of T T is strictly below Q+1/2 Q + 1 2 , then the current is regular at q q . The regularity is understood in the following sense: there is a neighborhood of q q in which T T consists of a finite number of regular minimal submanifolds meeting transversally at Γ Γ (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 $Q=1$ Q = 1 . As a corollary, if Ω ⊂ R²⁺ⁿ Ω ⊂ R 2 + n is a bounded uniformly convex set and Γ ⊂ ∂ Ω Γ ⊂ ∂ Ω a smooth 1-dimensional closed submanifold, then any area-minimizing current T T with ∂ T = Q [\!\![Γ ]\!\!] ∂ T = Q 〚 Γ 〛 is regular in a neighborhood of Γ Γ .