An Allard-type boundary regularity theorem for 2 d minimizing currents at smooth curves with arbitrary multiplicity
Camillo De Lellis, Stefano Nardulli, Simone Steinbrüchel · Publications mathématiques de l IHÉS · 2024
We consider integral area-minimizing 2-dimensional currents T in U ⊂ 𝐑 2 + n with ∂ T = Q [ Γ ] , where Q ∈ 𝐍 ∖ { 0 } and Γ is sufficiently smooth. We prove that, if q ∈ Γ 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 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 . As a corollary, if Ω ⊂ 𝐑 2 + n is a bounded uniformly convex set and Γ ⊂ ∂ Ω a smooth 1-dimensional closed submanifold, then any area-minimizing current T with ∂ T = Q [ Γ ] is regular in a neighborhood of Γ .