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

Abstract

AbstractWe consider integral area-minimizing 2-dimensional currents $T$ T in $U\subset \mathbf {R}^{2+n}$ U ⊂ R 2 + n with $\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]$ ∂ T = Q 〚 Γ 〛 , where $Q\in \mathbf {N} \setminus \{0\}$ Q ∈ N ∖ { 0 } and $\Gamma $ Γ is sufficiently smooth. We prove that, if $q\in \Gamma $ q ∈ Γ is a point where the density of $T$ T is strictly below $\frac{Q+1}{2}$ 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 $\Gamma $ 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 $\Omega \subset \mathbf {R}^{2+n}$ Ω ⊂ R 2 + n is a bounded uniformly convex set and $\Gamma \subset \partial \Omega $ Γ ⊂ ∂ Ω a smooth 1-dimensional closed submanifold, then any area-minimizing current $T$ T with $\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]$ ∂ T = Q 〚 Γ 〛 is regular in a neighborhood of $\Gamma $ Γ .