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

TL;DR

This paper proves a boundary regularity theorem for 2D area-minimizing currents near smooth curves with arbitrary multiplicity, extending classical results and establishing optimal conditions for regularity at boundary points.

Contribution

It generalizes Allard's boundary regularity theorem to currents with arbitrary multiplicity, providing new optimal regularity conditions at smooth boundary curves.

Findings

Regularity at points with density below (Q+1)/2

Currents consist of regular minimal submanifolds meeting transversally

Result is optimal and extends classical theorems

Abstract

We consider integral area-minimizing $2$-dimensional currents $T$ in $U\subset \mathbb R^{2+n}$ with $\partial T = Q[\![\Gamma]\!]$, where $Q\in \mathbb N \setminus \{0\}$ 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 $\Omega\subset \mathbb R^{2+n}$ is a bounded uniformly convex set and $\Gamma\subset \partial \Omega$ a smooth $1$-dimensional closed submanifold, then any area-minimizing current $T$ with $\partial T = Q [\![\Gamma]\!]$ is regular in a neighborhood of $\Gamma$.