Density of the boundary regular set of 2d area minimizing currents with arbitrary codimension and multiplicity

TL;DR

This paper investigates the boundary regularity of 2D area minimizing currents with arbitrary codimension and boundary multiplicity, establishing that the regular set is open and dense without convex barrier assumptions.

Contribution

It extends boundary regularity results to 2D area minimizing currents with arbitrary codimension and multiplicity, without requiring convex barriers.

Findings

The boundary regular set of 2D area minimizing currents is open and dense.

Results apply to currents with arbitrary boundary multiplicity.

No convex barrier assumptions are needed for the main regularity result.

Abstract

In the present work, we consider area minimizing currents in the general setting of arbitrary codimension and arbitrary boundary multiplicity. We study the boundary regularity of 2d area minimizing currents, beyond that, several results are stated in the more general context of $(C_0, \alpha_0, r_0)$-almost area minimizing currents of arbitrary dimension $m$ and arbitrary codimension taking the boundary with arbitrary multiplicity. Furthermore, we do not consider any type of convex barrier assumption on the boundary, in our main regularity result which states that the regular set, which includes one-sided and two-sided points, of any 2d area minimizing current $T$ is an open dense set in the boundary.