Allard-type regularity theory for area minimizing currents at boundaries with arbitrary multiplicity

TL;DR

This paper announces a series of works extending Allard's boundary regularity theorem to area minimizing currents with smooth boundaries of arbitrary multiplicity, with significant structural implications.

Contribution

It generalizes Allard's boundary regularity theorem to currents with arbitrary boundary multiplicity, advancing the understanding of boundary regularity in geometric measure theory.

Findings

Generalization of Allard's boundary regularity theorem

Structural consequences for area minimizing currents

Framework for future boundary regularity results

Abstract

This is an announcement of a series of upcoming works on boundary regularity for area minimizing currents, one of which is in collaboration with Reinaldo Resende. The setting we consider is that of an area minimizing current with a smooth boundary taken with arbitrary multiplicity. The main result is a generalization of Allard's boundary regularity theorem from which we derive important structural consequences.