On the fundamental regularity theorem for mass-minimizing flat chains

TL;DR

This paper establishes a precise condition on the coefficients of flat chains that guarantees a regularity theorem, ensuring mass-minimizing chains near a flat disk are smooth perturbations with the same multiplicity.

Contribution

It provides a simple necessary and sufficient condition on the group element for the fundamental regularity principle to hold in flat chains.

Findings

Identifies the exact condition on group elements for regularity.

Shows mass-minimizing chains are smooth perturbations under this condition.

Enhances understanding of regularity in mass-minimizing flat chains.

Abstract

In the theory of flat chains with coefficients in a normed abelian group, we give a simple necessary and sufficient condition on a group element $g$ in order for the following fundamental regularity principle to hold: if a mass-minimizing chain is, in a ball disjoint from the boundary, sufficiently weakly close to a multiplicity $g$ disk, then, in a smaller ball, it is a $C^{1,\alpha}$ perturbation with multiplicity $g$ of that disk.