Fine Structure of Singularities in Area-Minimizing Currents Mod$(q)$

TL;DR

This paper investigates the detailed structure of singularities in area-minimizing currents mod(q), revealing regularity and rectifiability properties of the singular set through excess decay and frequency function analysis.

Contribution

It establishes that certain singular points form a smooth submanifold and others are rectifiable, with unique tangent cones, advancing understanding of the interior regularity of these currents.

Findings

Points with translation-invariant tangent cones form a $C^{1,eta}$ submanifold.

The remaining singular set is countably $(m-2)$-rectifiable.

Almost monotonicity of a universal frequency function underpins the results.

Abstract

We study fine structural properties related to the interior regularity of $m$-dimensional area minimizing currents mod$(q)$ in arbitrary codimension. We show: (i) the set of points where at least one tangent cone is translation invariant along $m-1$ directions is locally a connected $C^{1,\beta}$ submanifold, and moreover such points have unique tangent cones; (ii) the remaining part of the singular set is countably $(m-2)$-rectifiable, with a unique flat tangent cone (with multiplicity) at $\mathcal{H}^{m-2}$-a.e. flat singular point. These results are consequences of fine excess decay theorems as well as almost monotonicity of a universal frequency function.