Structure of two-dimensional mod$(q)$ area-minimizing currents near flat singularities: the codimension one case

TL;DR

This paper provides a detailed structural analysis of two-dimensional mod(q) area-minimizing currents near flat singularities, showing they are perturbations of special multiple-valued harmonic functions, and characterizes the isolated nature of certain flat singularities.

Contribution

It introduces a new local regularity description for mod(q) currents near flat singularities and characterizes the isolated flat singularities of a specific density.

Findings

Currents are $C^{1,eta}$-perturbations of special multiple-valued harmonic functions.

Flat singularities of density q/2 are isolated points.

Provides groundwork for higher codimension regularity results.

Abstract

We obtain a fine structural result for two-dimensional mod$(q)$ area-minimizing currents of codimension one, close to flat singularities. Precisely, we show that, locally around any such singularity, the current is a $C^{1,\alpha}$-perturbation of the graph of a radially homogeneous special multiple-valued function that arises from a superposition of homogeneous harmonic polynomials. Additionally, as a preliminary step towards an analogous result in arbitrary codimension, we prove in general that the set of flat singularities of density $\frac{q}{2}$, where the current is ``genuinely mod$(q)$", consists of isolated points.