Published Paper
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Inserted: 24 jun 2025
Last Updated: 14 nov 2026

Journal: J. Math. Pures Appl.
Volume: 212
Number: 103918
Pages: 1-30
Year: 2025
Doi: https://doi.org/10.1016/j.matpur.2026.103918

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.

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• Structure 2D mod(q) minimizers in codimension 1