Improved generic regularity of codimension-1 minimizing integral currents

Ars Inveniendi Analytica (2024), Paper No. 3, 16 pp.

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Let Γ be a smooth, closed, oriented, (n−1)-dimensional submanifold of Rⁿ⁺¹. We show that there exist arbitrarily small perturbations Γ′ of Γ with the property that minimizing integral n- currents with boundary Γ′ are smooth away from a set of Hausdorff dimension ≤ n − 9 − ε_n, where ε_n ∈ (0,1] is a dimensional constant. This improves on our previous result (where we proved generic smoothness of minimizers in 9 and10 ambient dimensions). The key ingredients developed here are a new method to estimate the full singular set of the foliation by minimizers and a proof of superlinear decay of closeness (near singular points) that holds even across non-conical scales.

DOI: https://doi.org/10.15781/z70n-ed29