Improved generic regularity of codimension-1 minimizing integral currents

TL;DR

This paper demonstrates that small perturbations of a smooth boundary lead to minimizers that are mostly smooth, with singularities confined to a set of Hausdorff dimension at most n-9 minus a small constant, improving previous results.

Contribution

The authors introduce a new method to estimate the full singular set and prove superlinear decay of closeness, enhancing the regularity results for minimizing integral currents.

Findings

Singular set dimension is at most n-9 minus a small constant.

Small boundary perturbations yield mostly smooth minimizers.

New techniques improve understanding of singularities in minimal currents.

Abstract

Let $\Gamma$ be a smooth, closed, oriented, $(n-1)$-dimensional submanifold of $\mathbb{R}^{n+1}$. We show that there exist arbitrarily small perturbations $\Gamma'$ of $\Gamma$ with the property that minimizing integral $n$-currents with boundary $\Gamma'$ are smooth away from a set of Hausdorff dimension $\leq n-9-\varepsilon_n$, where $\varepsilon_n \in (0, 1]$ is a dimensional constant. This improves on our previous result (where we proved generic smoothness of minimizers in $9$ and $10$ 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.