Improved generic regularity of codimension-1 minimizing integral currents

Author:

Otis Chodosh, Christos Mantoulidis, Felix Schulze

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Analysis of PDEs (math.AP)

journal:

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date:

2023-06-21 16:00:00

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.