An upper Minkowski bound for the interior singular set of area minimizing currents

TL;DR

This paper proves that the interior singular set of area-minimizing currents has an upper Minkowski dimension at most m-2, strengthening previous Hausdorff dimension bounds and improving understanding of singularity persistence.

Contribution

It establishes a sharper Minkowski dimension bound for the singular set of area-minimizing currents, advancing prior Hausdorff dimension results.

Findings

Upper Minkowski dimension of singular set is at most m-2

Improved understanding of singularity persistence along blow-up scales

Strengthened bounds compared to previous Hausdorff dimension results

Abstract

We show that for an area minimizing $m$-dimensional integral current $T$ of codimension at least 2 inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most $m-2$. This provides a strengthening of the existing $(m-2)$-dimensional Hausdorff dimension bound due to Almgren and De Lellis & Spadaro. As a by-product of the proof, we establish an improvement on the persistence of singularities along the sequence of center manifolds taken to approximate $T$ along blow-up scales.