Hausdorff measure bounds for density-$Q$ flat singularities of minimizing integral currents

TL;DR

The paper addresses the size and structure of the singular set for area-minimizing integral currents in general codimension by focusing on flat singular points of locally maximal density. It develops a two-phase approach: first, a quantitative tilt-excess decay and frequency-based analysis yield that flat points with low singularity degree form an $\mathcal{H}^{m-2}$-null set, while second, a conical excess-decay framework with a stopping/restarting scheme provides local Minkowski-content bounds for flat points of higher degree. The main contributions are a local Hausdorff measure bound for the flat high-density singularities, a precise dichotomy between low- and high-degree flat points, and a robust method to obtain Minkowski bounds without extensive decompositions, advancing the quantitative understanding of the singular set in higher codimension. These results build on the Almgren–De Lellis–Spadaro program, Krummel–Wickramasekera's work, and recent Nab­er–Valtorta tools, with consequences such as almost-everywhere lower bounds on the singularity degree $\mathrm{I}(T,x) \ge 1+\frac{1}{Q}$ at flat points.

Abstract

In this article we prove that the set of flat singular points of locally highest density of area-minimizing integral currents of dimension $m$ and general codimension in a smooth Riemannian manifold $Σ$ has locally finite $(m-2)$-dimensional Hausdorff measure. In fact, the set of such flat singular points can be split into a union of two sets, one of which we show is locally $\mathcal{H}^{m-2}$-negligible, while for the other we obtain local $(m-2)$-dimensional Minkowski content bounds.

Theorems & Definitions (16)

• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Proposition 3.1: Uniform tilt excess decay
• Remark 3.2
• Proposition 3.3
• proof
• proof : Proof of Proposition \ref{['p:tilt-decay']}