Regularity of area minimizing currents mod $p$

TL;DR

This paper proves a general partial regularity theorem for area minimizing currents mod p, showing that their singular set has Hausdorff dimension at most m-1 and is rectifiable when p is odd.

Contribution

It establishes the first broad partial regularity result for area minimizing currents mod p across all dimensions and codimensions, including rectifiability for odd p.

Findings

Singular set Hausdorff dimension ≤ m-1

Singular set is (m-1)-rectifiable when p is odd

Interior singular set has locally finite (m-1)-dimensional measure

Abstract

We establish a first general partial regularity theorem for area minimizing currents $\mathrm{mod}(p)$, for every $p$, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an $m$-dimensional area minimizing current $\mathrm{mod}(p)$ cannot be larger than $m-1$. Additionally, we show that, when $p$ is odd, the interior singular set is $(m-1)$-rectifiable with locally finite $(m-1)$-dimensional measure.