Excess decay for minimizing hypercurrents mod $2Q$

TL;DR

This paper establishes an optimal excess-decay estimate for area-minimizing currents mod 2Q in Riemannian manifolds, crucial for understanding their singular set structure and regularity properties.

Contribution

It provides a refined excess-decay estimate with optimal dependence on the second fundamental form, advancing the regularity theory of mod 2Q currents.

Findings

Proves excess-decay estimates for currents mod 2Q in Riemannian manifolds.

Improves dependence on the second fundamental form in decay estimates.

Facilitates decomposition of the singular set into regular and lower-dimensional parts.

Abstract

We consider codimension $1$ area-minimizing $m$-dimensional currents $T$ mod an even integer $p=2Q$ in a $C^2$ Riemannian submanifold $\Sigma$ of the Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point $q\in \mathrm{spt} (T)\setminus \mathrm{spt}^p (\partial T)$ where at least one such tangent cone is $Q$ copies of a single plane. While an analogous decay statement was proved in arXiv:2111.11202 as a corollary of a more general theory for stable varifolds, in our statement we strive for the optimal dependence of the estimates upon the second fundamental form of $\Sigma$. This technical improvement is in fact needed in arXiv:2201.10204 to prove that the singular set of $T$ can be decomposed into a $C^{1,\alpha}$ $(m-1)$-dimensional submanifold and an additional closed remaining set of Hausdorff dimension at most $m-2$.