Interior regularity of area minimizing currents within a $C^{2,\alpha}$-submanifold

TL;DR

This paper proves that the interior singular set of area-minimizing currents within a $C^{2,eta}$-submanifold has Hausdorff dimension at most $m-2$, extending regularity results to arbitrary codimension.

Contribution

It establishes the interior regularity and singular set dimension bounds for area-minimizing currents in $C^{2,eta}$-submanifolds, generalizing previous hypercurrent results.

Findings

Hausdorff dimension of singular set ≤ m-2

Complete regularity theory in arbitrary codimension

Extension of interior regularity to $C^{2,eta}$-submanifolds

Abstract

Given an area-minimizing integral $m$-current in $\Sigma$, we prove that the Hausdorff dimension of the interior singular set of $T$ cannot exceed $m-2$, provided that $\Sigma$ is an embedded $(m+\bar{n})$-submanifold of $\mathbb{R}^{m+n}$ of class $C^{2,\alpha}$, where $\alpha>0$. This result establishes the complete counterpart, in the arbitrary codimension setting, of the interior regularity theory for area-minimizing integral hypercurrents within a Riemannian manifold of class $C^{2,\alpha}$.