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

Abstract

Given an area-minimizing integral $m$-current in $Σ$, we prove that the Hausdorff dimension of the interior singular set of $T$ cannot exceed $m-2$, provided that $Σ$ is an embedded $(m+\bar{n})$-submanifold of $\mathbb{R}^{m+n}$ of class $C^{2,α}$, where $α>0$. 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 $C^{2,α}$.