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

Abstract

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

3 figures, comments are welcome