Hausdorff measure bounds for density-$Q$ flat singularities of minimizing integral currents

Abstract

In this article we prove that the set of flat singular points of locally highest density of area-minimizing integral currents of dimension $m$ and general codimension in a smooth Riemannian manifold $\Sigma$ has locally finite $(m-2)$-dimensional Hausdorff measure. In fact, the set of such flat singular points can be split into a union of two sets, one of which we show is locally ${\mathcal{H}}^{m-2}$-negligible, while for the other we obtain local $(m-2)$-dimensional Minkowski content bounds.