An upper Minkowski dimension estimate for the interior singular set of area minimizing currents

Abstract

AbstractWe show that for an area minimizing m‐dimensional integral current T of codimension at least two inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most . This provides a strengthening of the existing ‐dimensional Hausdorff dimension bound due to Almgren and De Lellis & Spadaro. As a by‐product of the proof, we establish an improvement on the persistence of singularities along the sequence of center manifolds taken to approximate T along blow‐up scales.