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

Abstract

We show that for an area minimizing $m$-dimensional integral current $T$ of codimension at least 2 inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most $m-2$. This provides a strengthening of the existing $(m-2)$-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.

52 pages, proofs of Theorem 7.8 and Proposition 7.1 corrected, and some minor modifications made to the introduction

References in corpus (2)

• The Singular Structure and Regularity of Stationary and Minimizing Varifolds
• Fine properties of branch point singularities: Dirichlet energy minimizing multi-valued functions