The fine structure of the singular set of area-minimizing integral currents II: Rectifiability of flat singular points with singularity degree larger than $1$

Abstract

We consider an area-minimizing integral current $T$ of codimension higher than $1$ in a smooth Riemannian manifold $\Sigma$. In a previous paper we have subdivided the set of interior singular points with at least one flat tangent cone according to a real parameter, which we refer to as the ‘singularity degree’. In this paper, we show that the set of points for which the singularity degree is strictly larger than $1$ is $(m-2)$-rectifiable. In a subsequent work, we prove that the remaining flat singular points form a ${\mathcal{H}}^{m-2}$-null set, thus concluding that the singular set of $T$ is $(m-2)$-rectifiable.