The Fine Structure of the Singular Set of Area-Minimizing Integral Currents III: Frequency 1 Flat Singular Points and $\mathcal{H}^{m-2}$-a.e. Uniqueness of Tangent Cones

Abstract

We consider an area-minimizing integral current $T$ of codimension higher than 1 ins a smooth Riemannian manifold $\Sigma$. We prove that $T$ has a unique tangent cone, which is a superposition of planes, at $\mathcal{H}^{m-2}$-a.e. point in its support. In combination with works of the first and third authors, we conclude that the singular set of $T$ is countably $(m-2)$-rectifiable.