TitleThe fine structure of the singular set of area-minimizing integral currents III: Frequency 1 flat singular points and H^{m-2}-a.e uniqueness of tangent cones
Publication TypeJournal Article
Year of Publication2023
AuthorsDe Lellis C, Minter P, Skorobogatova A
Type of ArticleInterior regularity

We consider an area-minimizing integral current $T$ of codimension higher than $1$ in 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. The techniques in the present work can be seen as a counterpart for area-minimizers, in arbitrary codimension, to those developed by Simon ([29]) for multiplicity one classes of minimal surfaces and Wickramasekera ([32]) for stable minimal hypersurfaces.