On the uniqueness of tangent cones to area minimizing currents at boundaries with arbitrary multiplicity

Abstract

We consider an area minimizing current $T$ in a $C^2$ submanifold $\Sigma$ of $\mathbb{R}^{m+n}$, with arbitrary integer boundary multiplicity $\partial T = Q [\![ \Gamma ]\!]$ where $\Gamma$ is a $C^2$ submanifold of $\Sigma$. We show that at every density $Q/2$ boundary point the tangent cone to $T$ is unique and there is a power rate of convergence to the unique tangent cone. In particular, if $\Gamma$ is a closed manifold which lies at the boundary of a uniformly convex set $\Omega$ and $\Sigma=\mathbb{R}^{m+n}$ then $T$ has a unique tangent cone at every boundary point. As a structural consequence of the uniqueness of the tangent cone, we obtain a decomposition theorem which is the starting point of the boundary regularity theory we develop in another paper in collaboration with Reinaldo Resende. The regularity theory we obtain generalizes Allard's boundary regularity theorem to a higher multiplicity setting.