Uniqueness of boundary tangent cones for $2$-dimensional area-minimizing currents

Abstract

In this paper we show that, if $T$ is an area-minimizing $2$-dimensional integral current with $\partial T=Q[[\Gamma ]]$, where $\Gamma$ is a $C^{1,\alpha }$ curve for $\alpha >0$ and $Q$ an arbitrary integer, then $T$ has a unique tangent cone at every boundary point, with a polynomial convergence rate. The proof is a simple reduction to the case $Q=1$, studied by Hirsch and Marini.

9 pages, 2 figures