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

TL;DR

This paper proves that 2-dimensional area-minimizing currents with boundary on a smooth curve have unique tangent cones at all boundary points, with a polynomial rate of convergence, simplifying previous results by reducing to the case Q=1.

Contribution

The paper establishes boundary tangent cone uniqueness for 2D area-minimizing currents with arbitrary integer boundary multiplicity, extending prior work to more general boundary conditions.

Findings

Unique tangent cones at boundary points for all area-minimizing currents studied.

Polynomial convergence rate to tangent cones.

Reduction of the general case to the Q=1 case.

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.