Uniqueness of degree-one Ginzburg-Landau vortex in the unit ball in dimensions $N\ge 7$

For $ϵ>0$, we consider the Ginzburg-Landau functional for $\mathbb R^N$-valuedmapsdefinedintheunitball$B^N\subset \mathbb R^N$ with the vortex boundary data $x$ on $\mathrm{\partial }{B}^N$. In dimensions $N\ge 7$, we prove that for every $ϵ>0$, there exists a unique global minimizer $u_{ϵ}$ of this problem; moreover, $u_{ϵ}$ is symmetric and of the form $u_{ϵ}(x)=f_{ϵ}(\mathrm{\mid }x\mathrm{\mid })\frac{x}{\mathrm{\mid }x\mathrm{\mid }}$ for $x\in B^N$.