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

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