Uniqueness of free boundary minimal annuli

We prove that every smooth properly embedded free-boundary minimal annulus in the unit ball is congruent to the critical catenoid. We also classify positive solutions of $(Δ_{\mathbb{S}^2}+2)u=0$ on smooth spherical annuli with $u=0$ and $|\nabla u|=1$ on the boundary; their one-homogeneous extensio