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 $({\mathrm{\Delta }}_{{\mathbb{S}}^2}+2)u=0$ on smooth spherical annuli with $u=0$ and $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }=1$ on the boundary; their one-homogeneous extensions are the axially symmetric Alt-Caffarelli cones. Lastly, we also treat the case of free-boundary minimal annuli in spherical caps.