Remarks on the critical catenoid and an overdetermined eigenvalue problem in ${\mathbb{S}}^2$

It was recently proved that an embedded minimal annulus with free boundary in the unit ball ${\mathbb{B}}^3$ of the Euclidean space is, up to a rotation, the critical catenoid. We prove that an immersed, possibly branched, free boundary minimal annulus in ${\mathbb{B}}^3$ with embedded boundary components is necessarily free of branch points and is embedded. This shows the uniqueness of the critical catenoid holds under these weaker hypotheses. We then apply these results to prove that if $\mathrm{\Omega }$ is an annulus in ${\mathbb{S}}^2$ for which there exists a smooth function satisfying the overdetermined eigenvalue problem \begin{equation*} \begin{cases} Δu+ 2u = 0 \quad $\text{on}$ \quad Ω\ \qquad \quad u=0 \quad $\text{in}$\quad\partialΩ\ \quad,,,, |\nabla u|=1 \quad $\text{in}$\quad \partialΩ. \end{cases} \end{equation*} then $\mathrm{\Omega }$ is, up to rotation, a rotational annulus with equatorial symmetry.