An overdetermined eigenvalue problem and the Critical Catenoid conjecture

We consider the eigenvalue problem ${\mathrm{\Delta }}^{{\mathbb{S}}^2}\xi +2\xi =0$ in $\mathrm{\Omega }$ and $\xi =0$ along $\mathrm{\partial }\mathrm{\Omega }$, being $\mathrm{\Omega }$ the complement of a disjoint and finite union of smooth and bounded simply connected regions in the two-sphere ${\mathbb{S}}^2$. Imposing that $\mathrm{\mid }\mathrm{\nabla }\xi \mathrm{\mid }$ is locally constant along $\mathrm{\partial }\mathrm{\Omega }$ and that $\xi$ has infinitely many maximum points, we are able to classify positive solutions as the rotationally symmetric ones. As a consequence, we obtain a characterization of the critical catenoid as the only embedded free boundary minimal annulus in the unit ball whose support function has infinitely many critical points.