An overdetermined eigenvalue problem and the Critical Catenoid conjecture

We consider the eigenvalue problem $\Delta^{\mathbb{S}^2} \xi + 2 \xi=0$ in $\Omega$ and $\xi = 0$ along $\partial \Omega$, being $\Omega$ the complement of a disjoint and finite union of smooth and bounded simply connected regions in the two-sphere $\mathbb{S}^2$. Imposing that $|\nabla \xi|$ is locally constant along $\partial \Omega$ 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.