An overdetermined eigenvalue problem and the critical catenoid conjecture

Abstract

We consider the eigenvalue problem ${\Delta }^{{\mathbb{S}}^2}\xi +2\xi =0$ in $\Omega$ and $\xi =0$ along $\partial \Omega$, where $\Omega$ is the complement of a finite disjoint union of smooth, bounded, simply connected regions in the two-sphere ${\mathbb{S}}^2$. Assuming that $|\nabla \xi |$ is locally constant along $\partial \Omega$ and that $\xi$ has infinitely many maximum points, we classify positive solutions: every positive solution is rotationally symmetric. 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.