An overdetermined eigenvalue problem and the Critical Catenoid conjecture

TL;DR

This paper classifies positive solutions to a specific eigenvalue problem on the sphere under certain boundary conditions, leading to a characterization of the critical catenoid as a unique free boundary minimal surface with infinitely many support function critical points.

Contribution

It introduces a novel classification of solutions to an overdetermined eigenvalue problem on the sphere, linking it to the geometric characterization of the critical catenoid.

Findings

Positive solutions are rotationally symmetric.

The critical catenoid is uniquely characterized among free boundary minimal annuli.

The support function of the critical catenoid has infinitely many critical points.

Abstract

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.