A Characterization of the Critical Catenoid

TL;DR

This paper characterizes the critical catenoid as the unique embedded minimal annulus in the unit ball with orthogonal boundary intersection and symmetry, using nodal domain arguments and Steklov eigenvalue properties.

Contribution

It provides a new characterization of the critical catenoid based on symmetry, boundary conditions, and Steklov eigenvalues, extending previous results.

Findings

The critical catenoid is uniquely determined by symmetry and boundary conditions.

Embedded minimal annuli with certain symmetries have Steklov eigenvalue 1.

General criteria are given for free boundary minimal surfaces with symmetry to have Steklov eigenvalue 1.

Abstract

We show that an embedded minimal annulus $\Sigma^2 \subset B^3$ which intersects $\partial B^3$ orthogonally and is invariant under reflection through the coordinate planes is the critical catenoid. The proof uses nodal domain arguments and a characterization, due to Fraser and Schoen, of the critical catenoid as the unique free boundary minimal annulus in $B^n$ with lowest Steklov eigenvalue equal to 1. We also give more general criteria which imply that a free boundary minimal surface in $B^3$ invariant under a group of reflections has lowest Steklov eigenvalue 1.