Sufficient symmetry conditions for free boundary minimal annuli to be the critical catenoid

TL;DR

This paper investigates symmetry conditions that characterize the critical catenoid among free boundary minimal annuli in the unit ball, establishing uniqueness results based on boundary symmetry and congruence conditions.

Contribution

It introduces new symmetry-based criteria that uniquely identify the critical catenoid among free boundary minimal annuli, extending previous understanding of such surfaces.

Findings

Critical catenoid characterized by boundary symmetry

Uniqueness of free boundary minimal annuli with certain symmetries

Symmetry conditions imply congruence to the critical catenoid

Abstract

We first consider a uniqueness problem for embedded free boundary minimal annuli in the three-dimensional Euclidean unit half-ball. Then, we obtain symmetry properties for compact embedded free boundary minimal surfaces in the unit ball. Finally, we obtain several uniqueness results for the critical catenoid under symmetry conditions on the boundary. For example, we show that if an embedded free boundary minimal annulus whose boundary consists of two congruent components and has a reflection symmetry by a plane, then it is congruent to the critical catenoid.