Free boundary minimal surfaces and the reflection principle

TL;DR

This paper demonstrates that minimal surfaces intersecting a sphere orthogonally can be reflected across it, leading to a proof that the only embedded free boundary minimal annulus in a ball is the critical catenoid.

Contribution

It introduces a reflection principle for minimal surfaces meeting spheres orthogonally and proves the uniqueness of the critical catenoid as the only embedded free boundary minimal annulus.

Findings

Reflection principle for minimal surfaces at spheres

Uniqueness of the critical catenoid as free boundary minimal annulus

Characterization of embedded free boundary minimal annuli

Abstract

We show that a minimal surface meeting a sphere at a 90-degree angle can be reflected across the sphere. Using this reflection, we prove the uniqueness that every embedded free boundary minimal annulus in a ball is necessarily the critical catenoid.