Genus one critical catenoid

TL;DR

This paper constructs a genus one free boundary minimal surface in the 3D unit ball using advanced variational methods, extending min-max theory to include symmetries and discovering a novel sweepout.

Contribution

It introduces an extension of equivariant min-max theory to orientation-reversing symmetries and finds a new two-parameter sweepout for minimal surface construction.

Findings

Constructed a genus one free boundary minimal surface with two boundary components.

Extended min-max theory to include orientation-reversing isometries.

Discovered a nontrivial two-parameter sweepout for variational methods.

Abstract

We use variational methods to construct a free boundary minimal surface in the three-dimensional unit ball with genus one, two boundary components and prismatic symmetry. Key ingredients are an extension of the equivariant min-max theory to include orientation-reversing isometries and the discovery of a nontrivial two-parameter sweepout.