Genus two embedded minimal surfaces in $\mathbb{S}^3$ with bidihedral symmetry

TL;DR

This paper proves the uniqueness of a genus 2 embedded minimal surface in the 3-sphere with a specific symmetry group, expanding understanding of symmetric minimal surfaces in geometric analysis.

Contribution

It establishes the uniqueness of the genus 2 embedded minimal surface in $S^3$ with isometry group containing the bidihedral group $D_{4h}$, a new symmetry classification result.

Findings

Proves $\xi_{2,1}$ is the unique genus 2 minimal surface with $D_{4h}$ symmetry.

Identifies the isometry group of the classical Lawson surface.

Enhances classification of symmetric minimal surfaces in $S^3$.

Abstract

The isometry group of the classical Lawson embedded minimal surface $\xi_{2,1}\subset \mathbb{S}^3$ of genus 2 is isomorphic to the product $S_3\times D_4$ of the permutation group of three elements and the dihedral group of order 8 (symmetries of a square). $S_3\times D_4$ has a subgroup of index 3 isomorphic to the bidihedral group $D_{4h}=\mathbb{Z}_2\times D_4$, where $D_4$ is the dihedral group of order 8. We prove that $\xi_{2,1}$ is the unique closed embedded minimal surface of genus 2 in $\mathbb{S}^3$ whose isometry group contains $D_{4h}$.