Generalized Henneberg stable minimal surfaces

TL;DR

This paper introduces an infinite family of stable, non-orientable minimal surfaces in three-dimensional space, generalizing the classical Henneberg surface and exploring their symmetry and Björling problem solutions.

Contribution

It extends the classical Henneberg surface to a broader family with complex structures, symmetry groups, and unique Björling problem solutions depending on a complexity parameter.

Findings

Classical Henneberg surface is the simplest case with complexity 1.

For each complexity m, a family of minimal surfaces with specific symmetry groups is constructed.

Even and odd complexities have unique solutions to the Björling problem involving hypocycloids.

Abstract

We generalize the classical Henneberg minimal surface by giving an infinite family of complete, finitely branched, non-orientable, stable minimal surfaces in $\mathbb{R}^3$. These surfaces can be grouped into subfamilies depending on a positive integer (called the complexity), which essentially measures the number of branch points. The classical Henneberg surface $H_1$ is characterized as the unique example in the subfamily of the simplest complexity $m=1$, while for $m\geq 2$ multiparameter families are given. The isometry group of the most symmetric example $H_m$ with a given complexity $m\in \mathbb{N}$ is either isomorphic to the dihedral isometry group $D_{2m+2}$ (if $m$ is odd) or to $D_{m+1}\times \mathbb{Z}_2$ (if $m$ is even). Furthermore, for $m$ even $H_m$ is the unique solution to the Bj\"orling problem for a hypocycloid of $m+1$ cusps (if $m$ is even), while for $m$ odd the conjugate minimal surface $H_m^*$ to $H_m$ is the unique solution to the Bj\"orling problem for a hypocycloid of $2m+2$ cusps.