Generalized Henneberg stable minimal surfaces

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 $m\ge 2$ 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 Björling problem for a hypocycloid of $m+1$ cusps (if $m$ is even), while for $m$ odd the conjugate minimal surface $H_m^{\ast }$ to $H_m$ is the unique solution to the Björling problem for a hypocycloid of $2m+2$ cusps.