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 R3. 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 H1 is characterized as the unique example in the subfamily of the simplest complexity m = 1, while for m ≥ 2 multiparameter families are given. The isometry group of the most symmetric example Hm with a given complexity m ∈ N is either isomorphic to the dihedral isometry group D2m+2 (if m is odd) or to Dm+1 × Z2 (if m is even). Furthermore, for m even Hm 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 Hm is the unique solution to the Bj¨orling problem for a hypocycloid of 2m + 2 cusps.