Generalized Henneberg Stable Minimal Surfaces

We generalize the classical Henneberg minimal surface by giving an infinite family of complete, finitely branched, non-orientable, stable minimal surfaces in 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 >= 2 multiparameter families are given. The isometry group of the most symmetric example H-m with a given complexity m is an element of N is either isomorphic to the dihedral isometry group D2m+2 (if m is odd) or to Dm+1 x Z(2) (if m is even). Furthermore, for m even Hm is the unique solution to the Bjorling 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 Bjorling problem for a hypocycloid of 2m + 2 cusps.

Automated summary

We generalize the classical Henneberg minimal surface by providing an infinite family of complete, finitely branched, non-orientable, stable minimal surfaces in R-3. These surfaces can be classified into subfamilies based on a positive integer called the complexity, which measures the number of branch points. The most symmetric example H-m has an isometry group isomorphic to either the dihedral isometry group D2m+2 (if m is odd) or Dm+1 x Z(2) (if m is even). Moreover, for even m, Hm is the unique solution to the Bjorling problem for a hypocycloid with m + 1 cusps, while for odd m, the conjugate minimal surface H-m(*) to H-m is the unique solution to the Bjorling problem for a hypocycloid with 2m + 2 cusps.