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 surfaces in $$\mathbb {R}^3$$ R 3 . These can be grouped into subfamilies depending on a positive integer (called complexity ), which essentially measures number branch points. The $$H_1$$ H 1 is characterized as unique example subfamily simplest $$m=1$$ m = , while for $$m\ge 2$$ ≥ 2 multiparameter families are given. isometry group most symmetric $$H_m$$ with given $$m\in \mathbb {N}$$ ∈ N either isomorphic to dihedral $$D_{2m+2}$$ D + (if m odd) or $$D_{m+1}\times {Z}_2$$ × Z even). Furthermore, even solution Björling problem hypocycloid $$m+1$$ cusps even), odd conjugate $$H_m^*$$ ∗ $$2m+2$$ cusps.