The $$C_{m r}^{\kappa h}$$ minimal surfaces

Abstract

In this work is presented a family that generalizes Bour’s, Enneper’s, Richmond’s and Henneberg’s minimal surfaces. First of all is introduced a generalization for the well-known Bour’s family $$B_m$$ by including a new index r, determining a new bigger family $$B_{m r}$$ , which includes Bour’s, Enneper’s and Richmond’s families of minimal surfaces. After that, realizing some similarities between the Enneper–Weierstrass data of the Henneberg’s minimal surface and the $$B_{m r}$$ family, it is established a new family of minimal surfaces $$H_{m r}$$ that includes the classical Henneberg minimal surface. Finally, it is presented a bigger family $$C_ {m r}^{\kappa h}, \ m,r \in {\mathbb {R}}, \ \kappa \in {\mathbb {C}}$$ which includes as particular cases $$B_{m r}$$ and $$H_{m r}$$ families. Consequently, catenoid, helicoid, Enneper’s minimal surfaces, Bour’s minimal surfaces, Richmond’s minimal surfaces and Henneberg’s minimal surfaces are just some elements of the set of $$C_ {m r}^{\kappa h}$$ surfaces.