The higher-order Henneberg-type minimal surfaces family in ${\mathbb{R}}^4$

Abstract

We consider a higher-order Henneberg-type minimal surfaces family using the generalized Weierstrass--Enneper representation in four-dimensional space ${\mathbb{R}}^4$. We derive explicit parametric equations for the surface and determine its differential geometric characteristics, including the normal vector fields ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$, as well as the Gauss curvature. Furthermore, by projecting these parametric forms from four to three dimensions, we generate visualizations that reveal the geometric structure of the Henneberg-type minimal surface. In addition, we examine the integral-free form and derive the corresponding algebraic function for this family of surfaces.

16 pages, 4 figures