Parametric polynomial minimal surfaces of arbitrary degree

TL;DR

This paper introduces explicit parametric forms for polynomial minimal surfaces of any degree, generalizing classical surfaces like Enneper and exploring their geometric properties and deformations.

Contribution

It provides a new explicit parametric representation for polynomial minimal surfaces of arbitrary degree, including their conjugates and isometric deformations.

Findings

Includes classical Enneper surface as a special case

Classifies surfaces into four categories based on degree

Provides explicit forms for conjugate minimal surfaces

Abstract

Weierstrass representation is a classical parameterization of minimal surfaces. However, two functions should be specified to construct the parametric form in Weierestrass representation. In this paper, we propose an explicit parametric form for a class of parametric polynomial minimal surfaces of arbitrary degree. It includes the classical Enneper surface for cubic case. The proposed minimal surfaces also have some interesting properties such as symmetry, containing straight lines and self-intersections. According to the shape properties, the proposed minimal surface can be classified into four categories with respect to $n=4k-1$ $n=4k+1$, $n=4k$ and $n=4k+2$. The explicit parametric form of corresponding conjugate minimal surfaces is given and the isometric deformation is also implemented.