Polynomial minimal surfaces of degree five

TL;DR

This paper characterizes all polynomial minimal surfaces of degree five in isothermal parameters, revealing that they belong to three specific families, thus advancing understanding of higher-degree polynomial minimal surfaces.

Contribution

It provides a general form for polynomial minimal surfaces of any degree and classifies degree five surfaces into three families, extending classical results.

Findings

Classified degree five polynomial minimal surfaces into three families.

Derived a general form for polynomial minimal surfaces of arbitrary degree.

Extended classical minimal surface theory to higher degrees.

Abstract

The problem of finding all minimal surfaces presented in parametric form as polynomials of certain degree is discussed by many authors. It is known that the classical Enneper surface is (up to position in space and homothety) the only polynomial minimal surface of degree 3 in isothermal parameters. In higher degrees the problem is quite more complicated. Here we find a general form for the functions that generate a polynomial minimal surface of arbitrary degree via the Weierstrass formula and prove that any polynomial minimal surface of degree 5 in isothermal parameters may be considered as belonging to one of three special families.