(ζ−m, ζm)-Type Algebraic Minimal Surfaces in Three-Dimensional Euclidean Space

Erhan Güler,Ömer Kişi

doi:10.3390/axioms11010026

Abstract

We introduce the real minimal surfaces family by using the Weierstrass data (ζ−m,ζm) for ζ∈C, m∈Z≥2, then compute the irreducible algebraic surfaces of the surfaces family in three-dimensional Euclidean space E3. In addition, we propose that family has a degree number (resp., class number) 2m(m+1) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c).

Highlights

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
We consider the minimal surfaces family by using the Weierstrass data (ζ −m, ζ m ) for ζ ∈ C, and some integers m ≥ 2, and show that these kinds of surfaces are algebraic in E3
We present some findings on the Weierstrass data and the minimal curve to constuct the minimal surfaces used in the whole paper

Summary

Introduction

We consider the minimal surfaces family by using the Weierstrass data (ζ −m , ζ m ) for ζ ∈ C, and some integers m ≥ 2, and show that these kinds of surfaces are algebraic in E3. In E3 , our minimal surface is given by the following equation: Z. With the use of the elimination techniques, we compute the irreducible algebraic surface equations, the degrees, and the classes of the minimal surfaces family. By eliminating u and v, we can obtain an irreducible algebraic equation Q( a, b, c) = 0 of b s(u, v) in the inhomogeneous tangential coordinates.

Degree

Conclusions