The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space

Erhan Güler

doi:10.3390/axioms11010004

Abstract

We consider the Enneper family of real maximal surfaces via Weierstrass data (1,ζm) for ζ∈C, m∈Z≥1. We obtain the irreducible surfaces of the family in the three dimensional Minkowski space E2,1. Moreover, we propose that the family has degree (2m+1)2 (resp., class 2m(2m+1)) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c).

Highlights

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We consider the Enneper family of maximal surfaces Em for positive integers m ≥ 1 by using Weierstrass data (1, ζ m ) for ζ ∈ C, and show that these surfaces are algebraic in E2,1
We have maximal surfaces in the associated family of a maximal curve, given by the following Weierstrass representation theorem for ZMC surfaces, or maximal surfaces

Summary

Introduction

A minimal surface is a surface of vanishing mean curvature in three dimensional Euclidean space E3. Lie [9] studied the algebraic minimal surfaces and gave a table classifying these surfaces. Weierstrass-type representation for conformal spacelike surfaces with mean curvature identically 0, called maximal surfaces, in three dimensional Minkowski space E2,1. We consider the Enneper family of maximal surfaces Em for positive integers m ≥ 1 by using Weierstrass data (1, ζ m ) for ζ ∈ C, and show that these surfaces are algebraic in E2,1. See Güler [16] for a Euclidean case of Enneper’s algebraic minimal surfaces family.

Family of Enneper Maximal Surfaces

Degree and Class of Enneper Maximal Surfaces

Conclusions