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

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).

Representation of Maximal Surfaces in a 3-Dimensional Lightlike Cone

In this paper, the representation formula of maximal surfaces in a 3-dimensional lightlike cone Q3 is obtained by making use of the differential equation theory and complex function theory. Some particular maximal surfaces under a special induced metric are presented explicitly via the representation formula.