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

Weierstrass [13] revealed a representation for minimal surfaces in three dimensional Euclidean space E 3 . Almost one hundred years later, Kobayashi [18] gave an analogous Weierstrass-type representation for conformal spacelike surfaces with mean curvature identically 0, called maximal surfaces, in three dimensional Minkowski space E 2,1 .
In this paper, we consider the Enneper family of maximal surfaces E m for positive integers m ≥ 1 by using Weierstrass data (1, ζ m ) for ζ ∈ C, and then show that these surfaces are algebraic in E 2,1 . See Güler [16] for a Euclidean case of Enneper's algebraic minimal surfaces family.
In Section 2, we give this family of real maximal surfaces in (r, θ) and (u, v) coordinates by using Weierstrass representation in E 2,1 . In Section 3, we find irreducible algebraic equations defining surfaces E m (u, v) in terms of running coordinates x, y, z, and a, b, c, and also compute degrees and classes of E m (u, v). Finally, we summarize all findings in tables in the last section, then give some open problems.
A vector x ∈ E n,1 is called space-like if x, x > 0, time-like if x, x < 0, and light-like if x = 0 and x, x = 0. A surface in E n,1 is called space-like (resp. time-like, light-like) if the induced metric on the tangent planes is a Riemannian (resp. Lorentzian, degenerate) metric. Now, let E 2,1 be three dimensional Minkowski space with Lorentzian metric . , . = x 1 y 1 + x 2 y 2 − x 3 y 3 . We identify − → x and − → x t without further comment.
Let U be an open subset of C. A maximal curve is an analytic function ϑ : U → C n such that ϑ (ζ), ϑ (ζ) = 0, where ζ ∈ U , and ϑ := ∂ϑ ∂ζ . In addition, if ϑ , ϑ = |ϑ | 2 = 0, then ϑ is a regular maximal curve. We then have maximal surfaces in the associated family of a maximal curve, given by the following Weierstrass representation theorem for ZMC (zero mean curvature) surfaces, or maximal surfaces. Kobayashi [18] found a Weierstrass type representation for space-like conformal maximal surfaces in E 2,1 : Theorem 1. Let g(ω) be a meromorphic function and let f (ω) be a holomorphic function, f g 2 is analytic, defined on a simply connected open subset U ⊂ C such that f (ω) does not vanish on U except at the poles of g(ω). Then, is a space-like conformal immersion with mean curvature identically 0 (i.e., space-like conformal maximal surface). Conversely, any spacelike conformal maximal surface can be described in this manner.
Next, we give some facts about Weierstrass data, and a maximal curve to construct some maximal surfaces.

Definition 1.
A pair of a meromorphic function g and a holomorphic function f , ( f , g) is called Weierstrass data for a maximal surface.
Therefore, we have ε m , ε m = 0 by using (2). Hence, in E 2,1 , the Enneper maximal surface is given by where ζ = u + iv. Im ε m (ζ)dζ gives the adjoint minimal surface E * m (u, v) of the surface E m (u, v) in (3). Then, we get the following: Corollary 1. The Weierstrass data (1, ζ m ) of (3) is a representation of the Enneper maximal surface, where integer m ≥ 1.
Considering the findings above with ζ = re iθ , we get the following Enneper family of maximal surfaces: where m = −1, −1/2. See Figure 1 for Enneper maximal surfaces E 1 , E 2 , E 3 in (r, θ) coordinates. Hence, using the binomial formula, we obtain more clear representation of E m (u, v) in (3): We study surface E m (u, v) in (u, v) coordinates for m = 1, 2, . . . , 5, taking ζ = u + iv at Cartesian coordinates x, y, z, and also in inhomogeneous tangential coordinates a, b, c, by using Weierstrass representation equation.
Next, we give a theorem about maximality of surface E 1 (u, v) (see Figure 1, Left): which has Weierstrass data (1, ζ), is an Enneper maximal surface in E 2,1 .
Proof. The coefficients of the first fundamental form of the surface E 1 (u, v) (E 1 , for short) are given by That is, conformality holds. Then, the Gauss map e 1 (u, v) of E 1 is as follows where λ = 1. The coefficients of the second fundamental form of E 1 are given by Then, we obtain the following mean curvature and the Gaussian curvature of the surface E 1 : Hence, the Enneper surface is maximal surface with positive Gaussian curvature. Therefore, we obtain the following parametric equations of the higher order maximal Figure 2 Middle for E 2 , and Figure 2 Right for E 3 .):

Degree and Class of Enneper Maximal Surfaces
In this section, using some elimination techniques, we derive the irreducible algebraic surface equation, degree and class of Enneper maximal surfaces family E m (u, v) for integers 1 ≤ m ≤ 5 in three dimensional Minkowski space E 2,1 .
Let us see some basic notions of the surfaces.

Definition 2.
The set of roots of a polynomial Q(x, y, z) = 0 gives an algebraic surface equation. An algebraic surface s is said to be of degree d when d = deg(s).
, the tangent plane is given by where e = (X(u, v), Y(u, v), Z(u, v)) is the Gauss map, and P = P(u, v). Then, in inhomogeneous tangential coordinates a, b, c, we have the following surface: Therefore, we can obtain an algebraic equationQ(a, b, c) = 0 of s(u, v) in inhomogeneous tangential coordinates. See [6], for details of a Euclidean case. Hence, we obtain the following findings for degrees and classes of Enneper maximal surfaces that we use:
Then, its degree number is 9. Next, we continue our computations to find Q m (x, y, z) = 0 for integers m = 2, 3. We compute the following irreducible algebraic surface equations Q 2 (x, y, z) = 0 (see Figure 3, Middle) and Q 3 (x, y, z) = 0 (see Figure 3, Right) of the surfaces E 2 (u, v) and E 3 (u, v), respectively, Therefore, Q m (x, y, z) = 0 are the algebraic maximal surfaces of the surfaces E m (u, v), where m = 2, 3, and they have degree numbers 25 and 49, respectively.

Class
Now, we introduce the class of the surfaces E m (u, v) for integers 1 ≤ m ≤ 4. The case m = 5, marked with "*" presented in tables of Section 4. Computing the irreducible algebraic surface equationsQ m (a, b, c) = 0, we obtain the Gauss maps e m (u, v) (see Figure 4 for e 1 , e 2 , e 3 ) for integers 1 ≤ m ≤ 5 of the surfaces E m (u, v), and we also generalize them as follows: . . .
So far, we find surfaces E 1 and E 2 . By using E 3 − E 5 , e 3 − e 5 , and also (12), (13), we obtain the following surfaces: E m (u, v) = (a, b, c): We also generalize the above functions, and find the following results: Corollary 3. The surfaces S m≥1 (u, v) for integers m, are given by where integers k ≥ 1, ζ = u + iv and |ζ| = λ. Corollary 4. In E 2,1 , the relations between the Enneper maximal surface E m≥1 (u, v) in the inhomogeneous tangential coordinates and the Gauss map e m≥1 (u, v) of the Enneper maximal surface E m≥1 (u, v) in the cartesian coordinates are given by where integers k ≥ 1, ζ = u + iv and |ζ| = λ.

Conclusions
To reveal the irreducible algebraic surface equations of the Enneper maximal surfaces E m (u, v) in E 2,1 , we have tried a series of standard techniques in elimination theory: only Sylvester by hand for Q 1 (x, y, z) = 0, and then projective (Macaulay) and sparse multivariate resultants implemented in the Maple software [19] package multires for Q m (x, y, z) = 0 andQ m (a, b, c) = 0.
Maple's native implicitization command Implicitize, and implicitization based on Maple's native implementation of the Groebner Basis. For the latter, we implemented in Maple the method in [20] (Chapter 3, p. 128). Under reasonable time, we only succeed for m = 1, 2 in all above methods.
For m = 3, the successful method we have tried was to compute the equation defining the elimination ideal using the Groebner Basis package FGb of Faugère in [21].
The time required to output the irreducible algebraic surface equations Q m (x, y, z) = 0 (resp.Q m (a, b, c) = 0) for integers 1 ≤ m ≤ 3 and polynomials defining the elimination ideal was under reasonable seconds determined by the following Table 1 (resp. Table 2).
Hence, we propose the following: Proposition 1. For integers m ≥ 1, degree number of the irreducible algebraic surfaces Q m (x, y, z) = 0 in the Cartesian coordinates is of (2m + 1) 2 , and class number of irreducible algebraic surfaceŝ Q m (a, b, c) = 0 in inhomogeneous tangential coordinates is of 2m(2m + 1) of the (1, ζ m )-type real Enneper maximal surfaces E m (u, v).

Open Problems
Here, we give some problems that we could not find the answers in this paper: Problem 1. Find the irreducible Enneper algebraic maximal surface eq. Q m≥4 (x, y, z) = 0 in the cartesian coordinates by using the parametric equation of the Enneper maximal surface E m≥4 (u, v).

Problem 2.
Find the irreducible Enneper algebraic maximal surface eq.Q m≥5 (a, b, c) = 0 in the inhomogeneous tangential coordinates by using the parametric equation of the Enneper maximal surface E m≥5 (u, v).
Finally, we give all findings in Tables 1 and 2.