Family of Enneper Minimal Surfaces

We consider a family of higher degree Enneper minimal surface Em for positive integers m in the three-dimensional Euclidean space E3. We compute algebraic equation, degree and integral free representation of Enneper minimal surface for m = 1, 2, 3. Finally, we give some results and relations for the family Em.


Introduction
Minimal surfaces have an important role in the mathematics, physics, biology, architecture, etc.These kinds of surfaces have been studied over the centuries by many mathematicians and also geometers.A minimal surface in E 3 is a regular surface for which the mean curvature vanishes identically.
In this paper, we introduce a family of higher degree Enneper minimal surface E m for positive integers m in the three-dimensional Euclidean space E 3 .In Section 2, we give the family of Enneper minimal surfaces E m .We obtain the algebraic equation and degree of surface E 1 (resp., E 2 , E 3 ).Using the integral free form of Weierstrass, we find some algebraic functions for E m (m ≥ 1, m ∈ Z) in Section 3. Finally, we give some general findings for a family of higher degree Enneper minimal surface E m with a table in the last section.

The Family of Enneper Minimal Surfaces E m
We will often identify − → x and − → x t without further comment.Let E 3 be a three-dimensional Euclidean space with natural metric ., .= dx 2 + dy 2 + dz 2 .
Let U be an open subset of C. A minimal (or isotropic) curve is an analytic function Thus, let see the following lemma for complex minimal curves.
Lemma 1.Let Ψ : U → C 3 be a minimal curve and write Ψ = (ϕ 1 , ϕ 2 , ϕ 3 ) .Then, Therefore, we have minimal surfaces in the associated family of a minimal curve, as given by the following Weierstrass representation theorem [9] for minimal surfaces: Theorem 1.Let F and G be two holomorphic functions defined on a simply connected open subset U of C such that F does not vanish on U .Then, the map is a minimal, conformal immersion of U into C 3 , and x is called the Weierstrass patch.
We now consider the Enneper's curve of value m: Then, we have Lemma 3. The Weierstrass patch determined by the functions is a representation of Enneper's higher degree surfaces E m , where m ≥ 2.
Remark 1.Note that the catenoid and classical Enneper's surface are the only complete regular minimal surfaces in E 3 with finite total curvature −4π.
See [5] for details.Gray, Abbena and Salamon [26] gave the complex forms of the Enneper's curve and surface of value m.Therefore, the associated family of minimal surfaces is described by When α = 0 (resp.α = π/2), we have the Enneper's surface of value m (resp.the conjugate surface E * m ).
The parametric equation of E m , in polar coordinates, is Using the binomial formula, we obtain the following parametric equations of E m (u, v) : Next, we will focus on the algebraic equation and degree of surface E m .With R 3 = {(x, y, z) | x, y, z ∈ R}, the set of roots of a polynomial f (x, y, z) = 0 gives an algebraic surface.An algebraic surface is said to be of degree n, when n = deg( f ).
It is seen that deg (x Table 1  for details).Using polynomial eliminate methods, we calculate the algebraic equations and degrees of the surfaces E 1 , E 2 , E 3 .For the surface E 1 (i.e., classical Enneper surface), it is known that the surface has degree 9. Thus, it is also an algebraic minimal surface.For expanded results of E 1 , see [4].

Algebraic Equation of Enneper Minimal Surface E 1
The simplest Weierstrass representation (F , G) = (1, ζ) gives classical Enneper minimal surface of value 1.In polar coordinates, the parametric equation of E 1 is where where u, v ∈ R.
Lemma 4. A plane intersects an algebraic minimal surface in an algebraic curve [13].
See also [4] for details.Considering the above lemma, we find the algebraic equation of the curve on the xz-plane is as follows (see Figure 1, left): and its degree is deg(γ 1 ) = 3.Thus, xz-plane intersects the algebraic minimal surface E 1 in an algebraic curve γ 1 (u).
Using the polynomial eliminate method, we calculate the irreducible algebraic equation E 1 (x, y, z) = 0 of surface E 1 (u, v) by hand as follows (see Figure 1, right): Its degree is deg(E 1 ) = 9.Therefore, E 1 is an algebraic minimal surface.All of these results for classical Enneper surface E 1 were obtained first in [11] by Enneper.
Next, we study algebraic equations and degrees of the higher degree Enneper minimal surfaces for values m = 2 and m = 3.
Using the polynomial eliminate method, we find the algebraic equation of the curve on the xz-plane as follows (see Figure 2, left)
We get the algebraic equation of the curve on the xz-plane as follows: Its degree is deg(γ 3 ) = 7.Then, we see that the xz-plane intersects the algebraic minimal surface E 3 in an algebraic curve γ 3 (u).
Its degree is deg(E 3 ) = 49.Thus, E 3 is an algebraic minimal surface.
Corollary 1.The family of higher degree (also classical) Enneper minimal surfaces E m (u, v) are algebraic minimal surfaces, where m ∈ Z, m ≥ 1 (see Table 1).

Integral Free Form
Integral free form of the Weierstrass representation (see [15]) is where algebraic function φ(w) and the functions f i (w) are connected by the relation for w ∈ C. Integral free form is suitable for algebraic minimal surfaces.For instance, φ(w) = 1 6 w 3 gives rise to classical Enneper minimal surface E 1 (see [4] for details).
Hence, we have following lemma: Lemma 5.The algebraic function in the integral free form for a higher degree (also classical) Enneper minimal surfaces E m is as follows: where m ≥ 1, m ∈ Z.

Conclusions
Briefly, we give all findings, calculated in Sections 2 and 3 for the Enneper surface family, in Table 1 as follows.Looking at the table above, we also have the following results: where integers m ≥ 1.
Remark 2. For integers m ≥ 4, algebraic equations and also degrees of Enneper minimal surfaces E m can be calculated.However, calculation is a time problem for software programmes.