Enneper-Weierstrass Immersion Formula

• The Enneper-Weierstrass immersion formula is a classical representation that uses holomorphic and meromorphic functions to derive minimal surfaces in three-dimensional space.
• Its polynomial generalization provides explicit parametric forms for minimal surfaces of arbitrary degree, facilitating computational design and deeper geometric insights.
• The formulation also yields conjugate minimal surfaces and an associated isometric deformation family, enhancing applications in CAD and mathematical analysis.

The Enneper-Weierstrass immersion formula is a classical parametric representation for minimal surfaces in three-dimensional Euclidean space, based on holomorphic or meromorphic data. The formula serves as the analytic backbone for generating minimal surfaces and has far-reaching generalizations and explicit construction techniques. The work in "Parametric polynomial minimal surfaces of arbitrary degree" (Xu et al., 2010) provides an explicit closed-form polynomial analogue of the Enneper-Weierstrass immersion, yielding polynomial minimal surfaces of any degree.

1. Classical Enneper-Weierstrass Representation

The standard Enneper-Weierstrass representation produces a minimal surface in $\mathbb{R}^3$ from two holomorphic functions via: $$\vec{F}(w) = \operatorname{Re} \int \left( f(1 - g^2),\; i f(1 + g^2),\; 2 f g \right)\, dw$$ where $f$ is holomorphic, $g$ is meromorphic, and $fg^2$ is holomorphic on a simply connected domain. This yields surfaces like the catenoid, helicoid, and in particular the Enneper surface for specific polynomial choices.

2. Explicit Parametric Polynomial Minimal Surfaces

To surpass classical constraints requiring integration of holomorphic data, the formula in (Xu et al., 2010) constructs minimal surfaces directly as explicit polynomials in real variables $(u, v)$ for arbitrary degree $n$. This is achieved by systematically generating the parametric form, with all coordinate functions polynomials of $u$ and $v$.

Polynomial Basis Functions

The key components are the polynomials: $$P_n = \sum_{k=0}^{\lceil (n-1)/2 \rceil} (-1)^k {n \choose 2k} u^{n-2k} v^{2k}$$

$$Q_n = \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} (-1)^k {n \choose 2k+1} u^{n-2k-1} v^{2k+1}$$

These encode the generalized real polynomial behavior, extending the cubic structure of the classical Enneper surface.

Parametric Formula for Degree $n$

The resulting minimal surface is given explicitly as: $$\begin{aligned} X(u,v) & = -P_n + \omega P_{n-2} \ Y(u,v) & = Q_n + \omega Q_{n-2} \ Z(u,v) & = \frac{2\sqrt{n(n-2)\omega}}{n-1}\,P_{n-1} \end{aligned}$$ where $\omega$ is a real parameter. This construction guarantees that each coordinate function is a polynomial of degree at most $n$, and the surface is minimal.

3. Classical Enneper Surface and Cubic Case

For $n = 3$, the formula yields the classical Enneper surface: $$E(u,v) = \left( - (u^3 - 3 u v^2) + \omega u,\ - (v^3 - 3 v u^2) + \omega v,\ \sqrt{3\omega}(u^2 - v^2) \right)$$ The surface exhibits characteristic symmetry, straight lines, and self-intersections, features retained and generalized in higher-degree cases.

4. Conjugate Minimal Surfaces and Isometric Deformation

Given the explicit parametric form, the conjugate minimal surface is constructed as: $$\begin{aligned} X_s(u,v) &= -Q_n + \omega Q_{n-2} \ Y_s(u,v) &= -P_n - \omega P_{n-2} \ Z_s(u,v) &= \frac{2\sqrt{n(n-2)\omega}}{n-1} Q_{n-1} \end{aligned}$$ These surfaces are harmonic conjugates (related by the Cauchy-Riemann equations). An associated family of isometric minimal surfaces is realized by: $C_t(u, v) = \cos t\, r(u,v) + \sin t\, s(u,v)$ where $r(u,v)$ is the original surface, $s(u,v)$ is its conjugate, and $t \in [0, 2\pi)$. All members of the family are isometric and possess identical Gaussian curvature and first fundamental form.

5. Classification by Degree and Geometric Properties

Degree-based categorization reveals structural symmetries and special lines:

• For $n=4k-1$: symmetry about $X=0$ and $Y=0$; two straight lines $x = \pm y$ in $Z=0$.
• For $n=4k$: symmetry about $Z=0$ and $Y=0$.
• For $n=4k+1$: symmetry about $X=0$, $Y=0$, $X=Y$, $X=-Y$; self-intersections on symmetric planes.
• For $n=4k+2$: symmetry about $Z=0$ and $Y=0$.

Surfaces for each degree inherit and generalize features of the classical Enneper surface: symmetry lines, straight lines, and controlled self-intersections.

6. Explicit Examples and Key Formulas

For instance, the quintic ($n=5$) surface takes the form: $$\begin{aligned} X(u, v) &= -[u^5 - 10 u^3 v^2 + 5 u v^4] + \omega u(u^2 - 3 v^2) \ Y(u, v) &= -[v^5 - 10 v^3 u^2 + 5 v u^4] + \omega v(v^2 - 3 u^2) \ Z(u, v) &= \frac{\sqrt{15 \omega}}{2} (u^4 - 6u^2 v^2 + v^4) \end{aligned}$$

A summary table highlights the recursive structure:

Degree $n$	Parametric Form ($X, Y, Z$)	Conjugate Surface ($X_s, Y_s, Z_s$)	Example
Arbitrary $n$	$\begin{aligned} X&=-P_n+\omega P_{n-2}\ Y&=Q_n+\omega Q_{n-2}\ Z&=\frac{2\sqrt{n(n-2)\omega}{n-1} P_{n-1}\end{aligned}$	$\begin{aligned} X_s&=-Q_n+\omega Q_{n-2}\ Y_s&=-P_n-\omega P_{n-2}\ Z_s&=\frac{2\sqrt{n(n-2)\omega}{n-1}Q_{n-1}\end{aligned}$	Gen. Enneper surface
$n=3$	Cubic formulas as given above	Explicit as above	Enneper
$n=5$	Quintic formulas as above	Explicit as above	Quintic

7. Analytical and Computational Impact

The closed-form polynomial generalization is suitable for advanced computer-aided geometric design (CAD), symbolic computation, and further analytical investigation of minimal surfaces. The conjugate surfaces and associated isometric deformation family are given by explicit polynomial formulae, making these constructions amenable to both theoretical analysis and computational visualization.

These explicit structures allow for comprehensive analysis of self-intersection loci, symmetry, and stationary lines, as depicted in the paper's numerical figures. The framework encompasses the classical Enneper-Weierstrass immersion as a special case ($n=3$), while providing a systematic extension to surfaces of arbitrary algebraic degree.

8. Conclusion

The Enneper-Weierstrass immersion formula, in its explicit polynomial parametric form as developed in (Xu et al., 2010), systematically generates a family of minimal surfaces of any desired degree. This approach gives direct polynomial parameterizations, generalizes the classical cubic Enneper surface, preserves key geometrical properties via degree-based classification, and provides both conjugate surfaces and their isometric deformation families in analytical closed form. The result substantially broadens the analytic and computational toolkit for minimal surface construction and analysis.