Weierstrass–Enneper parameterization

In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry.

Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863.

Let ƒ and g be functions on either the entire complex plane or the unit disk, where g is meromorphic and ƒ is analytic, such that wherever g has a pole of order m, f has a zero of order 2m (or equivalently, such that the product ƒg² is holomorphic), and let c₁, c₂, c₃ be constants. Then the surface with coordinates (x₁,x₂,x₃) is minimal, where the x_k are defined using the real part of a complex integral, as follows:

${\begin{aligned}x_{k}(\zeta )&{}=\Re \left\{\int _{0}^{\zeta }\varphi _{k}(z)\,dz\right\}+c_{k},\qquad k=1,2,3\\\varphi _{1}&{}=f(1-g^{2})/2\\\varphi _{2}&{}={\mathbf {i}}f(1+g^{2})/2\\\varphi _{3}&{}=fg\end{aligned}}$

The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type.

For example, Enneper's surface has ƒ(z) = 1, g(z) = z^m.

Parametric surface of complex variables

The Weierstrass-Enneper model defines a minimal surface $X$ ($\mathbb {R} ^{3}$) on a complex plane ($\mathbb {C}$). Let $\omega =u+vi$ (the complex plane as the $uv$ space), we write the Jacobian matrix of the surface as a column of complex entries:

$\mathbf {J} ={\begin{bmatrix}\left(1-g^{2}(\omega )\right)f(\omega )\\i\left(1+g^{2}(\omega )\right)f(\omega )\\2g(\omega )f(\omega )\end{bmatrix}}$

Where $f(\omega )$ and $g(\omega )$ are holomorphic functions of $\omega$.

The Jacobian $\mathbf {J}$ represents the two orthogonal tangent vectors of the surface:

$\mathbf {X_{u}} ={\begin{bmatrix}\operatorname {Re} \mathbf {J} _{1}\\\operatorname {Re} \mathbf {J} _{2}\\\operatorname {Re} \mathbf {J} _{3}\end{bmatrix}}\;\;\;\;\mathbf {X_{v}} ={\begin{bmatrix}-\operatorname {Im} \mathbf {J} _{1}\\-\operatorname {Im} \mathbf {J} _{2}\\-\operatorname {Im} \mathbf {J} _{3}\end{bmatrix}}$

The surface normal is given by

$\mathbf {\hat {n}} =\mathbf {X_{u}} \times \mathbf {X_{v}} ={\frac {1}{|g|^{2}+1}}{\begin{bmatrix}2\operatorname {Re} g\\2\operatorname {Im} g\\|g|^{2}-1\end{bmatrix}}$

The Jacobian $\mathbf {J}$ leads to a number of important properties: $\mathbf {X_{u}} \cdot \mathbf {X_{v}} =0$, $\mathbf {X_{u}} ^{2}=\mathbf {X_{v}} ^{2}$, $\mathbf {X_{uu}} +\mathbf {X_{vv}} =0$. The proofs can be found in Sharma's essay: The Weierstrass representation always gives a minimal surface. The derivatives can be used to construct the first fundamental form matrix:

${\begin{bmatrix}\mathbf {X_{u}} \cdot \mathbf {X_{u}} &\;\;\mathbf {X_{u}} \cdot \mathbf {X_{v}} \\\mathbf {X_{v}} \cdot \mathbf {X_{u}} &\;\;\mathbf {X_{v}} \cdot \mathbf {X_{v}} \end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}$

and the second fundamental form matrix

${\begin{bmatrix}\mathbf {X_{uu}} \cdot \mathbf {\hat {n}} &\;\;\mathbf {X_{uv}} \cdot \mathbf {\hat {n}} \\\mathbf {X_{vu}} \cdot \mathbf {\hat {n}} &\;\;\mathbf {X_{vv}} \cdot \mathbf {\hat {n}} \end{bmatrix}}$

Finally, a point $\omega _{t}$ on the complex plane maps to a point $\mathbf {X}$ on the minimal surface in $\mathbb {R} ^{3}$ by

$\mathbf {X} ={\begin{bmatrix}\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{1}d\omega \\\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{2}d\omega \\\operatorname {Re} \int _{\omega _{0}}^{\omega _{t}}\mathbf {J} _{3}d\omega \end{bmatrix}}$

where $\omega _{0}=0$ for all minimal surfaces throughout this paper except for Costa's surface where $\omega _{0}=(1+i)/2$.

Embedded minimal surfaces and examples

The classical examples of embedded complete minimal surfaces in $\mathbb {R} ^{3}$ with finite topology include the plane, the catenoid, the helicoid, and the Costa's surface. Costa's surface involves Weierstrass's elliptic function $\wp$ :

Failed to parse (syntax error): {\displaystyle \begin{align*} g(\omega)=\frac{A}{\wp'(\omega)}\\ f(\omega)= \wp(\omega) \end{align*} }

where $A$ is a constant (Chapter 22).

See also

• Associate family
• Bryant surface, found by an analogous parameterization in hyperbolic space