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 $$f$$ and $$g$$ be functions on either the entire complex plane or the unit disk, where $$g$$ is meromorphic and $$f$$ 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 $$f g^2$$ is holomorphic), and let $$c_1,c_2,c_3$$ be constants. Then the surface with coordinates $$(x_1, x_2, x_3)$$ is minimal, where the $$x_k$$ are defined using the real part of a complex integral, as follows: $$\begin{align} x_k(\zeta) &{}= \mathrm{Re} \left\{ \int_{0}^{\zeta} \varphi_{k}(z) \, dz \right\} + c_k, \qquad k=1,2,3 \\ \varphi_1 &{}= f(1-g^2)/2 \\ \varphi_2 &{}= i f(1+g^2)/2 \\ \varphi_3 &{}= fg \end{align}$$

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 $f(z) = 1$, $g(z) = z^{m}$.

Parametric surface of complex variables
The Weierstrass-Enneper model defines a minimal surface $$X$$ ($$\Reals^3$$) on a complex plane ($$\Complex$$). Let $$\omega=u+v i$$ (the complex plane as the $$uv$$ space), the Jacobian matrix of the surface can be written 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 $$ \mathbf{\hat{n}} = \frac{\mathbf{X_u}\times \mathbf{X_v}}{|\mathbf{X_u}\times \mathbf{X_v}|} = \frac{1}{| g|^2+1} \begin{bmatrix} 2\operatorname{Re} g \\ 2\operatorname{Im} g \\ \end{bmatrix} $$
 * g|^2-1

The Jacobian $$\mathbf{J}$$ leads to a number of important properties: $$\mathbf{X_u} \cdot \mathbf{X_v}=0$$, $$\mathbf{X_u}^2 = \operatorname{Re}(\mathbf{J}^2)$$, $$\mathbf{X_v}^2 = \operatorname{Im}(\mathbf{J}^2)$$, $$\mathbf{X_{uu}} + \mathbf{X_{vv}}=0$$. 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 $$\R^3$$ 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 minimal 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 minimal surface. Costa's surface involves Weierstrass's elliptic function $$\wp $$: $$g(\omega)=\frac{A}{\wp' (\omega)}$$ $$f(\omega)= \wp(\omega)$$ where $$A$$ is a constant.

Helicatenoid
Choosing the functions $$f(\omega) = e^{-i \alpha}e^{\omega/A}$$ and $$g(\omega) = e^{-\omega/A}$$, a one parameter family of minimal surfaces is obtained.

$$\varphi_1 = e^{-i \alpha} \sinh\left(\frac{\omega}{A}\right)$$ $$\varphi_2 = i e^{-i \alpha} \cosh\left(\frac{\omega}{A}\right)$$ $$\varphi_3 = e^{-i \alpha}$$ $$ \mathbf{X}(\omega) = \operatorname{Re} \begin{bmatrix} e^{-i\alpha} A \cosh \left( \frac{\omega}{A} \right) \\ i e^{-i\alpha} A \sinh \left( \frac{\omega}{A} \right) \\ e^{-i\alpha} \omega \\ \end{bmatrix} = \cos(\alpha) \begin{bmatrix} A \cosh \left( \frac{\operatorname{Re}(\omega)}{A} \right) \cos \left( \frac{\operatorname{Im}(\omega)}{A} \right)\\ - A \cosh \left( \frac{\operatorname{Re}(\omega)}{A} \right) \sin \left( \frac{\operatorname{Im}(\omega)}{A} \right) \\ \operatorname{Re}(\omega) \\ \end{bmatrix} + \sin(\alpha) \begin{bmatrix} A \sinh \left( \frac{\operatorname{Re}(\omega)}{A} \right) \sin \left( \frac{\operatorname{Im}(\omega)}{A} \right)\\ A \sinh \left( \frac{\operatorname{Re}(\omega)}{A} \right) \cos \left( \frac{\operatorname{Im}(\omega)}{A} \right) \\ \operatorname{Im}(\omega) \\ \end{bmatrix} $$

Choosing the parameters of the surface as $$\omega = s + i(A \phi)$$: $$\mathbf{X}(s,\phi)= \cos(\alpha) \begin{bmatrix} A \cosh \left( \frac{s}{A} \right) \cos \left( \phi \right)\\ - A \cosh \left( \frac{s}{A} \right) \sin \left( \phi \right) \\ s \\ \end{bmatrix} + \sin(\alpha) \begin{bmatrix} A \sinh \left( \frac{s}{A} \right) \sin \left( \phi \right)\\ A \sinh \left( \frac{s}{A} \right) \cos \left( \phi \right) \\ A \phi \\ \end{bmatrix}$$

At the extremes, the surface is a catenoid $$(\alpha = 0)$$ or a helicoid $$(\alpha = \pi/2)$$. Otherwise, $$\alpha$$ represents a mixing angle. The resulting surface, with domain chosen to prevent self-intersection, is a catenary rotated around the $$\mathbf{X}_3$$ axis in a helical fashion.

Lines of curvature
One can rewrite each element of second fundamental matrix as a function of $$f$$ and $$g$$, for example $$ \mathbf{X_{uu}} \cdot \mathbf{\hat{n}} = \frac{1}{|g|^2+1} \begin{bmatrix} \operatorname{Re} \left(  ( 1- g^2 ) f' - 2gfg'\right)  \\ \operatorname{Re} \left( ( 1+ g^2 ) f'i+ 2gfg'i \right) \\ \operatorname{Re} \left(  2gf' +2fg' \right) \\ \end{bmatrix} \cdot \begin{bmatrix} \operatorname{Re} \left( 2g  \right) \\ \operatorname{Re} \left( -2gi  \right) \\ \operatorname{Re} \left( |g|^2-1  \right) \\ \end{bmatrix} = -2\operatorname{Re} (fg') $$

And consequently the second fundamental form matrix can be simplified as $$\begin{bmatrix} -\operatorname{Re} f g' & \;\;  \operatorname{Im} f g' \\ \operatorname{Im} f g' & \;\;  \operatorname{Re} f g' \end{bmatrix} $$ One of its eigenvectors is $$ \overline{\sqrt{ f g'} }$$ which represents the principal direction in the complex domain. Therefore, the two principal directions in the $$uv$$ space turn out to be $$ \phi = -\frac{1}{2} \operatorname{Arg}(f g') \pm k \pi /2$$