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:
$\begin x_k(\zeta) &= \Re \left\ + c_k , \qquad k=1,2,3 \\ \varphi_1 &= f(1-g^2)/2 \\ \varphi_2 &= \mathbf f(1+g^2)/2 \\ \varphi_3 &= fg \end$

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 , .

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:
$\mathbf = \begin \left( 1 - g^2(\omega) \right)f(\omega) \\ i\left( 1+ g^2(\omega) \right)f(\omega) \\ 2g(\omega) f(\omega) \end$
where $f(\omega)$ and $g(\omega)$ are holomorphic functions of $\omega$.

The Jacobian $\mathbf$ represents the two orthogonal tangent vectors of the surface:
$\mathbf = \begin \operatorname\mathbf_1 \\ \operatorname\mathbf_2 \\ \operatorname \mathbf_3 \end \;\;\;\; \mathbf = \begin -\operatorname\mathbf_1 \\ -\operatorname\mathbf_2 \\ -\operatorname \mathbf_3 \end$

The surface normal is given by
$\mathbf = \frac = \frac \begin 2\operatorname g \\ 2\operatorname g \\ , g, ^2-1 \end$

The Jacobian $\mathbf$ leads to a number of important properties: $\mathbf \cdot \mathbf=0$, $\mathbf^2 = \operatorname(\mathbf^2)$, $\mathbf^2 = \operatorname(\mathbf^2)$, $\mathbf + \mathbf=0$. 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:
$\begin \mathbf \cdot \mathbf & \;\; \mathbf \cdot \mathbf\\ \mathbf \cdot \mathbf & \;\;\mathbf \cdot \mathbf \end= \begin 1 & 0 \\ 0 & 1 \end$

and the second fundamental form matrix
$\begin \mathbf \cdot \mathbf & \;\; \mathbf \cdot \mathbf\\ \mathbf \cdot \mathbf & \;\; \mathbf \cdot \mathbf \end$

Finally, a point $\omega_t$ on the complex plane maps to a point $\mathbf$ on the minimal surface in $\R^3$ by
$\mathbf= \begin \operatorname \int_^\mathbf_1 d\omega\\ \operatorname \int_^ \mathbf_2 d\omega\\ \operatorname \int_^ \mathbf_3 d\omega \end$
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 $\mathbb^3$ with finite topology include the plane, the catenoid, the helicoid, and the Costa's minimal surface. Costa's surface involves Weierstrass's elliptic function $\wp$:
$g(\omega)=\frac$
$f(\omega)= \wp(\omega)$
where $A$ is a constant.

Helicatenoid

Choosing the functions $f(\omega) = e^e^$ and $g(\omega) = e^$, a one parameter family of minimal surfaces is obtained.

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

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

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 $\mathbf_3$ 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
$\mathbf \cdot \mathbf = \frac \begin \operatorname \left( ( 1- g^2 ) f' - 2gfg'\right) \\ \operatorname \left( ( 1+ g^2 ) f'i+ 2gfg'i \right) \\ \operatorname \left( 2gf' +2fg' \right) \\ \end \cdot \begin \operatorname \left( 2g \right) \\ \operatorname \left( -2gi \right) \\ \operatorname \left( , g, ^2-1 \right) \\ \end = -2\operatorname (fg')$

And consequently the second fundamental form matrix can be simplified as
$\begin -\operatorname f g' & \;\; \operatorname f g' \\ \operatorname f g' & \;\; \operatorname f g' \end$

One of its eigenvectors is $\overline$ which represents the principal direction in the complex domain. Therefore, the two principal directions in the $uv$ space turn out to be
$\phi = -\frac \operatorname(f g') \pm k \pi /2$

See also

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

References

{{DEFAULTSORT:Weierstrass-Enneper parameterization
Differential geometry
Surfaces
Minimal surfaces