User:Sam Derbyshire/Weierstrass-Enneper parametrization

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 &fnof; and g be functions on either the entire complex plane or the unit disk, where g is meromorphic and &fnof; 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 &fnof;g2 is holomorphic), and let c1, c2, c3 be constants. Then the surface with coordinates (x1,x2,x3) is minimal, where the xk are defined using the real part of a complex integral, as follows:


 * $$\begin{align}

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{align}$$

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

To explain why this works, remember than given the Gauss map ν of our surface M to the sphere S2, we can consider the composite map $$G \colon M \overset{\nu}{\to} S^2 \overset{\pi}{\to} \mathbb{C}$$, where π is just the stereographic projection. We can then consider the map $$G \circ s$$ obtained by taking a point in $$\mathbb{C}$$ to the corresponding point on M and then applying G. The surface M is then minimal if and only if $$G \circ s$$ is conformal (holomorphic).

It is in fact possible to explicitly write out G for a minimal surface, and we obtain $$ G = \frac{-dx_1 + i dx_2}{dx_3}$$. This allows us to find x1, x2 and x3 by integrating, and we get that $$ \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \Re \int \begin{pmatrix} \frac{1}{2} \left (\frac{1}{G} - G \right ) \\ \frac{i}{2} \left (\frac{1}{G} + G \right ) \\ 1 \end{pmatrix} dx_3$$. This is totally analoguous to the previous formula where we can just see dx3 as fgdz and G as g, and we obtain the same formula. dx3 is often written as dh, and called the height differential.

This method allows us to give parametrizations for many minimal surfaces:

Enneper surface: G = z, dh = z dz

Catenoid: G = z, dh = dz/z

Helicoid: G = z, dh = i dz/z

k-noid: G = zk-1, dh = (zk + z-k -2)-1 dz/z

This is also the method that lead to the construction of Costa's minimal surface and Riemann's minimal surface, using the Weierstrass p function.