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 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.

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 $G={\frac {-dx_{1}+idx_{2}}{dx_{3}}}$. 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 = z^k-1, dh = (z^k + 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.

References

Category:Differential geometry Category:surfaces