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:

$\begin{align} x_k(\zeta) &{}= \Re \left\{ \int_{0}^{\zeta} \varphi_{k}(z) \, dz \right\} %2B c_k , \qquad k=1,2,3 \\ \varphi_1 &{}= f(1-g^2)/2 \\ \varphi_2 &{}= \bold{i} f(1%2Bg^2)/2 \\ \varphi_3 &{}= fg \end{align}$

The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type.^[1]

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

References

^ Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O. Minimal surfaces, vol. I, p. 108. Springer 1992. ISBN 3540531696