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:

x k ( ζ ) = ℜ { ∫ 0 ζ φ k ( z ) d z } + c k , k = 1 , 2 , 3 φ 1 = f ( 1 − g 2 ) / 2 φ 2 = i f ( 1 + g 2 ) / 2 φ 3 = f g

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 ƒ(z) = 1, g(z) = z^m.