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 ${\displaystyle f}$ and ${\displaystyle g}$ be functions on either the entire complex plane or the unit disk, where ${\displaystyle g}$ is meromorphic and ${\displaystyle f}$ is analytic, such that wherever ${\displaystyle g}$ has a pole of order ${\displaystyle m}$, ${\displaystyle f}$ has a zero of order ${\displaystyle 2m}$ (or equivalently, such that the product ${\displaystyle fg^{2}}$ is holomorphic), and let ${\displaystyle c_{1},c_{2},c_{3}}$ be constants. Then the surface with coordinates ${\displaystyle (x_{1},x_{2},x_{3})}$ is minimal, where the ${\displaystyle x_{k}}$ are defined using the real part of a complex integral, as follows:

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 f(z) = 1, g(z) = z^m.