EnneperWeierstrassMinimalSurfaceRepresentation

Enneper-Weierstrass Representation of a Minimal Surface

The Enneper-Weierstrass representation is a mathematical formula used to describe minimal surfaces. Minimal surfaces are those that locally minimize their area under small variations, and they play a crucial role in various fields such as mathematics, physics, and engineering. The Enneper-Weierstrass representation is named after two mathematicians, Alfred Enneper and Karl Weierstrass, who contributed to the development of this theory.

In the Enneper-Weierstrass representation, a minimal surface can be parameterized by two complex-valued functions, typically called $g(z)$ and $h(z)$, where $z$ is a complex number. These functions are holomorphic (i.e., complex differentiable) and satisfy the so-called "Weierstrass-Enneper compatibility conditions," which ensure that the resulting surface is minimal. The representation is given by the following integral formula:

$$X(z) = \operatorname{Re} \left( \int (1 - g(z)^2) dz, \int (1 + g(z)^2) dz, \int (2g(z)) dz \right)$$

where

$$X(t) = (x(t), y(t), z(t))$$

is a parametrization of the minimal surface, and $\operatorname{Re}$ denotes the real part of the complex integrals.

Here, $g(z)$ is called the Weierstrass-Gauss map, and it represents the complex stereographic projection of the Gauss map of the minimal surface (which describes the orientation of the surface's normals). The function $h(z)$ is called the Weierstrass height differential and encodes information about the mean curvature of the surface.

The Enneper-Weierstrass representation is powerful because it allows for the systematic construction and study of minimal surfaces by manipulating the $g(z)$ and $h(z)$ functions. It also reveals deep connections between minimal surfaces, complex analysis, and algebraic geometry.