Building minimal surfaces from complex analysis — the (g, η) construction
The Weierstrass representation is a remarkable bridge between complex analysis and differential geometry. It shows that every minimal surface can be constructed from a pair of complex-analytic data: a meromorphic function g (the stereographic Gauss map) and a holomorphic 1-form η.
This construction is extraordinarily powerful: by choosing different (g, η) pairs, you can generate every known minimal surface — and discover new ones.
Select a Weierstrass data preset to see the corresponding minimal surface alongside the domain coloring of its Gauss map g(z). The hue encodes the argument of g(z), and the brightness encodes its modulus.
Given a meromorphic function g and a holomorphic 1-form η = f(z)dz, the minimal surface is:
X₁ = Re ∫ (1 - g²) η
X₂ = Re ∫ i(1 + g²) η
X₃ = Re ∫ 2g η
The orange dot traces a point on the Enneper surface (left) and its corresponding unit normal on the Gauss map sphere (right). The function g in the Weierstrass representation is exactly the stereographic projection of this normal — a deep connection between complex analysis and geometry.
Multiplying η by e^(iθ) rotates through the associate family — a continuous family of isometric minimal surfaces. The most famous example morphs the catenoid (θ = 0) into the helicoid (θ = π/2).