Building minimal surfaces from complex analysis — the (g, η) construction
Feed two ingredients from complex analysis into one formula and out comes a minimal surface. Change the ingredients and you get a different surface.
Pick a preset — catenoid, helicoid, Enneper, or Scherk — to see the surface beside a coloring of its Gauss map g(z): hue shows the angle of g(z), brightness its size.
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.
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
Follow the orange dot across the Enneper surface on the left and watch its normal direction trace a path on the sphere on the right.
The function g in the Weierstrass representation is exactly the stereographic projection of this normal — a deep connection between complex analysis and geometry.
Press Animate to watch the catenoid bend into the helicoid without stretching, then try the Enneper family.
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).
Spot a mistake or have an idea?