Minimal Surfaces

Surfaces that minimize area - soap films and beyond

Minimal Surfaces

Dip a wire loop in soapy water and the film snaps into the smallest area it can. These minimal surfaces bend equally up and down at every point, so their mean curvature is zero.

Minimal Surface Gallery

Flip through the catenoid, helicoid, Enneper, and Scherk surfaces in soap-film shading. Each looks different, yet every point is a balanced saddle with H = 0.

The surface of revolution of a catenary curve. The only minimal surface of revolution.

Go deeper:Why Mean Curvature Zero?

A minimal surface is a surface with zero mean curvature(H = 0) everywhere. This means that at every point, the surface curves equally in opposite directions — like a saddle.

Minimal surfaces model soap films: when a soap film spans a wire frame, it naturally finds the shape that minimizes surface area, which always has H = 0.

A surface minimizes area (locally) if and only if its mean curvature vanishes. The mean curvature measures the “net bending” at each point:

H = (k₁ + k₂) / 2
Average of principal curvatures
H = 0 ⟺ k₁ = -k₂
Equal and opposite curvatures (saddle-like)

The first variation formula shows that H = 0 is equivalent to being a critical point of the area functional.

Go deeper:Famous Minimal Surfaces
Catenoid (1744)

Discovered by Euler. The only minimal surface of revolution. Formed by rotating a catenary curve.

Helicoid (1776)

Discovered by Meusnier. A ruled minimal surface shaped like a spiral staircase. The only ruled minimal surface besides the plane.

Enneper (1864)

A self-intersecting minimal surface with no boundary. Simple parametrization but beautiful saddle geometry.

Scherk (1834)

Doubly periodic minimal surface. Can be viewed as a deformation of two infinite orthogonal planes joined by saddles.

Go deeper:Soap Films and Plateau's Problem

Plateau's problem asks: given a closed curve in space, does there exist a minimal surface spanning that curve? This was answered affirmatively by Jesse Douglas and Tibor Radó in 1930.

Physical soap films solve this problem naturally: surface tension causes the film to minimize its area, always producing a surface with H = 0 (or forming bubbles where pressure differences create constant H ≠ 0).

Catenoid-Helicoid Transformation

Press play or drag θ to bend a catenoid into a helicoid. Every frame along the way is still a minimal surface.

Catenoidθ = 0.0°

The catenoid and helicoid are related by the associate family — a continuous 1-parameter deformation through minimal surfaces. They share the same Gauss map!

Go deeper:The Associate Family

The associate family: the catenoid and helicoid can be continuously deformed into each other through a family of minimal surfaces.

Every minimal surface belongs to a 1-parameter family of isometric minimal surfaces called the associate family. The transformation is:

Xθ = cos(θ) · X + sin(θ) · X*

where X* is the conjugate surface. The catenoid (θ = 0) and helicoid (θ = π/2) are conjugate pairs!

Mean Curvature Comparison

Switch between minimal and ordinary surfaces. Green marks H ≈ 0, so the catenoid and helicoid turn uniformly green while the sphere and torus do not.

Min H
-0.000
Avg H
-0.000
Max H
0.000
H ≈ 0 (minimal)
H > 0
H < 0

H = 0 everywhere

Key Takeaways

  • Minimal surfaces have zero mean curvature: k₁ = -k₂ at every point.
  • Soap films form minimal surfaces because they minimize area.
  • Minimal surfaces are critical points of the area functional.
  • The catenoid and helicoid are conjugate, linked by the associate family.

Spot a mistake or have an idea?

Search