Minimal Surfaces

Zero mean curvature: soap films, the catenoid, and the helicoid.

The idea

A minimal surface is a surface whose mean curvature vanishes at every point: $H = \tfrac{1}{2}(\kappa_1 + \kappa_2) = 0,$ where $\kappa_1$ and $\kappa_2$ are the principal curvatures.

The condition is a derivative test for area. Deform the surface slightly in the normal direction; a computation shows that the first variation of area is controlled by $H$, so $H = 0$ everywhere is exactly the condition that the area is stationary. Soap films realize it physically: a film spanning a wire loop pulls itself to the least area with that boundary. As with a vanishing derivative, stationary area is necessary for least area but does not by itself guarantee it.

Minimality is extrinsic: it describes how the surface sits in space, not distances measured inside it. It does not, however, depend on the choice of unit normal, since reversing the normal reverses the sign of $H$ and leaves $0$ fixed. The plane is minimal because both principal curvatures vanish; the catenoid and the helicoid are minimal because their principal curvatures cancel at every point without vanishing.

Ways to work on it

• Walkthrough. Learn the vanishing-mean-curvature condition that defines a minimal surface and what it forces on the principal curvatures.
• Practice. Compute the mean curvature and decide whether a point is minimal.
• Hardest. Verify that a classical surface of revolution is minimal and find its Gaussian curvature.

Not sure where to start? Take the ten-question placement test.