Skip to content

Minimal Surfaces

Source lecture(s): PHY622 Lec1

Intuition

A soap film spanning a wire loop minimizes its surface area, giving a fun physical realization of a variational problem.

Formal Definition

A surface with zero mean curvature (\(H=0\)) at every point.

Mathematical Formulation

Mean curvature: \(H=\frac{1}{2}(\kappa_1+\kappa_2)=0\). Area functional: \(A=\int\sqrt{EG-F^2}\,du\,dv\).

Derivation

Apply variational calculus to the area functional with Dirichlet boundary conditions; the extremal satisfies \(H=0\).

Worked Example

Between two parallel rings, the catenoid is a minimal surface.

Common Mistakes

  • Confusing minimal surface with minimum-energy surface under tension.
  • Assuming \(H=0\) implies planar.

Quiz

Q1. What is the defining equation for a minimal surface?

Answer

Zero mean curvature \(H=0\).