The area functional, first variation, and why minimal surfaces are critical points
The calculus of variations is the mathematical framework for finding functions that minimize (or maximize) a functional — a "function of functions." For minimal surfaces, the functional is the area:
A[S] = ∬ √(EG - F²) du dv
A surface is minimal when it is a critical point of this functional — meaning the first variation vanishes for all compactly supported normal perturbations.
Perturb a catenoid (minimal surface) by adding a bump. The left panel shows the deformed surface; the right panel plots the area as a function of bump height. Notice the parabolic minimum at h = 0 — the signature of a critical point where dA/dh = 0.
If we deform S by a normal variation S + tφN:
This vanishes for all φ if and only if H = 0.
The Jacobi operator governs stability:
A minimal surface is stable if the smallest eigenvalue of L is non-negative.
Pull the boundary circles apart. As the separation increases, the smallest Jacobi eigenvalue decreases toward zero. At the critical separation, the catenoid becomes unstable and collapses into two flat disks — the globally area-minimizing configuration.