Calculus of Variations

The area functional, first variation, and why minimal surfaces are critical points

Calculus of Variations

Calculus finds the lowest point of a curve. The calculus of variations finds the best shape among all possible shapes — here, the surface with the least area.

Area Functional Explorer

Raise and lower the bump on the catenoid and watch the area curve on the right. Its flat-bottomed minimum at h = 0 marks a critical point.

Catenoid + perturbation φ
Area functional A[S + hφ]
Go deeper:The Area Functional

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.

Go deeper:First Variation

If we deform S by a normal variation S + tφN:

δA = -∬ 2Hφ dA

This vanishes for all φ if and only if H = 0.

Catenoid Stability & Collapse

Pull the boundary circles apart and watch the eigenvalue bar shrink. When it hits zero, the catenoid collapses into two flat disks.

Jacobi Operator Eigenvalueλ₁ ≈ -0.010
unstable (λ₁ < 0)stable (λ₁ > 0)
Go deeper:Second Variation

The Jacobi operator governs stability:

L = Δ + |A|²

A minimal surface is stable if the first eigenvalue of −L = −Δ − |A|² is non-negative.

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.

Key Takeaways

  • Area functional: Minimal surfaces are critical points of area, not necessarily minima.
  • First variation → H = 0: The Euler–Lagrange equation for area is mean curvature = 0.
  • Jacobi operator: L = Δ + |A|² determines stability of a minimal surface.
  • Catenoid collapse: Beyond a critical separation, the catenoid is unstable and two disks win.

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