Minimal Surfaces

Plateau's Problem — Zero Mean Curvature & Soap Film Physics

Surface Type

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Properties

Mean curvature H≡ 0
Gauss curvature K
Topology χ
Self-intersections

Theory

Minimal surface: H = 0
H = (κ₁ + κ₂)/2 = 0
(principal curvatures cancel)

Weierstrass-Enneper:
x + iy = ∫f(1−g²) + if(1+g²) + 2fg dw

Physical: soap film minimizes area between wire boundaries — Laplace pressure ΔP = 2γH = 0