Plateau's problem: find the surface of minimum area spanning a wire boundary
Soap films solve Plateau's problem (Plateau 1849, mathematical solution by Douglas and Radó 1930-31 — both received Fields medals). A minimal surface has zero mean curvature H=(κ₁+κ₂)/2=0 everywhere. The discrete Laplace-Beltrami operator drives each interior vertex toward the mean of its neighbors, implementing surface tension γ∇²z=0. The catenoid is the only minimal surface of revolution: r=a·cosh(z/a).