Surface tension minimization — mean curvature H=0 at equilibrium
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Physics
Soap films minimize surface area (surface energy = γ·Area). Mathematically, the equilibrium condition is mean curvature H = 0 (Euler-Lagrange equation for area functional). The catenoid (surface of revolution of cosh) is the only minimal surface of revolution. The helicoid is a ruled minimal surface. Plateau's problem: find a minimal surface spanning a given boundary — solved for any closed curve (Douglas-Rado, 1931). Soap bubbles: internal pressure ΔP = 4γ/R (two surfaces), so smaller bubbles have higher pressure.