Surfaces that minimize area — mean curvature H = 0 everywhere. Soap films naturally adopt these shapes, solving Plateau's problem.
These surfaces satisfy H = (κ₁+κ₂)/2 = 0: principal curvatures cancel. The catenoid is the only minimal surface of revolution; it and the plane are the only complete embedded minimal surfaces with finite topology (Meeks-Yau, López-Ros).