Mean curvature H = 0 everywhere — soap film between two rings
22
50
5
40°
A minimal surface satisfies H = (κ₁+κ₂)/2 = 0 at every point (principal curvatures equal and opposite). Catenoid: r(t) = c·cosh(z/c), the unique minimal surface of revolution. Meusnier 1776; Euler 1740. Helicoid is the only other embedded minimal surface besides plane (do Carmo–Dajczer 1983). Costa surface (1982): first new complete embedded minimal surface in decades — three ends, genus 1.