Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 24H / i / Solution

A minimal surface has zero mean curvature $H=(k_1+k_2)/2$; equivalently its area is stationary under compactly supported normal variations. Its principal curvatures satisfy $k_2=-k_1$, so its Gaussian curvature isEquality occurs precisely where its second fundamental form vanishes.