= Minimal surface
{wiki}
A <minimal surface> has vanishing <mean curvature vector>. For a <smooth surface> in three-dimensional <Euclidean space>, this means $H=0$, equivalently $k_1+k_2=0$. Thus <Gaussian curvature> is $K=-k_1^2\leq0$, but neither <principal curvature> needs to vanish. A <plane> and a <catenoid> both satisfy the condition, with very different <Gaussian curvatures>.
By the <first variation of area formula>, a <minimal surface> is a <stationary point> of <surface area> for every smooth variation that fixes its boundary, or has <compact support> in its interior. The word “minimal” specifies this differential equation; it does not guarantee a <global minimum> of <surface area>, or even a <local minimum>. A <stable minimal surface> imposes a further condition on the <second variation>.
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