minimal surface

Among the surfaces  $F(x,y,z)=0$,  with $F$ twice continuously differentiable, a minimal surface is such that in every of its points, the mean curvature vanishes.  Because the mean curvature is the arithmetic mean of the principal curvatures ${\varkappa }_1$ and ${\varkappa }_2$, the equation

$${\varkappa }_2=-{\varkappa }_1$$

is valid in each point of a minimal surface.

A minimal surface has also the property that every sufficiently little portion of it has smaller area than any other regular surface with the same boundary curve.

Trivially, a plane is a minimal surface.  The catenoid is the only surface of revolution which is also a minimal surface.