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.