minimal surfaceMinimalSurface22008-06-15 13:44:462008-06-15 13:44:46Definitionmean curvature at surface point% this is the default PlanetMath preamble. as your knowledge
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Among the surfaces \,$F(x,\,y,\,z) = 0$,\, with $F$ twice continuously differentiable, a {\em 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.