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Homeminimal surface

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# 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.

Related:

PlateausProblem, LeastSurfaceOfRevolution

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## Mathematics Subject Classification

53A05*no label found*26B05

*no label found*26A24

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