# minimal surface

Among the surfaces$F(x,\,y,\,z)=0$,  with $F$ twice continuously differentiable, a 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.

Title minimal surface MinimalSurface 2013-03-22 18:08:56 2013-03-22 18:08:56 pahio (2872) pahio (2872) 5 pahio (2872) Definition msc 53A05 msc 26B05 msc 26A24 PlateausProblem LeastSurfaceOfRevolution