PlanetMath: minimal surface

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minimal surface

(Definition)

Among the surfaces $F(x,\,y,\,z) = 0$ , with 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

$\displaystyle \varkappa_2 = -\varkappa_1$

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.

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"minimal surface" is owned by pahio.

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See Also: Plateau's Problem

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Cross-references: surface of revolution, catenoid, plane, curve, boundary, regular, area, property, equation, principal curvatures, arithmetic mean, vanishes, mean curvature, points, continuously differentiable, surfaces
There is 1 reference to this entry.

This is version 1 of minimal surface, born on 2008-06-15.
Object id is 10705, canonical name is MinimalSurface2.
Accessed 296 times total.

Classification:

AMS MSC:	26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
	26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)
	53A05 (Differential geometry :: Classical differential geometry :: Surfaces in Euclidean space)

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