Minimal Surface(redirected from Minimal submanifold)
minimal surface[′min·ə·məl ′sər·fəs]
a surface whose mean curvature at any point is zero. Minimal surfaces arise in solving the following variational problem. In the class of surfaces passing through a given closed curve in space find the surface such that its part included within the given curve has the least area (minimal area). If the given curve is a plane curve, then the part of the plane bounded by this curve will evidently be the solution.
In the case of a space curve, a surface with minimal area must satisfy a certain necessary condition. This condition was established by J. Lagrange in 1760 and was interpreted geometrically somewhat later by J. Meusnier in a form equivalent to the requirement that the mean curvature vanish. Although this condition is not sufficient, that is, it does not guarantee a minimum area, the term “minimal surface” has subsequently been preserved for every surface with zero mean curvature. If a surface is given by an equation of the form z = f(x,y) then setting the expression for the mean curvature equal to zero leads to the second-order partial differential equation
(1 + q2)r − 2pqs + (1 + p2)t = 0
p = ∂z/∂x, q = ∂z/∂y, r = ∂2z/∂x2
s = ∂2z/∂x∂y, t = ∂2z/∂y2
Many mathematicians, beginning with Lagrange and G. Monge, have investigated different forms of this equation. Examples of minimal surfaces are an ordinary spiral surface; the catenoid, which is the only real minimal surface of revolution; and the surface of Scherk, defined by the equation z = ln (cos y/cos x).
A minimal surface has nonpositive total curvature at any point. The Belgian physicist J. Plateau proposed a method for experimentally realizing a minimal surface using soap films stretched across a wire frame.
REFERENCESKagan, V. F. Osnovy teorii poverkhnostei v tenzornom izlozhenii,part 1. Moscow-Leningrad, 1947.
Courant, R., and H. Robbins. Chto takoe matematika, 2nd ed. Moscow, 1967. (Translated from English.)
Blaschke, W. Vvedenie v differentsial’nuiu geometriiiu. Moscow, 1957. (Translated from German.)