# Minimal Surface

(redirected from*Minimal submanifold*)

## minimal surface

[′min·ə·məl ′sər·fəs]## Minimal Surface

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 + *q*^{2})*r* − 2*pqs* + (1 + *p*^{2})*t* = 0

where

*p* = *∂z/∂x*, *q* = *∂z/∂y*, *r* = *∂*^{2}*z*/*∂x*^{2}

*s* = *∂ ^{2}z/∂x∂y, t = ∂*

^{2}

*z*/

*∂y*

^{2}

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.

### REFERENCES

Kagan, 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.)