# Minimal Surface

(redirected from Minimal surfaces)

## minimal surface

[′min·ə·məl ′sər·fəs]
(mathematics)
A surface that has assumed a geometric configuration of least area among those into which it can readily deform.

## 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 + q2)r − 2pqs + (1 + p2)t = 0

where

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.

### 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.)
References in periodicals archive ?
Introduction to Global Analysis: Minimal Surfaces in Riemannian Manifolds
In this paper we discuss the curvature ellipse of minimal surfaces in [N.
In the classical theories of minimal surfaces in three dimensional Euclidean space [E.
Their topics include minimal surfaces with finite topology and more than one end, limits of embedded minimal surfaces without local area or curvature bounds, conformal structure of minimal surfaces, embedded minimal surfaces of finite genus, topological aspects of minimal surfaces, and Calabi-Yau problems.
Minimal surfaces are one of the most important surface classes in differential geometry.
These surfaces are strongly related to minimal surfaces [1]-[4].
Key words: harmonic univalent function, minimal surfaces distortion bounds, neighborhood.
An animated plot showing the minimal surfaces as m is varied in equal increments from -5 to 5
The classical Weierstrass formulae for minimal surfaces immersed in the three-dimensional Euclidean space [R.
These proceedings of the January 2006 conference include 31 papers covering a wide range of topics in the geometric theory of functions of one and several complex variables, including univalent functions, conformal and quasi-conformal mappings, minimal surfaces and dynamics in infinite-dimensional spaces along with other related subjects.
The second method deals with efficient approximation of minimal surfaces and geodesics.
The exhibition is a memorial to Prof Willmore, whose life's work, which had international influence, was in a very different sphere ( the minimal surfaces of n-dimensional geometry.

Site: Follow: Share:
Open / Close