# Minimal Surface

(redirected from Minimal submanifold)

## 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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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 ?
Then M is semisymmetric if and only if M is minimal submanifold (in which case M is (n - 2)-ruled), or M is a round hypercone in some totally geodesic subspace [E.sup.n+1] of [E.sup.n+m].
Chen, Some pinching and classification theorems for minimal submanifolds, Arch.
If H = 0 on M we say that M is a minimal submanifold of [??].
If M satisfies the equality case of (6.1) identically, then M is a minimal submanifold and:
If M satisfies the equality case of (6.3) identically, then M is a minimal submanifold and:
If a Legendrian submanifold [M.sup.n] of a Sasakian space form [[??].sup.2n+1] (c) satisfies identically the equality case, then it is a minimal submanifold.
Tuliga, "Tuliga Discrete, multitime recurrences and discrete minimal submanifolds," Balkan Journal of Geometry and Its Applications, vol.
In the case of vanishing A, we recover the case of minimal submanifolds, which of course are the stationary points of the MCF.
Key words and phrases: para-Kahler geometry - Lagrangian submanifolds - Minimal submanifolds, Self-similar solutions to the Mean Curvature Flow.
Li: An intrinsic rigidity theorem for minimal submanifolds in a sphere, Arch.
Among the topics covered are computational aspects of discrete minimal surfaces, conjugate plateau constructions, parabolicity and minimal surfaces, the isoperimetric problem, the genus-one helicoids as a limit of screw-motion invariant helicoids with handles, isoperimetric inequalities of minimal submanifolds, embedded minimal disks, minimial surfaces of finite topology, conformal structures and necksizes of embedded constant mean curvature surfaces, and variational problems in Lagrangian geometry.

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