Minimal Surface

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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 ?
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.
In three dimensions, minimal surfaces divide spaces efficiently without intersecting--a critical attribute for cells since they can't function without a clear separation between inside and outside.
Duran's topics include general properties of harmonic mappings, harmonic mappings into convex regions, harmonic self-mappings of the disk, harmonic univalent functions, external problems, mapping problems, minimal surfaces and curvature of minimal surfaces, with particular attention to the Weierstrass-Enneper representation.
The finding, made by David Hoffman and Fusheng Wei of the University of Massachusetts at Amherst and Hermann Karcher of the University of Bonn in Germany, is the latest in a series of discoveries that have greatly expanded the number of known examples of minimal surfaces of various types (SN: 3/16/85, p.
Topics include a review of preliminaries such as continuous and Holder continuous functions, Sobolev spaces and convex analysis; classical methods such as Euler-Lagrange equations; direct methods such as the Dirichlet integral; regularity, such as the one-dimensinal case; minimal surfaces such as in the Douglas- Courant-Tonelli method; and isoperimetric inequality.
Given suitable geometric starting points, the surface evolver computes and displays a wide range of minimal surfaces.
The question of the geometry of the interface between two linked but repelling polymers turns out to be closely related to the mathematical problem of defining minimal surfaces -- surfaces that take up the least possible area within a certain boundary.
Using her theory, Taylor can now compute and display the different types of minimal surfaces that a crystal surface assumes within a given boundary -- the solid analogs of soap films confined within a certain wire loop.
Taylor's foray into "cubic" bubbles andrelated anisotropic forms is an extension of centuries of research done on minimal surfaces, as inspired by soap film studies.
Hoffman of the University of Massachusetts in Amherst, "The collection of all possible minimal surfaces is extremely rich, complicated and not yet completely understood.
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.

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