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.

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 ?
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Kobayashi: Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields (1970), 59-75.
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Matsuyama: On some pinchings of minimal submanifolds, Geometry and its applications, edited by Tadashi Nagano et al.
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.
His major contributions include several work on conjectures, such as the Calabi conjecture, positive mass conjecture and existence of black holes, Smith conjecture, Hermitian Yang-Mills connection and stable vector bundles, Frankel conjecture and Mirror conjecture, as well as new methods and concepts of gradient estimates and Harnack inequalities, uniformization of complex manifolds, harmonic maps and rigidity, minimal submanifolds, and also open problems in geometry, covering harmonic functions with controlled growth, rank rigidity of nonpositively curved manifolds, Kahler-Einstein metrics and stability of manifolds and Mirror symmetry.