Editing Minimal surface (section)

==Definitions==
[[File:Saddle Tower Minimal Surfaces.png|thumb|[[Saddle tower]] minimal surface. While any small change of the surface increases its area, there exist other surfaces with the same boundary with a smaller total area.]]

Minimal surfaces can be defined in several equivalent ways in <math>\R^3</math>. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially [[differential geometry]], [[calculus of variations]], [[potential theory]], [[complex analysis]] and [[mathematical physics]].<ref>{{cite journal
 | first1 = William H. III
 | last1 = Meeks
 | first2 = Joaquín
 | last2 = Pérez
 | year = 2011
 | title = The classical theory of minimal surfaces
 | journal =  [[Bull. Amer. Math. Soc.]]
 | volume = 48
 | issue = 3
 | pages = 325–407
 | doi = 10.1090/s0273-0979-2011-01334-9
 | mr = 2801776 
| doi-access = free
 }}</ref>

:'''Local least area definition''': A surface <math>M \subset \R^3</math> is minimal if and only if every point ''p'' ∈ ''M'' has a [[neighbourhood (topology)|neighbourhood]], bounded by a simple closed curve, which has the least area among all surfaces having the same boundary.

This property is local: there might exist regions in a minimal surface, together with other surfaces of smaller area which have the same boundary. This property establishes a connection with soap films; a soap film deformed to have a wire frame as boundary will minimize area.

:'''Variational definition''': A surface <math>M \subset \R^3</math> is minimal if and only if it is a [[Critical point (mathematics)|critical point]] of the area [[Functional (mathematics)|functional]] for all compactly supported [[Calculus of variations|variations]].

This definition makes minimal surfaces a 2-dimensional analogue to [[geodesics]], which are analogously defined as critical points of the length functional.

[[File:Minimal surface curvature planes-en.svg|thumb|Minimal surface curvature planes.  On a minimal surface, the curvature along the principal curvature planes are equal and opposite at every point. This makes the mean curvature zero.]]

:'''Mean curvature definition''': A surface <math>M \subset \R^3</math> is minimal if and only if its [[mean curvature]] is equal to zero at all points.

A direct implication of this definition is that every point on the surface is a [[saddle point]] with equal and opposite [[principal curvatures]]. Additionally, this makes minimal surfaces into the static solutions of [[mean curvature flow]]. By the [[Young–Laplace equation]], the [[mean curvature]] of a soap film is proportional to the difference in pressure between the sides. If the soap film does not enclose a region, then this will make its mean curvature zero. By contrast, a spherical [[soap bubble]] encloses a region which has a different pressure from the exterior region, and as such does not have zero mean curvature.

:'''Differential equation definition''': A surface <math>M \subset \R^3</math> formed by the image of a region <math>X \subset \R^2</math> under function <math> \mathbf{f} : X \to M </math>, <math>(x, y) \mapsto (x, y, u(x, y)) </math>, where <math>u: X \to \R</math> is a real valued function, is minimal if and only if <math>u</math> satisfies

::<math>(1+u_x^2)u_{yy} - 2u_xu_yu_{xy} + (1+u_y^2)u_{xx}=0</math>

:The [[partial differential equation]] in this definition was originally found in 1762 by [[Lagrange]],<ref name="Lagrange1760">J. L. Lagrange. Essai d'une nouvelle methode pour determiner les maxima et les minima des formules integrales indefinies. Miscellanea Taurinensia 2, 325(1):173{199, 1760.</ref> and [[Jean Baptiste Meusnier]] discovered in 1776 that it implied a vanishing mean curvature.<ref name="Meusnier1785">J. B. Meusnier. Mémoire sur la courbure des surfaces. Mém. Mathém. Phys. Acad. Sci. Paris, prés. par div. Savans, 10:477–510, 1785. Presented in 1776.</ref> This equation gives an asymmetric definition in the sense that the position on the <math>z</math>-axis is determined as a function <math>u</math> of <math>x</math> and <math>y</math>. Not all surfaces are conveniently represented this way. An alternative definition based on the more general representation <math>\mathbf{x} : \R^{2} \to \R^{3}, (u,v) \mapsto (x,y,z)</math> is

:<math>\frac{\partial}{\partial u} \frac{\frac{\partial \mathbf{x}}{\partial v} \boldsymbol{\times} (\frac{\partial \mathbf{x}}{\partial u} \boldsymbol{\times} \frac{\partial \mathbf{x}}{\partial v} )}{\sqrt{(\frac{\partial \mathbf{x}}{\partial u} \boldsymbol{\times} \frac{\partial \mathbf{x}}{\partial v} ) \boldsymbol{\cdot} (\frac{\partial \mathbf{x}}{\partial u} \boldsymbol{\times} \frac{\partial \mathbf{x}}{\partial v} )} } = \frac{\partial}{\partial v} \frac{\frac{\partial \mathbf{x}}{\partial u} \boldsymbol{\times} (\frac{\partial \mathbf{x}}{\partial u} \boldsymbol{\times} \frac{\partial \mathbf{x}}{\partial v} )}{\sqrt{(\frac{\partial \mathbf{x}}{\partial u} \boldsymbol{\times} \frac{\partial \mathbf{x}}{\partial v} ) \boldsymbol{\cdot} (\frac{\partial \mathbf{x}}{\partial u} \boldsymbol{\times} \frac{\partial \mathbf{x}}{\partial v} )} }</math>.
 
:'''Energy definition''': A [[Conformal map|conformal]] immersion <math>X: M \rightarrow \R^3</math> is minimal if and only if it is a critical point of the [[Dirichlet energy]] for all compactly supported variations, or equivalently if any point <math>p \in M</math> has a neighbourhood with least energy relative to its boundary.

This definition ties minimal surfaces to [[harmonic functions]] and [[potential theory]].

:'''Harmonic definition''': If <math>X = (x_1, x_2, x_3) : M \rightarrow \R^3</math> is an [[Isometry|isometric]] [[Immersion (mathematics)|immersion]] of a [[Riemann surface]] into 3-space, then <math>X</math> is said to be minimal whenever <math>x_i</math> is a [[harmonic function]] on <math>M</math> for each <math>i</math>.

A direct implication of this definition and the [[Harmonic functions#Maximum principle|maximum principle for harmonic functions]] is that there are no [[compact space|compact]] [[Complete metric space|complete]] minimal surfaces in <math>\R^3</math>.

:'''Gauss map definition''': A surface <math>M \subset \R^3</math> is minimal if and only if its [[Stereographic projection|stereographically]] projected [[Gauss map]] <math>g: M \rightarrow \C \cup {\infty}</math> is [[meromorphic]] with respect to the underlying [[Riemann surface]] structure, and <math>M</math> is not a piece of a sphere.

This definition uses that the mean curvature is half of the [[Trace (linear algebra)|trace]] of the [[Shape operator#Shape operator|shape operator]], which is linked to the derivatives of the Gauss map. If the projected Gauss map obeys the [[Cauchy–Riemann equations]] then either the trace vanishes or every point of ''M'' is [[Umbilical point|umbilic]], in which case it is a piece of a sphere.

The local least area and variational definitions allow extending minimal surfaces to other [[Riemannian manifolds]] than <math>\R^3</math>.<ref>See {{ harv | Nishikawa | 2002 }} about variational definition.</ref>