Editing Minimal surface

{{Short description|Surface that locally minimizes its area}}
{{Distinguish|text= [[minimal rational surface]] or [[Enriques–Kodaira classification#Minimal models and blowing up|minimal algebraic surface]]}}
[[Image:Bulle de savon hélicoïde.PNG|thumb|180px|right|A [[helicoid]] minimal surface formed by a soap film on a helical frame]]
In [[mathematics]], a '''minimal surface''' is a surface that locally minimizes its area. This is equivalent to having zero [[mean curvature]] (see definitions below).

The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a [[soap film]], which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may [[Immersed submanifold#Immersed submanifolds|self-intersect]] or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see [[minimal surface of revolution]]): the standard definitions only relate to a [[local optimum]], not a [[global optimum]].

==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>

==History==
Minimal surface theory originates with [[Lagrange]] who in 1762 considered the variational problem of finding the surface <math>z=z(x,y)</math> of least area stretched across a given closed contour. He derived the [[Euler–Lagrange equation]] for the solution

:<math>\frac{d}{dx}\left(\frac{z_x}{\sqrt{1+z_x^2+z_y^2}}\right ) + \frac{d}{dy}\left(\frac{z_y}{\sqrt{1+z_x^2+z_y^2}}\right )=0</math>

He did not succeed in finding any solution beyond the plane. In 1776 [[Jean Baptiste Marie Meusnier]] discovered that the [[helicoid]] and [[catenoid]] satisfy the equation and that the differential expression corresponds to twice the [[mean curvature]] of the surface, concluding that surfaces with zero mean curvature are area-minimizing.

By expanding Lagrange's equation to

:<math>\left(1 + z_x^2\right)z_{yy} - 2z_xz_yz_{xy} + \left(1 + z_y^2\right)z_{xx} = 0</math>

[[Gaspard Monge]] and [[Adrien-Marie Legendre|Legendre]] in 1795 derived representation formulas for the solution surfaces. While these were successfully used by [[Heinrich Scherk]] in 1830 to derive his [[Scherk surface|surfaces]], they were generally regarded as practically unusable. [[Eugène Charles Catalan|Catalan]] proved in 1842/43 that the helicoid is the only [[ruled surface|ruled]] minimal surface.

Progress had been fairly slow until the middle of the century when the [[Björling problem]] was solved using complex methods. The "first golden age" of minimal surfaces began. [[Hermann Schwarz|Schwarz]] found the solution of the [[Plateau problem]] for a regular quadrilateral in 1865 and for a general quadrilateral in 1867 (allowing the construction of his periodic [[Schwarz minimal surface|surface families]]) using complex methods. [[Weierstrass]] and [[Alfred Enneper|Enneper]] developed more useful [[Weierstrass–Enneper parameterization|representation formulas]], firmly linking minimal surfaces to [[complex analysis]] and [[harmonic functions]]. Other important contributions came from Beltrami, Bonnet, Darboux, Lie, Riemann, Serret and Weingarten.

Between 1925 and 1950 minimal surface theory revived, now mainly aimed at nonparametric minimal surfaces. The complete solution of the Plateau problem by [[Jesse Douglas]] and [[Tibor Radó]] was a major milestone. [[Bernstein's problem]] and [[Robert Osserman]]'s work on complete minimal surfaces of finite total curvature were also important.

Another revival began in the 1980s. One cause was the discovery in 1982 by Celso Costa of [[Costa's minimal surface|a surface]] that disproved the conjecture that the plane, the catenoid, and the helicoid are the only complete embedded minimal surfaces in <math>\R^3</math> of finite topological type. This not only stimulated new work on using the old parametric methods, but also demonstrated the importance of computer graphics to visualise the studied surfaces and numerical methods to solve the "period problem" (when using the [[Associate family|conjugate surface method]] to determine surface patches that can be assembled into a larger symmetric surface, certain parameters need to be numerically matched to produce an embedded surface). Another cause was the verification by H. Karcher that the [[triply periodic minimal surface]]s originally described empirically by Alan Schoen in 1970 actually exist. This has led to a rich menagerie of surface families and methods of deriving new surfaces from old, for example by adding handles or distorting them.

Currently the theory of minimal surfaces has diversified to minimal submanifolds in other ambient geometries, becoming relevant to mathematical physics (e.g. the [[positive mass conjecture]], the [[Riemannian Penrose inequality|Penrose conjecture]]) and three-manifold geometry (e.g. the [[Smith conjecture]], the [[Poincaré conjecture]], the [[Thurston Geometrization Conjecture]]).

==Examples==
[[File:Costa's Minimal Surface.png|thumb|[[Costa's minimal surface]]]]

Classical examples of minimal surfaces include:
* the [[plane (geometry)|plane]], which is a [[trivial (mathematics)|trivial]] case
* [[catenoid]]s: minimal surfaces made by rotating a [[catenary]] once around its directrix
* [[helicoid]]s: A surface swept out by a line rotating with uniform velocity around an axis perpendicular to the line and simultaneously moving along the axis with uniform velocity

Surfaces from the 19th century golden age include:
* [[Schwarz minimal surface]]s: [[Triply periodic minimal surface|triply periodic surfaces]] that fill <math>\R^3</math>
* [[Riemann's minimal surface]]: A posthumously described periodic surface
* the [[Enneper surface]]
* the [[Henneberg surface]]: the first non-orientable minimal surface
* [[Bour's minimal surface]]
* the [[Neovius surface]]: a triply periodic surface

Modern surfaces include:
* the [[Gyroid]]: One of Schoen's surfaces from 1970, a triply periodic surface of particular interest for [[liquid crystal]] structure
* the [[Saddle tower]] family: generalisations of [[Scherk surface|Scherk's second surface]]
* [[Costa's minimal surface]]: Famous conjecture disproof. Described in 1982 by [[Celso Costa]] and later visualized by [[James Hoffman|Jim Hoffman]]. Jim Hoffman, David Hoffman and William Meeks III then extended the definition to produce a family of surfaces with different rotational symmetries.
* the [[Chen–Gackstatter surface]] family, adding handles to the Enneper surface.

==Generalisations and links to other fields==
Minimal surfaces can be defined in other [[manifolds]] than <math>\R^3</math>, such as [[hyperbolic space]], higher-dimensional spaces or [[Riemannian manifolds]].

The definition of minimal surfaces can be generalized/extended to cover [[constant-mean-curvature surface]]s: surfaces with a constant mean curvature, which need not equal zero.

The curvature lines of an isothermal surface form an isothermal net.<ref>{{Cite web |title=Isothermal surface - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Isothermal_surface#:~:text=A%20surface%20whose%20curvature%20lines,Rotation%20surface;%20Minimal%20surface). |access-date=2022-09-04 |website=encyclopediaofmath.org}}</ref>

In [[discrete differential geometry]] discrete minimal surfaces are studied: [[simplicial complex]]es of triangles that minimize their area under small perturbations of their vertex positions.<ref>{{cite journal | first1=Ulrich | last1=Pinkall | first2=Konrad | last2=Polthier | title=Computing Discrete Minimal Surfaces and Their Conjugates | journal=[[Experimental Mathematics (journal)|Experimental Mathematics]] | volume=2 | issue=1 | pages=15–36 | year=1993 | mr=1246481 | url=http://projecteuclid.org/euclid.em/1062620735| doi=10.1080/10586458.1993.10504266 }}</ref> Such discretizations are often used to approximate minimal surfaces numerically, even if no closed form expressions are known.

[[Wiener process|Brownian motion]] on a minimal surface leads to probabilistic proofs of several theorems on minimal surfaces.<ref>{{cite journal | first=Robert | last=Neel | year=2009 | arxiv=0805.0556 | title=A martingale approach to minimal surfaces | journal=Journal of Functional Analysis | volume=256 | issue=8 | pages=2440–2472 | doi=10.1016/j.jfa.2008.06.033 | mr=2502522| s2cid=15228691 }}</ref>

Minimal surfaces have become an area of intense scientific study, especially in the areas of [[molecular engineering]] and [[materials science]], due to their anticipated applications in [[self-assembly]] of complex materials.<ref>{{Cite journal |last1=Han |first1=Lu |last2=Che |first2=Shunai |date=April 2018 |title=An Overview of Materials with Triply Periodic Minimal Surfaces and Related Geometry: From Biological Structures to Self-Assembled Systems |url=https://onlinelibrary.wiley.com/doi/10.1002/adma.201705708 |journal=Advanced Materials |language=en |volume=30 |issue=17 |page=1705708 |doi=10.1002/adma.201705708|pmid=29543352 |bibcode=2018AdM....3005708H |s2cid=3928702 }}</ref> The [[endoplasmic reticulum]], an important structure in cell biology, is proposed to be under evolutionary pressure to conform to a nontrivial minimal surface.<ref>{{Cite journal|last1=Terasaki|first1=Mark|last2=Shemesh|first2=Tom|last3=Kasthuri|first3=Narayanan|last4=Klemm|first4=Robin W.|last5=Schalek|first5=Richard|last6=Hayworth|first6=Kenneth J.|last7=Hand|first7=Arthur R.|last8=Yankova|first8=Maya|last9=Huber|first9=Greg|date=2013-07-18|title=Stacked endoplasmic reticulum sheets are connected by helicoidal membrane motifs|journal=Cell|volume=154|issue=2|pages=285–296|doi=10.1016/j.cell.2013.06.031|issn=0092-8674|pmc=3767119|pmid=23870120}}</ref>

In the fields of [[general relativity]] and [[Lorentzian manifold|Lorentzian geometry]], certain extensions and modifications of the notion of minimal surface, known as [[apparent horizon]]s, are significant.<ref>Yvonne Choquet-Bruhat. General relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2009. xxvi+785 pp. {{ISBN|978-0-19-923072-3}} (page 417)</ref> In contrast to the [[event horizon]], they represent a [[curvature]]-based approach to understanding [[black hole]] boundaries.

[[File:CircusTent02.jpg|thumb|Circus tent approximates a minimal surface.]]
Structures with minimal surfaces can be used as tents.

Minimal surfaces are part of the [[Generative Design|generative design]] toolbox used by modern designers. In architecture there has been much interest in [[tensile structure]]s, which are closely related to minimal surfaces. Notable examples can be seen in the work of [[Frei Otto]], [[Shigeru Ban]], and [[Zaha Hadid]]. The design of the [[Olympiastadion (Munich)|Munich Olympic Stadium]] by Frei Otto was inspired by soap surfaces.<ref>{{Cite web |date=2011-02-11 |title=AD Classics: Olympiastadion (Munich Olympic Stadium) / Behnisch and Partners & Frei Otto |url=https://www.archdaily.com/109136/ad-classics-munich-olympic-stadium-frei-otto-gunther-behnisch |access-date=2022-09-04 |website=ArchDaily |language=en-US}}</ref> Another notable example, also by Frei Otto, is the German Pavilion at [[Expo 67 pavilions|Expo 67]] in Montreal, Canada.<ref>{{Cite web |title=Expo 67 German Pavilion |url=https://architectuul.com/architecture/expo-67-german-pavilion |access-date=2022-09-04 |website=Architectuul}}</ref>

In the art world, minimal surfaces have been extensively explored in the sculpture of [[Robert Engman]] (1927–2018), [[Robert Longhurst]] (1949– ), and [[Charles O. Perry]] (1929–2011), among others.

==See also==
{{div col|colwidth=20em}}
* [[Bernstein's problem]]
* [[Bilinear interpolation]]
* [[Bryant surface]]
* [[Curvature]]
* [[Enneper–Weierstrass parameterization]]
* [[Harmonic map]]
* [[Harmonic morphism]]
* [[Plateau's problem]]
* [[Schwarz minimal surface]]
* [[Soap bubble]]
* [[Surface Evolver]]
* [[Stretched grid method]]
* [[Tensile structure]]
* [[Triply periodic minimal surface]]
* [[Weaire–Phelan structure]]
{{Div col end}}

==References==
{{Reflist}}

==Further reading==
'''Textbooks'''
* [[R. Courant]]. ''Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces.'' Appendix by M. Schiffer. Interscience Publishers, Inc., New York, N.Y., 1950. xiii+330 pp.
* [[H. Blaine Lawson, Jr.]] ''Lectures on minimal submanifolds. Vol. I.'' Second edition. Mathematics Lecture Series, 9. Publish or Perish, Inc., Wilmington, Del., 1980. iv+178 pp. {{ISBN|0-914098-18-7}}
* [[Robert Osserman]]. ''A survey of minimal surfaces.'' Second edition. Dover Publications, Inc., New York, 1986. vi+207 pp. {{ISBN|0-486-64998-9}}, {{mr|0852409}}
* Johannes C.C. Nitsche. ''Lectures on minimal surfaces. Vol. 1. Introduction, fundamentals, geometry and basic boundary value problems.'' Translated from the German by Jerry M. Feinberg. With a German foreword. Cambridge University Press, Cambridge, 1989. xxvi+563 pp. {{ISBN|0-521-24427-7}}
* {{ cite book | last = Nishikawa | first = Seiki | year = 2002 | title = Variational problems in geometry | series = Translations of mathematical monographs; Iwanami series in modern mathematics | volume = 205 | isbn = 0821813560 | issn = 0065-9282 <!--| ref = harv --> | translator-last = Abe | translator-first = Kinetsu | publisher = Providence, R. I. : [[American Mathematical Society]]  | postscript = , translated from: }}
:* {{ cite book | author = 西川青季 | year = 1998 <!-- | date = 1998-01- --> | title = 幾何学的変分問題  | series = 岩波講座現代数学の基礎 | volume = 28 | publisher = [[Iwanami Shoten |岩波書店]] | location = Tokyo | isbn = 4-00-010642-2 | language = Japanese }}
* Ulrich Dierkes, Stefan Hildebrandt, and Friedrich Sauvigny. ''Minimal surfaces.'' Revised and enlarged second edition. With assistance and contributions by A. Küster and R. Jakob. Grundlehren der Mathematischen Wissenschaften, 339. Springer, Heidelberg, 2010. xvi+688 pp. {{ISBN|978-3-642-11697-1}}, {{doi|10.1007/978-3-642-11698-8}} {{closed access}}, {{mr|2566897}}
* [[Tobias Holck Colding]] and [[William Minicozzi|William P. Minicozzi]], II. ''A course in minimal surfaces.'' Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011. xii+313 pp. {{ISBN|978-0-8218-5323-8}}
'''Online resources'''
* {{cite web
 | first1 = Hermann
 | last1 = Karcher
 | first2 = Konrad
 | last2 = Polthier
 | url = http://page.mi.fu-berlin.de/polthier/booklet/intro.html
 | title= Touching Soap Films - An introduction to minimal surfaces
 | year  = 1995
 | access-date = December 27, 2006 }} ''(graphical introduction to minimal surfaces and soap films.)''
* {{cite web
 | title=Periodic Minimal Surfaces Gallery
 | url=http://www-klinowski.ch.cam.ac.uk/pmsgal1.html
| author = Jacek Klinowski
| access-date = February 2, 2009 }} ''(A collection of minimal surfaces with classical and modern examples)''
* {{cite web
 | title=Grape Minimal Surface Library
 | url=http://numod.ins.uni-bonn.de/grape/EXAMPLES/AMANDUS/amandus.html
 | author = Martin Steffens and Christian Teitzel
 | access-date = October 27, 2008 }} ''(A collection of minimal surfaces)''
* {{cite web
 | author = Various
 | url = http://www.eg-models.de/models.html
 | title= EG-Models
 | year  = 2000
 | access-date = September 28, 2004 }} ''(Online journal with several published models of minimal surfaces)''

== External links==
* {{springer|title=Minimal surface|id=p/m063920}}
* [http://3d-xplormath.org/j/index.html 3D-XplorMath-J Homepage &mdash; Java program and applets for interactive mathematical visualisation]
* [http://xahlee.org/surface/gallery_m.html Gallery of rotatable minimal surfaces]
* [http://www.princeton.edu/~rvdb/WebGL/minsurf.html WebGL-based Gallery of rotatable/zoomable minimal surfaces]

{{Minimal surfaces}}
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[[Category:Differential geometry]]
[[Category:Differential geometry of surfaces]]
[[Category:Minimal surfaces| ]]