Editing Minimal surface (section)

==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.