View source for Catenoid

{{Short description|Surface of revolution of a catenary}}
[[Image:Catenoid.svg|thumb|right|alt=three-dimensional diagram of a catenoid|A catenoid]]
[[Image:Catenoid.gif|thumb|right|alt=animation of a catenary sweeping out the shape of a catenoid as it rotates about a central point|A catenoid obtained from the rotation of a catenary]]

In [[geometry]], a '''catenoid''' is a type of [[Surface (mathematics)|surface]], arising by rotating a [[catenary]] curve about an axis (a [[surface of revolution]]).<ref>{{cite book|last1=Dierkes|first1=Ulrich|last2=Hildebrandt|first2=Stefan|last3=Sauvigny|first3=Friedrich|title=Minimal Surfaces|date=2010|publisher=[[Springer Science & Business Media]]|isbn=9783642116988|page=141|url=https://books.google.com/books?id=9YhBOg6vO-EC&pg=PA141|language=en}}</ref> It is a [[minimal surface]], meaning that it occupies the least area when bounded by a closed space.<ref name=Gullberg>{{cite book|last1=Gullberg|first1=Jan|title=Mathematics: From the Birth of Numbers|date=1997|publisher=[[W. W. Norton & Company]]|isbn=9780393040029|page=[https://archive.org/details/mathematicsfromb1997gull/page/538 538]|url=https://archive.org/details/mathematicsfromb1997gull|url-access=registration|language=en}}</ref> It was formally described in 1744 by the mathematician [[Leonhard Euler]].

[[Soap film]] attached to twin circular rings will take the shape of a catenoid.<ref name=Gullberg/> Because they are members of the same [[associate family]] of surfaces, a catenoid can be bent into a portion of a [[helicoid]], and vice versa.

== Geometry ==
The catenoid was the first non-trivial minimal [[surface (topology)|surface]] in 3-dimensional Euclidean space to be discovered apart from the [[plane (geometry)|plane]]. The catenoid is obtained by rotating a catenary about its [[Directrix (conic section)|directrix]].<ref name=Gullberg/> It was found and proved to be minimal by [[Leonhard Euler]] in 1744.<ref>{{cite book |last1=Euler |first1=Leonhard |author-link=Leonard Euler |editor-last=Carathéodory |editor-first=Constantin |editor-link=Constantin Carathéodory |title=Methodus inveniendi lineas curvas: maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti |date=1952 |orig-year=reprint of 1744 edition |publisher=Springer Science & Business Media |isbn=3-76431-424-9 |language=Latin |url=https://books.google.com/books?id=zNDdVFZalSAC}}</ref><ref name=Colding06>{{cite journal|last1=Colding|first1=T. H.|last2=Minicozzi|first2=W. P.|title=Shapes of embedded minimal surfaces|journal=Proceedings of the National Academy of Sciences|date=17 July 2006|volume=103|issue=30|pages=11106–11111|doi=10.1073/pnas.0510379103|pmc=1544050|pmid=16847265|bibcode=2006PNAS..10311106C|doi-access=free}}</ref>

Early work on the subject was published also by [[Jean Baptiste Meusnier]].<ref name=salvert>{{cite book|url=https://archive.org/details/mmoiresurlathor00salvgoog|format=PDF|last1=Meusnier|first1=J. B.|title=Mémoire sur la courbure des surfaces|trans-title=Dissertation on the curvature of surfaces |date=1881|publisher=F. Hayez, Printer of the Royal Academy of Belgium|location=Brussels|language=fr|isbn=9781147341744|pages=477–510}}</ref><ref name=Colding06/>{{rp|11106}} There are only two [[minimal surfaces of revolution]] ([[surfaces of revolution]] which are also minimal surfaces): the [[plane (geometry)|plane]] and the catenoid.<ref>{{cite web|title=Catenoid|url=http://mathworld.wolfram.com/Catenoid.html|website=Wolfram MathWorld|accessdate=15 January 2017|language=en}}</ref>

The catenoid may be defined by the following parametric equations:
{{NumBlk|::|<math display=block>\begin{align}
x &= c \cosh \frac{v}{c} \cos u \\
y &= c \cosh \frac{v}{c} \sin u \\
z &= v
\end{align}</math>|{{EquationRef|1}}}}
where <math>u \in [-\pi, \pi)</math> and <math>v \in \mathbb{R}</math> and <math>c</math> is a non-zero real constant.

In cylindrical coordinates:
<math display=block>\rho =c \cosh \frac{z}{c},</math>
where <math>c</math> is a real constant.

A physical model of a catenoid can be formed by dipping two [[circle|circular]] rings into a soap solution and slowly drawing the circles apart.

The catenoid may be also defined approximately by the [[stretched grid method]] as a facet 3D model.

== Helicoid transformation ==

[[Image:helicatenoid.gif|thumb|right|256px|alt=Continuous animation showing a right-handed helicoid deforming into a catenoid, a left-handed helicoid, and back again|Deformation of a right-handed [[helicoid]] into a left-handed one and back again via a catenoid]]

Because they are members of the same [[associate family]] of surfaces, one can bend a catenoid into a portion of a [[helicoid]] without stretching.  In other words, one can make a (mostly) [[continuous function|continuous]] and [[Isometry|isometric]] deformation of a catenoid to a portion of the [[helicoid]] such that every member of the deformation family is [[Minimal surface|minimal]] (having a [[mean curvature]] of zero).  A [[Parametric equation|parametrization]] of such a deformation is given by the system
<math display=block>\begin{align}
x(u,v) &= \sin \theta \,\cosh v \,\cos u + \cos \theta \,\sinh v \,\sin u \\
y(u,v) &= \sin \theta \,\cosh v \,\sin u - \cos \theta \,\sinh v \,\cos u \\
z(u,v) &=  v \sin \theta + u \cos \theta
\end{align}</math>
for <math>(u,v) \in (-\pi, \pi] \times (-\infty, \infty)</math>, with deformation parameter <math>-\pi < \theta \le \pi</math>, where:
* <math>\theta = \pi</math> corresponds to a right-handed helicoid, 
* <math>\theta = \pm \pi / 2</math> corresponds to a catenoid, and
* <math>\theta = 0</math> corresponds to a left-handed helicoid.

== The critical catenoid conjecture ==
A ''critical'' catenoid is a catenoid in the unit ball that meets the boundary sphere orthogonally. Up to rotation about the origin, it is given by rescaling {{EquationNote|Eq. 1}} with <math> c=1</math> by a factor <math> (\rho_0\cosh\rho_0)^{-1} </math>, where <math>\rho_0\tanh\rho_0=1 </math>. It is an embedded annular solution of the [[free boundary problem]] for the area functional in the unit ball and the ''critical catenoid conjecture'' states that it is the unique such annulus. 

The similarity of the critical catenoid conjecture to [[Hsiang-Lawson's conjecture]] on the Clifford torus in the 3-sphere, which was proven by [[Simon Brendle]] in 2012,<ref>{{cite journal |last=Brendle |first=Simon |title=Embedded minimal tori in S<sup>3</sup> and the Lawson conjecture | journal = Acta Mathematica | volume = 211 | pages = 177–190 | year = 2013 |issue=2 | doi=10.1007/s11511-013-0101-2|s2cid=119317563 |doi-access=free |arxiv=1203.6597 }}</ref> has driven interest in the conjecture,<ref name=Dev19>{{cite journal|last=Devyver |first=B.|title=Index of the critical catenoid |journal= Geometriae Dedicata |date=2019|volume=199|pages=355–371|doi=10.1007/s10711-018-0353-2}}</ref><ref name="FL14"/> as has its relationship to the Steklov eigenvalue problem.<ref name=FS11>{{cite journal|last1= Fraser |first1=Ailana |authorlink1=Ailana Fraser |last2= Schoen |first2=Richard |authorlink2=Richard Schoen|title= The first Steklov eigenvalue, conformal geometry, and minimal surfaces |journal=[[Advances in Mathematics]] |date=2011|volume=226|issue=5|pages=4011–4030|doi= 10.1016/j.aim.2010.11.007|doi-access=free |arxiv=0912.5392}}</ref>

Nitsche proved in 1985 that the only immersed minimal disk in the unit ball with free boundary is an equatorial totally geodesic disk.<ref name=Nitsche85>{{cite journal|last=Nitsche|first=J. C. C.|title= Stationary partitioning of convex bodies |journal= Archive for Rational Mechanics and Analysis |date=1985|volume=89|issue=1 |pages=1–19|doi= 10.1007/BF00281743 |bibcode=1985ArRMA..89....1N }}</ref> 
Nitsche also claimed without proof in the same paper that any free boundary constant mean curvature annulus in the unit ball is rotationally symmetric, and hence a catenoid or a parallel surface. Non-embedded counterexamples to Nitsche’s claim have since been constructed.<ref name=Wente93>{{cite conference |first=H. C. |last=Wente |title=Tubular capillary surfaces in a convex body |page=288 |editor1-first=P. |editor1-last=Concus |editor2-first=K. |editor2-last=Lancaster |book-title=Advances in Geometric Analysis and Continuum Mechanics |conference=Proceedings of a conference held at Stanford University on August 2–5, 1993, in honor of the seventieth birthday of Robert Finn |publisher=International Press |year=1993}}</ref><ref name=FHM23>{{cite journal|last1= Fernández |first1=I. |last2= Hauswirth |first2=L.|last3=Mira |first3=P.|title= Free boundary minimal annuli immersed in the unit ball |journal= Archive for Rational Mechanics and Analysis |date=2023|volume=247|issue=6|pages=108|doi= 10.1007/s00205-023-01943-z|doi-access=free |arxiv=2208.14998|bibcode=2023ArRMA.247..108F }}</ref>

The critical catenoid conjecture is stated in the embedded case by Fraser and Li<ref name=FL14>{{cite journal|last1= Fraser |first1=A. |last2=Li |first2=M. M.|title= Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary |journal= Journal of Differential Geometry |date=2014|volume=96|issue=6|pages=183–200|doi=10.4310/jdg/1393424916|doi-access=free  |arxiv=1204.6127}}</ref> and has been proven by McGrath with the extra assumption that the annulus is reflection invariant through coordinate planes,<ref name= McGrath18>{{cite journal|last= McGrath |first=P.| title= A characterization of the critical catenoid |journal= Indiana University Mathematics Journal |date=2018|volume=67|issue=2|pages=889–897|doi=10.1512/iumj.2018.67.7251 |jstor=26769410 |url= https://www.jstor.org/stable/26769410|arxiv=1603.04114}}</ref> and by Kusner and McGrath when the annulus has antipodal symmetry.<ref name=KM24>{{cite journal|last1= Kusner |first1=R.|last2= McGrath |first2=P.|title= On Steklov eigenspaces for free boundary minimal surfaces in the unit ball |journal= American Journal of Mathematics |date=2024|volume=146|issue=5|pages=1275–1293|doi=10.1353/ajm.2024.a937942 |arxiv=2011.06884}}</ref>  

As of 2025 the full conjecture remains open.

== References ==
{{reflist}}

== Further reading ==
* {{cite book |last1=Krivoshapko |first1=Sergey |last2=Ivanov |first2=V. N. |title=Encyclopedia of Analytical Surfaces |date=2015 |publisher=Springer |isbn=9783319117737 |chapter=Minimal Surfaces |chapter-url=https://books.google.com/books?id=cXTdBgAAQBAJ&pg=PA427 |language=en}}

== External links ==
* {{springer|title=Catenoid|id=p/c020800}}
* [http://www.princeton.edu/~rvdb/WebGL/catenoid.html Catenoid – WebGL model]
* [http://posner.library.cmu.edu/Posner/books/book.cgi?call=517.4_E88M_1744 Euler's text describing the catenoid] at Carnegie Mellon University
* [https://www.youtube.com/watch?v=31Om4VrSzb8 Calculating the surface area of a Catenoid]
* [https://mathworld.wolfram.com/MinimalSurfaceofRevolution.html Minimal Surface of Revolution]

{{Minimal surfaces}}

[[Category:Minimal surfaces]]

[[de:Minimalfläche#Das Katenoid]]