Catenoid

The catenoid was the first non-trivial minimal surface in 3-dimensional Euclidean space to be discovered apart from the plane. The catenoid is obtained by rotating a catenary about its directrix. It was found and proved to be minimal by Leonhard Euler in 1744.{{cite book|last1=Helveticae|first1=Euler, Leonhard |editor= Carathëodory Constantin |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}}{{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}}

Early work on the subject was published also by Jean Baptiste Meusnier.{{cite book|url=https://archive.org/details/mmoiresurlathor00salvgoog|format=PDF|last1=Meusnier|first1=J. B|title=Mémoire sur la courbure des surfaces|trans-title=Memory on the curvature of surfaces.|date=1881|publisher=F. Hayez, Imprimeur De L'Acdemie Royale De Belgique|location=Bruxelles|language=French|isbn=9781147341744|pages=477–510}}{{rp|11106}} There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.{{cite web|title=Catenoid|url=http://mathworld.wolfram.com/Catenoid.html|website=Wolfram MathWorld|accessdate=15 January 2017|language=en}}

The catenoid may be defined by the following parametric equations:

{{NumBlk|::|$$\backslash begin\{align\}$$

x &= c \cosh \frac{v}{c} \cos u \\

y &= c \cosh \frac{v}{c} \sin u \\

z &= v

\end{align}|{{EquationRef|1}}}}

where $u\; \backslash in\; [-\backslash pi,\; \backslash pi)$ and $v\; \backslash in\; \backslash mathbb\{R\}$ and $c$ is a non-zero real constant.

In cylindrical coordinates:

$$\backslash rho\; =c\; \backslash cosh\; \backslash frac\{z\}\{c\},$$

where $c$ is a real constant.

A physical model of a catenoid can be formed by dipping two 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.

Image:helicatenoid.gif 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 and isometric deformation of a catenoid to a portion of the helicoid such that every member of the deformation family is minimal (having a mean curvature of zero). A parametrization of such a deformation is given by the system

$$\backslash begin\{align\}$$

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}

for $(u,v)\; \backslash in\; (-\backslash pi,\; \backslash pi]\; \backslash times\; (-\backslash infty,\; \backslash infty)$, with deformation parameter , where:

• $\backslash theta\; =\; \backslash pi$ corresponds to a right-handed helicoid,
• $\backslash theta\; =\; \backslash pm\; \backslash pi\; /\; 2$ corresponds to a catenoid, and
• $\backslash theta\; =\; 0$ corresponds to a left-handed helicoid.

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 $c=1$ by a factor $(\backslash rho\_0\backslash cosh\backslash rho\_0)^\{-1\}$, where $\backslash rho\_0\backslash tanh\backslash rho\_0=1$. 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,{{cite journal |last=Brendle |first=Simon |title=Embedded minimal tori in S³ 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 }} has driven interest in the Conjecture,{{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}} as has its relationship to the Steklov eigenvalue problem.{{cite journal|last1= Fraser |first1=A.|last2= Schoen |first2=R.|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}}

Nitsche proved in 1985 that the only immersed minimal disk in the unit ball with free boundary is an equatorial totally geodesic disk.{{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|pages=1–19|doi= 10.1007/BF00281743 }}

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.H.C. Wente, "Tubular capillary surfaces in a convex body," p288, in P. Concus, K. Lancaster and R. Finn, Advances in Geometric Analysis and Continuum Mechanics, International Press 1993.{{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}}

The Critical Catenoid Conjecture is stated in the embedded case by Fraser and Li {{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}} and has been proven by McGrath with the extra assumption that the annulus is reflection invariant through coordinate planes,{{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|url= https://www.jstor.org/stable/26769410}} and by Kusner and McGrath when the annulus has antipodal symmetry.{{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}}

As of 2025 the full Conjecture remains open.

{{reflist}}

• {{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}}

• {{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

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