Catenoid

A catenoid

A catenoid obtained from the rotation of a catenary

In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). ^[1] It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. ^[2] 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. ^[2] 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 in 3-dimensional Euclidean space to be discovered apart from the plane. The catenoid is obtained by rotating a catenary about its directrix. ^[2] It was found and proved to be minimal by Leonhard Euler in 1744. ^{[3] [4]}

Early work on the subject was published also by Jean Baptiste Meusnier. ^{[5] [4] : 11106} There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid. ^[6]

The catenoid may be defined by the following parametric equations:

$${\displaystyle {\begin{aligned}x&=c\cosh {\frac {v}{c}}\cos u\\y&=c\cosh {\frac {v}{c}}\sin u\\z&=v\end{aligned}}}$$

1

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

In cylindrical coordinates: $${\displaystyle \rho =c\cosh {\frac {z}{c}},}$$ where ${\displaystyle c}$ 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.

Helicoid transformation

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 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 $${\displaystyle {\begin{aligned}x(u,v)&=\sin \theta \,\cosh v\,\cos u+\cos \theta \,\sinh v\,\sin u\\y(u,v)&=\sin \theta \,\cosh v\,\sin u-\cos \theta \,\sinh v\,\cos u\\z(u,v)&=v\sin \theta +u\cos \theta \end{aligned}}}$$ for ${\displaystyle (u,v)\in (-\pi ,\pi ]\times (-\infty ,\infty )}$, with deformation parameter ${\displaystyle -\pi <\theta \leq \pi }$, where:

• ${\displaystyle \theta =\pi }$ corresponds to a right-handed helicoid,
• ${\displaystyle \theta =\pm \pi /2}$ corresponds to a catenoid, and
• ${\displaystyle \theta =0}$ 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 Eq. 1 with ${\displaystyle c=1}$ by a factor ${\displaystyle (\rho _{0}\cosh \rho _{0})^{-1}}$, where ${\displaystyle \rho _{0}\tanh \rho _{0}=1}$. 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, ^[7] has driven interest in the conjecture, ^{[8] [9]} as has its relationship to the Steklov eigenvalue problem. ^[10]

Nitsche proved in 1985 that the only immersed minimal disk in the unit ball with free boundary is an equatorial totally geodesic disk. ^[11] 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. ^{[12] [13]}

The critical catenoid conjecture is stated in the embedded case by Fraser and Li ^[9] and has been proven by McGrath with the extra assumption that the annulus is reflection invariant through coordinate planes, ^[14] and by Kusner and McGrath when the annulus has antipodal symmetry. ^[15]

As of 2025 the full conjecture remains open.

References

1. Dierkes, Ulrich; Hildebrandt, Stefan; Sauvigny, Friedrich (2010). Minimal Surfaces. Springer Science & Business Media. p. 141. ISBN   9783642116988.
2. Gullberg, Jan (1997). Mathematics: From the Birth of Numbers . W. W. Norton & Company. p.  538. ISBN   9780393040029.
3. Euler, Leonhard (1952) [reprint of 1744 edition]. Carathéodory, Constantin (ed.). Methodus inveniendi lineas curvas: maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti (in Latin). Springer Science & Business Media. ISBN   3-76431-424-9.
4. Colding, T. H.; Minicozzi, W. P. (17 July 2006). "Shapes of embedded minimal surfaces". Proceedings of the National Academy of Sciences. 103 (30): 11106–11111. Bibcode:2006PNAS..10311106C. doi: 10.1073/pnas.0510379103 . PMC   1544050 . PMID   16847265.
5. Meusnier, J. B. (1881). Mémoire sur la courbure des surfaces [Dissertation on the curvature of surfaces](PDF) (in French). Brussels: F. Hayez, Printer of the Royal Academy of Belgium. pp. 477–510. ISBN   9781147341744.
6. "Catenoid". Wolfram MathWorld. Retrieved 15 January 2017.
7. Brendle, Simon (2013). "Embedded minimal tori in S³ and the Lawson conjecture". Acta Mathematica. 211 (2): 177–190. arXiv: 1203.6597 . doi: 10.1007/s11511-013-0101-2 . S2CID   119317563.
8. Devyver, B. (2019). "Index of the critical catenoid". Geometriae Dedicata. 199: 355–371. doi:10.1007/s10711-018-0353-2.
9. Fraser, A.; Li, M. M. (2014). "Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary". Journal of Differential Geometry. 96 (6): 183–200. arXiv: 1204.6127 . doi: 10.4310/jdg/1393424916 .
10. Fraser, A.; Schoen, R. (2011). "The first Steklov eigenvalue, conformal geometry, and minimal surfaces". Advances in Mathematics. 226 (5): 4011–4030. arXiv: 0912.5392 . doi: 10.1016/j.aim.2010.11.007 .
11. Nitsche, J. C. C. (1985). "Stationary partitioning of convex bodies". Archive for Rational Mechanics and Analysis. 89 (1): 1–19. Bibcode:1985ArRMA..89....1N. doi:10.1007/BF00281743.
12. Wente, H. C. (1993). "Tubular capillary surfaces in a convex body". In Concus, P.; Lancaster, K. (eds.). Advances in Geometric Analysis and Continuum Mechanics. Proceedings of a conference held at Stanford University on August 2–5, 1993, in honor of the seventieth birthday of Robert Finn. International Press. p. 288.
13. Fernández, I.; Hauswirth, L.; Mira, P. (2023). "Free boundary minimal annuli immersed in the unit ball". Archive for Rational Mechanics and Analysis. 247 (6): 108. arXiv: 2208.14998 . Bibcode:2023ArRMA.247..108F. doi: 10.1007/s00205-023-01943-z .
14. McGrath, P. (2018). "A characterization of the critical catenoid". Indiana University Mathematics Journal. 67 (2): 889–897. doi:10.1512/iumj.2018.67.7251. JSTOR   26769410.
15. Kusner, R.; McGrath, P. (2024). "On Steklov eigenspaces for free boundary minimal surfaces in the unit ball". American Journal of Mathematics. 146 (5): 1275–1293. arXiv: 2011.06884 . doi:10.1353/ajm.2024.a937942.

Further reading

• Krivoshapko, Sergey; Ivanov, V. N. (2015). "Minimal Surfaces". Encyclopedia of Analytical Surfaces. Springer. ISBN   9783319117737.

External links

• "Catenoid", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
• Catenoid – WebGL model
• Euler's text describing the catenoid at Carnegie Mellon University
• Calculating the surface area of a Catenoid
• Minimal Surface of Revolution