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.^[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.

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.

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.