Two equal circular rings of <radius> $R$ in parallel <planes> at $z=\pm L$ provide a boundary for an open film. In the ideal axisymmetric model, a stationary <connected> film has the <catenoid> profile
$$
r(z)=a\cosh(z/a),\qquad R=a\cosh(L/a).
$$
The <catenoid existence threshold for equal rings> explains why this equation can have two solutions, one solution, or none. The cancellation of <principal curvatures> holds for both solutions whenever they exist; the <mean curvature> equation alone does not select a <stable minimal surface>.

\Image[/catenoid-soap-film-between-rings.png]
{title=A catenoid spanning equal coaxial rings and two stationary profiles at the same separation}

This ideal geometry uses $R=1$ and $L/R=0.60$. The two profiles have different waist <radii> $a$, although both have $H=0$. For either profile the <surface area> follows by <integration>:
$$
A_{\mathrm{cat}}=2\pi a^2\left(U+\sinh U\cosh U\right),\qquad U=L/a.
$$
At this illustrated separation both profiles have greater <surface area> than the two disconnected disks, whose total is $2\pi R^2$. This comparison disproves a claim of global area minimization in a class allowing disconnected competitors; it does not determine the sign of the <second variation> for small variations of a <connected> film.

A reported experiment used coaxial wire rings of <radius> $4\,\mathrm{cm}$, drawn from a <surfactant> solution and slowly separated until collapse. The solution contained <tetradecyltrimethylammonium bromide> in deionized <water>, with <glycerol> added to vary <viscosity>. <Fluorescein> enabled <fluorescence> imaging under cyan LED arrays; a Phantom V641 camera recorded 4,000 or 5,600 frames per second. Extracted contours gave neck <radius>, attachment <angle>, and inferred <surface area>. They showed neck narrowing and eventual breakup, rather than indefinite persistence of a stationary <catenoid>. Comparing solutions across <viscosities> helped distinguish liquid dissipation from the surrounding air's <inertia>, but the ring displacement beyond the last stable state was not precisely controlled. These are collapse observations, not direct measurements of equilibrium $H$. <Geometry of catenoidal soap film collapse induced by boundary deformation by Raymond E. Goldstein and others>[Geometry of Catenoidal Soap Film Collapse Induced by Boundary Deformation, §II, manuscript pp. 3–4, Figs. 1–3: apparatus, contour readouts, viscosity comparison, and displacement limitation]{ref}

A proposed classroom measurement would instead hold two equal rings coaxial on a translation stage, leave both sides open to the same <gas>, and photograph the settled silhouette beside a scale. Fit $r(z)=a\cosh(z/a)$ and inspect fit residuals before inferring cancellation of the <principal curvatures>. Repeat at several separations and with the camera rotated about the axis to test <rotational symmetry>. Keep solution composition, ring wetting, and elapsed drainage time comparable, and shield the film from drafts. Film weight, drainage, evaporation, <surfactant> gradients, and rupture can invalidate the ideal <soap film pressure balance>. A silhouette fit alone neither measures the <pressure> jump nor tests all possible perturbations of a <stable minimal surface>.


 Soap film between coaxial rings (source code)