 Catenoid existence threshold for equal rings

An axisymmetric smooth connected soap film spanning equal radius-$R$ rings at heights $\pm H$ has stationary profile $r=k^{-1}\cosh (kz)$, $k>0$. The boundary equation is $R/H=\cosh u/u$, $u=kH$. Its unique minimum occurs at $u\tanh u=1$, because the derivative numerator is $u\sinh u-\cosh u$. If $A$ is this root, the minimum is $\cosh A/A=\sinh A$. There are no catenoids below that ratio, one at equality, and two above it. Existence of stationary connected films does not assert global minimality against pinching or disconnected competitors.