= Solution
Fixed endpoint variations yield the Euler–Lagrange equation
$$
F_r-\frac d{dz}F_{r'}=0.
$$
When $F$ is independent of $z$, differentiation gives
$$
\frac d{dz}(F-r'F_{r'})=r'\left(F_r-\frac d{dz}F_{r'}\right)=0.
$$
This is the <Beltrami identity>. Denote the constant by $1/k$ when nonzero; for a general Lagrangian a zero constant is also possible.
The film area is $S=2\pi\int_{-H}^Hr\sqrt{1+r'^2}\,dz$. Suppressing its harmless positive factor $2\pi$, take $F=r\sqrt{1+r'^2}$. The Euler–Lagrange equation reduces to
$$
\boxed{rr''=1+r'^2,\qquad r(-H)=r(H)=R.}
$$
The <first integral> is $r/\sqrt{1+r'^2}=1/k$ with $k>0$. Hence $1+r'^2=k^2r^2$ and, using the <differential equation> without dividing by $r'$, $r''=k^2r$. The positive solution obeying that <first integral> is $r=k^{-1}\cosh(k(z-z_0))$. Equal endpoint radii force $z_0=0$, giving
$$
\boxed{r(z)=k^{-1}\cosh(kz),\qquad R=k^{-1}\cosh(kH).}
$$
Thus every smooth positive stationary film in this axisymmetric class is a <catenoid>.
Put $u=kH>0$. The boundary equation is $R/H=f(u)=\cosh u/u$. The <function> tends to infinity at both endpoints of $(0,\infty)$, and
$$
f'(u)=\frac{u\sinh u-\cosh u}{u^2}.
$$
Its minimum occurs when $u=\coth u$. The <function> $u-\coth u$ is strictly increasing, with <derivative> $1+\operatorname{csch}^2u>0$, and has limits $-\infty$ and $+\infty$. It therefore has exactly one positive zero $A$. At this point $1/A=\tanh A$, so $f(A)=\sinh A$. This proves the <catenoid existence threshold for equal rings>:
$$
\boxed{R/H<\sinh A\Longrightarrow\text{no smooth connected catenoid solution}.}
$$
At equality there is one stationary solution and above it two. The equation describes candidates for an area minimum, not a guarantee that both branches minimize. The thinner-neck branch is unstable to suitable axisymmetric variations; topology-changing collapse can also compete with <connected> films. In particular two separate planar disks have area $2\pi R^2$. The derived threshold concerns existence of the smooth annular stationary film; it does not exclude such disconnected competitors.
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