= Negative area direction at an extended catenoid
A <catenoid> has zero <mean curvature>, but need not minimize area for its fixed boundary rings. For the profile $y(x)=\cosh x$ on $[-2,2]$, take an endpoint-zero variation $\eta(x)=\cosh x-\cosh2$. The <second variation> of the rotational area is
$$
\delta^2J[\eta]=2\pi\int_{-2}^2\operatorname{sech}^2x\,(\eta'^2-\eta^2)\,dx
=2\pi\{4C\arctan(\sinh2)-2(1+C^2)\tanh2\},\qquad C=\cosh2.
$$
To derive the integral, differentiate $y\sqrt{1+y'^2}$ twice, then integrate its mixed term by parts; the endpoint term vanishes. To evaluate it, substitute $\eta'^2-\eta^2=-1-C^2+2C\cosh x$, using the antiderivatives of $\operatorname{sech}x$ and $\operatorname{sech}^2x$. The result is negative because $\arctan(\sinh2)<\pi/2$ and $(C+C^{-1})\tanh2>\pi$.
Small positive and negative perturbations remain positive and preserve the boundary values. The first variation vanishes, while the negative second variation gives nearby smaller areas. Thus being a <minimal surface> in the zero-mean-curvature sense is a stationarity condition, not a guarantee of area minimization for arbitrary boundary separations.
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