= Axisymmetric second variation of a catenoid
{title2=$\delta^2F=2\pi\int(\xi_z^2-\xi^2)\operatorname{sech}^2z\,dz$}
For the <catenoid> profile $y(x)=E\cosh(x/E)$ spanning fixed-endpoint rings, the <second variation> of $F[y]=2\pi\int y\sqrt{1+y'^2}\,dx$ is
$$
\delta^2F[y,\xi]=2\pi\int_{-L/E}^{L/E}(\xi_z^2-\xi^2)\operatorname{sech}^2z\,dz,
\qquad z=x/E,\quad \xi(\pm L/E)=0.
$$
This refers to axisymmetric graph variations, not all possible changes in surface topology. Strict positivity for every nonzero admissible variation gives a strict local minimum in the $C^1$ topology: the positive lowest <Dirichlet eigenvalue> of the regular <Sturm-Liouville problem> gives <coercivity>, and sufficiently small coefficient perturbations preserve it. The zero variation must be excluded from a strict-positivity hypothesis.
Back to article page
