= Stable minimal surface
{title2=$\delta^2\operatorname{Area}[f]\geq0$}
A <stable minimal surface> has nonnegative <second variation> of <surface area> for every admissible normal variation $fN$ fixing the boundary. For a two-sided <minimal surface> in three-dimensional <Euclidean space>, the quadratic form is
$$
\delta^2\operatorname{Area}[f]=\int_M\bigl(|\nabla_M f|^2-(k_1^2+k_2^2)f^2\bigr)\,dA.
$$
Here $\nabla_M f$ is the <gradient> for the induced <Riemannian metric>. The <gradient> term penalizes rapid changes in the displacement; the <principal curvature> term can make the <second variation> negative. A <plane> has only the nonnegative <gradient> term. Nonnegativity alone, especially with zero modes, does not prove a strict <local minimum> or a <global minimum>.
For a finite <catenoid> between fixed rings, the answer depends on the ring separation, waist <radius>, and allowed variations. The <axisymmetric second variation of a catenoid> tests a specified restricted class. Comparing with two disconnected disks tests a different class of competitors, including a change in <topology>. Consequently, a stationary <catenoid> can have greater <surface area> than two disks without immediately collapsing under every small perturbation.
Back to article page
