Outermost Minimal Surface in a Manifold

An outermost minimal surface is a minimal surface which does not contain another minimal surface within it. Interestingly, outer minimal surfaces are always spheres, because the only other possibility (that they might be tori) is removed via a stability argument.

If a Riemannian $3$-manifold $(M,g)$ contains an outermost minimal surface, will a small perturbation of that manifold (ie. the metric is $g+h$, where $h$ is some small perturbation) also have that same outermost minimal surface, or will that surface also have a perturbation added to it?