Let $\Sigma$ be a strictly stable minimal surface. Can we show that $\Sigma$ is in fact strictly area minimizing amongst surfaces that are close in a certain topology? (say $W^{2,2}$ for instance) It seems to me that this should be the case but I cannot find any proof of this statement. More generally, does the same conclusion hold true for a strictly stable critical point of a functional $$ \Sigma \mapsto \int_\Sigma f(x,A,\nabla A,\dots)d\mu $$ where $f$ is a smooth function and $A$ the second fundamental form of $\Sigma$?
Perhaps this is a bit of a stupid question, I just get stuck at the following point: if we have a small perturbation $\Sigma_t$ then it is clear that $|\Sigma_t|>|\Sigma|$ for some small $t\in(0,\epsilon)$, where $\epsilon$ depends on the perturbation. In a finite dimensional problem we could use compactness and be done, however how can we proceed in an infinite dimensional problem?