Quantitative minimality of strictly stable extremal submanifolds in a flat neighbourhood

In this paper we extend the results of A strong minimax property of nondegenerate minimal submanifolds, by White, where it is proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in a certain geodesic tubular neighbourhood. We prove a similar result, replacing the tubular neighbourhood with one induced by the flat distance and we provide quantitative estimates. Our proof is based on the introduction of a penalized minimization problem, in the spirit of A selection principle for the sharp quantitative isoperimetric inequality, by Cicalese and Leonardi, which allows us to exploit the regularity theory for almost minimizers of elliptic parametric integrands.