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.