Flat currents modulo $p$ in metric spaces and filling radius inequalities

Abstract

We adapt the theory of currents in metric spaces, as developed by the first-mentioned author in collaboration with B. Kirchheim, to currents with coefficients in $Z_p$ . We obtain isoperimetric inequalities $\mathrm{mod}(p)$ in Banach spaces and we apply these inequalities to provide a proof of Gromov’s filling radius inequality which applies also to nonorientable manifolds. With this goal in mind, we use the Ekeland principle to provide quasi-minimizers of the mass $\mathrm{mod}(p)$ in the homology class, and use the isoperimetric inequality to give lower bounds on the growth of their mass in balls.