Flat currents modulo \(p\) in metric spaces and filling radius inequalities. (English) Zbl 1222.49057

Summary: 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 \(\mathbb Z_{p}\). We obtain isoperimetric inequalities \(\text{mod}(p)\) in Banach spaces and we apply these inequalities to provide a proof of Gromov’s filling radius inequality which can be applied also to nonorientable manifolds. With this goal in mind, we use Ekeland’s principle to provide quasi-minimizers of the mass \(\text{mod}(p)\) in the homology class, and use the isoperimetric inequality to give lower bounds on the growth of their mass in balls.


49Q20 Variational problems in a geometric measure-theoretic setting
53C65 Integral geometry
49J52 Nonsmooth analysis
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