Commentarii Mathematici Helvetici


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Volume 86, Issue 3, 2011, pp. 557–591
DOI: 10.4171/CMH/234

Flat currents modulo p in metric spaces and filling radius inequalities

Luigi Ambrosio[1] and Mikhail G. Katz[2]

(1) Classe di Scienze M F N, Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, PISA, ITALY
(2) Department of Mathematics and Statistics, Bar-Ilan University, 52 900, RAMAT-GAN, ISRAEL

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 ℤp . We obtain isoperimetric inequalities 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 mod (p) in the homology class, and use the isoperimetric inequality to give lower bounds on the growth of their mass in balls.

Keywords: Filling radius, currents, isoperimetric inequality

Ambrosio L, Katz M. Flat currents modulo p in metric spaces and filling radius inequalities. Comment. Math. Helv. 86 (2011), 557-591. doi: 10.4171/CMH/234