Filling radius

Abstract: If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most c(n)Volume(M,hyp), then we prove that the universal cover of (M,g) contains a unit ball with volume greater than the volume of a unit ball in hyperbolic n-space.

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. Building on S. Wenger's work in the orientable case, we obtain isoperimetric inequalities mod(p) in Banach spaces and we apply these inequalities to provide a proof of Gromov's filling radius inequality (and therefore also the systolic inequality) which applies to nonorientable manifolds, as well. 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.