Flat Convergence for Integral Currents in Metric Spaces

It is well known that in compact local Lipschitz neighborhood retracts in Rn flat convergence for Euclidean integer rectifiable currents amounts just to weak convergence. The purpose of the present paper is to extend this result to integral currents in complete metric spaces admitting a local cone type inequality. This includes for example all Banach spaces and complete CAT(κ)-spaces, κ ∈ R. The main result can be used e.g. to prove the existence of minimal elements in a fixed Lipschitz homology class in compact metric spaces admitting local cone type inequalities or to conclude that integral currents which are weak limits of sequences of absolutely area minimizing integral currents are again absolutely area minimizing.