*Published Paper*

**Inserted:** 12 apr 1999

**Last Updated:** 19 dec 2001

**Journal:** Acta Mathematica

**Number:** 185

**Pages:** 1-80

**Year:** 2000

**Abstract:**

We develop a theory of currents in metric spaces which extends the classical theory of Federer-Fleming in Euclidean spaces and in Riemannian manifolds. The main idea, suggested by De Giorgi, is to replace the duality with differential forms with the duality with (k+1)-ples of Lipschitz functions, where k is the dimension of the current. We show, by a metric proof which is new even for currents in euclidean spaces, that the closure theorem and the boundary rectifiability theorem for integral currents hold in any complete metric space E. Moreover, we prove some existence results for a generalized Plateau problem in compact metric spaces and in some classes of Banach spaces, not necessarily finite dimensional.

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