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MiS Preprint
21/2000

Currents in metric spaces

Luigi Ambrosio and Bernd Kirchheim

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 by the duality with (k+1)-tuples 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.

Received:
Mar 8, 2000
Published:
Mar 8, 2000

Related publications

inJournal
2000 Repository Open Access
Luigi Ambrosio and Bernd Kirchheim

Currents in metric spaces

In: Acta mathematica, 185 (2000) 1, pp. 1-80