Currents in metric spaces
Luigi Ambrosio and Bernd Kirchheim
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Submission date: 08. Mar. 2000
published in: Acta mathematica, 185 (2000) 1, p. 1-80
DOI number (of the published article): 10.1007/BF02392711
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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.