Calculus of Variations and Geometric Measure Theory

L. Ambrosio - B. Kirchheim

Currents in Metric Spaces

created on 12 Apr 1999
modified on 19 Dec 2001


Published Paper

Inserted: 12 apr 1999
Last Updated: 19 dec 2001

Journal: Acta Mathematica
Number: 185
Pages: 1-80
Year: 2000


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.