Currents in metric spaces, isoperimetric inequalities, and applications to area minimization problems

Integral currents were introduced by H. Federer and W. H. Fleming in 1960

as a suitable generalization of surfaces in connection with the study of area

minimization problems in Euclidean space. L. Ambrosio and B. Kirchheim have

recently extended the theory of currents to arbitrary metric spaces. The new

theory provides a suitable framework to formulate and study area minimization

and isoperimetric problems in metric spaces.

The aim of the talk is to discuss such problems for Banach spaces and for

spaces with an upper curvature bound in the sense of Alexandrov. We present

some techniques which lead to isoperimetric inequalities, solutions to

Plateau's problem, and to other results such as the equivalence of flat and

weak convergence for integral currents.