Flat convergence

Integral currents

A k-dimensional current T is a multilinear real-valued operator on smooth k-forms. For example, given a Lipschitz map from a manifold into Euclidean space, F: N^k → R^N, one has an integral current T(ω) defined by integrating the pullback of the differential k-form, ω, over N. Currents have a notion of boundary ∂ T (which is the usual boundary when N is a manifold with boundary) and a notion of mass, M(T), (which is the volume of the image of N). An integer rectifiable current is defined as a countable sum of currents formed in this respect. An integral current is an integer rectifiable current whose boundary has finite mass. It is a deep theorem of Federer-Fleming that the boundary is then also an integral current.

Flat norm and flat distance

The flat norm |T| of a k-dimensional integral current T is the infimum of M(A) + M(B), where the infimum is taken over all integral currents A and B such that T = A + ∂ B .

The flat distance between two integral currents is then d_F(T,S) = |T − S|.

Compactness theorem

Federer-Fleming proved that if one has a sequence of integral currents T i whose supports lie in a compact set K with a uniform upper bound on M ( T i ) + M ( ∂ T i ) , then a subsequence converges in the flat sense to an integral current.

This theorem was applied to study sequences of submanifolds of fixed boundary whose volume approached the infimum over all volumes of submanifolds with the given boundary. It produced a candidate weak solution to Plateau's problem.