A Generalized Stokes' Theorem on integral currents

Abstract

The purpose of this paper is to study the validity of Stokes' Theorem for singular submanifolds and differential forms with singularities in Euclidean space. The results are presented in the context of Lebesgue Integration, but their proofs involve techniques from gauge integration in the spirit of R.~Henstock, J.~Kurzweil and W.~F.~Pfeffer. We manage to prove a generalized Stokes' Theorem on integral currents of dimension $m$ whose singular sets have finite $m-1$ dimensional intrinsic Minkowski content. This condition applies in particular to codimension $1$ mass minimizing integral currents with smooth boundary and to semi-algebraic chains. Conversely, we give an example of integral current of dimension $2$ in $\mathbb{R}^3$, with only one singular point, to which our version of Stokes' Theorem does not apply.