# A Generalized Stokes' Theorem on integral currents

## TL;DR

This paper extends Stokes' Theorem to certain singular submanifolds using Lebesgue and gauge integration techniques, covering integral currents with controlled singularities, but not all singular cases.

## Contribution

It establishes a generalized Stokes' Theorem for integral currents with finite Minkowski content singular sets, broadening applicability to singular geometric objects.

## Key findings

- Proves Stokes' Theorem for integral currents with finite Minkowski content singularities.
- Identifies limitations with certain singular currents, providing a counterexample.
- Connects geometric measure theory with gauge integration techniques.

## 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.