---
tag: EG4AJVEFT2
kind: concept
name: Integral current
msc: [49Q15, 28A75]
specializes: [360H7MD9RD]
sources:
  - url: https://en.wikipedia.org/wiki/Current_(mathematics)
---

## Summary

An integral current is a [[current|360H7MD9RD]] that behaves like an oriented surface with integer multiplicities.
It is given by integration over a [[rectifiable set|7N0GFMW05R]] with an orientation and a positive integer multiplicity.
Both it and its boundary have finite mass.
Integral currents include all compact oriented smooth submanifolds with boundary.
By the [[Federer–Fleming compactness theorem|TPXYARH912]], they are closed under the limits that matter for area minimisation.
They are the setting of the Plateau problem in all dimensions and codimensions.

## Explanations

### L1 human

An integral current is a generalised surface that can have corners and singularities, and parts counted twice or three times, but never a fractional number of times.
You integrate a differential form over it as over an ordinary oriented surface, counting each part with its multiplicity.
For example, the unit disc in the plane, traversed once, is an integral $2$-current with boundary the unit circle.
The same disc counted twice is also one, but the disc counted half a time is not.