---
tag: 4NKP73KVVW
kind: concept
name: Area-minimizing current
msc: [49Q15, 49Q05, 49Q20]
specializes: [EG4AJVEFT2]
sources:
  - url: https://en.wikipedia.org/wiki/Current_(mathematics)
---

## Summary

An area-minimizing current is an [[integral current|EG4AJVEFT2]] whose mass is not larger than that of any other integral current with the same boundary.
It is the solution concept of the Plateau problem in geometric measure theory, and it generalises oriented [[minimal surfaces|41X0YW2R9N]] that minimise area.
Such currents exist for every boundary that bounds some integral current.
Their regularity is a central topic: they are smooth away from a singular set of small dimension.

## Explanations

### L1 human

Take a closed wire loop and ask for the surface of least area spanning it, counting area with orientation and multiplicity.
An area-minimizing current is such a least-area surface in a generalised sense, so that one exists even when no smooth one does.
For a round circle in space, the flat disc is area-minimizing; a hemisphere with the same rim is not, because it has twice the area.
In high dimensions minimisers can have singular points, for example cones over products of spheres.