---
tag: N3QA80M44M
kind: concept
name: Varifold
msc: [49Q15, 49Q20]
sources:
  - url: https://en.wikipedia.org/wiki/Varifold
same_as:
  wikidata: Q7915890
  wikipedia: Varifold
---

## Summary

An $m$-dimensional varifold in an open set $U \subseteq \R^n$ is a [[Radon measure|HZNEQQBCW4]] on a product space.
The product is $U$ times the space of $m$-dimensional linear subspaces of $\R^n$.
It records where a generalised surface lies and which tangent planes it has there, but not its orientation.
L. C. Young introduced them in 1951 as generalised surfaces.
Almgren gave them the name varifold in the 1960s, and Allard developed them as a setting for minimal surfaces with singularities.
Their compactness properties make them a main tool in the calculus of variations for area.

## Explanations

### L1 human

Think of a surface as a cloud of tiny flat pieces.
Each piece has a position and a tilt, given by its tangent plane.
A varifold is a mass distribution over all pairs of position and tilt.
A smooth surface gives such a distribution: at each point of the surface, its area sits on the tangent plane at that point.
Unlike a current, a varifold has no orientation, so two sheets on top of each other add up instead of cancelling.