---
tag: JW431VDN08
kind: concept
name: Geometric measure theory
msc: [49Q15, 28A75, 49Q20]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_measure_theory
same_as:
  wikidata: Q3526918
  wikipedia: Geometric measure theory
---

## Summary

Geometric measure theory studies the geometry of sets and measures in Euclidean space and in manifolds with the tools of measure theory.
Its basic objects are [[Hausdorff measure|Q1TGZK8RN6]], [[rectifiable sets|7N0GFMW05R]], [[currents|360H7MD9RD]] and [[varifolds|N3QA80M44M]], which act as surfaces that may be singular, have multiplicities, or arise as limits.
It provides compactness theorems for such generalised surfaces, which give existence results for variational problems such as [[Plateau's problem|DVYA6GD8RN]] in every dimension and codimension.
It also studies the regularity of the minimisers and the structure of their singular sets.

## Explanations

### L1 human

Soap films can have edges where three sheets meet, and limits of smooth surfaces can develop folds and corners.
Classical calculus works with smooth surfaces and struggles with such objects.
Geometric measure theory replaces a surface by something more flexible, such as a measure that records area, so that limits always exist.
For example, the graph of the function $\abs{x}$ has a corner, but it still has a well-defined length and an approximate tangent line at almost every point.
Once a best generalised surface is found, one proves that it is smooth except on a small set.