---
tag: 6KVJHG1KRZ
kind: concept
name: Almgren–Pitts min-max theory
msc: ["58E12", 49Q05, 53A10]
sources:
  - url: https://en.wikipedia.org/wiki/Almgren–Pitts_min-max_theory
same_as:
  wikidata: Q19903256
  wikipedia: Almgren–Pitts min-max theory
---

## Summary

Almgren–Pitts min-max theory constructs closed [[minimal hypersurfaces|41X0YW2R9N]] in a closed Riemannian manifold by a min-max over families of hypersurfaces that sweep out the manifold.
It is the higher-dimensional analogue of the [[Birkhoff min-max method|1R5X9HW6D3]] for closed geodesics.
Frederick Almgren set up the topology of spaces of cycles, and Jon Pitts proved the regularity of the min-max hypersurface in low dimensions.
Fernando Marques and André Neves developed the theory further and used it to prove the Willmore conjecture.
It is also the main tool behind [[Yau's conjecture on infinitely many minimal surfaces|VAP45DFXW3]].

## Explanations

### L1 human

Fill a closed curved space with a family of soap films that starts as nothing, grows to fill the whole space, and shrinks back to nothing.
At some moment the film in the family has the largest area.
Choose the family so that this largest area is as small as possible.
The film at the critical moment cannot be pushed down further, and it settles into a minimal hypersurface, a film whose mean curvature is zero.
For a round $3$-sphere, sweeping by parallel $2$-spheres gives the equatorial $2$-sphere as the min-max surface.