---
tag: 1R5X9HW6D3
kind: concept
name: Birkhoff min-max method
msc: ["58E10", 53C22, 49J35]
sources:
  - url: https://en.wikipedia.org/wiki/Closed_geodesic
---

## Summary

The Birkhoff min-max method finds a [[closed geodesic|4MCCP8GG4J]] on a Riemannian sphere by sweeping the sphere with a one-parameter family of closed curves.
One takes, over all sweepouts, the length of the longest curve, and then the infimum of this number over all sweepouts.
George Birkhoff showed that this min-max value, the width, is positive and is the length of a closed geodesic.
The method is the prototype of min-max constructions in geometry, including [[Almgren–Pitts min-max theory|6KVJHG1KRZ]] for minimal hypersurfaces.

## Explanations

### L1 human

Slide a rubber band over a sphere from the south pole to the north pole, so that it passes over every point.
Somewhere along the way the band is longest.
Now try to choose the sweep so that this longest moment is as short as possible.
The best possible longest band cannot be shrunk further without the sweep getting stuck, and it sits on a closed geodesic.
On a round unit sphere, sweeping by circles of latitude gives longest band the equator, of length $2\pi$, and this is the width.