Birkhoff min-max method

Summary

The Birkhoff min-max method finds a closed geodesic 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 for minimal hypersurfaces.