In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces.
The theory started with the efforts for generalizing Birkhoff's method for the construction of simple closed geodesics on the sphere, to allow the construction of embedded minimal surfaces in arbitrary 3-manifolds.
It has played roles in the solutions to a number of conjectures in geometry and topology found by F. Almgren and J. Pitts themselves and also by other mathematicians, such as M. L. Gromov, R. Schoen, S.-T. Yau, F. C. Marques, A. A. Neves, I. Agol, among others.
Description and basic concepts
The theory allows the construction of embedded minimal hypersurfaces though variational methods.
References
Almgren–Pitts min-max theory Wikipedia(Text) CC BY-SA