Almgren–Pitts min-max theory

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.^[1]

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.^{[2][3][4][5][6][7][8][9][10]}

Description and basic concepts

The theory allows the construction of embedded minimal hypersurfaces though variational methods.^[11]

See also

• Varifold
• Geometric measure theory
• Geometric analysis
• Minimal surface
• Freedman–He–Wang conjecture
• Willmore conjecture

Original references

• Frederick J. Almgren (1964). The Theory of Varifolds: A Variational Calculus in the Large for the K-dimensional Area Integrand. Institute for Advanced Study. 
• Jon T. Pitts (1981). Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. Princeton University Press. ISBN 978-0-691-08290-5. 

Further reading

• Memarian, Yashar (2013). "A Note on the Geometry of Positively-Curved Riemannian Manifolds". arXiv:1312.0792 [math.MG]. 
• Le Centre de recherches mathématiques, CRM Le Bulletin, Automne/Fall 2015 — Volume 21, No 2, pp. 10–11 Iosif Polterovich (Montréal) and Alina Stancu (Concordia), "The 2015 Nirenberg Lectures in Geometric Analysis: Min-Max Theory and Geometry, by André Neves"

References