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

1. Tobias Colding & Camillo De Lellis: "The min-max construction of minimal surfaces", Surveys in Differential Geometry
2. Giaquinta, Mariano; Mucci, Domenico (2006). "The BV-energy of maps into a manifold : relaxation and density results". Annali della Scuola Normale Superiore di Pisa – Classe di Scienze, Sér. 5, 5. pp. 483–548.
3. Helge Holden, Ragni Piene – The Abel Prize 2008-2012, p. 203.
4. Robert Osserman – A Survey of Minimal Surfaces, p. 160.
5. "Content Online - CDM 2013 Article 1". Intlpress.com. Retrieved 2015-05-31.
6. Fernando C. Marques; André Neves. "Applications of Almgren-Pitts Min-max theory" (PDF). F.imperial.ac.uk. Retrieved 2015-05-31.
7. Daniel Ketover. "Degeneration of Min-Max Sequences in Three-Manifolds". arXiv:1312.2666.
8. Xin Zhou. "Min-max hypersurface in manifold of positive Ricci curvature" (PDF). Arvix.org. Retrieved 2015-05-31.
9. Stephane Sabourau. "Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature" (PDF). Arvix.org. Retrieved 2015-05-31.
10. Davi Maximo; Ivaldo Nunes; Graham Smith. "Free boundary minimal annuli in convex three-manifolds". arXiv:1312.5392.
11. Zhou, Xin. "Min-max minimal hypersurface in $(M^{n+1}, g)$ with $Ric \geq 0$ and $2 \leq n \leq 6$". J. Differential Geom. 100 (2015), no. 1, 129–160. http://projecteuclid.org/euclid.jdg/1427202766.