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 George David 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 Almgren and Pitts themselves and also by other mathematicians, such as Mikhail Gromov, Richard Schoen, Shing-Tung Yau, Fernando Codá Marques, André Neves, Ian 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 through variational methods.[11]

In his PhD thesis, Almgren proved that the m-th homotopy group of the space of flat k-dimensional cycles on a closed Riemannian manifold is isomorphic to the (m+k)-th dimensional homology group of M. This result is a generalization of the Dold–Thom theorem, which can be thought of as the k=0 case of Almgren's theorem. Existence of non-trivial homotopy classes in the space of cycles suggests the possibility of constructing minimal submanifolds as saddle points of the volume function, as in Morse theory. In his subsequent work Almgren used these ideas to prove that for every k=1,...,n-1 a closed n-dimensional Riemannian manifold contains a stationary integral k-dimensional varifold, a generalization of minimal submanifold that may have singularities. Allard showed that such generalized minimal submanifolds are regular on an open and dense subset.

In the 1980s Almgren's student Jon Pitts greatly improved the regularity theory of minimal submanifolds obtained by Almgren in the case of codimension 1. He showed that when the dimension n of the manifold is between 3 and 6 the minimal hypersurface obtained using Almgren's min-max method is smooth. A key new idea in the proof was the notion of 1/j-almost minimizing varifolds. Richard Schoen and Leon Simon extended this result to higher dimensions. More specifically, they showed that every n-dimensional Riemannian manifold contains a closed minimal hypersurface constructed via min-max method that is smooth away from a closed set of dimension n-8.

By considering higher parameter families of codimension 1 cycles one can find distinct minimal hypersurfaces. Such construction was used by Fernando Codá Marques and André Neves in their proof of the Willmore conjecture.[12][13]

See also

• Almgren isomorphism theorem
• Varifold
• Geometric measure theory
• Geometric analysis
• Minimal surface
• Freedman–He–Wang conjecture
• Willmore conjecture
• Yau's conjecture

References

1. Tobias Colding and 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. Archived from the original on 2015-06-10. Retrieved 2015-05-02.
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 (2013). "Degeneration of Min-Max Sequences in Three-Manifolds". arXiv:1312.2666 [math.DG].
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 (2013). "Free boundary minimal annuli in convex three-manifolds". arXiv:1312.5392 [math.DG].
11. Zhou Xin (2015). "Min-max minimal hypersurface in ${\displaystyle (M^{n+1},g)}$ with ${\displaystyle Ric\geq 0}$ and ${\displaystyle 2\leq n\leq 6}$". J. Differential Geom. 100 (1): 129–160. doi:10.4310/jdg/1427202766.
12. White, Brian (1998). "The mathematics of F. J. Almgren, Jr" (PDF). Journal of Geometric Analysis. 8 (5): 681–702. doi:10.1007/BF02922665. S2CID 122083638.
13. Marques, Fernando & Neves, André. (2020). Applications of Min–Max Methods to Geometry. 10.1007/978-3-030-53725-8_2.

Further reading

• 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.
• 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"