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.
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.[1][2][3][4][5][6][7][8][9]
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].
References
- ↑ 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.
- ↑ Helge Holden, Ragni Piene – The Abel Prize 2008-2012, p. 203.
- ↑ Robert Osserman – A Survey of Minimal Surfaces, p. 160.
- ↑ "Content Online - CDM 2013 Article 1". Intlpress.com. Retrieved 2015-05-31.
- ↑ Fernando C. Marques; André Neves. "Applications of Almgren-Pitts Min-max theory" (PDF). F.imperial.ac.uk. Retrieved 2015-05-31.
- ↑ Daniel Ketover. "Degeneration of Min-Max Sequences in Three-Manifolds". arXiv:1312.2666.
- ↑ Xin Zhou. "Min-max hypersurface in manifold of positive Ricci curvature" (PDF). Arvix.org. Retrieved 2015-05-31.
- ↑ Stephane Sabourau. "Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature" (PDF). Arvix.org. Retrieved 2015-05-31.
- ↑ Davi Maximo; Ivaldo Nunes; Graham Smith. "Free boundary minimal annuli in convex three-manifolds". arXiv:1312.5392.
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