Almgren–Pitts min-max theory

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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[edit]

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

See also[edit]

References[edit]

  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. Cite uses deprecated parameter |dead-url= (help)
  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 (2015). "Min-max minimal hypersurface in with and ". J. Differential Geom. 100 (1): 129–160. doi:10.4310/jdg/1427202766.

Further reading[edit]