Almgren–Pitts min-max theory

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, like in the 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 was able to greatly improve 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]

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

• 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"