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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
728:
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
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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.
171:, 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
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175:. 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
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179:, a generalization of minimal submanifold that may have singularities. Allard showed that such generalized minimal submanifolds are regular on an open and dense subset.
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By considering higher parameter families of codimension 1 cycles one can find distinct minimal hypersurfaces. Such construction was used by
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Marques, Fernando & Neves, André. (2020). Applications of Min–Max
Methods to Geometry. 10.1007/978-3-030-53725-8_2.
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299:. Annali della Scuola Normale Superiore di Pisa – Classe di Scienze, Sér. 5, 5. pp. 483–548. Archived from
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Davi Maximo; Ivaldo Nunes; Graham Smith (2013). "Free boundary minimal annuli in convex three-manifolds".
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The Theory of
Varifolds: A Variational Calculus in the Large for the K-dimensional Area Integrand
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Memarian, Yashar (2013). "A Note on the
Geometry of Positively-Curved Riemannian Manifolds".
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found by
Almgren and Pitts themselves and also by other mathematicians, such as
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436:"Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature"
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Daniel
Ketover (2013). "Degeneration of Min-Max Sequences in Three-Manifolds".
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297:"The BV-energy of maps into a manifold : relaxation and density results"
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Existence and
Regularity of Minimal Surfaces on Riemannian Manifolds
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411:"Min-max hypersurface in manifold of positive Ricci curvature"
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Helge Holden, Ragni Piene – The Abel Prize 2008-2012, p. 203.
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of the space of flat k-dimensional cycles on a closed
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The theory started with the efforts for generalizing
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group of M. This result is a generalization of the
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It has played roles in the solutions to a number of
148:minimal hypersurfaces through variational methods.
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47:'s method for the construction of simple closed
365:"Applications of Almgren-Pitts Min-max theory"
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8:
295:Giaquinta, Mariano; Mucci, Domenico (2006).
283:The min-max construction of minimal surfaces
51:on the sphere, to allow the construction of
187:notion of 1/j-almost minimizing varifolds.
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163:is isomorphic to the (m+k)-th dimensional
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332:– A Survey of Minimal Surfaces, p. 160.
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624:"The mathematics of F. J. Almgren, Jr"
144:The theory allows the construction of
343:"Content Online - CDM 2013 Article 1"
7:
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285:", Surveys in Differential Geometry
757:. You can help Knowledge (XXG) by
363:Fernando C. Marques; André Neves.
14:
482:"Min-max minimal hypersurface in
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182:In the 1980s Almgren's student
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105:Description and basic concepts
1:
677:Frederick J. Almgren (1964).
631:Journal of Geometric Analysis
584:{\displaystyle 2\leq n\leq 6}
683:Institute for Advanced Study
22:Almgren–Pitts min-max theory
829:Differential geometry stubs
520:{\displaystyle (M^{n+1},g)}
246:Freedman–He–Wang conjecture
221:Almgren isomorphism theorem
151:In his PhD thesis, Almgren
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696:Princeton University Press
552:{\displaystyle Ric\geq 0}
26:Frederick J. Almgren, Jr.
231:Geometric measure theory
819:Calculus of variations
753:-related article is a
608:10.4310/jdg/1427202766
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207:in their proof of the
751:differential geometry
690:Jon T. Pitts (1981).
622:White, Brian (1998).
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201:Fernando Codá Marques
91:Fernando Codá Marques
45:George David Birkhoff
595:J. Differential Geom
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32:) is an analogue of
434:Stephane Sabourau.
251:Willmore conjecture
209:Willmore conjecture
161:Riemannian manifold
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370:. F.imperial.ac.uk
236:Geometric analysis
124:. You can help by
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480:Zhou Xin (2015).
279:Camillo De Lellis
169:Dold–Thom theorem
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155:that the m-th
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87:Shing-Tung Yau
83:Richard Schoen
79:Mikhail Gromov
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120:This section
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58:in arbitrary
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38:hypersurfaces
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24:(named after
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759:expanding it
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443:. Retrieved
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305:. Retrieved
301:the original
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173:Morse theory
150:
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126:adding to it
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42:
34:Morse theory
30:Jon T. Pitts
21:
15:
441:. Arvix.org
416:. Arvix.org
205:André Neves
95:André Neves
67:conjectures
60:3-manifolds
18:mathematics
798:Categories
445:2015-05-31
420:2015-05-31
409:Xin Zhou.
374:2015-05-31
349:2015-05-31
307:2015-05-02
262:References
193:Leon Simon
720:1312.0792
651:122083638
576:≤
570:≤
544:≥
466:1312.5392
395:1312.2666
184:Jon Pitts
49:geodesics
809:Geometry
804:Topology
226:Varifold
215:See also
177:varifold
165:homology
146:embedded
133:May 2015
99:Ian Agol
75:topology
71:geometry
53:embedded
729:Neves"
702:
649:
153:proved
20:, the
749:This
715:arXiv
647:S2CID
627:(PDF)
527:with
461:arXiv
439:(PDF)
414:(PDF)
390:arXiv
368:(PDF)
755:stub
700:ISBN
559:and
277:and
203:and
191:and
73:and
36:for
639:doi
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599:100
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16:In
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131:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
