The min--max construction of minimal surfaces

By T. Colding, Camillo De Lellis. 2003

184 citations recorded by the source corpus.

In this paper we survey with complete proofs some well-known, but hard to find, results about constructing closed embedded minimal surfaces in a closed 3-dimensional manifold via min-max arguments. This includes results of J. Pitts, F. Smith, and L. Simon and F. Smith.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5a043bead64c3c63a9136ba90d2068c63c4c7afe

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Minimal surfaces in germs of hyperbolic 3{manifolds (2004, background): “More recently, Pitts and Rubinstein [11, 10, 12], (see also [3]) proved that all compact, hyperbolic 3–manifolds have at least one unstable, immersed, minimal surface.”
Existence of hypersurfaces with prescribed mean curvature I – generic min-max (2018, background): “8, we see that blowups of an h-almost minimizing varifolds have the good replacement property of [13, 16].”
The Willmore Conjecture (2014, background): “The prototype theorem was proven by Pitts [47] (see also [13] for a nice exposition), a student of Almgren at the time, and states the following.”
Min-max theory for minimal hypersurfaces with boundary (2016, methodology): “Part of this program was carried out later in the papers [8], [9], [19] and [7], whereas several other questions raised in [28,29] constitute an active area of research right now (see for instance [25]).”
Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman (2003, methodology): “This well–known approach has been considered by many people, including Freedman, Meeks, Pitts, Rubinstein, Schoen, Simon, Smith, and Yau; see [2].”
Genus bounds for minimal surfaces arising from min-max constructions (2009, methodology): “The paper [8] reports a proof of the following regularity result.”
Minimal submanifolds (2005, unknown): “We will highlight a few key results below but refer to [13] for a thorough treatment.”
Minimal 2-spheres in 3-spheres (2017, background): “Appendix A in [10]) it follows that ω1(M) > 0.”
On the Multiplicity One Conjecture in min-max theory (2019, unknown): “We refer to [39, 6, 10, 17, 9, 23, 7, 47, 50, 32] for other variants of this theory.”
Min-Max theory and the Willmore conjecture (2012, methodology): “There are other treatments of the min-max theory, such as [41], [11].”
Comparing the Morse index and the first Betti number of minimal hypersurfaces (2016, methodology): “Still in three dimensions, Ketover [18] proved that the genus of the min-max surface is controlled in an effective way by the genus of the surfaces in the sweepouts (see also the previous work by De Lellis and Pellandini [11]).”
Morse index of multiplicity one min-max minimal hypersurfaces (2015, methodology): “153, see also [11, 20]), we can define a continuous map”
One-sided complete stable minimal surfaces (2006, background): “The second homology group of M is H2(M, Z2) = Z2⊕Z2⊕Z2 and a basis of it is given by α1 = [1, 3, 6, 4], α2 = [1, 3, 9, 7] and α3 = [3, 6, 12, 9] (note that the first surface is a Klein bottle while the two other ones are tori).”
A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary (2012, methodology): “Recently T. Colding and C. De Lellis [5] wrote a detailed account of the minimax construction, and they were able to simplify some of the proofs significantly.”
The existence of embedded minimal hypersurfaces (2009, background): “Its proof is given in Section 5, where we basically follow [18] (see also [6]).”