Applications of Almgren-Pitts Min-max theory

Abstract

Abstract. We survey recent applications of Almgren-Pitts Min-max theory found by the authors. Namely, the proof of the Willmore conjecture, the proof of the Freedman-He-Wang conjecture on the Möbius energy of links (jointly with Ian Agol), and the proof that manifolds with positive Ricci curvature metrics admit an infinite number of minimal hypersurfaces.

Weyl law for the volume spectrum

“…Note that since lim j→∞ M(Q j (x)) = 0, by the coarea formula we can choose {r j i } such that (20) x∈I 0 (m,l) 0 v i=1 M( Q j (x), d i , r j i ) ≤ a(n)δ < δ 2(n + 1) for every sufficiently large j. Furthermore, using (16), (17), (18), and Proposition 2.4, we get that (21) ||φ j (x)||(B r j i (p i )) < δ 3 , ||T ||(B r j i (p i ) \ M ) < δ 3 ,…”

“…We will be succinct and just describe the main concepts. The reader can see [1,16,17,18] for some background and explicit constructions.…”

Morse index and multiplicity of min-max minimal hypersurfaces

“…In Section 3, we explain how to use the original discretized setting to obtain a min-max theory for maps that are continuous in the F-topology as above. A similar construction for the mass topology was done by the authors in [22].…”

The regularity of parametrized integer stationary varifolds in two dimensions

“…[10] and [31]) and has been responsible for many important applications (see e.g. [30]). For a survey on a simplified approach, which is a combination of the contributions of Almgren, Meeks, Pitts, Simon, Smith and Yau, we refer the reader to the paper [8] by Colding and De Lellis (which deals for simplicity with 1-parameter sweepouts).…”