New applications of Min-max Theory

Abstract

I will talk about my recent work with Fernando Marques where we used Almgren-Pitts Min-max Theory to settle some open questions in Geometry: The Willmore conjecture, the Freedman-He-Wang conjecture for links (jointly with Ian Agol), and the existence of infinitely many minimal hypersurfaces in manifolds of positive Ricci curvature. Some open questions are suggested in the last section.

arXiv commentAppears in Proceedings of the International Congress of Mathematicians, Seoul 2014

References in corpus (1)

• Lower bounds for nodal sets of eigenfunctions

Cited by in corpus (12)

• Free boundary minimal surfaces in the unit ball : recent advances and open questions
• Multiplicity one and strictly stable Allen-Cahn minimal hypersurfaces
• Minmax Hierarchies and Minimal Surfaces in Manifolds
• Existence of minimal surfaces of arbitrary large Morse index
• Comparing the Morse index and the first Betti number of minimal hypersurfaces
• Sweeping out 3-manifold of positive Ricci curvature by short 1-cycles via estimates of min-max surfaces
• Index one minimal surfaces in positively curved $3$-manifolds
• A note on the index of closed minimal hypersurfaces of flat tori
• The systole of large genus minimal surfaces in positive Ricci curvature
• The (p;m)-width of Riemannian manifolds and its realization
• Index Estimates for Surfaces with Constant Mean Curvature in $3$-dimensional Manifolds
• On the Willmore problem for surfaces with symmetry