New applications of Min-max Theory

TL;DR

This paper discusses recent advances using Almgren-Pitts Min-max Theory to resolve key open problems in geometry, including the Willmore conjecture, link conjectures, and minimal hypersurfaces in positively curved manifolds.

Contribution

The work applies Min-max Theory to settle several longstanding open questions in geometric analysis and minimal surface theory.

Findings

Resolved the Willmore conjecture.

Proved the Freedman-He-Wang conjecture for links.

Established the existence of infinitely many minimal hypersurfaces in manifolds with positive Ricci curvature.

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.