The Willmore conjecture

TL;DR

This paper surveys the history of the Willmore conjecture, discusses its recent proof using min-max methods, and explores open questions in the field of conformal geometry and geometric analysis.

Contribution

It presents a comprehensive overview of the Willmore conjecture and details the recent solution achieved through a min-max approach, advancing understanding in geometric analysis.

Findings

Proof of the Willmore conjecture using min-max methods

Connection of the conjecture to conformal geometry and PDEs

Open questions in geometric measure theory and related fields

Abstract

The Willmore conjecture, proposed in 1965, concerns the quest to find the best torus of all. This problem has inspired a lot of mathematics over the years, helping bringing together ideas from subjects like conformal geometry, partial differential equations, algebraic geometry and geometric measure theory. In this article we survey the history of the conjecture and our recent solution through the min-max approach. We finish with a discussion of some of the many open questions that remain in the field.