Min-Max theory and the Willmore conjecture

TL;DR

This paper proves the Willmore conjecture, which states that the integral of the square of the mean curvature of a torus in Euclidean three-space is at least 2π^2, using min-max theory of minimal surfaces.

Contribution

The paper provides a proof of the long-standing Willmore conjecture employing min-max theory, a novel approach in this context.

Findings

Confirmed the Willmore conjecture for tori in Euclidean space.

Established min-max theory as a tool for solving geometric conjectures.

Demonstrated the minimal surface approach in curvature integral problems.

Abstract

In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in Euclidean three-space is at least 2\pi^2. We prove this conjecture using the min-max theory of minimal surfaces.