Min-Max theory and the Willmore conjecture


In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in $\mathbb{R}^3$ is at least $2\pi^2$. We prove this conjecture using the min-max theory of minimal surfaces.


Fernando C. Marques

IMPA, Rio de Janeiro, Brazil

André Neves

Imperial College London, South Kensington Campus, London, UK