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Two years ago, F.C. Marques and A.A. Neves implemented, in the framework ofclosed rectifiable 2-dimensional currents of the 3-dimensional sphere, amin-max method in geometric measure theory due to F. Almgren and J. Pitts.Using this approach they succeeded in proving that the famous Clifford torusminimizes the area among all closed minimal surfaces of non-zero genus in$S^3$. Another spectacular consequence of their work is to provide a proof ofthe Willmore conjecture. The goal of this talk is to discuss first the generalframework of these two theorems of Marques and Neves. We shall then present thestructures and some key details of their proofs. We will then address the scopeof this remarkable contribution to the calculus of variations on surfaces in a3-dimensional space.

Bourbaki Seminar 1081 : Min-max methods and the Willmore conjecture, after Fernando Cod\'a Marques and Andr\'e Arroja Neves