In 1982, Li and Yau resolved the Willmore conjecture in the non-embedded case. More precisely, they established that, given any smooth immersion of a closed surface in the 3-sphere which fails to be an embedding, the Willmore energy is bounded below by 8π. This is complemented by a 2012 result of Fernando Marques and André Neves, which says that in the alternative case of a smooth embedding of the 2-dimensional torus , the Willmore energy is bounded below by 2π2.Marques, Fernando C.; Neves, André. Min-max theory and the Willmore conjecture. Ann. of Math. (2) 179 (2014), no. 2, 683–782. Together, these results comprise the full Willmore conjecture, as originally formulated by Thomas Willmore in 1965. Although their assumptions and conclusions are quite similar, the methods of Li−Yau and Marques−Neves are distinct. Nonetheless, they both rely on structurally similar minimax schemes. Marques and Neves made novel use of the Almgren–Pitts min-max theory of the area functional from geometric measure theory; Li and Yau's approach depended on their new "conformal invariant", which is a min-max quantity based on the Dirichlet energy. The main work of their article is devoted to relating their conformal invariant to other geometric quantities.
