Willmore conjecture

[2] A proof by Fernando Codá Marques and André Neves was announced in 2012 and published in 2014.

In particular, calculation of W(M) for tori with various symmetries led Willmore to propose in 1965 the following conjecture, which now bears his name In 1982, Peter Wai-Kwong Li and Shing-Tung Yau proved the conjecture in the non-embedded case, showing that if

[4] In 2012, Fernando Codá Marques and André Neves proved the conjecture in the embedded case, using the Almgren–Pitts min-max theory of minimal surfaces.

[3][1] Martin Schmidt claimed a proof in 2002,[5] but it was not accepted for publication in any peer-reviewed mathematical journal (although it did not contain a proof of the Willmore conjecture, he proved some other important conjectures in it).

Prior to the proof of Marques and Neves, the Willmore conjecture had already been proved for many special cases, such as tube tori (by Willmore himself), and for tori of revolution (by Langer & Singer).