The Willmore Conjecture

Every immersed torus M ⊂ ℝ³ satisfies ∫ H² dA ≥ 2π², with equality exactly for the Clifford torus (up to conformal transformations).

Willmore's energy dates to Poisson-era elasticity but his 1965 inequality question crystallised the field. Partial results accumulated (Shiohama–Xu for high-area tori; Lamm–Li classes) before Fernando Codá Marques and André Neves proved it in 2012 via the Almgren–Pitts min-max theory they systematically revitalised — the same paper dispatched Lawson's 1970 conjecture that the Clifford torus is the unique embedded minimal genus-one surface in S³. Published in Annals 2014; both authors are now central figures in the minimal-surfaces renaissance.