Every immersed torus \(M \subset \mathbb{R}^3\) satisfies \(\int H^2 \, dA \geq 2\pi^2\), with equality exactly for the Clifford torus (up to conformal transformations). Proved by Marques–Neves (2012) via min–max minimal surface theory; published 2014.
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.
Fernando Codá Marques and André Neves proved the Willmore conjecture via min–max minimal surface theory: every immersed torus in \(R^{3}\) has Willmore energy at least 2π², with equality only for the Clifford torus up to conformal maps. Announced 2012, published in the Annals of Mathematics 2014.
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