Willmore conjecture

Remarks

Willmore stated the conjecture in 1965, and Marques and Neves proved it; their proof appeared in 2014. The torus in the equality case is the stereographic image of the Clifford torus in the three-sphere. If $f$ is not an embedding, a theorem of Li and Yau gives $\mathcal{W}(f) \ge 8\pi > 2\pi^2$, so only embedded tori need work. For those, Marques and Neves used conformal invariance to move the problem to the three-sphere and applied Almgren–Pitts min-max theory to a five-parameter family of surfaces built from the torus. A key step is that every closed embedded minimal surface of positive genus in the three-sphere has area at least $2\pi^2$. Equality holds there only for the Clifford torus, up to isometry.

For every closed surface $\mathcal{W} \ge 4\pi$, so the conjecture says that being a torus costs at least $2\pi^2 - 4\pi$ extra energy.

Going deeper

For a student

Because $\mathcal{W}$ is invariant under Möbius transformations, the minimizers come in whole families, and the natural setting is conformal geometry. The proof is one of the main applications of min-max theory for minimal surfaces. It belongs to a larger program of finding the least Willmore energy for each genus.