Willmore's conjecture for tori in the three-sphere

Let $\mathcal W(\Sigma)=\int_{\Sigma}(1+H^2)\\,d\mu$ denote the Willmore energy of a compact surface $\Sigma\subset S^3$, where $H$ is its mean curvature. Willmore's conjecture. Every compact surface $\Sigma\subset S^3$ of genus one must satisfy

$$\mathcal W(\Sigma)\geq 2\pi^2.$$

This is the stereographic-projection-equivalent formulation of Willmore's conjecture and was proved by Marques and Neves; equality occurs only for the Clifford torus, up to conformal transformations and the corresponding rigidity described in the paper.