On the Willmore energy of flat $n$-tori in ${\mathbb{R}}^N$ and Chen's conjecture for $n$-tori

This paper establishes the sharp lower bound $(4n{\pi }^2{)}^{n\mathrm{/}2}$ for the Willmore energy $\mathcal{W}$ of flat $n$-tori in the Euclidean space. Up to Möbius transformations, the Clifford $n$-torus ${\mathbb{S}}^1\left(\sqrt{1\mathrm{/}n}\right)\times \cdots \times {\mathbb{S}}^1\left(\sqrt{1\mathrm{/}n}\right)\subset {\mathbb{S}}^{2n-1}\subset {\mathbb{R}}^{2n}$ is shown to be the unique minimizer attaining this bound. This also confirms Chen’s conjecture for flat $n$-tori. However, when $n\ge 3$, we show that Chen’s conjecture fails on the total mean curvature of general immersed $n$-tori: certain Möbius transformations of the Clifford $n$-torus strictly decrease the total mean curvature.