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

Abstract: This paper establishes the sharp lower bound $(4nπ<sup>2)<sup>{n/2}$ for the Willmore energy $\mathcal{W}$ of flat $n$-tori in the Euclidean space. Up to Möbius transformations, the Clifford $n$-torus $$\mathbb{S}^{1\bigl(\sqrt{1/n}\,\bigr)} \times \cdots \times \mathbb{S}^{1\bigl(\sqrt{1/n}\,\bigr)} \subset \mathbb{S}^{2n-1} \subset \mathbb{R}<sup>{2n}$$ is shown to be the unique minimizer attaining this bound. This also confirms Chen's conjecture for flat $n$-tori. However, when $n \geq3$, 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.