On the Willmore energy of flat $n$-tori in $\mathbb{R}^N$ and Chen's conjecture for $n$-tori
Year: 2026
Type: preprint
Abstract: This paper establishes the sharp lower bound $(4nπ^2)^{n/2}$ for the Willmore energy $\mathcal{W}$ of flat $n$-tori in the Euclidean space. Up to Möbius transformations, the Clifford $n$-torus $\mathbb{S}^1\bigl(\sqrt{1/n}\,\bigr) \times \cdots \times \mathbb{S}^1\bigl(\sqrt{1/n}\,\bigr) \subset \mathbb{S}^{2n-1} \subset \mathbb{R}^{2n}$ 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.
Source: arXiv (Cornell University)
Authors: Ni, Ruijie, Wang, Peng, Xie, Zhenxiao
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1220981610, G3912457081
OpenAlex ID: W7214976707